Seminari del Dipartimento di Matematica

Luis Ferroni Rivetti (Università di Bologna)

Ehrhart Positivity of Certain Polytopes

algebra e geometria

The Ehrhary Polynomial of an Integral Polytope P counts the number of lattice points inside the integer dilations of P. There are several families of polytopes for which the Ehrhart Polynomial has been extensively studied. In this sense, it is a very hard problem to understand what the coefficients of this polynomial represent. In fact, a relevant issue is to determine if they are positive or not. We will discuss some of the known results and conjectures on this topic.

Fine Boundary Properties in Complex Analysis and Discrete Potential Theory

nel ciclo di seminari: COMPLEX ANALYSIS LAB

PhD thesis defense

Disuguaglianze differenziali e disuguaglianze integrali

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

Il filo di questo seminario, del tutto espositivo, segue, in una particolare direzione, la relazione che intercorre tra disuguaglianze differenziali e integrali: a partire dalla nota disuguaglianza di Jensen (e, tempo permettendo, gli spazi di Orlicz), attraverso classici argomenti di subarmonicità nello studio di alcuni integrali singolari, quindi il metodo di Burkholder per stime di operatori su spazi di martingale, e arrivare infine al metodo delle funzioni di Bellman di Nazarov, Treil e Volberg, che ha le sue radici nella teoria del controllo stocastico ottimale.

Some Perspectives on Numerical Methods and Preconditioning for PDE-Constrained Optimization

analisi numerica

In recent years, a field that has risen to prominence within mathematics and engineering is that of PDE-constrained optimization, which has numerous applications within these subject areas. In this talk, we consider the development of preconditioned iterative methods for a number of PDE-constrained optimization problems. In particular, we discuss challenges arising from the interior point solution of problems with additional box constraints, the solution of nonlinear problems arising from chemotaxis, and the application of deferred correction approaches to improve the discretization error obtained in the time variable.

Counting problems in infinite measure.

SEMINARIO INTERDISCIPLINARE

Given a group Γ acting properly discontinuously and by isometries on a metric space X, one can wonder how grows the orbit of a given point. More precisely, given two points x, y ∈ X and ρ > 0, we define the orbital function as NΓ(x, y, ρ) := #(Γ · y ∩ B(x, ρ)) , where B(x, ρ) denotes the ball centred at x of radius ρ. A counting problem con- sists to estimate the orbital function when ρ → ∞. In the setting of groups acting on hyperbolic spaces this question was widely in- vestigated for decades, with mainly two different approaches: an analytical one relying on Selberg’s pre-trace formula, due to Huber in the 50’s, and a dynami- cal one relying on the mixing of the geodesic flow, due to Margulis in the late 60’s. During the talk, we shall describe Margulis’ dynamical method in order to mo- tivate the introduction of the Brownian motion. Combined with the use of the pre-trace formula, we shall establish a counting theorem linking the heat kernel of the quotient manifold and the orbital function. If the time allows it, we also shall review a couple of corollaries of the approach.

Characterization of the Palais-Smale sequences for the fractional CR Yamabe functional

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

In this talk we consider the functional whose critical points are solutions of the fractional CR Yamabe-type equation on the CR sphere. Due to the lack of compactness for the associated critical Sobolev embedding, the functional does not satisfy the Palais-Smale condition. By adapting a classical arguments by Struwe and by making use of some recent commutator estimates, which allow us to deal with our non-local setting, we obtain a characterization of the Palais-Smale sequences. Then, as an application, we prove a multiplicity result for the related equation. This is joint work with A.Maalaoui and V.Martino.

You don’t fool me! Disclosing visual illusions via Wilson-Cowan-type dynamics

analisi matematica

In this talk, I will discuss the reproduction of visual perception phenomena, specifically visual illusions, by means of Wilson-Cowan-type models of neuronal dynamics. In particular, we show that the Wilson-Cowan equations can reproduce a number of brightness and orientation-dependent illusions, and that the latter type of illusions require that the neuronal dynamics equations consider explicitly the orientation, coherently with the architecture of V1. I’ll then focus on a slightly different modification of the Wilson-Cowan equations which makes such model consistent with the efficient representation principle, that is it can be interpreted as the gradient flow of a suitable functional. This shows that this model minimises redundant information, making it capable of replicating more visual illusions than the original Wilson-Cowan formulation.

Dicembre

03

Curve di genere 2 su superfici abeliane

Margherita Lelli Chiesa

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

Nella prima parte del seminario richiamerò alcune nozioni di base sulle varietà abeliane e sulla varietà Jacobiana di una curva liscia e irriducibile. Motiverò inoltre lo studio delle curve di genere geometrico 2 su superfici abeliane. Nella seconda parte del seminario parlerò di un lavoro in collaborazione con A.L.Knutsen. Sia (S,L) una superficie abeliana generale con polarizzazione di tipo (d_1,d_2). Nel sistema lineare |L| sono presenti un numero finito di curve di genere geometrico 2. In analogia con il risultato di Chen riguardante curve razionali su superfici K3, è naturale chiedersi se tutte le curve di genere 2 in |L| sono nodali. Dimostreremo che questo è vero se e solo se 4 non divide d_2.

Conferenzieri e organizzatori del convegno Two Days on CalcVar&PDEs

Tavola rotonda II parte: regolarità nelle PDEs

analisi matematica

Nicola Fusco (U. Napoli "Federico II")

Stabilità asintotica per il flusso gradiente di energie non locali

analisi matematica

Nel seminario verranno presentati alcuni recenti risultati sulla stabilità asintotica del flusso gradiente rispetto alla norma $H^{-1}$ del funzionale perimetro (surface diffusion) e di funzionali del tipo somma di un perimetro e di un termine non locale di volume che intervengono in alcuni modelli variazioni di Scienza dei Materiali.

Stable minimizers of functionals of the gradient

Giulia Treu (U. Padova)

analisi matematica

Let $\Omega\subset \mathbb{R}^n$ be a bounded Lipschitz domain. Let $L:\mathbb{R}^n\rightarrow \bar{\mathbb{R}}=\mathbb{R}\cup \{+\infty\}$ be a continuous function with superlinear growth at infinity, and consider the functional $\mathcal{I}(u)=\int_\Omega L(Du)$, $u\in W^{1,1}(\Omega)$. We provide necessary and sufficient conditions on $L$ under which, for all $f\in W^{1,1}(\Omega)$ such that $\mathcal{I}(f)<+\infty$, the problem of minimizing $\mathcal{I}(u)$ with the boundary condition $u_{|\partial\Omega}=f$ has a solution which is stable, or -- alternatively -- is such that all of its solutions are stable. By stability of $\mathcal{I}$ at $u$ we mean that $u_k\rightharpoonup u$ weakly in $W^{1,1}(\Omega)$ together with $\mathcal{I}(u_k)\to \mathcal{I}(u)$ imply $u_k\rightarrow u$ strongly in $W^{1,1}(\Omega)$. This extends to general boundary data some results obtained by Cellina and Cellina and Zagatti. Furthermore, with respect to the preceding literature on existence results for scalar variational problems, we drop the assumption that the relaxed functional admits a continuous minimizer. The results are contained in a joint paper with G. Colombo, Dipartimento di Matematica, University of Padova and M. Sychev, Sobolev Institute of Mathematics, Novosibirsk.

Carlo Mariconda (U. Padova)

The basic problem of the calculus of variations: some recent results for nonautonomous Lagrangians that are highly discontinuous in the state or velocity variable

analisi matematica

Ermanno Lanconelli (U. Bologna)

On the stability of the Gauss mean value formula for harmonic functions

analisi matematica

Among the various rigidity properties of the Euclidean balls one of the best known examples is the Gauss mean value formula for harmonic functions. This property raises the question of its stability. i.e. : if $D$ is an open set with finite measure and $x_0$ is a point of $D$ such that $u(x_0)$ is close to the average of $u$ on $D$ for every integrable harmonic functions $u$ in $D$, is it true that $D$ is close to a ball centered at $x_0$? In this talk we present some positive answers to this question, obtained in collaboration with Giovanni Cupini, Nicola Fusco and Xiao Zhong

Cristina Marcelli (U. Politecnica delle Marche)

Coercivity of integral functionals of non-everywhere superlinear lagrangians

analisi matematica

We consider the following non-autonomous variational problem: \[\textrm{minimize\,} \left\{F(v)=\int_a^b f(x,v(x),v'(x))\ \mathrm dx\,:\,v\in \Omega \right\} \] where $\Omega:=\{v\in W^{1,1}(a,b),\ v(a)=A, \ v(b)=B,\ v(x)\in I \}$. The Lagrangian $F$ is assumed to have just a "non-everywhere" superlinear growth, being allowed to vanish at some $x_0\in [a,b]$, or $s_0\in I$. We prove some sufficient conditions ensuring the coercivity of the functional $F$. As a consequence, when $f$ is convex with respect to the last variable, the existence of the minimum can be immediately derived.

Abstract

Michela Eleuteri (U. Modena e Reggio Emilia)

Lipschitz regularity results for variational models with non standard growth

analisi matematica

In this talk we are going to present some recent Lipschitz regularity results for functionals and systems with non standard growth, without further structure conditions on the integrand, both in the scalar and in the vector case. This is a joint project with Paolo Marcellini and Elvira Mascolo. In the last part of the talk, we will complement these results by dealing with obstacle problems. This is a joint project with Antonia Passarelli di Napoli and Michele Caselli.

Antonia Passarelli di Napoli (U. Napoli "Federico II")

Regularity for minimizers of a class of degenerate convex functionals

analisi matematica

I will present some regularity results for vectorial minimizers of integral functionals of the type $$\int_\Omega f(x,Du(x))\,dx$$ with energy densities $f(x,\xi)=\tilde f (x,|\xi|)$ that are degenerate convex with respect to the $\xi$ variable and satisfy non standard growth conditions. Assuming that the partial map $x\mapsto f(x,\xi)$ belongs to a suitable Sobolev class, we establish the higher differentiability and the higher integrability of the gradient of the minimizers.

Luca Lussardi (Politecnico di Torino)

On the Kirchhoff–Plateau problem

analisi matematica

The Kirchhoff–Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments. In this seminar I will present some results concerning the Kirchhoff-Plateau problem obtained in collaboration with Eliot Fried, Giulio Giusteri, Giulia Bevilacqua and Alfredo Marzocchi.

Cristiana De Filippis (U. Oxford)

Recent results for a class of non-uniformly elliptic problems

analisi matematica

I will discuss some recent results on a class of non-uniformly elliptic problems modelled upon the double phase energy

Abstract

Time behaviour for solutions to an evolution pde with nonstandard growth

analisi matematica

TBA

Symplectic factorization, Darboux theorem and ellipticity

analisi matematica

Paolo Marcellini (U. Firenze)

Some remarks in the calculus of variations

analisi matematica

We will discuss some examples, problems and remarks related to the calculus of variations and to partial differential equations.

Conferenzieri e organizzatori del convegno Two Days on CalcVar&PDEs

Tavola rotonda I parte: regolarità nel calcolo delle variazioni

analisi matematica

Diego Alberici (Università di Bologna)

Statistical Mechanics & Complex Zeros: it's a match!

SEMINARIO INTERDISCIPLINARE

Localizing the complex zeros of certain large degree polynomials is crucial in Statistical Mechanics in order to identify the phase transitions of the system. I will introduce this beautiful bridge across disciplines mainly through two examples: the matching model and the Ising model on graphs. The meaning of "phase transition" and its connection with complex zeros will be explained. Then I will show how to localize the zeros of the matching polynomials in order to prove the absence of phase transitions in matching models (Heilmann-Lieb). In the end I will mention that a similar approach is succesfull to localize the phase transitions in Ising models (Lee-Yang).

Abelian arrangements, 4

algebra e geometria

Euler number for measurable cocycles of surface groups

nell'ambito della serie: TOPICS IN MATHEMATICS 2019/2020

algebra e geometria

A quite useful philosophy in mathematics is to use the sharpness of an inequality regarding the "shape" of a topological space in order to detect a precise geometry on it: more precisely, the maximal value of the inequality usually allows to identify a specific geometric structure. Think for instance either to the applications of arithmetic/geometric mean inequality or to the isoperimetric inequality on the plane. Something similar happens in the world of Zimmer's cocycle theory. In this seminar we are going to focus our attention on Zimmer's cocycles associated to the fundamental group a surface S with genus bigger than or equal to 2. If such a measurable cocycle admits a (generalized) boundary map, one can define the notion of Euler number. The latter well behaves along cohomology classes and its absolute value is bounded by the modulus of the Euler characteristic of S. Remarkably the maximal value is attained if and only if the cocycle is cohomologous to a hyperbolization. The first part of the talk will be a gentle introduction to measurable cocycles and boundary theory. Then, we are going to introduce the orientation cocycle on the circle. Finally we will define the Euler number of a measurable cocycle and we will discuss its rigidity property. This is a joint work with Marco Moraschini.

Complexity of representations of quivers

SEMINARIO INTERDISCIPLINARE

Complexity of representations of quivers

nell'ambito della serie: COLLOQUIO DI DIPARTIMENTO

SEMINARIO INTERDISCIPLINARE

Rigour and aesthetics: Japanese traditional mathematics

SEMINARIO INTERDISCIPLINARE

14

Nicolò Forcillo (Università di Bologna)

A gentle introduction to free boundary problems

analisi matematica

I introduce free boundary problems, providing first some examples of them. Then, I try to motivate the using of viscosity solution for this kind of problems, briefly mentioning the main contributions given in this field. Finally, I speak about some of the main tools which are useful to study the regularity of the free boundary.

BACKWARD SDES, MARTINGALE PROBLEMS AND APPLICATIONS TO MATHEMATICAL FINANCE

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

The aim of this talk consists in introducing a formalism for the deterministic analysis associated with backward stochastic differential equations driven by general càdlàg martingales, coupled with a forward process. When the martingale is a standard Brownian motion, the natural deterministic analysis is provided by the solution u of a semilinear PDE of parabolic type coupled with a function v which is associated with the gradient ∇u, when u is of class C1 in space. When u is only a viscosity solution of the PDE, the link associating v to u is not completely clear: sometimes in the literature it is called the identification problem. We introduce in particular the notion of a decoupled mild solution of a PDE, a IPDE, a path-dependent PDE or more generally a deterministic problem associated with a BSDE. The idea is to introduce a suitable analysis to investigate the equivalent of the identification problem first in a general Markovian setting with a class of examples. An interesting application concerns the hedging problem under basis risk of a contingent claim g(XT,ST ), where S (resp. X) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated Föllmer-Schweizer decomposition. We revisit the case when the couple of price processes (X,S) is a diffusion and we provide explicit expressions when (X,S) is an exponential of additive processes. Extensions to non-Markovian (path-dependent) cases are discussed.

Alex Casarotti (Università di Ferrara)

Difettività: dalla geometria del contact locus all'identificabilità

algebra e geometria

Sia X una varietà proiettiva irriducibile non degenere in P^N. X è detta h-identificabile se il punto generico della Secante h-esima di X ammette un'unica decomposizione in h punti di X. Dopo una piccola overview dei principali risultati inerenti la difettività e la geometria dei contact locus, descriveremo un risultato che lega la non difettività di X all'identificabilità. Questo ci permetterà, sfruttando risultati noti sulla difettività di particolari varietà quali Segre, Segre-Veronese e Grassmanniane, di migliorare i risultati fino ad ora noti in letteratura riguardo l'identificabilità.

Alex Casarotti (Università di Ferrara)

Difettività: dalla geometria del contact locus all'identificabilità

algebra e geometria

Sia X una varietà proiettiva irriducibile non degenere in P^N. X è detta h-identificabile se il punto generico della Secante h-esima di X ammette un'unica decomposizione in h punti di X. Dopo una piccola overview dei principali risultati inerenti la difettività e la geometria dei contact locus, descriveremo un risultato che lega la non difettività di X all'identificabilità. Questo ci permetterà, sfruttando risultati noti sulla difettività di particolari varietà quali Segre, Segre-Veronese e Grassmanniane, di migliorare i risultati fino ad ora noti in letteratura riguardo l'identificabilità.

Alex Casarotti (Università di Ferrara)

Difettività: dalla geometria del contact locus all'identificabilità

algebra e geometria

Sia X una varietà proiettiva irriducibile non degenere in P^N. X è detta h-identificabile se il punto generico della Secante h-esima di X ammette un'unica decomposizione in h punti di X. Dopo una piccola overview dei principali risultati inerenti la difettività e la geometria dei contact locus, descriveremo un risultato che lega la non difettività di X all'identificabilità. Questo ci permetterà, sfruttando risultati noti sulla difettività di particolari varietà quali Segre, Segre-Veronese e Grassmanniane, di migliorare i risultati fino ad ora noti in letteratura riguardo l'identificabilità.

On stochastic differential equations

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

Strutture geometriche e rappresentazioni

algebra e geometria

Sia S una superficie chiusa e orientable, una (G,X)-structura su $ è definita come il dato di un atlante massimale su S le cui carte prendono valori in X ed i cambi di carta sono restrizioni di elementi di G. Ogni struttura definisce una rappresentazione R dal gruppo fondamentale di S in G, detta olonomia, che codifica i dati geometrici della struttura. Viceversa, data una generica rappresentazione R esiste una struttura geometrica avente olonomia R? Se esiste, tale geometria è unica? In questo seminario risponderemo a queste domande fornendo risultati noti e recenti.

Commento al film "Gifted - Il dono del talento"

SEMINARIO INTERDISCIPLINARE

Singularities and Robotics

nell'ambito della serie: TOPICS IN MATHEMATICS 2019/2020

The idea of singularity is found in many parts of mathematics, capturing the idea of a position where some regular behavior breaks down. A standard situation is in linear algebra where for a linear transformation or matrix rank deficiency corresponds to singularity. This provides a basic model for other settings, especially for differentiable functions between Euclidean spaces. Results of Marston Morse (the Morse Lemma) and Hassler Whitney (stable singularities of mappings from the plane to the plane) led to pioneering work in differential topology by René Thom, whose interest in biological morphogenesis gave rise to Elementary Catastrophe Theory and a wide interest in mathematical models of singularity. The foundations of singularity theory were developed, in which the concepts of transversality and stratification played an important role. In the first part of the seminar I will outline some of this history and ideas from singularity theory. In robotics, kinematic mappings relate inputs, outputs and constraints. The impact of singularities on robotic control systems was recognised in the 1960s. Subsequent interest in the variety of ways that singular phenomena occur in robot kinematics has led to a large literature on the subject. In the second part of the seminar, I will discuss some of my research on kinematic singularities.

11

Claudia Lederman, Università di Buenos Aires

About some regularity properties of the p(x) Laplace operator

analisi matematica

Abelian arrangements, 4

algebra e geometria

05

Peter Donelan (Victoria University of Wellington, New Zealand)

The Euclidean Group, Invariants and Robotics

SEMINARIO INTERDISCIPLINARE

Robot manipulators are generally modelled as a collection of rigid bodies or links connected pairwise by joints. The Euclidean group of proper isometries of 3-space and its subgroups therefore play a central role in kinematic and dynamic modelling of robot systems. Engineers frequently use a system called Denavit-Hartenberg (DH) parameters for the description of serial robot manipulators – the type most commonly used in large-scale industrial applications. From a mathematical point of view, these should be invariant quantities with respect to an action of the Euclidean group. The Euclidean group is closely related to the associative algebra of dual quaternions and this relation provides an approach to identifying polynomial invariants that validate the DH parametrization. This is joint work with Mohammed Daher and Petros Hadjicostas.

DIFFERENTIAL CALCULUS FOR JORDAN ALGEBRAS AND JORDAN MODULES

Alessandro Carotenuto

algebra e geometria

"The notion of differential calculus for Jordan algebras, as well as the theory of connections for Jordan modules, was recentely introduced by M. Dubois-Violete. In this talk I will give an overview of the representation theory for Jordan algebras and then I will present some results we obteined in the context of differential calculus for Jordan algebras and modules. If the time allows, I will also review two proposals from M. Dubois-Violette in which the use of Jordan algebra allows for a mathematical formalization of the Standard model of particle physics"

Weight filtration in the cohomology of Shimura varieties

algebra e geometria

Shimura varieties are algebraic varieties which are initially constructed, as complex analytic objects, as locally symmetric varieties with respect to a well-chosen semisimple algebraic group G. The first aim of the talk is to introduce these objects and to motivate the interest of the structures appearing in their cohomology. In a second part, we will explain some results which describe how the representation theory of the underlying group G controls the Hodge-theoretic weight filtration in the cohomology of Shimura varieties.

24

Giorgia Franchini (Università di Modena e Reggio Emilia)

Artificial Intelligence: ma l’intelligenza esattamente dov’è?

Negli ultimi decenni aziende pubbliche e private hanno concentrato i loro investimenti nello sviluppo di tecniche di Machine Learning (ML) o Artificial Intelligence (AI). In questa chiacchierata cercheremo di entrare nella profondità di questi algoritmi ed in particolare nella parte in cui il metodo apprende, quindi nella sua intelligenza, anche se in realtà intelligenza non è. Tutto però sta nel come si definisce intelligenza: Stephen Hawking scriveva ‘L’intelligenza è la capacità di adattarsi al cambiamento’ e questo è esattamente quello che l’ottimizzazione matematica cerca di insegnare ai metodi di apprendimento da esempi: la generalizzazione, cioè la capacità di adattarsi al cambiamento.

Optimization results for the higher eigenvalues of the p-Laplacian

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

In this talk we study the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.

Motivic zeta functions for degenerating Calabi-Yau varieties

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Zeta functions are historically important objects in mathematics, and continue to motivate a wealth of work in research today. Zeta functions often arise at the crossroads between different mathematical fields, such as analysis, number theory and algebraic geometry. This talk will mainly focus on so-called "motivic zeta functions", which are of a more geometric nature. These objects can be attached to degenerating families of Calabi-Yau varieties, and provide an interesting link between arithmetic and geometric aspects of such varieties. I will give an intuitive introduction to some of the key properties of motivic zeta functions, and mention a few influential open questions. I will also present some of my own results, both old and new, in this area.

A Free Boundary Problem Arising from Hedge Funds Flows

probabilità

CV

When the value of a hedge fund approaches its running maximum, performance fees are paid and the fund typically experiences positive flows, while large drawdowns routinely lead to negative flows. In a model where investors' flows are a function of the drawdown, we solve in closed form the stochastic optimal control problem of a manager who maximizes the expected value of future fees and anticipates investors' response to the fund’s performance. The problem involves a nonlinear HJB equation which can be linearized though a duality approach, at the cost of replacing it with a free-boundary problem. The verification result relies on an upper bound for the lifetime maximum of a diffusion with negative drift. In contrast to models where outflows are performance-insensitive, a higher drawdown induces the manager to take more risk. Such risk-shifting incentive increases as flows' sensitivity to performance increase, and as managerial ability decreases.

16

Pierre Bousquet (U. Tolosa)

ON THE ORTHOTROPIC LAPLACIAN

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

We present some new regularity results for the orthotropic harmonic functions, which are the minimizers of a egenerate and anisotropic variant of the Dirichlet functional. These results have been obtained in collaboration with L. Brasco (Ferrara), V. Julin (Jyvaskyla), C. Leone (Naples) and A. Verde (Naples).

Hodge structures of K3 type in Fano varieties

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type (FK3), for their deep links with hyperkähler geometry. In this talk we will present some examples of recently discovered FK3 varieties, and a general procedure that allows us to spread a (Hodge) K3 structure as a component of the Hodge structure of different varieties. This is in collaboration with Giovanni Mongardi and Marcello Bernardara--Laurent Manivel.

Matematica e Fisica: fake news...anche ad insaputa

SEMINARIO INTERDISCIPLINARE

Locandina dell'incontro

Il termine fake news indica informazioni inventate, ingannevoli o distorte. Tradizionalmente esse vengono veicolate attraverso i mass media ai quali si è, oggi e con forte impatto, aggiunto Internet, essenzialmente attraverso i social. Le fake news concernenti le discipline scientifiche sono quelle che colpiscono i soggetti meno colti della popolazione. Paradossalmente sono anche le più... involontarie a causa della generale “ignoranza” delle cose della matematica da parte di chi detiene molto del potere mediatico. Involontarie ma non meno “pericolose".

Matematica e Fisica: fake news...anche ad insaputa

SEMINARIO INTERDISCIPLINARE

Locandina evento

Il termine fake news indica informazioni inventate, ingannevoli o distorte. Tradizionalmente esse vengono veicolate attraverso i mass media ai quali si è, oggi e con forte impatto, aggiunto Internet, essenzialmente attraverso i social. Le fake news concernenti le discipline scientifiche sono quelle che colpiscono i soggetti meno colti della popolazione. Paradossalmente sono anche le più... involontarie a causa della generale “ignoranza” delle cose della matematica da parte di chi detiene molto del potere mediatico. Involontarie ma non meno “pericolose". Il seminario rientra nell'incontro "La cultura scientifica contro le fake news", organizzato insieme a tutte le aree del PLS. Ogni relatore avrà a disposizione 30 minuti, poi dovrà rispondere alle domande degli studenti.

10

Davide Bolognini (Università di Bologna)

Un'introduzione (informale) alla combinatoria algebrica

algebra e geometria

In questo seminario vogliamo dare una panoramica su temi di Combinatoria Algebrica. Allo scopo, introduciamo brevemente alcune nozioni di Topologia (l'omologia singolare), Algebra Commutativa (i numeri di Betti graduati) e Combinatoria (proprietà dei complessi simpliciali), stabilendo le connessioni esistenti tra di esse. Come esempio concreto di un tipico risultato di Combinatoria Algebrica, enunciamo un noto teorema di Fröberg, ponendo anche un problema aperto ad esso correlato.

Minimizing convex quadratics with variable precision conjugate gradients

analisi numerica

We investigate the method of conjugate gradients, exploiting inaccurate matrix-vector products, for the solution of convex quadratic optimization problems. Theoretical performance bounds are derived, and the necessary quantities occurring in the theoretical bounds estimated, leading to a practical algorithm. Numerical experiments suggest that this approach has significant potential, including in the steadily more important context of multi-precision computations.

10

JAC Weideman, Stellenbosch University, South Africa

Locating Complex Singularities: Numerics and Applications

analisi numerica

For numerical computation, a power series representation of a function is typically truncated to a polynomial. Being an entire function, a polynomial cannot reveal much of the singularity structure of the underlying function other than perhaps the distance to the nearest singularity in the complex plane. The same remarks apply to Fourier series and truncated Fourier series. In the first part of the talk we survey some numerical strategies for uncovering additional singularity information. This includes methods based on a theorem of Darboux, as well as Padé and Fourier-Padé approximations. We discuss numerical implementations, stability, and pitfalls. Our test examples include meromorphic functions as well as functions with branch point singularities. In the second part of the talk, we apply these techniques to the computation of some special functions and to a nonlinear PDE.

A comparison between kinetic theory and extended thermodynamics models of a monatomic gas mixture

fisica matematica

The models of extended thermodynamics can be constructed through different approaches. In this talk we will compare them in the case of a monatomic rarefied gas mixture.

Su una congettura di Voisin

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

In collaborazione con R. Laterveer (Strasburgo) e G. Pacienza (Nancy) diamo un criterio per verificare una congettura di Voisin sui cicli algebrici 0-dimensionali di una varietà complessa, proiettiva, liscia, di dimensione al più cinque, e di genere geometrico uno.

Stationary flow and heat transfer in extended thermodynamics

fisica matematica

Recently, in extended thermodynamics, attention was devoted to stationary flows and stationary heat transfer problems in bounded domains. It was shown that even the 13-moment extended thermodynamics model is able to predict results different from those predicted by Navier-Stokes-Fourier thermodynamics and closer to experimental data. The differences are more evident when non-planar geometries and/or velocity fields are present. Here general results and future perspectives will be presented.

The K(pi, 1) conjecture for affine Artin groups

algebra e geometria

Artin groups are a generalization of braid groups, and arise as the fundamental groups of configuration spaces associated with Coxeter groups. A long-standing open problem, called the K(pi, 1) conjecture, states that the higher homotopy groups of these configuration spaces are trivial. For finite Coxeter groups, this was proved by Deligne in 1972. In this talk I will present a recent proof of the K(pi, 1) conjecture in the affine case, which is a joint work with Mario Salvetti. The first part of the talk will be dedicated to introducing Coxeter groups, Artin groups, and the K(pi, 1) conjecture, so that only few topological and combinatorial prerequisites are needed.

RISULTATI RECENTI DI FANO 4-FOLDS TRAMITE CONIC BUNDLES

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In questo talk presenteremo alcuni risultati recenti riguardanti Fano 4-folds, usando particolari strutture di conic bundles che tali varietà ammettono. Nella prima parte del talk ci focalizzeremo su alcuni oggetti geometrici preliminari che ci serviranno per discutere i risultati principali. In particolare, richiameremo un invariante per tali varietà, introdotto da Casagrande, chiamato Lefschetz defect. Rivisiteremo la letteratura esistente nel caso in cui tale invariante sia maggiore o uguale a quattro e discuteremo il legame che sussiste tra il Lefschetz defect e le strutture di conic bundles delle varietà in questione. Successivamente daremo una caratterizzazione delle Fano 4-folds con Lefschetz defect uguale a 3 in termini di strutture di conic bundles da cui dedurremo risultati di classificazione per tali varietà.

Sampling, Marcinkiewicz-Zygmund sets, approximation, and quadrature rules

analisi matematica

Suppose you are given n samples of a function f on the torus: what can you say on f?

Settembre

26

Alberto Viscardi (Università di Bologna)

WTF is a WTF? A short introduction to Wavelets, (Tight) Frames and some applications

analisi numerica

Frames were introduced by Duffin and Schaffer in 1952 as a generalization of orthonormal basis. On the other hand, while the first wavelet was introduced by Haar in 1909, Wavelets became popular among both mathematicians and engineers at the end of the eighties, with the works of Daubechies, Meyer and many others. The children of these two concepts, namely Wavelet Frames, were born in the following years. They provide useful representations of functions that led to fast and reliable algorithms for signal analysis, compression, edge detection, denoising, inpainting and more. This is intended to be an introductory talk about the topic and it is aimed to illustrate which mathematical concepts are behind them and how they can be exploited in some applications.

analisi matematica

Measure-valued Solutions for Some Problems Arising in Biology and Social Sciences

analisi matematica

In this talk I will present some results concerning the well-posedness and asymptotic behaviour of some evolution equations arising in social sciences naturally involving measure-valued functions. More precisely I will present some models of opinion formation. Imagine a group of individuals debating over some question. As a result of interactions among individuals, the distribution of opinion, a priori a general probability measure, is not static but evolves in time. The problem is then to study the long-time behaviour of this distibution assuming some knowledge about the way people interact. This amounts to study the aymptotic behaviour of some transport equation for measure-valued function. We obtain the asymptotic behaviour of the solution and an estimation of the rate of convergence in terms of a Monge-Kantorovich distance.

Measure-valued Solutions for Some Problems Arising in Biology and Social Sciences

analisi matematica

In this talk I will present some results concerning the well-posedness and asymptotic behaviour of some evolution equations arising in social sciences naturally involving measure-valued functions. More precisely I will present some models of opinion formation. Imagine a group of individuals debating over some question. As a result of interactions among individuals, the distribution of opinion, a priori a general probability measure, is not static but evolves in time. The problem is then to study the long-time behaviour of this distibution assuming some knowledge about the way people interact. This amounts to study the aymptotic behaviour of some transport equation for measure-valued function. We obtain the asymptotic behaviour of the solution and an estimation of the rate of convergence in terms of a Monge-Kantorovich distance.

The two-phase problem for harmonic measure in VMO

analisi matematica

We consider the two-phase problem for harmonic measure in VMO.

Distortion and Distribution of Sets under Inner Functions

analisi matematica

Distortion and distribution of sets under inner functions. It is a classical result that Lebesgue measure on the unit circle is invariant under inner functions xing the origin and in this setting, the distortion of Hausdor contents has also been studied by Fernández and Pestana. similar results focusing on inner functions with xed points on the unit circle. We will present In particular, our results yield information not only on the size of preimages of sets under inner functions, but also on their distribution with respect to a given boundary point. Join work with Matteo Levi and Oí Soler iGibert.

Settembre

24

Differential calculus for Jordan algebras and Jordan modules

Alessandro Carotenuto

Abstract: "The notion of differential calculus for Jordan algebras, as well as the theory of connections for Jordan modules, was recentely introduced by M. Dubois-Violete. In this talk I will give an overview of the representation theory for Jordan algebras and then I will present some results we obteined in the context of differential calculus for Jordan algebras and modules. If the time allows, I will also review two proposals from M. Dubois-Violette in which the use of Jordan algebra allows for a mathematical formalization of the Standard model of particle physics"

On the interpolation of Hardy spaces of Dirichlet series

analisi matematica

In this talk, we will prove that, for the usual methodsof interpolation, the interpolated space between two Hardy spaces ofDirichlet series is not the Hardy space of Dirichlet series we wouldexpect (based on a joint work with M. Mastylo).

Image reconstruction from x-ray radiography

analisi numerica

China-Italy scientific collaboration meeting on numerical methods on image processing

analisi numerica

Luglio

16

Lorenzo Ruffoni (Florida State University)

Strict hyperbolization and special cubulation

Gromov introduced some procedures to turn a given polyhedron into a new one endowed with a piecewise Euclidean metric of non-positive curvature, while preserving some of its original topological features. Charney and Davis have proposed a refinement of Gromov's construction in which the new space carries a strictly negatively curved metric, and thus has hyperbolic fundamental group. After reviewing these constructions, we will discuss the problem of cubulating these groups, and some applications. This is joint work with J. Lafont.

Luglio

10

Gianmarco Giovannardi (Università di Bologna)

Variational problems for curves

In this talk I will focus on the first variational formula for the length functional. First of all I will start by showing the geodesic equation for curves immersed in a Riemannian manifold. Then, I will describe the behavior of the length functional for curves immersed in a graded manifold equipped with a Riemannian metric. The first variation can be computed only if the curve can be deformed in a suitable sense. This condition is given by the surjection of the holonomy map that expresses the controllability of a differential system along the curve. This map gives us the possibility to define when a curve is regular or singular. Finally, I will show the Euler-Lagrange equation for immersed regular curves.

The conformal Geometry of Embedded hypersurfaces with Boundary

algebra e geometria

For an embedded conformal hypersurface with boundary, we construct critical order local invariants and their canonically associated differential operators. These are obtained by a construction that uses a singular Yamabe problem and a corresponding minimal hypersurface with boundary. They include an extrinsic Q-curvature for the boundary of the embedded conformal manifold and, for its interior, the Q-curvature and accompanying boundary transgression curvatures. This gives universal formulae for extrinsic analogs of Branson Q-curvatures that simultaneously generalize the Willmore energy density, including the boundary transgression terms required for conformal invariance.

Finite symmetries of complex K3 surfaces

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Named by André Weil after the mathematicians Kodaira, Kummer and Kähler and the mountain K2, K3 surfaces are one of the prominent classes of complex surfaces. Their (finite) symmetries are closely related to the Mathieu group M_{24}. This group, discovered in the 19th century, is one of the finite sporadic simple groups.

Simplicial volume for non-compact manifolds

nell'ambito della serie: SEMINARI BAD

algebra e geometria

\emph{Simplicial volume} is a homotopy invariant of compact manifolds introduced by Gromov in the early '80s. It measures the complexity of manifolds in terms of (real) singular chains. Despite its topological meaning, simplicial volume has many applications in geometry. For instance it provides useful information about the Riemannian volume of negatively curved manifolds. However, as soon as we consider non-compact manifolds its geometric meaning is much more mysterious. Indeed, one may extend the notion of simplicial volume to non-compact manifolds by considering locally finite homology, but its behaviour is not yet well understood. Among the key ingredients for studying the simplicial volume of (non-)compact manifolds, \emph{amenable groups} play a fundamental role. Recall that amenable groups are groups carrying invariant means. The aim of this talk is to investigate the relation between simplicial volume and amenable groups. More precisely, after having introduced the notion of \emph{amenable covering} of compact manifolds, we will discuss a classical vanishing result for the simplicial volume. Later we will construct special amenable coverings of non-compact manifolds. This will allow us to obtain the corresponding vanishing result in this setting. If there will be enough time, we will discuss a striking application of these results: the simplicial volume of the product of at least three non-compact manifolds always vanishes. Some results presented in this talk are part of a joint work with Roberto Frigerio.

Tame CP1 structure on the thrice punctures sphere with elliptic holonomy.

SEMINARIO INTERDISCIPLINARE

CP1 structures are geometric structures modelled on the complex projective line, acted on by the projective group PSL(2,C). These structures are not as rigid as Riemmannian structure (like Euclidean, hyperbolic or spherical), nor as flexible as conformal structures. For example they still allow the notion of circles and therefore can be used to study circle packings. In this talk we show that all CP1 structures on the thrice punctured sphere -with elliptic holonomy- that are tame (i.e. the developing map extends continuously to the ends) can be constructed by elementary cutting and gluing (i.e. grafting) on simple triangular structures. The talk will be accessible to non-experts, with minimal background in topology.

Viscosity solutions approach to variational problems

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

In this talk we discuss some extensions of the classical Krylov-Safonov Harnack inequality. After reviewing the standard regularity theory, we will introduce a weaker notion of viscosity solutions. The novelty is that we consider functions that do not necessarily satisfy an infinitesimal equation but rather exhibit a two-scale behavior. Roughly, our viscosity solutions satisfy comparison in a neighborhood of a touching point whose size depends on the properties of the test functions. As an application, we recover the C^{1,\alpha} estimates of Almgren and Tamanini for quasi-minimizers of the perimeter functional. We also establish the regularity of the free boundary for almost minimizers of one-phase type problems.

Luglio

04

Ovidiu Savin, Columbia University, New York

The singular set in the fully nonlinear obstacle problem

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

For the Obstacle Problem involving a convex fully nonlinear elliptic operator, we show that the singular set of the free boundary stratifies. The top stratum is locally covered by a $C^{1,\alpha}$-manifold, and the lower strata are covered by $C^{1,\log^\eps}$-manifolds. This essentially recovers the regularity result obtained by Figalli-Serra when the operator is the Laplacian.

Homotopical decompositions of Vietoris-Rips complexes

algebra e geometria

In this talk we will investigate the idea of decomposing a simplicial complex and we will study it from a homotopical point of view. Since in general the initial simplicial complex and its decompositions do not have the same homotopy type, a natural question that might arise is under which conditions they are weakly equivalent. Furthermore, we will use the general machinery developed in the context of Vietoris-Rips complexes and their decompositions.

Edoardo Fossati (SNS - Pisa)

Che cos'è un riempimento simplettico

algebra e geometria

CV

In questo seminario verranno introdotti alcuni concetti che stanno alla base della topologia di contatto. Vedremo come rappresentare oggetti 4-dimensionali e studiare problemi complessi armati solamente di matite colorate. Cosa aspettarsi da questo seminario? Il taglio sarà puramente topologico: si parlerà di varietà di contatto, ma non di "foliazioni caratteristiche su superfici convesse", si parlerà di varietà di Stein senza coinvolgere né "funzioni pluri-subarmoniche esaustive" né "coomologia di fasci analitici coerenti".

The solution of the multiscale SK model

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

fisica matematica

We consider a mean field spin system with random interaction in a multiscale thermal equilbrium, namely when the thermodynamic equilibrium is reached through successive hierarchical thermalization. This model apperead in physical literature un the 90's as a possible framework describing thermodynamic properties of systems out of standard Gibbs equilibrium. We show how to construct a variational problem that gives a representation of the thermodynamic limit of the pressure density (generating functional). We present the main ideas behind the proof and discuss possible applications of the result.

25

Oriented first passage percolation on the hypercube

Marius Alexander Schmidt

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

We will discuss why the extremal process of oriented FPP on the hypercube converges to a Cox process with exponential intensity (arXiv:1804.03117 and 1808.04598)

Phase transitions in high-dimensional generalized linear models

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

fisica matematica

I will present a framework for rigorously establishing the information-theoretic limits in high-dimensional generalized linear models (GLMs). The GLM includes as special cases plethora of important models in signal processing (compressed-sensing, phase retrieval etc), communications, but also in learning such as the famous perceptron neural network. Many instances of GLMs have been analyzed in the statistical physics literature, in particular thanks to the heuristic replica method developed in the context of spin glasses. I will discuss a recent technique called « adaptive interpolation method » that allows to vindicate the statistical approach in a unified manner, as well as recent findings about the rich algorithmic behaviors encountered in such models.

Direct/Inverse Hopfield model and Restricted Boltzmann Machines

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

fisica matematica

Mean-field methods fail to reconstruct the parameters of the model when the dataset is clusterized. This situation is found at low temperatures because of the emergence of multiple thermodynamic states. The paradigmatic Hopfield model is considered in a teacher student scenario as a problem of unsupervised learning with Restricted Boltzmann Machines (RBM). For different choices of the priors on units and weights, the replica symmetric phase diagram of random RBM’s is analyzed and in particular the paramagnetic phase boundary is presented as directly related to the optimal size of the training set necessary for a good generalization. The connection between the direct and inverse problem is pointed out by showing that inference can be efficiently performed by suitably adapting both standard learning technique and standard approaches to the direct problem.

Quenched and annealed Ising models on random graphs

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

fisica matematica

The ferromagnetic Ising model is a paradigmatic model of statistical physics used to study phase transitions in lattice systems. In this talk I shall consider the setting where the regular spatial structure is replaced by a random graph, which is often used to model complex networks. I shall treat both the case where the graph is essentially frozen (quenched setting) and the case where instead it is rapidly changing (annealed setting). I shall prove that quenched and annealed may have different critical temperatures, provided the graph has sufficient inhomogeneity. I shall also discuss how universal results (law of large numbers, central limit theorems, critical exponents) are affected by the disorder in the spatial structure. The picture that I will present emerges from several joint works, involving V.H. Can, S. Dommers, C. Giberti, R.van der Hofstad and M.L.Prioriello.

The generalized TAP free energy

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

fisica matematica

Spin glasses are disordered spin systems initially invented by theoretical physicists with the aim of understanding some strange magnetic properties of certain alloys. In particular, over the past decades, the study of the Sherrington-Kirkpatrick (SK) mean-field model via the replica method has received great attention. In this talk, I will discuss another approach to studying the SK model proposed by Thouless-Anderson-Palmer (TAP). I will explain how the generalized TAP correction appears naturally and give the corresponding generalized TAP representation for the free energy. Based on a joint work with D. Panchenko and E. Subag.

The replica trick in the frame of the interpolation scheme

nel ciclo di seminari: MATHEMATICAL PHYSICS OF DISORDERED SYSTEMS

fisica matematica

We give a short review about the replica trick in the frame of a simple interpolation scheme on the number of replicas. In this way we avoid any problem with analytic continuation and recognise that replica symmetry breaking is in fact characterised by a phase transition. We give application of the method to the case of the Random Energy Model, and the Sherrington-Kirkpatrick mean field spin glass model.

On a mathematical theory of repeated quantum measurements

fisica matematica

The statistics of the (finite alphabet) outcomes of repeated quantum measurements is studied by methods of thermodynamic formalism. Viewed as one-dimensional spin systems with long range interactions, repeated quantum measurements exhibit very rich (and sometimes very singular) thermodynamic behaviour. We will describe general thermodynamical formalism of these systems and illustrate its unexpected features on a number of examples.

Differential calculus on quantum flags

algebra e geometria

We describe a unique and with minimal dimension, differential calculus over quantum flags

Cyclic Gerstenhaber-Schack cohomology

algebra e geometria

Cyclic Gerstenhaber-Schack cohomology" "In this talk, we answer a long-standin question by explaining how the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Gerstenhaber bracket, and that allows to define cyclic Gerstenhaber-Schack cohomology. In case the Hopf algebra in question is finite dimensional, the Gerstenhaber bracket turns out to be zero in cohomology and hence the interesting structure is not given by this e2-algebra structure but rather by the resulting e3-algebra structure, which is expressed in terms of the cup product and B."

Cell complex structure of quantum projective spaces

algebra e geometria

I will explain how the graph C*-algebras of a trimmable graph can be decomposed as U(1)-equivariant pullback of two simpler C*-algebras. A main example is given by the algebra of Vaksman-Soibelman quantum sphere, that can be realized as pushout of a lower dimensional quantum sphere and the product of a quantum ball with a circle (we understand the pullback of C*-algebras as pushout of the underlying "noncommutative spaces"). The U(1)-invariant part of this pullback diagram gives a "CW complex" realization of quantum projective spaces that allows to give an explicit description of the K-theory generators. Further examples include quantum lens spaces, one-loop extensions of Cuntz algebras of the Toeplitz algebra. This is a joint work with Francesca Arici, Piotr M. Hajac and Mariusz Tobolski.

Higher brackets on cyclic and negative cyclic (co)homology

algebra e geometria

Both the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups and the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups are examples of the brackets arising from the general noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in which this BV bracket vanishes, one obtains an e_3-algebra structure on Hochschild cohomology. The results are given in the general and unifying setting of (opposite) cyclic modules over (cyclic) operads. Joint work with Niels Kowalzig (arXiv:1712.09717)

Toric generalised Kahler manifolds and elliptic functions

algebra e geometria

In this talk I shall sketch a construction of generalised Kahler 4-manifold for toric fano manifolds. It has been shown that certain type of generalised Kahler structures can be encoded in terms of a Morita equivalence of two holomorphic Poisson structures. We combine this with a construction due to Hitchin of bi-hermitian 4 manifolds in the toric setting, where, thanks to the action angle coordinates, one can be very explicit. We show that one obtains a generalised Kahler potential expressed in terms of Weistrass elliptic functions.

Koszul complex on Supervarieties

algebra e geometria

Symmetric pairs for Nichols algebras of diagonal type

algebra e geometria

The theory of quantum symmetric pairs provides coideal subalgebras of quantum groups which give rise to braided module categories over braided monoidal categories. In this talk I will outline a program to extend the theory of quantum symmetric pairs to a setting of (pre-)Nichols algebras (of diagonal type). I will explain how the resulting coideal subalgebras are obtained via star products on partial bosonizations. This new perspective allows a conceptual, bar-involution free interpretation of the quasi K-matrix, which is the crucial ingredient in the construction of the braiding on the corresponding module category. The talk is based on joint work with Milen Yakimov.

Compact Quantum Homogeneous Spaces

Abstract: The notion of a covariant Hermitian structure was recently introduced as an algebraic framework in which to perform noncommutative Hermitian geometry on quantum homogeneous spaces. Combining covariant Hermitian structures with Woronowicz's theory of compact quantum groups produces a canonical Hilbert completion carrying a beautiful interaction of analysis, geometry, and algebra. We highlight two aspects of this completion: Firstly, the associated $*$-algebra of smooth functions, and secondly the interaction of Dirac operator index theory with noncommutative Dolbeault cohomology and noncommutative Fano structures. Time permitting, we will discuss the relationship of these structures with Connes' notion of a spectral triple. Throughout, the irreducible quantum flag manifolds, endowed with their Heckenberger--Kolb differential calculus, are presented as motivating examples, focusing in particular on the quantum Grassmannians.

Discrete groups and the geometry of compact quantum groups

algebra e geometria

I will present some reformulation of geometric-topological notions in terms of representation theory of compact topological groups, which may be used to extend such notions to the quantum setting. I will then explain how some natural issues in compact quantum group theory are related to difficult problems in the context of discrete groups.

Derivations based differential calculi on non commutative algebras deforming a class of three dimensional classical spaces

algebra e geometria

In this talk we shall describe how it is possible to define a differential calculus on a Lie type non commutative spaces deforming spaces which are classically given by the foliation given by the coadjoint action of a 3 dimensional Lie algebra on R^3.

Introduction to Poisson geometry and the BV formalism

fisica matematica

We introduce Poisson geometry and we discuss how to use it within the context of BV formalism

Derivations based differential calculi on non commutative algebras deforming a class of three dimensional classical spaces

algebra e geometria

In this talk we shall describe how it is possible to define a differential calculus on a Lie type non commutative spaces deforming spaces which are classically given by the foliation given by the coadjoint action of a 3 dimensional Lie algebra on R^3.

17

Andras, Domokos (California State University, Sacramento)

PDEs properties in Carnot groups

analisi matematica

Derivations based differential calculi on non commutative algebras deforming a class of three dimensional classical spaces

algebra e geometria

In this talk we shall describe how it is possible to define a differential calculus on a Lie type non commutative spaces deforming spaces which are classically given by the foliation given by the coadjoint action of a 3 dimensional Lie algebra on R^3.

17

Juan Manfredi, Pittsburgh University

analisi matematica

Francesca Colasuonno, Università di Torino

Some supercritical problems with Neumann boundary conditions

analisi matematica

Some recent results in the study of fractional mean curvature flow

analisi matematica

Recent results regarding the regularity of p-harmonic functions on Carnot groups

analisi matematica

Enrico Valdinoci, University of Western Australia

Nonlocal minimal graphs in the plane are generically sticky

analisi matematica

Italo Capuzzo Dolcetta

TBA (in the Workshop: Something about nonlinear problems)

analisi matematica

Eugenio Vecchi, Università di Trento

TBA (in the Workshop: Something about nonlinear problems)

analisi matematica

The twisted cotangent bundle of a hyperkahler manifold

algebra e geometria

Let X be a complex projective Hyperk ̈ahler manifold. By a recent result of H ̈oring and Peternell the cotangent bundle of X is not pseudoeffective. One way to measure this negativity more precisely is to give sufficient conditions on an ample line bundle A such that the twist ΩX ⊗ A is pseudoeffective. I will give a sufficient condition that depends only on the Segre classes and the Beauville–Fujiki form of X. Then I will discuss when this sufficent condition is also necessary. This is a joint work with Andreas H ̈oring.

Giulio Tralli, Università di Padova

Some global Sobolev inequalities related to Kolmogorov-type operators

analisi matematica

Serena Dipierro, University of Western Australia

A free boundary problem driven by the biharmonic operators

analisi matematica

TBA (in the Workshop: Something about nonlinear problems)

analisi matematica

Juan Manfredi, Pittsburgh University

Random walks and random tug of war in the Heisenberg group

analisi matematica

Mattia G. Bergomi, Pietro Vertechi

Persistenza topologica e non - 2

algebra e geometria

Slides M. Ferri

Slides A. Mella

Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other combinatorial, based on arbitrary set-valued functors. To unify the study of topological and combinatorial persistence in a common categorical framework, we give axioms for a generalized rank function on objects in a target category, so that functors to that category induce persistence functions. We port the interleaving and bottleneck distances to this novel framework and generalize classical equalities and inequalities. Unlike sets and vector spaces, in many categories the rank of an object does not identify it up to isomorphism: to preserve information about the structure of persistence modules, we define colorable ranks, persistence diagrams and prove the equality between multicolored bottleneck distance and interleaving distance in semisimple Abelian categories. To illustrate our framework in practice, we give examples of multicolored persistent homology on filtered topological spaces with a group action and labeled point cloud data. SLIDES - The ones of the present talk are here: https://mgbergomi.github.io/slideshow_bologna/ The ones of last week are attached.

Laboratorio PLS "Tecniche matematiche per l'animazione" - III incontro

nel ciclo di seminari: LABORATORIO PLS "TECNICHE MATEMATICHE PER L'ANIMAZIONE"

SEMINARIO INTERDISCIPLINARE

Laboratorio PLS "Tecniche matematiche per l'animazione" - II incontro

nel ciclo di seminari: LABORATORIO PLS "TECNICHE MATEMATICHE PER L'ANIMAZIONE"

SEMINARIO INTERDISCIPLINARE

Introduction to scattering resonances 4

Scattering resonances replace bound states/eigenvalues for spectral problems in which escape (scattering) to infinity is possible. These states have rates of oscillation and decay and that information is elegantly encoded in considering the corresponding ``eigenvalues" as poles of the meromorphic continuation of Green functions. The most famous ``pure maths" example is given by zeros of the Riemann zeta function which can be interpreted as resonances for scattering on the modular surface. In ``applied maths" they appear anywhere from gravitational waves to MEMS (Micro-Electro-Mechanical Systems). The mini course will provide a gentle introduction in the setting of potential scattering in dimension three. Only basic functional analysis will be a prerequisite. 1. One dimensional scattering: intuition behind outgoing and incoming waves and the definition of scattering resonances. 2. Analytic Fredholm theory and, as application, meromorphic continuation of Green's function for potentials scattering in dimension three. 3. Resonance free regions and expansion of waves in terms of resonances. 4. Counting resonances: upper bounds and existence (and some open problems). Complex valued potentials with no resonances. Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math Sci '17) will provide a reference with a more detailed presentation in the forthcoming book http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).

Laboratorio PLS "Tecniche matematiche per l'animazione" - I incontro

nel ciclo di seminari: LABORATORIO PLS "TECNICHE MATEMATICHE PER L'ANIMAZIONE"

SEMINARIO INTERDISCIPLINARE

Introduction to scattering resonances 3

Scattering resonances replace bound states/eigenvalues for spectral problems in which escape (scattering) to infinity is possible. These states have rates of oscillation and decay and that information is elegantly encoded in considering the corresponding ``eigenvalues" as poles of the meromorphic continuation of Green functions. The most famous ``pure maths" example is given by zeros of the Riemann zeta function which can be interpreted as resonances for scattering on the modular surface. In ``applied maths" they appear anywhere from gravitational waves to MEMS (Micro-Electro-Mechanical Systems). The mini course will provide a gentle introduction in the setting of potential scattering in dimension three. Only basic functional analysis will be a prerequisite. 1. One dimensional scattering: intuition behind outgoing and incoming waves and the definition of scattering resonances. 2. Analytic Fredholm theory and, as application, meromorphic continuation of Green's function for potentials scattering in dimension three. 3. Resonance free regions and expansion of waves in terms of resonances. 4. Counting resonances: upper bounds and existence (and some open problems). Complex valued potentials with no resonances. Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math Sci '17) will provide a reference with a more detailed presentation in the forthcoming book http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).

04

Xavier Cabré, ICREA and Universitat Politècnica de Catalunya (Barcelona)

A gradient estimate for nonlocal minimal graphs

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

The talk will be concerned with s-minimal surfaces, that is, hypersurfaces of R^n with zero nonlocal mean curvature. These are the equations associated to critical points of the fractional s-perimeter. We will present a recent result in collaboration with M. Cozzi in which we establish, in any dimension, a gradient estimate for nonlocal minimal graphs. It leads to their smoothness, a result that was only known for n=2 and 3 (but without a quantitative bound); in higher dimensions only their continuity had been established. We will also present a work with E. Cinti and J. Serra in which we prove that half spaces are the only stable s-minimal cones in R^3 for s sufficiently close to 1.

Persistenza topologica e non - 1

algebra e geometria

Persistent Homology is one of the main tools of Topological Data Analysis. It consists in comparing, by homology, all pairs of sublevel sets of a pair (X, f) where X is a topological space (or a simplicial complex) and f is a real valued function on it. This produces a Persistence Diagram, a standard object which turns very useful in shape analysis and classification. But is the topological setup necessary for getting persistence diagrams? Massimo Ferri will introduce (classical) persistent homology in the first part. Alessandro Mella will then show a wide generalization of it and some initial applications.

Introduction to scattering resonances 2

Scattering resonances replace bound states/eigenvalues for spectral problems in which escape (scattering) to infinity is possible. These states have rates of oscillation and decay and that information is elegantly encoded in considering the corresponding ``eigenvalues" as poles of the meromorphic continuation of Green functions. The most famous ``pure maths" example is given by zeros of the Riemann zeta function which can be interpreted as resonances for scattering on the modular surface. In ``applied maths" they appear anywhere from gravitational waves to MEMS (Micro-Electro-Mechanical Systems). The mini course will provide a gentle introduction in the setting of potential scattering in dimension three. Only basic functional analysis will be a prerequisite. 1. One dimensional scattering: intuition behind outgoing and incoming waves and the definition of scattering resonances. 2. Analytic Fredholm theory and, as application, meromorphic continuation of Green's function for potentials scattering in dimension three. 3. Resonance free regions and expansion of waves in terms of resonances. 4. Counting resonances: upper bounds and existence (and some open problems). Complex valued potentials with no resonances. Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math Sci '17) will provide a reference with a more detailed presentation in the forthcoming book http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).

Introduction to scattering resonances 1

Scattering resonances replace bound states/eigenvalues for spectral problems in which escape (scattering) to infinity is possible. These states have rates of oscillation and decay and that information is elegantly encoded in considering the corresponding ``eigenvalues" as poles of the meromorphic continuation of Green functions. The most famous ``pure maths" example is given by zeros of the Riemann zeta function which can be interpreted as resonances for scattering on the modular surface. In ``applied maths" they appear anywhere from gravitational waves to MEMS (Micro-Electro-Mechanical Systems). The mini course will provide a gentle introduction in the setting of potential scattering in dimension three. Only basic functional analysis will be a prerequisite. 1. One dimensional scattering: intuition behind outgoing and incoming waves and the definition of scattering resonances. 2. Analytic Fredholm theory and, as application, meromorphic continuation of Green's function for potentials scattering in dimension three. 3. Resonance free regions and expansion of waves in terms of resonances. 4. Counting resonances: upper bounds and existence (and some open problems). Complex valued potentials with no resonances. Section 2 of https://math.berkeley.edu/~zworski/revres.pdf (Bull Math Sci '17) will provide a reference with a more detailed presentation in the forthcoming book http://math.mit.edu/~dyatlov/res/ (AMS '19, to appear).

Nonlocal minimal surfaces

nell'ambito della serie: COLLOQUIO DI DIPARTIMENTO

SEMINARIO INTERDISCIPLINARE

CV

In 2010 Caffarelli, Roquejoffre & Savin started the study of nonlocal minimal surfaces, that is, of hypersurfaces in Euclidean space with zero nonlocal mean curvature. This is the equation associated to critical points of the fractional perimeter. Among other motivations (such as image processing), they are relevant in phase-transition phenomena in the presence of long range interactions. Since their introduction, nonlocal minimal surfaces have attracted much attention, first and foremost to understand their regularity and to make progress towards their classification. We will describe the results obtained up to date, as well as the remaining open problems. As we will see, there is a remarkable resemblance with the classical theory of minimal surfaces.

Maggio

31

Un'applicazione del teorema di perturbazione di Miyadera a un problema misto iperbolico

Dott. Amedeo Altavilla

analisi matematica

analisi matematica

Consideriamo un problema misto lineare iperbolico del secondo ordine, con una condizione al contorno dinamica contenente anche un operatore ellittico sulla frontiera del dominio. Utilizzando un certo teorema di perturbazione per semigruppi fortemente continui (solitamente attribuito a Miyadera), estendiamo un caso particolare (essenzialmente gia' noto) a situazioni molto più' generali. Questi risultari sono applicabili anche al caso di condizioni al contorno di Wentzell.

E-cigarette smoking with peer pressure - II

fisica matematica

A model is presented involving non-smokers, tobacco cigarette smokers, and those who smoke electronic cigarettes. The transfer from the tobacco smoker class to the e-cigarette class is via a peer pressure term. It is shown that there are three distinct equilibria. One involves no smokers of any kind. A second has only non-smokers and smokers of tobacco cigarettes. The third has a steady state with all three categories. Conditions are derived under which each equilibrium state will be stable. Numerical simulations are given that show the convergence to steady state for the third equilibrium. Simulations are also performed for a more general model where the peer pressure term in the tobacco to e-cigarette transition involves also a conformity bias.

An introduction to score-driven models

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

In recent years, dynamic conditional score-driven (DCS) models have attracted lot of interest in Economics, Finance, and Econometrics. However, their potential extends far beyond. The reason lies in the simplicity of the approach to time-series modelling and the easiness in parameter estimation. After presenting some motivating examples, in the first part of the talk I review the main theoretical properties of DCS models and discuss the flexibility of the approach in empirical applications. In the second part, I detail an application to high-frequency financial data. The approach has proved to be very effective in disentangling the fundamental price dynamics from micro-structure noise and in recovering the seasonal behaviour of prices at intra-day level.

The Lattice of Subracks of a Rack is Atomic and Complemented

algebra e geometria

A rack is a set R together with a binary operation ▷ such that • For each x, y, z ∈ R, x ▷ (y ▷ z) = (x ▷ y) ▷ (x ▷ z), and • for each x, y ∈ R, there exists a unique element z ∈ R with x ▷ z = y. If we have the extra condition x ▷ x = x for each x ∈ R, then R is called a quandle. For an example, a group G together with the operation x ▷ y = xyx−1 is a quandle. The study of racks and quandles dates back to 1943 when Takasaki used a certain algebraic structure to study reflections in finite geometries [?]. Since then, Racks and quandles have been used in some branches of mathematics such as knot theory for encoding knot diagrams. In 2015, I. Heckenberger et al. started the study of racks in a combined perspective of combinatorics and group theory. Indeed, they considered the lattice of subracks of a rack and obtained some interesting results [?]. Moreover, they posed some important questions in the last section of their paper. Two of these questions have been solved in [?] and [?]. Actually, it has been shown that the lattice of subracks of a rack is atomic, and this lattice for finite racks is complemented but there are some infinite racks whose lattices are not complemented.

E-cigarette smoking with peer pressure - I

fisica matematica

A model is presented involving non-smokers, tobacco cigarette smokers, and those who smoke electronic cigarettes. The transfer from the tobacco smoker class to the e-cigarette class is via a peer pressure term. It is shown that there are three distinct equilibria. One involves no smokers of any kind. A second has only non-smokers and smokers of tobacco cigarettes. The third has a steady state with all three categories. Conditions are derived under which each equilibrium state will be stable. Numerical simulations are given that show the convergence to steady state for the third equilibrium. Simulations are also performed for a more general model where the peer pressure term in the tobacco to e-cigarette transition involves also a conformity bias.

Spaces of Entire Functions in C^{n+1}

analisi matematica

Locandina

A renowned space of entire functions of one complex variable is the Paley–Wiener space P W A , that is, the space of entire functions of exponential type A whose restriction to the real line is square integrable. In this talk I will present a generalization of P W A in several complex variables. In particular, I will consider entire functions which satisfy a suitable exponential growth condition and whose restriction to the boundary of the Siegel half-space satisfy some integrability conditions. For this space I will provide a Paley–Wiener type characterization and a sampling result. This is a joint work with Marco Peloso and Maura Salvatori.

algebra e geometria

interdisciplinare

I politopi (come cubi, piramidi, tetraedri...), studiati fin dagli albori della matematica, sono tuttora di moda anche grazie all'avvento della digitalizzazione e dell'ottimizzazione lineare. Il grafo di un politopo è semplicemente la struttura formata dai suoi vertici e dai suoi lati. Il diametro e la connettività di questi grafi sono di particolare interesse per le applicazioni. Parleremo di un approccio metrico (con K.Adiprasito) e di un approccio algebrico (con M. Varbaro, M. Dimarca, B.Bolognese) che stanno dando risultati promettenti.

Prescribing the \bar{Q}'-Curvature on three dimensional Pseudo-Einstein Manifolds

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Exponential convergence of Parabolic Optimal Transport on bounded domains

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Topics in Global Analysis - 6

nel ciclo di seminari: TOPICS IN GLOBAL ANALYSIS.

SEMINARIO INTERDISCIPLINARE

A quantitative isoperimetric inequality in the plane

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Tridispersive thermal convection

fisica matematica

We derive the linear instability and nonlinear stability thresholds for a problem of thermal convection in a tridispersive porous medium with a single temperature. Importantly we demonstrate that the nonlinear stability threshold is the same as the linear instability one. The significance of this is that the linear theory is capturing completely the physics of the onset of thermal convection.

Minkowski inequality for mean convex domains

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

On the 1/H flow via p-Laplace approximation under Ricci lower bounds

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Γ-convergence for integral functionals depending on vector fields

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Harnack inequality for a class of degenerate elliptic PDEs in non-divergence form

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Towards the critical exponents for fully nonlinear degenerate Lane-Emden type equations

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Asymptotic decay results for critical growth equations

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Symmetry in R^2 of global minimizers of a general type of nonlocal energy

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Sobolev immersions and Torsion functions

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Symmetry results for critical p-Laplace equations

analisi matematica

Seminario nell'ambito della conferenza: Variational and PDE problems in Geometric Analysis, II

Topics in Global Analysis - 5

nel ciclo di seminari: TOPICS IN GLOBAL ANALYSIS.

SEMINARIO INTERDISCIPLINARE

Thermal convection in double porosity media

fisica matematica

A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores due to cracks or fissures in the solid skeleton. The linear instability and nonlinear stability thresholds for a problem of thermal convection in bidispersive porous media with a single temperature are obtained and we show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is able to capture completely the physics of the onset of thermal convection.

Elementi totalmente commutativi nei gruppi di Coxeter affini

algebra e geometria

Sia W un gruppo di Coxeter. Un elemento w di W è totalmente commutativo (TC) se prese due qualsiasi delle sue espressioni ridotte si può passare dall’una all’altra soltanto con una serie di scambi di generatori che commutano. Gli elementi TC sono stati studiati nel caso finito da Stembridge e indicizzano una base dell’algebra di Temperley-Lieb generalizzata associata a W. In questo talk daremo una classificazione degli elementi TC nel caso dei gruppi di Coxeter affini e mostreremo nel caso finito di tipo A e in quello affine di tipo \tilde{A} delle interpretazioni combinatorie che permettono il calcolo della funzione generatrice degli elementi TC secondo la lunghezza di Coxeter.

Horizontally isotropic bidispersive thermal convection

fisica matematica

A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores due to cracks or fissures in the solid skeleton. We present general equations for thermal convection in a bidispersive porous medium when the permeabilities, interaction coefficient and thermal conductivity are anisotropic but symmetric tensors. In this case, we show exchange of stabilities holds and fluid movement will commence via stationary convection, and additionally we show the global nonlinear stability threshold is the same as the linear instability one. Attention is then focused on the case where the interaction coefficient and thermal conductivity are isotropic, and the permeability is isotropic in the horizontal directions, although the permeability in the vertical direction is different. The nonlinear stability threshold is calculated in this case and numerical results are presented and discussed in detail.

Topics in Global Analysis - 4

nel ciclo di seminari: TOPICS IN GLOBAL ANALYSIS.

SEMINARIO INTERDISCIPLINARE

Derivations base differential calculi on a class of Lie type non commutative spaces.

algebra e geometria

In this talk we shall present how the embedding of 3 dimensional Lie algebras g within the 4 dimensional Moyal space allows to define a differential calculus on non commutative spaces deforming the foliation coming from the coadjoint action of g on its dual. This differential calculus has a frame, so the spaces turn to be parallelisable.

Symmetric pairs for Nichols algebras of diagonal type

algebra e geometria

The theory of quantum symmetric pairs provides coideal subalgebras of quantum groups which give rise to braided module categories over braided monoidal categories. In this talk I will outline a program to extend the theory of quantum symmetric pairs to a setting of (pre-)Nichols algebras (of diagonal type). I will explain how the resulting coideal subalgebras are obtained via star products on partial bosonizations. This new perspective allows a conceptual, bar-involution free interpretation of the quasi K-matrix, which is the crucial ingredient in the construction of the braiding on the corresponding module category. The talk is based on joint work with Milen Yakimov.

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

Topics in Global Analysis - 3

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

nel ciclo di seminari: TOPICS IN GLOBAL ANALYSIS.

SEMINARIO INTERDISCIPLINARE

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

Poincaré and Sobolev inequalities for differential forms on Euclidean spaces and Heisenberg groups.

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

In this talk we present some recent results obtained in collaboration with B. Franchi and P. Pansu about Poincaré and Sobolev inequalities in Heisenberg groups (some results are new also for Euclidean spaces). For $L^p$, $p>1$, the estimates are consequence of singular integral estimates. I would like to concentrate the seminar, in particular, to the limiting case $L^1$, where the exterior Rumin-differential of a differential form is measured in $L^1$ norm. Unlike for $L^p$, $p>1$, the estimates are doomed to fail in top degree. In the limiting case, the singular integral estimates are replaced with inequalities which go back to Bourgain-Brezis and Lanzani-Stein in Euclidean spaces, and to Chanillo-Van Schaftingen and Baldi-Franchi-Pansu in Heisenberg groups.

“Mathematical imaging models for digital art conservation and restoration”

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

analisi numerica

Nonlinear Schroedinger equations: theory and applications

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

In this talk we discuss some recent results for a class of nonlinear models in Quantum Mechanics. In particular, in the first lecture we review some general results concerning the nonlinear Schroedinger equation; while in the second lecture we discuss in detail an explicit model: the one-dimensional nonlinear Schroedinger equation with a symmetric double-well potential.

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

Spring School in Finance 2019 on XVA modeling

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

nel ciclo di seminari: SEMINARI DI FINANZA MATEMATICA

probabilità

Topics in Global Analysis - 2

nel ciclo di seminari: TOPICS IN GLOBAL ANALYSIS.

SEMINARIO INTERDISCIPLINARE

Approssimazioni diofantee, dimensione e formalismo termodinamico per gruppi Fuchsiani

SEMINARIO INTERDISCIPLINARE

Nella teoria classica delle approssimazioni diofantee l'insieme "Bad" è costituito dai numeri reali che sono male approssimabili dai razionali: si tratta di un insieme di misura zero e dimensione uno nella retta reale. Proprietà metriche più fini sono state studiate in dettaglio, sia nel caso classico che in varie generalizzazioni, in contesti legati alla dinamica su spazi omogenei ed altri spazi di moduli. L'insieme Bad ammette un'esaustione in sottoinsiemi Bad(c), la cui dimensione converge a 1 quando il parametro c>0 tende a zero. Nel caso classico D. Hensley ha ottenuto il primo ordine in c nello sviluppo asintotico della dimensione, attraverso un'analoga stima della dimensione dell'insieme dei numeri reali la cui frazione continua ha tutti i coefficienti parziali uniformemente limitati. Presenterò una generalizzazione della formula asintotica di Hensley nel contesto dei gruppi Fuchsiani, considerando l'insieme dei punti del bordo dello spazio iperbolico che sono male approssimabili per l'azione di un lattice non-uniforme G in PSL(2,R) ed un'esaustione di tale insieme in sottoinsiemi Bad(G,c), in termini di un parametro c>0. L'"espansione al bordo" di Bowen e Series permette di approssimare Bad(G,c) con insiemi di Cantor definiti dinamicamente, la cui dimensione può essere stimata con grande precisione tramite tecniche di formalismo termodinamico introdotte da Ruelle e Bowen. Un'analisi perturbativa del raggio spettrale dell'operatore di trasferimento fornisce la dimensione di Bad(G,c) al primo ordine in c.

Topics in Global Analysis - 1

nel ciclo di seminari: TOPICS IN GLOBAL ANALYSIS.

SEMINARIO INTERDISCIPLINARE

Abstract del ciclo di seminari

Maggio

Tiziano De Angelis (The University of Leeds)

Probabilistic results concerning smoothness of the value function and of the free boundary in optimal stopping

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

I will present probabilistic proofs of some regularity properties for the value function of general optimal stopping problems and for the associated optimal boundaries. In particular this talk focusses on C^1 regularity of the value function and Lipschitz continuity of the optimal boundary. Most of our arguments rely on fundamental concepts from the theory of Markov processes and bridge the probabilistic and the analytical strands of the literature on free boundary problems. I will also illustrate situations in which our work improves or complements known facts from PDE theory.

Dalle coordinate alla equivalenza: la conquista di von Neumann: II parte

abstract esteso

Da uno spazio vettoriale V si ricava il sistema dei sottospazi lineari L, ordinato per inclusione, e l'anello degli endomorfismi R. Escludendo dim(V)<3, e con cura se dim(V)=3, seguendo von Staudt da L si ricostruisce V, e seguendo Birkhoff / Menger si caratterizza L, ma NON si ha una equivalenza. Generalizzando a moduli (o oggetti abeliani) V tali che ogni endomorfismo ha nucleo e immagine sommandi diretti, seguendo von Neumann si ha una equivalenza tra R e L, che si caratterizzano come anelli con inversa generalizzata (ogni x ha un y tale che xyx=x) e reticoli modulari complementati, con un sistema di unità matriciali di ordine n>2 in R e una base omogenea di ordine n in L (arguesiano se n=3). Il contesto generale chiarisce che V si ricostruisce solo a meno di equivalenze di Morita. L'equivalenza di von Neumann è archetipica per studiare altri casi. Due sono notevoli: [1] Partendo da uno spazio di Hilbert H, si ha una equivalenza tra gli anelli con involuzione A di operatori lineari continui tali che A=A'' (dove X' indica l'anello degli operatori che commutano con ogni x in X) e gli associati poset con involuzione P(A) delle proiezioni (idempotenti autoaggiunti, ordinati per divisibilità ef=e, con 1-e ortocomplemento di e), sempre escludendo i casi in cui una immagine omomorfa sia del tipo escluso sopra per V. Per i fondamenti logici della meccanica quantistica (i casi esclusi hanno intepretazione logico - quantistica della loro esclusione), von Neumann era particolarmente interessato al sottocaso indecomponibile (e=0,1 le uniche proiezioni che danno una decomposizione diretta) e ``finito'' (xy=1 implica yx=1 in A, ovvero P(A) modulare): von Neumann caratterizza P(A) come geometria continua con ortogonalità che permette libera mobilità e univocamente determina una probabilità di transizione; A è l'anello degli elementi limitati (sottoanello generato dalle proiezioni) dentro l'anello R associato a L=P(A). [2] Altri tipi di moduli (o oggetti abeliani) V ammettono una equivalenza come sopra, per esempio il caso di Baer - Inaba - J\'onsson / Monk dei moduli su anelli artiniani a ideali principali, caso che include i gruppi abeliani finiti e gli spazi vettoriali finito dimensionali con l'azione di una trasformazione lineare o antilineare. Se l'equivalenza per un singolo V è rara, accade invece sempre che una catagoria abeliana di vari V si ricostruisca dal reticolo associato. Il risutato finale (combinando Freyd - Mitchell per categorie abeliane e il teorema di G. Hutchinson per i reticoli) è che tre teorie in tre diversi linguaggi permettono di fare le stesse cose: (algebra lineare classica) moduli su un anello (algebra lineare moderna) categorie abeliane (geometria d'incidenza sintetica moderna) reticoli modulari con 0 in cui gli elementi sono raddoppiabili: $\forall x\exists y,z$: $x\vee y=y\vee z=z\vee x$ & $x\wedge y=y\wedge z=z\wedge x=0.$

Mean value formulas and Liouville theorems for linear second order PDEs

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

We present several Mean Value formulas for solutions to linear second order PDEs endowed with smooth ''local fundamental solutions''. We then show how these formulas can be used to obtain Liouville Theorems for entire solutions. Our formulas are, in general, weighted average formulas. The relevant weights are ''densities with the mean value property'' a notion playing a central role in rigidity and stability problems. The results we present, related to the Mean value formulas, are obtained in collaboration with Giovanni Cupini. The ones related to the Liouville Theorems are joint works with Alessia Kogoj.

Metodi variazionali e PDE per l’elaborazione delle immagini

analisi numerica

Corso di Dottorato "Metodi variazionali e PDE per l’elaborazione delle immagini"- Cicli di seminari In this course we will present some classical and recent approaches for some problems in image reconstruction (denoising, deblurring, inpainting, shadow-removal…) formulated in terms of appropriate minimisation problems in infinite-dimensional functional spaces. We will further draw connections between these minimisation problems and parabolic Partial Differential Equations (PDEs) based on non-linear diffusion and possibly combined with transport terms. For the practical implementation of the models above, we will review standard finite difference stencils discussing their extensions to anisotropic diffusion and diffusion-transport problems. The course will be complemented by some practical MATLAB classes where simple exemplar problems will be solved by means of some reference iterative algorithms. Classical examples of imaging problems (denoising, deblurring, inpainting, segmentation..). Formulation as ill-posed inverse problems. Variational regularisation methods: regularisation term VS data fitting. Statistical interpretation: MAP estimation (2h) Sobolev spaces, standard methods in calculus of variations: a review. Total variation, the space of functions of bounded variations (2h) Second-order parabolic PDEs for image processing: heat equation, mean-curvature flow. Applications to image processing: linear VS non-linear PDEs. Regularisation of non-smoothness: lagged diffusivity. Anisotropic diffusion and diffusion-transport problems. (4h) Finite differences stencils for PDE-based imaging models. (2h) Numerical implementation and simulations in MATLAB for PDE-based models for image reconstruction (deblurring, inpainting, face fusion). (5h) Gli orari e le aule saranno specificati alla pagina web del dottorato ed inviati di volta in volta secondo il calendario 9/5: 2h (Teoria), mattina - 10/5: 2h (Teoria), mattina - 13/5: 2h (Teoria), mattina 14/5: 2h (Laboratorio), mattina 15/5: 2h (Teoria), mattina + 1h (Laboratorio), pomeriggio 16:5: 2h (Teoria), mattina - 17/5: 2h (Laboratorio), mattina

Interior Point Methods for Linear, Quadratic and Nonlinear Programming

nel ciclo di seminari: MARGHERITA PORCELLI

analisi numerica

SEMINARIO INTERDISCIPLINARE

La scienza in generale (e la matematica in particolare) sono sempre state presenti nei fumetti Disney. Nei 70 anni di storia di Topolino libretto sono apparse quasi ventimila storie sul principale fumetto Disney italiano, e ci sono quindi centinaia di storie in cui appare la matematica. In questo seminario analizzeremo i vari diversi usi della matematica all'interno delle storie di topi e paperi. Passeremo poi a descrivere la recente collana di storie scientifiche Topolino Comic&Science, a cui per la matematica hanno collaborato Roberto Natalini del CNR di Roma e il sottoscritto. Infine analizzeremo un possibile utilizzo laboratoriale nelle scuole del fumetto "Paperino e i ponti di Quackenberg" per l'apprendimento di alcuni rudimenti di teoria dei grafi, combinatoria e del concetto di dimostrazione.

Cobordismi razionali e omologia intera II

algebra e geometria

Nella prima parte introdurremo alcune nozioni basilari della "topologia in dimensione 3.5", e in particolare il gruppo di concordanza per nodi, i gruppi di cobordismo intero e razionale, e le loro relazioni. Richiameremo quindi la classificazione degli spazi lenticolari a meno di cobordismo razionale. Nella seconda parte enunceremo alcuni nostri risultati; in particolare dimostreremo che all'interno del sottogruppo dei lenticolari e' sempre possibile trovare rappresentanti la cui omologia a coefficienti interi si inietta nell'omologia di ogni altro elemento nella classe. Discuteremo infine alcune conseguenze. Tra queste, alcuni risultati di struttura per il gruppo di cobordismo razionale, criteri di divisibilita' per concordanze per nodi a 2 ponti e stime per il problema di Berge razionale ottenute applicando l'omologia di Heegaard Floer per nodi.

Interior Point Methods for Linear, Quadratic and Nonlinear Programming

nel ciclo di seminari: MARGHERITA PORCELLI

analisi numerica

Cobordismi razionali e omologia intera Parte I

algebra e geometria

Nella prima parte introdurremo alcune nozioni basilari della "topologia in dimensione 3.5", e in particolare il gruppo di concordanza per nodi, i gruppi di cobordismo intero e razionale, e le loro relazioni. Richiameremo quindi la classificazione degli spazi lenticolari a meno di cobordismo razionale. Nella seconda parte enunceremo alcuni risultati; in particolare dimostreremo che all'interno del sottogruppo dei lenticolari e' sempre possibile trovare rappresentanti la cui omologia a coefficienti interi si inietta nell'omologia di ogni altro elemento nella classe. Discuteremo infine alcune conseguenze. Tra queste, alcuni risultati di struttura per il gruppo di cobordismo razionale, criteri di divisibilita' per concordanze per nodi a 2 ponti e stime per il problema di Berge razionale ottenute applicando l'omologia di Heegaard Floer per nodi.

Dalle coordinate alla equivalenza: la conquista di von Neumann: I parte

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

abstract esteso

Da uno spazio vettoriale V si ricava il sistema dei sottospazi lineari L, ordinato per inclusione, e l'anello degli endomorfismi R. Escludendo dim(V)<3, e con cura se dim(V)=3, seguendo von Staudt da L si ricostruisce V, e seguendo Birkhoff / Menger si caratterizza L, ma NON si ha una equivalenza. Generalizzando a moduli (o oggetti abeliani) V tali che ogni endomorfismo ha nucleo e immagine sommandi diretti, seguendo von Neumann si ha una equivalenza tra R e L, che si caratterizzano come anelli con inversa generalizzata (ogni x ha un y tale che xyx=x) e reticoli modulari complementati, con un sistema di unità matriciali di ordine n>2 in R e una base omogenea di ordine n in L (arguesiano se n=3). Il contesto generale chiarisce che V si ricostruisce solo a meno di equivalenze di Morita. L'equivalenza di von Neumann è archetipica per studiare altri casi. Due sono notevoli: [1] Partendo da uno spazio di Hilbert H, si ha una equivalenza tra gli anelli con involuzione A di operatori lineari continui tali che A=A'' (dove X' indica l'anello degli operatori che commutano con ogni x in X) e gli associati poset con involuzione P(A) delle proiezioni (idempotenti autoaggiunti, ordinati per divisibilità ef=e, con 1-e ortocomplemento di e), sempre escludendo i casi in cui una immagine omomorfa sia del tipo escluso sopra per V. Per i fondamenti logici della meccanica quantistica (i casi esclusi hanno intepretazione logico - quantistica della loro esclusione), von Neumann era particolarmente interessato al sottocaso indecomponibile (e=0,1 le uniche proiezioni che danno una decomposizione diretta) e ``finito'' (xy=1 implica yx=1 in A, ovvero P(A) modulare): von Neumann caratterizza P(A) come geometria continua con ortogonalità che permette libera mobilità e univocamente determina una probabilità di transizione; A è l'anello degli elementi limitati (sottoanello generato dalle proiezioni) dentro l'anello R associato a L=P(A). [2] Altri tipi di moduli (o oggetti abeliani) V ammettono una equivalenza come sopra, per esempio il caso di Baer - Inaba - J\'onsson / Monk dei moduli su anelli artiniani a ideali principali, caso che include i gruppi abeliani finiti e gli spazi vettoriali finito dimensionali con l'azione di una trasformazione lineare o antilineare. Se l'equivalenza per un singolo V è rara, accade invece sempre che una catagoria abeliana di vari V si ricostruisca dal reticolo associato. Il risutato finale (combinando Freyd - Mitchell per categorie abeliane e il teorema di G. Hutchinson per i reticoli) è che tre teorie in tre diversi linguaggi permettono di fare le stesse cose: (algebra lineare classica) moduli su un anello (algebra lineare moderna) categorie abeliane (geometria d'incidenza sintetica moderna) reticoli modulari con 0 in cui gli elementi sono raddoppiabili: $\forall x\exists y,z$: $x\vee y=y\vee z=z\vee x$ & $x\wedge y=y\wedge z=z\wedge x=0.$

Metodi di collocazione RBF con applicazioni

analisi numerica

The Radial Basis Function (RBF) numerical methods are widely adopted methodologies for solving Partial Differential Equations (PDEs) via collocation schemes. These approaches do not require data structures and are generally known as meshfree methods. In this research field, an important issue to be addressed is related to the accuracy of the computed solution that may suffers from instability due to the ill-conditioning of the interpolation matrices. In this talk, the RBF collocation methods will be discussed for three case studies: i) the source-type ows in porous media problems; ii) a financial application to option pricing; iii) the implicit surface reconstruction.

Subdivision schemes and numerical linear algebra

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

This talk is devoted to the topic of subdivision schemes, a special class of iterative methods for generating continuous curves and surfaces via the recursive application of suitable local refinement rules to a coarse initial set of prescribed control points. Due to their efficiency and simplicity of implementation, subdivision schemes are ones of the most used representation models in computer graphics and animation. Recently, they have shown their usefulness also in different areas of application like biomedical imaging and isogeometric analysis. Important tools for both the construction of linear subdivision schemes and the analysis of their properties are provided by classical numerical linear algebra techniques or adequate modifications of them. In particular, the construction of interpolatory subdivision schemes capable of generating curves and surfaces that pass through the initial set of prescribed control points, relies on algebraic strategies that differ according to the symmetry properties of the underlying refinement rules. The goal of this talk is to show some of the constructive strategies proposed in the literature for the subclass of stationary, odd- and even-symmetric, interpolatory subdivision schemes of arbitrary arity.

Collocation Methods based on Radial Basis Functions with Applications

analisi numerica

The Radial Basis Function (RBF) numerical methods are widely adopted methodologies for solving Partial Differential Equations (PDEs) via collocation schemes. These approaches do not require data structures and are generally known as meshfree methods. In this research field, an important issue to be addressed is related to the accuracy of the computed solution that may suffers from instability due to the ill-conditioning of the interpolation matrices. In this talk, the RBF collocation methods will be discussed for three case studies: i) the source-type flows in porous media problems; ii) a financial application to option pricing; iii) the implicit surface reconstruction.

Elementi centrali in U(gl(n)) , funzioni Shifted Simmetriche, Basi di Sahi/Okounkov/Schur, immananti quantici e immananti di Capelli (parte seconda)

algebra e geometria

ll centro ζ(n) dell’inviluppo universale U(gl(n)) è isomorfo all’algebra ∧∗(n) dei polinomi shifted-simmetrici, via l’isomorfismo di Harish Chandra. L’algebra ∧∗(n)ammette una base (lineare) molto rilevante, costituita dai polinomi di Schur shifted-simmetrici, scoperti e caratterizzati da Kostant e Sahi, poi studiati sistematicamente da Okounkov, Olshanski et al. ll problema di descrivere/studiare gli elementi centrali in U(gl(n)) che corrispondono ai polinomi di Schur shifted-simmetrici è stato studiato da Okounkov in una serie lavori, tramite la nozione di immanante quantico. Gli immananti quantici sono elementi di ζ(n) assai difficilmente trattabili, come commentato dallo stesso Okounkov. Si è sviluppata una teoria sistematica degli immananti quantici e del centro ζ(n) basata sulla nuova nozione di immanante di Capelli in U(gl(n)). Gli immananti di Capelli sono una generalizzazione sia degli immananti classici (Littlewood/Richardson, 1934) di una matrice ad entries commutativi, sia del celebre determinante di Capelli in U(gl(n)), sono efficacemente trattabili, e formano un sistema di generatori lineari (compatibili con la filtrazione naturale di U(gl(n)). Gli immananti quantici di Okounkov risultano semplici combinazioni lineari di immananti di Capelli. ll passaggio al limite n→ ∞ per ζ(n) e ∧∗(n) si descrive esplicitamente come limite diretto rispetto alla decomposizione/proiezione Olshanski.

Interior Point Methods for Linear, Quadratic and Nonlinear Programming

nel ciclo di seminari: MARGHERITA PORCELLI

analisi numerica

Symmetrization method in statistical mechanics

fisica matematica

In this talk we discuss the so called symmetrization method and its applications to statistical mechanics.

Calculus, heat flow, optimal transport and curvature-dimension bounds in metric measure spaces

nell'ambito della serie: COLLOQUIO DI DIPARTIMENTO

scarica allegato

In this survey lecture, modelled in large part on the ICM2018 one, I will illustrate the most recent developments on calculus in metric measure spaces and the key role played by the theory of optimal transport in the derivation of synthetic lower bounds on Ricci curvature and upper bounds on dimension, for metric measure structures. If time permits, I will also illustrate the emerging role of optimal transport and gradient flows in the field of machine learning.

Un’indagine numerica sulla funzione dei cumulanti dell’osservabile triangoli nel modello denso di Erdӧs-Rényi

probabilità

Il calcolo della probabilità degli eventi rari è l’obiettivo principale della teoria delle grandi deviazioni. Per esempio, in un caso semplice, si può considerare l’evento in cui una somma di variabili aleatorie di Bernoulli raggiunge un valore che è più grande della sua media. Un problema completamente differente e più complesso, è il calcolo delle grandi deviazioni di funzionali non lineari di variabili Bernoulliane, come per esempio i polinomi cubici. Un ambito in cui un problema di questo tipo insorge è, per esempio, lo studio delle reti complesse. In questo seminario presenterò il comportamento della funzione dei cumulanti (scaled cumulant generating function) del numero dei triangoli nel contesto del modello denso di Erdӧs-Rényi. La funzione dei cumulanti è strettamente connessa alla teoria delle grandi deviazioni in quanto, quando è possibile applicare il teorema di Gärtner-Ellis, essa risulta essere la trasformata di Legendre della funzione delle grandi deviazioni. L’obiettivo di questa comunicazione è duplice: da un lato, descrivere l’estensione di un noto metodo Monte Carlo, chiamato algoritmo Cloning, formalizzata per approssimare la funzione dei cumulanti di un’osservabile additiva nel contesto dei grafi random. Dall’altro, mantenendo il focus sull’osservabile triangoli, presentare l’indagine numerica che è stata svolta nella regione dei parametri dove l’espressione analitica di tale funzione non è nota (regime di rottura delle repliche).

On weakly monotone functions

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

The notion of weakly monotone functions was introduced, in the setting of Sobolev spaces, by J.Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. We propose a criterion for the continuity of weakly monotone functions in terms of the decreasing rearrangement of their gradient. We also prove the continuity of weakly monotone functions whose gradient is in suitable rearrangement-invariant spaces. In particular, weakly monotone functions with gradient belonging to an Orlicz space or to a Lorentz space are discussed. These results are contained in joint works with Andrea Cianchi.

Limit Laws and Conformal Ensembles in the Planar Ising Model

fisica matematica

In the last twenty years there has been tremendous progress in the mathematical understanding of phase transitions for models of statistical mechanics defined on planar lattices. Much of that progress is related to the study of scaling limits, obtained by sending the lattice spacing to zero. In this talk I will give a brief introduction to scaling limits and present some recent results in the mathematical theory of phase transitions. I will focus on the case of the Ising model, which was introduced in the 1920s to study ferromagnetism and is one of the most studied models of statistical mechanics. I will discuss the convergence of the Ising magnetization to a random field (i.e., a random generalized function) with interesting properties of conformal covariance, and the connection with Euclidean field theory and the associated quantum field theory. (Based on collaborations with Rene Conijn, Christophe Garban, Jianping Jiang, Demeter Kiss, and Chuck Newman.)

Elementi centrali in U(gl(n)) , funzioni Shifted Simmetriche, Basi di Sahi/Okounkov/Schur, immananti quantici e immananti di Capelli (parte prima)

algebra e geometria

ll centro ζ(n) dell’inviluppo universale U(gl(n)) è isomorfo all’algebra ∧∗(n) dei polinomi shifted-simmetrici, via l’isomorfismo di Harish Chandra. L’algebra ∧∗(n)ammette una base (lineare) molto rilevante, costituita dai polinomi di Schur shifted-simmetrici, scoperti e caratterizzati da Kostant e Sahi, poi studiati sistematicamente da Okounkov, Olshanski et al. ll problema di descrivere/studiare gli elementi centrali in U(gl(n)) che corrispondono ai polinomi di Schur shifted-simmetrici è stato studiato da Okounkov in una serie lavori, tramite la nozione di immanante quantico. Gli immananti quantici sono elementi di ζ(n) assai difficilmente trattabili, come commentato dallo stesso Okounkov. Si è sviluppata una teoria sistematica degli immananti quantici e del centro ζ(n) basata sulla nuova nozione di immanante di Capelli in U(gl(n)). Gli immananti di Capelli sono una generalizzazione sia degli immananti classici (Littlewood/Richardson, 1934) di una matrice ad entries commutativi, sia del celebre determinante di Capelli in U(gl(n)), sono efficacemente trattabili, e formano un sistema di generatori lineari (compatibili con la filtrazione naturale di U(gl(n)). Gli immananti quantici di Okounkov risultano semplici combinazioni lineari di immananti di Capelli. ll passaggio al limite n→ ∞ per ζ(n) e ∧∗(n) si descrive esplicitamente come limite diretto rispetto alla decomposizione/proiezione Olshanski.

Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

Maximum Principles on unbounded domains play a crucial role in several problems related to linear second-order PDEs of elliptic and parabolic type. In this seminar we consider a class of sub-elliptic operators L in R^N and we establish some criteria for an unbounded open set to be a Maximum Principle set for L.

Algebre di Lie linearmente compatte e pseudoalgebre di Lie

algebra e geometria

Presenterò brevemente la teoria delle algebre di Lie linearmente compatte, ricordando la classificazione delle strutture semplici e delle loro rappresentazioni irriducibili discrete. Introdurrò poi le pseudoalgebre di Lie (possibilmente super) spiegando come queste strutture solo legate alle (super)algebre di Lie linearmente compatte e reinterpretando la teoria della rappresentazione in termini del complesso di de Rham e di alcune sue generalizzazioni.

Twisted deformations vs. cocycle deformations for quantum groups

algebra e geometria

We study two deformation procedures for quantum groups — namely, quantum universal enveloping algebras — those realized as twist deformations (that modify the coalgebra structure, while keeping the algebra one), called “twisted quantum groups” (=TwQGp’s), and those realized as 2–cocycle deformations (that deform the algebra structure, but save the coalgebra one), called “multiparameter quantum groups” (=MpQG’s). Up to technicalities, we show that the two methods actually are equivalent, in that they eventually provide isomorphic outputs.

Twisted deformations vs. cocycle deformations for quantum groups

algebra e geometria

We study two deformation procedures for quantum groups — namely, quantum universal enveloping algebras — those realized as twist deformations (that modify the coalgebra structure, while keeping the algebra one), called “twisted quantum groups” (=TwQGp’s), and those realized as 2–cocycle deformations (that deform the algebra structure, but save the coalgebra one), called “multiparameter quantum groups” (=MpQG’s). Up to technicalities, we show that the two methods actually are equivalent, in that they eventually provide isomorphic outputs.

Twisted deformations vs. cocycle deformations for quantum groups

algebra e geometria

We study two deformation procedures for quantum groups — namely, quantum universal enveloping algebras — those realized as twist deformations (that modify the coalgebra structure, while keeping the algebra one), called “twisted quantum groups” (=TwQGp’s), and those realized as 2–cocycle deformations (that deform the algebra structure, but save the coalgebra one), called “multiparameter quantum groups” (=MpQG’s). Up to technicalities, we show that the two methods actually are equivalent, in that they eventually provide isomorphic outputs.

Aprile

01

An introduction to optimal switching problems

Marco Alessandro Fuhrman

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

Il seminario si pone lo scopo di presentare il problema di switching (commutazione) ottimale e alcune delle tecniche per risolverlo, sia quelle più classiche sia altre che sono state introdotte solo recentemente. Inizialmente verrà formulato il problema per un'equazione differenziale stocastica controllata, governata dal moto Browniano. Si considererà dapprima l'approccio basato sulla programmazione dinamica e le equazioni di Hamilton-Jacobi-Bellman, che in questo caso costituiscono un sistema di equazioni differenziali a derivate parziali accoppiate per mezzo di una condizione di ostacolo. In seguito si passerà a considerare una tecnica più probabilistica basata sulle equazioni differenziali stocastiche "backward": anche in questo caso si tratta di un sistema con condizioni di riflessione. Verrà presentato anche un approccio alternativo basato su una singola equazione con un vincolo di tipo differente. La presentazione sarà di tipo pedagogico ma avanzata. In particolare il problema di arresto ottimale verrà presentato come un caso più semplice per introdurre il caso di switching generale.

Tikhonov regularization, GMRES, and preconditioning for linear discrete ill-posed problems

analisi numerica

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as boundary value problems for elliptic partial differential equations. The method is also applied to the iterative solution of linear systems of equations that are obtained by discretizing linear ill-posed problems, such as many inverse problems. However, GMRES does not always perform well when applied to the latter kind of problems. This talk seeks to shed some light on reasons for the poor performance of GMRES in certain situations, and discusses some remedies based on specific kinds of preconditioning. The standard implementation of GMRES is based on the Arnoldi process, which also can be used to define a solution subspace for Tikhonov or TSVD regularization, giving rise to the Arnoldi-Tikhonov and Arnoldi-TSVD methods, respectively. The performance of the GMRES and the latter methods is discussed.

Mirella Manaresi (Università di Bologna)

Paolo Salmon a Bologna

nel ciclo di seminari: PAOLO SALMON E L'ALGEBRA COMMUTATIVA IN ITALIA

SEMINARIO INTERDISCIPLINARE

Claudio Pedrini (Università di Genova)

La "crisi" della geometria algebrica in Italia e l’algebra commutativa: l'attività scientifica di Paolo Salmon a Genova

nel ciclo di seminari: PAOLO SALMON E L'ALGEBRA COMMUTATIVA IN ITALIA

algebra e geometria

Carlo Traverso (Università di Pisa)

Seminormalità e gruppo di Picard

nel ciclo di seminari: PAOLO SALMON E L'ALGEBRA COMMUTATIVA IN ITALIA

algebra e geometria

Silvio Greco (Politecnico di Torino)

Paolo Salmon a Pisa: l'esordio di un maestro

nel ciclo di seminari: PAOLO SALMON E L'ALGEBRA COMMUTATIVA IN ITALIA

algebra e geometria

Paolo Valabrega (Politecnico di Torino)

Paolo Salmon, un matematico e un amico

nel ciclo di seminari: PAOLO SALMON E L'ALGEBRA COMMUTATIVA IN ITALIA

SEMINARIO INTERDISCIPLINARE

Claudio Procesi (Università Roma La Sapienza)

L'ALGEBRA IN ITALIA NEGLI ANNI '60

nel ciclo di seminari: PAOLO SALMON E L'ALGEBRA COMMUTATIVA IN ITALIA

algebra e geometria

Subspace acceleration techniques in gradient-projection methods for Quadratic Programming

analisi numerica

Gradient projection (GP) methods have proved to be very efficient in the solution of optimization problems in which the projection onto the feasible set can be computed cheaply. In this seminar we, at first, review the theoretical results and algorithmic techniques developed for the case of bound constrained quadratic programming problems (BQPs). Then, we propose a gradient-based framework, called "Proportionality-based Subspace Accelerated framework for Quadratic Programming" (PSAQP), for quadratic programming problems. Inspired by the gradient projection conjugate gradient (GPCG) algorithm for convex BQPs [J. J. Moré and G. Toraldo, SIAM J. Optim., 1 (1991), pp. 93{113], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase. The proposed framework differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. Indeed, thanks to a component-wise reformulation of the first-order KKT conditions, we introduce a way to estimate the Lagrange multipliers which is exploited to formulate an efficient criterion to switch between the two phases. If the objective function is bounded, every method fitting in the framework converges to a stationary point thanks to a suitable application of the GP method in the identification phase. For strictly convex problems, finite convergence is proved even in the case of degeneracy of the solution. Numerical experiments show that practical algorithms in this framework are competitive with reference algorithms for the solution of synthetic and real-life problems subject to bounds only or to bounds and a single linear constraint.

Tips for science communication

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

È matematicamente certo. Fate l’ipotesi che voi siate matematici e il vostro partner sia, per esempio, neurochirurgo. Vi presentate a cena, con un gruppo di nuovi amici. Fate quattro chiacchiere e dopo un po’ viene fuori che siete un ricercatore in matematica. Un attimo di sconcerto, sguardi di divertito stupore, e poi vi chiedono: «ma perché, cosa c'è da ricercare, in matematica?» Guardandosi tra loro, insistono: «non è già tutto scoperto?» E sicuramente almeno uno affermerà, con orgoglio: «io, la matematica, non l'ho mai capita!» Ma il peggio è che, mentre voi cercate di spiegare cosa mai giustifichi il vostro, peraltro misero, stipendio, i commensali scopriranno la professione del vostro partner. Fine dei vostri tre minuti di protagonismo: con sguardo stavolta sognante, il vicino di tavola si rivolgerà alla vostra metà, innanzi tutto convinto di essere in grado di intavolare una conversazione su argomenti di comune interesse, poi mentalmente calcolando il suo stipendio assolutamente rispettabile, per poi perdersi definitivamente dietro al fascino che il camice bianco evoca nella mente di chiunque. Voi scomparirete inesorabilmente, a far compagnia a ricordi tendenzialmente sgradevoli di numeri, equazioni, formule e simili inutilità. Insomma, nell'immaginario comune, il medico è un’attrazione, il matematico è un nerd. Sottili varianti si possono ottenere sostituendo a vostro piacimento "neurochirurgo" con “ingegnere elettronico", "magistrato", "architetto", "promotore finanziario", "biologo marino" e, addirittura, "fisico". C’è poco da fare, esiste un problema di rappresentazione della matematica e dei matematici nell’immaginario popolare, ed è un problema serio. In questo seminario ci proponiamo di chiarirne i termini, provando anche a esplorare possibili soluzioni. Purtroppo, anche alla fine di questa chiacchierata, non sarà ancora chiaro se queste esistano: in caso positivo, però, sappiamo già che certamente non saranno uniche!

26

Vladimir Druskin, Mathematical Sciences Dept, Worcester Polytechnic Institute, MA (USA)

Embedding properties of network realizations of reduced order models with applications to inverse scattering and data science

analisi numerica

Symmetry gaps for geometric structures

algebra e geometria

For a given type of differential geometric structure, there is often a gap between the maximal and "submaximal" infinitesimal symmetry dimensions. This was first observed in the 19th century for Riemannian metrics and such symmetry gaps were subsequently classified for various other geometric structures on a case-by-case basis. I will describe joint work with Boris Kruglikov that gave a uniform approach to the symmetry gap problem for the class of parabolic geometries. This is a diverse class of geometric structures that include conformal, projective, CR, 2nd order ODE systems, and large classes of generic distributions. A priori, submaximally symmetric structures need not even be homogeneous, but remarkably, in many cases this geometric problem reduces ultimately to Dynkin diagram combinatorics, and some submaximally symmetric models can be "immediately" found (in a sense that I will make precise).

Perpetuants: a lost treasure

algebra e geometria

Perpetuant is one of the several concepts invented (in 1882) by J. J. Sylvester in his investigations of covariants for binary forms. It appears in one of the first issues of the American Journal of Mathematics which he had founded a few years before. It is a name which will hardly appear in a mathematical paper of the last 70 years, due to the complex history of invariant theory which was at some time declared dead only to resurrect several decades later. I learned of this word from Gian-Carlo Rota who pronounced it with an enigmatic smile. In this talk I want to explain the concept, a Theorem of Stroh, and some new explicit description.

algebra e geometria

TBA

Title: The Arnoldi decomposition, Tikhonov regularization, GMRES, and preconditioning for linear discrete ill-posed problems

analisi numerica

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as boundary value problems for elliptic partial differential equations. The method is also applied to the iterative solution of linear systems of equations that are obtained by discretizing linear ill-posed problems, such as many inverse problems. However, GMRES does not always perform well when applied to the latter kind of problems. This talk seeks to shed some light on reasons for the poor performance of GMRES in certain situations, and discusses some remedies based on specific kinds of preconditioning. The standard implementation of GMRES is based on the Arnoldi process, which also can be used to define a solution subspace for Tikhonov or TSVD regularization, giving rise to the Arnoldi-Tikhonov and Arnoldi-TSVD methods, respectively. The performance of the GMRES and the latter methods is discussed. This talk presents joint work with Silvia Gazzola, Silvia Noschese, Paolo Novati, and Ronny Ramlau.

Homogeneous Levi non-degenerate hypersurfaces in C3

algebra e geometria

TBA

On the existence of continuous processes with given one-dimensional distribution

probabilità

Let $\mathcal{P}$ be the collection of Borel probability measures on $\mathbb{R}$, equipped with the weak* topology, and let $\mu:[0,1]\rightarrow\mathcal{P}$ be a continuous map. Say that $\mu$ is presentable if $X_t\sim\mu_t$, $t\in [0,1]$, for some real process $X$ with continuous paths. It may be that $\mu$ fails to be presentable. Conditions for presentability are given in this note. For instance, $\mu$ is presentable if $\mu_t$ is supported by an interval for all but countably many $t$. In addition, assuming $\mu$ presentable, we investigate whether there is a continuous process $X$ with the same finite dimensional distributions as the quantile process $Q$ induced by $\mu$. The latter is defined, on the probability space $((0,1),\mathcal{B}(0,1),\,$Lebesgue measure$)$, by \begin{gather*} Q_t(\alpha)=\inf\,\bigl\{x\in\mathbb{R}:\mu_t(-\infty,x]\ge\alpha\bigl\}\quad\quad\text{for all }t\in [0,1]\text{ and }\alpha\in (0,1). \end{gather*} Various open problems are stated as well.

Thermodynamic limit and phase transitions in non-cooperative games: some mean-ﬁeld examples

nell'ambito della serie: SEMINARI DI PROBABILITÀ E STATISTICA MATEMATICA

probabilità

In stochastic dynamics inspired by Statistical Mechanics the interaction between diﬀerent particles, or agents, is usually expressed as a given function of their states. The behavior of the system, in the limit of inﬁnitely many particles (thermodynamic limit), may change dramatically by small changes in the parameters of the model: when this occurs we say there is a phase transition. In many applications the interaction cannot be given a priori but it is rather a result of agents’ strategy, aimed at optimizing a given performance. Using the simplest models of this nature, mean ﬁeld games, we illustrate some examples of phase transitions, pointing to diﬃculties in the proof of the thermodynamic limit.

21

Antonio Vitolo, Università di Salerno

Maximum principles with low ellipticity

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

A class of examples of singular irreducible symplectic varieties.

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

By the Bogomolov decomposition theorem, irreducible holomorphic symplectic manifolds play a central role in the classification of compact Kähler manifolds with numerically trivial canonical bundle. Very recently, Höring and Peternell completed the proof of the existence of a singular analogue of the Bogomolov decomposition theorem. In view of this result, singular irreducible symplectic varieties (following Greb, Kebekus and Peternell) are singular analogue of irreducible holomorphic symplectic manifolds. In a joint work with Arvid Perego, still in progress, we show that all moduli spaces of sheaves on projective K3 surfaces are singular irreducible symplectic varieties. We compute their Beauville form and the Hodge decomposition of their second integral cohomology, generalizing previous results, in the smooth case, due to Mukai, O'Grady and Yoshioka.

A gentle introduction to matroids

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

Matroids axiomatise the notion of linear dependence for a list of vectors. While the applications to computer science and optimization are known since long time, surprising interactions of matroid theory with algebra and algebraic geometry were recently discovered, leading to the proof of important combinatorial conjectures and to the introduction of new matroids invariants. In the first part of the talk I will give an elementary introduction to the topic, focusing on examples arising from graphs and from families of hyperplanes in a vector space. In the second part of the talk I will show that the set of (isomorphism classes of) matroids has a natural structure of Hopf algebra. Then I will introduce a class of matroid-like objects, called minor systems, and describe the related bialgebras. This machinery allows to give rise to a wide number of invariants, old and new. (Partially based on joint work with Alex Fink and Clement Dupont)

Stefano Pagliarani - DIES, Università di Udine

Fixed-point theorems and Picard iteration methods for McKean-Vlasov mean-field SDEs with jumps

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

We consider a prototype class of Lévy-driven SDEs with McKean-Vlasov (mean-field) interaction in the drift. The coefficient is assumed to be affine in the state-variable and only measurable in the law. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK-V SDE, and derive a discretized Picard iteration method that approximates the law of the solution. Numerical illustrations show the effectiveness of the method, which appears to be appropriate to handle multi-dimensional settings. We finally describe possible extensions and generalizations to more general settings. This talk is based on joint work with Ankush Agarwal.

Toric arrangements: an introduction between Algebra, Topology and Combinatorics

algebra e geometria

In the first part, we will introduce the theory of hyperplane arrangements with particular attention to the cohomology algebra of the complement of the arrangement. The talk will start from the basic definitions of the topological and combinatorial objects involved. We will exhibit the connections between hyperplane arrangements and other branch oh Mathematics, e.g. knot theory and graph theory. The second part will focus on toric arrangements, a generalization of hyperplane arrangements. We will give a presentation of the cohomology algebra of the complement of a toric arrangement (this is a joint work with F. Callegaro, M. D'Adderio, E. Delucchi, and L. Migliorini) and we will discuss its dependency on the combinatorial data of the arrangement.

algebra e geometria

Elena Bandini (Università degli Studi di Milano-Bicocca)

BSDEs driven by a general random measure and optimal control for piecewise deterministic Markov processes

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

We consider an optimal control problem for piecewise deterministic Markov processes (PDMP) on a bounded state space. Here a pair of controls acts continuously on the deterministic flow and on the transition measure describing the jump dynamics of the process. For this class of control problems, the value function can be characterized as the unique viscosity solution to the corresponding integro-differential Hamilton-Jacobi-Bellman equation with a non-local type boundary condition. We are able to provide a probabilistic representation for the value function in terms of a suitable backward stochastic differential equation, known as nonlinear Feynman-Kac formula. The jump mechanism from the boundary entails the presence of predictable jumps in the PDMP dynamics, so that the associated BSDE turns out to be driven by a random measure with predictable jumps. Existence and uniqueness results for such a class of equations are non-trivial and are related to recent works on well-posedness for BSDEs driven by non quasi-left-continuous random measures.

Computational modeling of the motor cortex and the cerebellum

nel ciclo di seminari: NEUROMATEMATICA

SEMINARIO INTERDISCIPLINARE

This talk summarizes two modeling studies on the motor cortex and the cerebellum. The motor cortex is the final cortical pathway to motor circuits in the spinal cord, but its functional role has long been debated, particularly whether the motor cortex represents movement kinematics or dynamics. To resolve this issue, I modeled the visuomotor transformation using Newton-Euler equations of motion that has been used in robotics, and proposed that neural activities in the motor cortex represent vector cross products in the equations. This model explains a wide variety of the characteristics reported in the motor cortex in a unified manner. The cerebellum is hypothesized to predict a future state of the body from a current state and a corollary discharge, the computation known as an internal forward model. Although this hypothesis has been supported from a number of clinical, psychophysical and neuroimaging studies, a direct neurophysiological evidence is missing. I analyzed firing rates of mossy fibers (cerebellar inputs), Purkinje cells (outputs from cerebellar cortex), and dentate cells (cerebellar outputs) recorded from a behaving monkey. I found that the cerebellar outputs provided predictive information about future inputs to the cerebellum, providing direct neurophysiological evidence for the forward-model hypothesis of the cerebellum. [1] Tanaka, H., & Sejnowski, T. J. (2013). Computing reaching dynamics in motor cortex with Cartesian spatial coordinates. Journal of Neurophysiology, 109(4), 1182-1201. [2] Tanaka, H., & Sejnowski, T. J. (2015). Motor adaptation and generalization of reaching movements using motor primitives based on spatial coordinates. Journal of Neurophysiology, 113(4), 1217-1233. [3] Tanaka, H., Ishikawa, T., & Kakei, S. (2019). Neural Evidence of the Cerebellum as a State Predictor. The Cerebellum, 1-23.

Viscosity solutions for PDEs on Wasserstein space

probabilità

McKean-Vlasov stochastic optimal control problem

probabilità

Quenched mass transport of particles towards a target

probabilità

McKean-Vlasov stochastic control and Hamilton-Jacobi-Bellman equations on Wasserstein space

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

probabilità

An introduction to market impact modeling and optimal execution in financial markets

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

Market impact is the response of prices to trades and is a fundamental quantity to understand how supply and demand affect price, but also an important component of transaction costs. In this talk, I introduce the still open problem of mathematical modeling of market impact in a way which is consistent with data but also lacking dynamical arbitrage opportunities. I review some models proposed in the mathematical finance literature and discuss their comparison with data, deriving necessary conditions for the absence of dynamical arbitrage. I then focus on the optimal execution problem in continuous and discrete time, deriving the solution under different specifications of the impact model and of the chosen benchmark.

"Penalty formulations for mixed integer and PDE constrained optimization problems"

analisi numerica

Seminario nell'ambito del progetto MIUR-DAAD 2018-2020, Universita- di Bologna e TU-Chemnitz

Results and problems in Algebraic Combinatorics

algebra e geometria

In the first part of the seminar, I recall some basic definitions from Commutative Algebra and Combinatorics. In particular, I consider classes of homogeneous ideals arising from discrete structures. The main goal of my research is to describe algebraic properties of these ideals in combinatorial terms. A typical example of this approach is to establish relations between the graded Betti numbers of monomial ideals and the structure of associated simplicial complexes or hypergraphs. In this flavour, I present various results, suggesting also possible future developments. Time permitting, in the last part of the seminar I focus on new directions of research, involving arithmetic matroids and a generalization of flag varieties.

04

Piergiacomo Sabino (Uniper Global Commodities SE, Dusseldorf)

Forward or backward simulation? A comparative study

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

The aim of this study is to present algorithms for the backward simulation of standard processes that are commonly used in financial applications. We extend the works of Ribeiro and Webber and Avramidis and L’Ecuyer on gamma bridge and obtain the backward construction of a Gamma process. Moreover, we are able to write a novel acceptance-rejection algorithm to simulate Inverse Gaussian (IG) processes backward in time. Therefore, using the time-change approach, we can easily get the backward generation of the Compound Poisson with infinitely divisible jumps, the Variance–Gamma the Normal–Inverse–Gaussian processes and then the time-changed version of the OU process (SubOU) introduced by Li and Linetsky. We then compare the computational costs of the sequential and backward path generation of such processes and show the advantages of adopting the latter one, in particular in the context of pricing American options or energy facilities like gas storages.

Claudia Bucur, Università Cattolica del Sacro Cuore

Behaviour of nonlocal sets for small values of the fractional parameter

nell'ambito della serie: SEMINARI DI ANALISI MATEMATICA BRUNO PINI

analisi matematica

Paolo Pigato (Weierstrass Institute for Applied Analysis and Stochastics, Berlino)

Density and tube estimates for diffusion processes under Hormander-type conditions

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

We recall some classic results on the regularity of solutions of stochastic differential equations. Then we consider two specific diffusion processes satisfying hypoellipticity conditions of Hormander type. Using Malliavin Calculus techniques recently developed to deal with degenerate problems, we find estimates for the density of the law of the solution, which we use to prove exponential bounds for the probability that the diffusion remains in a small tube, around a deterministic path, up to a given time. We then present some work in progress on asymptotic sharp estimates for the density and its derivatives for a similar, higher dimensional system.

Introduction to minimax methods

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

We will introduce the basic tools for the topological methods in critical points theory: the Palais-Smale compactness condition and the deformation lemma. Starting from the finite dimensional case, we will illustrate how the minimax argument works. Eventually, we will show some applications to problems in infinite dimensional setting.

The monodromy group of IHS manifolds of OG10-type

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The monodromy group of an IHS manifold is one of the most important tools to investigate their geometry. In the first part of the talk, I will recall the main definitions, giving some motivation. In the second half I will focus on the OG10-type. This is the only type (among the known ones) for which the monodromy group is still a mystery. We explain how to construct new monodromy operators using two families, the O'Grady and the Laza-Saccà-Voisin ones, exhibiting an explicit subgroup of the monodromy group, that we conjecture being all. Time permitting, we will also discuss a geometric constraint to the fact that the monodromy group is smaller than the group of orientation preserving isometries.

20

Annalisa Panati, Centre de Physique Théorique, Luminy et Université de Toulon.

Heat fluctuations in the two-time measurement framework and ultraviolet regularity

fisica matematica

Since Kurchan’s seminal work (2000), two-time measurement statistics (also known as full counting statistics) has been shown to have an important theoretical role in the context of quantum statistical mechanics, as they allow for an extension of the celebrated fluctuation relation to the quantum setting. In this contribution, we consider two-time measurement statistics of heat for a locally perturbed system, and we show that the description of heat fluctuation differs considerably from its classical counterpart, in particular a crucial role is played by ultraviolet regularity conditions. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation. This phenomenon has no classical analogue.

Random Walks in Finite Groups

nell'ambito della serie: SEMINARI BAD

algebra e geometria

As the name declares, the theory of random walks in groups is somehow in the middle of group theory and probability theory. The central question is “how to generate a group efficiently?” In this seminar I will present the notions of diameter, mixing time and expander graphs, as well as some explicit constructions. The second part will be dedicated to the Bourgain-Gamburd Machine, a recent technique to show expansion in quasirandom groups.

18

Tensor decomposition and secant varieties

Alessandro Gimigliano

nell'ambito della serie: TOPICS IN MATHEMATICS 2018/2019

Determining secant varieties (i.e. varieties which are the union of secant lines, planes, etc.) for a given projective variety X is a classical problem in Algebraic Geometry. When X is a Segre Variety, its secant varieties parameterize tensors with assigned tensor rank, and their study is related to the study of tensor decompositions, a quite relevant issue in applied math. In a similar way, secant varieties of Veronese varieties are related to symmetric tensors. In this talk a sketchy view of the state of the art on these problems will be given.

15

Alessandro Della Corte (Sapienza Università di Roma)

The Kolakoski sequence: open problems and new approaches

fisica matematica

The Kolakoski sequence S is the unique sequence on the alphabet {1,2} starting with 1 and coinciding with its own run length encoding: S = 122112122122112… The (few) known properties and the open problems concerning S will be described and some new approaches will be proposed.

BGG sequences from infinite-rank bundles

algebra e geometria

I will present a generalization of Calderbank-Diemer construction that works for bundles whose fiber is unitarizable highest weight module. These modules exists only when $(G, K)$ is Hermitian symmetric pair. The resulting BGG sequences of Verma modules are (after a twist) generalizations of minimal free resolutions of determinantal ideals. This suggests that these sequences of differential operators are in fact resolutions for interesting differential operators such as Yamabe or Dirac on $G^\mathb{C}/P$. Moreover these differential operators still obey $A_\infty$ relations as in the classical case of finite-dimensional bundles.

Parabolic geometries: Overdetermined differential equations, holonomy reductions and compactifications/ Part 2

algebra e geometria

Parabolic geometries provide a uniform framework to describe, treat and analyse a number of differential geometric structures, most prominently projective structures, conformal structures & CR-structures. I will give an introduction to the most important features of parabolic geometries, most importantly in the area of conformal (spin) structures and how this framework can be used to treat interesting geometric differential equations via the BGG-machinery. A major advance in this area was a uniform holonomy reduction theorem, known as 'curved orbit decompositions'. I will explain via some examples how curved orbit decompositions can be used to understand the geometric implications of the existence of solutions to BGG-equations and in particular sheds light on 'singularity sets'. A final topic of this talk will be compactifications of parabolic geometries which are again intimately related with the concept of holonomy reductions and curved orbit decompositions.

Division algebras and the brane bouquet

nel ciclo di seminari: JOHN HUERTA

algebra e geometria

In the last talk, we met some "higher" algebraic structures associated to strings. We expand on this idea, to give the Fiorenza-Sat-Schreiber "brane bouquet" of L-infinity algebras. Then we describe our most recent work with Sati and Schreiber: in the 11 dimensional spacetime famous from supergravity and M-theory, we review the classification of finite group actions that preserve some supersymmetry, and show how we can extend the brane bouquet to this equivariant setting.

Representation theory and BGG

algebra e geometria

We discuss the link between representation theory and invariant operators

Parabolic geometries: Overdetermined differential equations, holonomy reductions and compactifications

algebra e geometria

Parabolic geometries provide a uniform framework to describe, treat and analyse a number of differential geometric structures, most prominently projective structures, conformal structures & CR-structures. I will give an introduction to the most important features of parabolic geometries, most importantly in the area of conformal (spin) structures and how this framework can be used to treat interesting geometric differential equations via the BGG-machinery. A major advance in this area was a uniform holonomy reduction theorem, known as 'curved orbit decompositions'. I will explain via some examples how curved orbit decompositions can be used to understand the geometric implications of the existence of solutions to BGG-equations and in particular sheds light on 'singularity sets'. A final topic of this talk will be compactifications of parabolic geometries which are again intimately related with the concept of holonomy reductions and curved orbit decompositions.

Homogeneous sub-Riemannian manifolds, part 1

algebra e geometria

There are many equivalent definitions of Riemannian geodesics The definitions can be divided into two classes : geodesics as "shortest curves defined by a variational principle, and geodesics as "straightest curves defined by a connection. All definitions are naturally generalised to sub-Riemannian manifolds, but become non-equivalent. A. Vershik and L. Faddeev showed that for a generic sub-Riemannian manifold (Q, D, g) shortest geodesics ( used in control theory) are different from straightest geodesics (used in non-holonomic mechanics)) on a open dense submanifold. They gave first example (compact Lie group with the bi-invariant metric) when shortest geodesics coincides with straightest geodesics and stated the problem to describe more general class of sub-Riemannian manifolds with this property . We generalised the Vershik-Faddeev example and consider a big class of sub-Riemannian manifolds associated with principal bundle over a Riemannian manifolds, for which shortest geodesics coincides with straightest geodesics. Using the geometry of flag manifolds, we describe some classes of compact homogeneous sub-Riemannian manifolds ( including contact sub-Riemannian manifolds and symmetric sub-Riemannian manifolds ) where straightest geodesics coincides with shortest geodesics. Construction of geodesics in these cases reduces to description of Riemannian geodesics of the Riemannian homogeneous manifold or left-invariant metric on a Lie group.

Super Jordan triple systems

algebra e geometria

In this seminar we will introduce the supersymmetric Jordan triple systems: a new algebraic structure which generalizes the class of Jordan triple systems as well as the class of N=6 3-algebras. We will describe their relation with graded Lie superalgebras with involutions via the Tits-Kantor-Koecher construction. Their classification, obtained via the TKK construction, will be discussed and the explicit realizations of the systems related to the special and to the exceptional Lie superalgebras will be given. The infinite-dimensional linearly-compact case will also be presented.

Division algebras and supersymmetry

nel ciclo di seminari: JOHN HUERTA

algebra e geometria

Abstract: In this introduction suitable for graduate students, we use the four normed division algebras to introduce some basic elements of supersymmetry. Namely, from the four normed division algebras (the real numbers, the complex numbers, the quaternions and the octonions), a uniform construction yields the super-Minkowski spacetimes on which the classical superstring can be defined. We review this construction and show how the alternativity of the division algebras allows us to define a class in the third cohomology on super-Minkowski spacetime, which in turn allows us to write the classical action of the superstring. In conclusion, we describe how this degree three class defines a "higher" algebraic structure.

Forms in Supergeometry, part 1

fisica matematica

I will review the theory of superforms, integral forms and inverse forms in supermanifolds from a sheaf-theoretical point of view.. The formal "distributional" properties of forms and of Picture Changing Operators in superstring field theory are recovered geometrically, providing a mathematical foundation for the concept of Large Hilbert Space. Finally, I will discuss a new A-infinity algebra structure emerging (more or less naturally...) on supermanifolds.

Lie groups and combinatorics of the exterior algebra.

nell'ambito della serie: SEMINARI BAD

algebra e geometria

Let G be a simple Lie group over C and let Lie(G) be its Lie algebra. The exterior algebra of Lie(G) is extensively studied in literature for its link with the geometry of G and the combinatorics of the Weyl group W(G). In this talk I will present an overview of some classical results about the exterior algebra in a "representation theory"-flavour, with a particular attention to two open conjectures due to Kostant and Reeder.

The de Branges theory of Hilbert spaces of entire functions and its applications to spectral theory of differential operators 6/6

nel ciclo di seminari: CORSO DI DOTTORATO INDAM/UNIBO: ANTON BARANOV SU THE DE BRANGES THEORY OF HILBERT SPACES OF ENTIRE FUNCTIONS AND ITS APPLICATIONS TO SPECTRAL THEORY OF DIFFERENTAIL OPERATORS

See https://phd.unibo.it/matematica/it/didattica/2018-2019

06

The Clark-Ocone formula and the Poincare' inequality for point processes

Giovanni Luca Torrisi

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

Clark-Ocone formulas are powerful results in stochastic analysis with a variety of applications. In the talk we provide the Clark-Ocone formula for square-integrable functionals of point processes with stochastic intensity. Then we present two applications of the formula: the Poincare' inequality and a concentration bound for those functionals. Our results generalize the corresponding ones on the Poisson space. The talk is based on joint works with Ian Flint and Nicolas Privault (NTU, Singapore)

The de Branges theory of Hilbert spaces of entire functions and its applications to spectral theory of differential operators 5/6

nel ciclo di seminari: CORSO DI DOTTORATO INDAM/UNIBO: ANTON BARANOV SU THE DE BRANGES THEORY OF HILBERT SPACES OF ENTIRE FUNCTIONS AND ITS APPLICATIONS TO SPECTRAL THEORY OF DIFFERENTAIL OPERATORS

analisi matematica

See https://phd.unibo.it/matematica/it/didattica/2018-2019

Fano varieties of K3 type and IHS manifolds

algebra e geometria

Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperk\"ahler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi.

Gennaio

30