Archivio 2019

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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

TBA

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

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).

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.

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.

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.

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.

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.

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.

"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"

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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

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.

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"

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).

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.

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.

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.

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.

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.

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 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.

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

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."

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.

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)

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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).

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).

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.

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).

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).

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.

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.

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.

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.

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.

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.

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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.

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.

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.

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.

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.

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: $\forall x\exists y,z$: $x\vee y=y\vee z=z\vee x$ & $x\wedge y=y\wedge z=z\wedge x=0.$

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

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.

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.

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.

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.

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.

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.

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

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).

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.)

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.

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.

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.

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.

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.

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.

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.

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).

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.

TBA

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.

TBA

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.

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.

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

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.

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.

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.

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 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.

We discuss the link between representation theory and invariant operators

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.

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.

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.

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.