Seminari del Dipartimento di Matematica

Carlangelo Liverani (Università di Roma Tor Vergata)

Locating Ruelle-Pollicott resonances

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

We study the spectrum of transfer operators associated to various dynamical systems. Our aim is to obtain precise information on discrete spectrum. To this end we propose a unitary approach. We consider various settings where new information can be obtained following different branches along the proposed path. These settings include affine expanding Markov maps, uniformly expanding Markov maps, non-uniformly expanding maps, hyperbolic diffeomorphisms. We believe this to be the germ of a general theory. Joint work with O. Butterley and N. Kiamari.

A new Tensor-Train based nonlinear optimization method for dictionary learning

analisi numerica

The interest around a sparse representation of data is growing in these years due to its several applications such as image classification, image denoising and image compression. Dictionary learning is one of the most important techniques to address this task. With such an approach, data are represented using a large matrix D and a sparse matrix X. Moreover, to deal with multidimensional data, a tensor formulation of the dictionary learning problem has been recently introduced. Within this framework,we propose a new Tensor-Train based nonlinear optimization algorithm and we compare its performance with well established dictionary learning algorithms such as K-SVD, Ho-SuKro and K-HOSVD.

Dicembre

03

Dimitris Xatzakos (Institute Mathematics De Bordeaux)

An introduction to the analytic theory of automorphic forms with a glimpse to some recent discoveries.

nel ciclo di seminari: SEMINARI BAD

SEMINARIO INTERDISCIPLINARE

I will give a short introduction to the analytic theory of automorphic forms. I will also discuss some important recent equidistribution results in the interface of number theory and quantum chaos.

On boundary quantum groups

nel ciclo di seminari: ALGEBRA E GEOMETRIA

algebra e geometria

In the last ten years, there has been a renewed interest in the theory of quantum symmetric pairs (QSPs). As the name suggests, a QSP is an algebraic datum which quantizes the notion of symmetric space. It consists of a Drinfeld-Jimbo quantum group of a simple Lie algebra and a distinguished coideal subalgebra quantizing the fixed point subalgebra of an involution. Although their representation theory remains quite mysterious, it is becoming more and more evident that QSPs possess an incredibly rich structure with their own theory of canonical basis and braid group actions yet adapted to a particular “boundary behavior”. In the first part of the talk, I will give an overview of this emerging theory, while in the second part I will report on ongoing joint work with B. Vlaar and T. Przezdziecki devoted to the construction of a meromorphic boundary Yang-Baxter operator for quantum affine symmetric pairs and a quantum Schur-Weyl duality which arises from the study of its poles.

26

Anna Miriam Benini (Università di Parma)

Infinite entropy for transcendental entire functions

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

Defining entropy on noncompact metric spaces is a tricky business, since there are several natural and nonequivalent generalizations of the usual notions of entropy for continuous maps on compact spaces. By defining entropy for transcendental maps on the complex plane as the sup over the entropy restricted to compact forward invariant subsets, we prove that with this definition the entropy of such functions is infinite. The proof relies on covering results which are distinctive to holomorphic maps.

The relative canonical embedding of curves with automorphisms

nel ciclo di seminari: SEMINARI BAD

algebra e geometria

scarica allegato

Despite the progress made in the recent years, the list open problems in characteristic p algebraic arithmetic geometry remains extensive. One of the strategies that has proven to be succesful, initially proposed by J. P. Serre in his Mexico paper, is the technique of lifting to characteristic 0: problems like the Galois module structure of (poly)differentials and Green’s syzygy conjecture are well understood in characteristic 0 but remain open in characteristic p. The above problems share a second interesting property: they involve the canonical sheaf Ω, which appears prominently in the classical theorem of M. Noether, F. Enriques and K. Petri. In this talk, following a review of the theory of lifting curves with automorphisms and the Noether-Enriques-Petri theorem, we will present joint work with H. Charalambous and A. Kontogeorgis, in which we study the relative canonical embedding of the flat family of curves obtained from lifting an Artin-Schreier curve to a Kummer curve. Combining elements of Gröbner theory with deformation-theoretic arguments we will give an explicit set of generators for the relative canonical ideal, obtaining in the process a relative version of Petri’s theorem.

Deep Plug-and-Play framework enhanced by Total Variation for image restoration

analisi numerica

Plug-and-Play (PnP) is an image restoration framework which leverages on a set of powerful denoisers to induce a priori information on the solution of a general image restoration problem. By using splitting techniques, such as the Half Quadratic Splitting (HQS), the PnP framework solves the classical regularized optimization problem where the regularizer-related step is replaced by a denoiser. In this talk, I will introduce HQS - Deep PnP which is mainly focus on Convolutional Neural Network (CNN) denoisers. I will show how increase the performances of PnP by considering a general framework where the denoiser act on a transformation of the image (e.g. the discrete gradient) and a further handcrafted regularization term is added (e.g. Total Variation). Moreover, I will prove that for the HQS - Deep PnP algorithm a fixed point convergence is guaranteed under certain assumptions. The good performances of the proposed approach for the task of non-blind denoising and deblurring are addressed through several experiments both on synthetic and real data.

Intrinsic Taylor formula in fractional Holder space

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

analisi matematica

We consider a class of non-local ultraparabolic Kolmogorov operators and we study suitable fractional Holder spaces that take into account the intrinsic sub-riemannian geometry induced by the operator. We prove a characterization relating the regularity along the vector fields to the existence of appropriate instrinsic Taylor formulas which extends in the non-local context the characterization given in the diffusive setting.

Topics in stochastic differential equations and their applications

nell'ambito della serie: TOPICS IN MATHEMATICS 2020/2021

probabilità

analisi matematica

We present a selection of research topics concerning the study of stochastic differential equations (SDEs) that arise in social, natural and physical sciences. We discuss two macro-classes: (i) jump-diffusion equations with degenerate behavior of their coefficients, and (ii) mean-field (McKean-Vlasov) diffusion equations. For the class (i) we first describe the main features of the models and the general connection with ultra-parabolic differential operators of Kolmogorov type. We then present some recent developments regarding regularity and asymptotic properties of the transition densities for some specific models. For the class (ii) we first describe the interplay between McKean-Vlasov SDEs, mean-field interacting particle systems, and non-linear Fokker-Planck equations. We then discuss some of the problems related to the existence, uniqueness, asymptotic properties of the solutions, as well as to their numerical approximations. We finally present some recent results in particular cases.

Birational sheets in linear algebraic groups

nel ciclo di seminari: ALGEBRA E GEOMETRIA

algebra e geometria

If G is an algebraic group acting on a variety X, the sheets of X are the irreducible components of subsets of elements of X with equidimensional G-orbits. For G complex connected reductive, the sheets for the adjoint action of G on its Lie algebra g were studied by Borho and Kraft in 1979. More recently, Losev has introduced finitely-many subvarieties of g consisting of equidimensional orbits, called birational sheets: their definition is less immediate than the one of a sheet, but they enjoy better geometric and representation-theoretic properties and are central in Losev's proposal to give an Orbit method for semisimple Lie algebras. In the first part of the seminar we give an historical overview on sheets and recall some basics about algebraic groups and Lusztig-Spaltenstein induction in terms of the so-called Springer generalized map and analyse its interplay with birationality. This will allow us to introduce Losev's birational sheets. The last part is aimed at defining an analogue of birational sheets of conjugacy classes in G, under the hypothesis that the derived subgroup of G is simply connected. We will conclude with an overview of the main features of these varieties, which mirror some of the properties enjoyed by the objects defined by Losev.

16

Isaia Nisoli (Universidade Federal do Rio de Janeiro, Brasile)

A simple system presenting Noise Induced Order

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

In this talk I will present a family of one dimensional systems with random additive noise such that, as the noise size increases, the Lyapunov exponent of the stationary measure transitions from positive to negative. This phenomena is known in literature as Noise Induced Order, and was first observed in a model of the Belosouv-Zhabotinsky reaction and its existence was proven only recently by Galatolo-Monge-Nisoli. In the talk I will show how this phenomena is strictly connected with non-uniform hyperbolicity and the coexistence of regions of expansion and contraction in phase space; the result is attained through a result on the continuity of the Lyapunov exponent of the stationary measure with respect to the size of the noise.

12

Stefano Marò (Università di Pisa)

Chaotic motion in the breathing circle billiard

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory. Joint work with Claudio Bonanno.

10

Alberto Calabri (Università di Ferrara)

Recent results on plane Cremona transformations

algebra e geometria

The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane The group of plane Cremona transformations is endowed with a natural topology and the set of maps of some bounded degree is closed. We will review some known properties of the quasi-projective variety which parametrizes plane Cremona maps of fixed degree. Then, we will address the question of determining which plane Cremona maps of small degree are limits of maps of higher degree. This talk is mainly based on joint works with Jérémy Blanc and with Cinzia Bisi and Massimiliano Mella.

05

Marco Zamparo (Politecnico di Torino)

Large Fluctuations and Transport Properties of the Lévy-Lorentz gas

fisica matematica

The Lévy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this seminar I review the model and discuss its large fluctuations and resulting transport properties, both annealed and quenched, under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity. In particular, by presenting large deviation estimates and the asymptotics of moments for the particle displacement, I show that the motion is superdiffusive in the annealed framework, whereas a normal diffusive behavior characterizes the motion conditional on a typical realization of the scatterers arrangement.

Dalle curve ellittiche alle superfici ellittiche

nel ciclo di seminari: SEMINARI BAD

algebra e geometria

Abbiamo tutti visto una curva ellittica in qualche momento nella nostra vita matematica. Oggi presenterò una versione della definizione di curva ellittica vista dagli occhi di un geometra algebrico, e colgo l'occasione per introdurre alcune nozioni di geometria algebrica che nel caso delle curve ellittiche diventano più facili da intuire, sempre motivate dalla generalizzazione di quello che corrisponderebbe ad una superficie ellittica. Niente panico e ricordate: “Complex curves (=compact Riemann surfaces) appear across a whole spectrum of math problems, from Diophantine arithmetic through complex function theory and low dimensional topology to differential equations of math physics. So go out and buy a complex curve today”- Miles Reid (trying to make a commercial break).

Il G-ordine di Bruhat fra sistemi locali sulle B-orbite di una varietà hermitiana simmetrica

algebra e geometria

Sia G un gruppo di Lie complesso e semisemplice. Sia P un sottogruppo parabolico di G tale che il radicale unipotente P^u sia abeliano e B ⊆ P un sottogruppo di Borel. In questo caso, se P = LP^u è una decomposizione di Levi di P, diciamo che G/L è una varietà hermitiana simmetrica. Il sottogruppo di Borel B agisce su G/L con un numero finito di orbite. L'insieme di tali orbite può essere ordinato in maniera naturale tramite l’ordine di Bruhat. Per ognuna di queste orbite possiamo considerare l'insieme dei C-sistemi locali di rango 1 B-equivarianti a meno di isomorfismo. Si ottiene così un insieme di coppie orbita-sistema locale a cui possiamo dare un ordine che estende l’ordine di Bruhat e che Lusztig e Vogan chiamano G-ordine di Bruhat In questo seminario presenterò una caratterizzazione combinatorica del G-ordine nel caso in cui il sistema di radici di G sia simply-laced. Questo risultato può essere utile per migliorare l’efficienza nel calcolo dei polinomi di Kazhdan-Lusztig. Mostrerò inoltre un risultato simile per sistemi di radici di tipo B e alcuni risultati parziali per sistemi di radici di tipo C.

A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations

analisi numerica

Nonlinear differential matrix equations generally stem from the semi-discretization on a rectangular grid of nonlinear partial differential equations (PDEs). The two main challenges related to approximating the solution of such matrix equations includes the high computational cost of time integrating the system when the matrices have large dimensions, as well as the cost related to evaluating the time-dependent nonlinear term at each timestep. In the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to an effective, structure aware low order approximation of the original problem. The nonlinear term is also reduced by means of a fully matricial interpolation using left and right projections onto two distinct reduction spaces, giving rise to a new two-sided version of DEIM. Several numerical experiments based on typical benchmark problems illustrate the effectiveness of the new matrix-oriented setting.

Diagonal coinvariants: old and new

algebra e geometria

The study of invariants and coinvariants is a classical topic that goes back to the early days of modern algebra. About 25 year ago, Garsia and Haiman initiated the study of diagonal coinvariants of the symmetric group in relation to what has become known as the "n! conjecture". In particular, the combinatorics behind these objects has been intensively studied in recent years, in connection with certain operators defined via the Macdonald symmetric functions. In this talk we review what is known on this subject, and present some of the surprising conjectures that arose in the last two years.

Rappresentazioni di (super)algebre di Lie linearmente compatte

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

algebra e geometria

fisica matematica

Spiegherò la costruzione dei cosiddetti moduli di Verma ed illustrerò attraverso esempi alcuni problemi, aperti e non, nello studio di questi moduli.

Recent trends in model theory

SEMINARIO INTERDISCIPLINARE

Model theory is a branch of mathematical logic that uses tools from logic in order to explore mathematical structures (models). When those structures are of geometric nature, we tend to call this study "tame geometry". We will give a broad introduction to tame geometry, and survey some topics at the nexus of model theory and: (a) Diophantine geometry, focusing on recovering algebraic curves from "definable" sets with many rational points, (b) group theory, on a definable version of Hilbert's 5th problem, and (c) extremal combinatorics, around the Vapnik-Chervonenkis theory and its links to statistical learning (time permitting).

A Cheeger-like inequality for 1-forms on closed 3-manifolds.

analisi matematica

It is well known that, on a closed Riemannian manifold, the Laplace operator has discrete spectrum. One can wonder if its first positive eigenvalue has some geometric meaning. Cheeger's seminal work, which is now referred to as Cheeger's inequality, asserts that one can give to this first eigenvalue a geometric lower bound, called the Cheeger's constant. A natural question is to find an analogous statement for differential forms of any degree. During the talk we will review Cheeger's theorem and propose a generalization of Cheeger constant for (coexact) 1-forms on closed 3-manifolds. We shall refer to this constant as the open book constant. If time allows it, we will give some elements of the proof of our main theorem which may be thought as a Cheeger inequality for 1-forms on 3-manifolds: the first eigenvalue of the Hodge Laplacian acting on (coexact) 1-form is bounded from below by the open book constant. This is a joint work with Gilles Courtois (IMJ-PRG, Sorbonne Université).

Alcuni problemi in analisi

analisi matematica

In questo seminario offriro' un panorama dei problemi di cui mi sono occupata nel corso degli anni (problemi al contorno per il laplaciano su domini lipschitziani; diseguaglianze di tipo ``div-curl’’; regolarità in L^p di vari integrali singolari con nucleo olomorfo associati a domini non lisci). Poi mi concentrero' sui miei interessi piu' recenti, tra i quali: la proiezione di Szego associata ai ``worm domains’’ di Diederich e Fornaess; il commutatore della proiezione di Szego associata a domini strettamente pseudoconvessi non lisci; definizone degli spazi di Hardy per una classe di domini singolari e costruzione di nuclei di riproduzione per funzioni olomorfe nei suddetti, con applicazioni al triangolo di Hartogs. Tempo permettendo, descrivero’ brevemente la tesi del mio piu’ recente studente di dottorato (Dicembre 2019).

Bayesian hierarchical models for sparse recovery problems

analisi numerica

A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining \ell_1-penalized least squares optimization with an appropriate numerical scheme to accomplish the task. A computationally efficient alternative for finding sparse solutions to linear inverse problems is provided by Bayesian hierarchical models, in which the sparsity is encoded by defining a conditionally Gaussian prior model with the prior parameter obeying a generalized gamma distribution. In this talk, we are discussing the analytic properties of this class of hypermodels, together with their sparsity promoting effects. The typical benefits of non-convex penalty terms will be coupled with the pleasant convexity guarantees thus making the way for a hybrid solver which allows to have the best of both worlds.

Degenerations of Hilbert schemes of points

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Let S be a smooth projective surface. The Hilbert schemes Hilb^n(S) of n points on S are well-understood and central objects in geometry. Less is known, however, about degenerations of such varieties. In this talk, I will present joint work with M. Gulbrandsen, K. Hulek and Z. Zhang, where we study how the Hilbert scheme degenerates "along" with the surface S. I will also discuss a few applications of our construction, and some open problems.

The story how analysis met geometric measure theory and what they were doing together?

SEMINARIO INTERDISCIPLINARE

Starting with the early 1990's several outstanding problems of complex analysis were solved by new methods that involved non-homogeneous harmonic analysis, geometric measure theory and random geometric constructions. I mean here the problems of Painlev\'e, Ahlfors, Vitushkin, Denjoy and solved by Tolsa, David--Mattila and Nazarov--Treil--Volberg. This lucky encounter continued for the next 20 years, where those methods were used to solve several free boundary problems of Bishop and Guy David. I will try to give the exposition of ideas behind this development.

The basic problem of the calculus of variations: new range of validity of the Du Bois - Reymond equation and applications to regularity

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

analisi matematica

Higher rank K-theoretic Donaldson-Thomas theory of points

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Recently Okounkov proved Nekrasov’s conjecture expressing the partition function of K-theoretic DT invariants of the Hilbert scheme of points Hilb(C^3,points) on affine 3-space as an explicit plethystic exponential. The higher rank analogue of Nekrasov’s formula is a conjecture in String Theory by Awata-Kanno. We state this conjecture and sketch how to prove it mathematically via Quot schemes. Specialising from K-theoretic to cohomological invariants, we obtain the statement of a conjecture of Szabo. This is joint work with Nadir Fasola and Sergej Monavari.

Fast, matrix-free matrix-vector product with the Löewner matrix

analisi numerica

The Löwner framework is one of the most successful data-driven model order reduction techniques. Given k right interpolation data and h left interpolation data, the standard layout of this approach is composed of two stages. First, the kh x kh Löwner matrix L and shifted Löwner matrix S are constructed. Then, an SVD of L-ζS, ζ belonging to one of the data sets, provides the projection matrices used to compute the sought reduced model. These two steps become numerically challenging for large k and h in terms of both computational time and storage demand. We show how the structure of L and S can be exploited to reduce the cost of performing (L-ζS)x while avoiding the explicit allocation of L and S.

Random models for temporal networks

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

finanza matematica

Many real systems in biology, economics, finance, social sciences can be represented as temporal networks, i.e. graphs whose structure is not constant but new links are formed and old ones are destroyed at each time. In the first part of my talk, I will introduce a general class of random models for networks, the exponential random graph (ERG) family, I show the connection with the maximum entropy principle and with the latent variables models, and I describe the inference problem when data are available. In the second part, I show some recent advancements to the use of ERGs to the modeling of temporal networks highlighting different mechanisms which are responsible for the memory in links dynamics. I present some applications to financial problems both for the static and for the dynamic case.

Partiamo representations of finite groups

Basi per algebre cluster di rango 2

algebra e geometria

An Approximate Tour in Approximation Theory

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

analisi numerica

Applying mathematics to real life problems often requires different layers of approximation. For example, to study very complex phenomena we rely on simplified models (e.g. differential equations) that usually do not have explicit solutions. To estimate such solutions we pass trough discretizations and algorithms whose results depend on the amount of invested resources (e.g. time, computational power). In this context, approximation theory takes care of how functions can be approximated using simpler ones and how the approximation error behaves with respect to the properties of the functions involved, exploiting knowledges from different areas of mathemat- ics. In this talk we review fundamental notions and results of constructive approximation and use them to introduce wavelets, frames and subdivision schemes, while showing examples linking also to other topics.

Random Networks with Macroscopic Structures: Inference, Detectability and Belief Propagation

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

fisica matematica

A network is a good representation of a system with many interacting agents and networks with macroscopic structures (communities, hierarchies, cores, ...) naturally emerge from interactions regularities at a microscopic level. In many applications ( financial networks, biological networks, social networks) much of the information hidden in the data can be extracted from the detection of macroscopic structures wich are robust against microscopic noise. Starting from motivations and possible applications I will introduce the inference framework based on the Stochastic Block Model and the statistical mechanics approach to the associated detectability problem. This approach at the same time allows to depict the problem complexity in terms of detectability phase transitions and offers an efficient solution through a Belief Propagation algorithm.

A problem in combinatorics: Simon's conjecture

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

algebra e geometria

Simon's conjecture is a long-standing open problem in Combinatorics. In the first part of the seminar, we provide its motivation and all the necessary definitions to formulate it. In the second part, we present some recent improvements on this topic, pointing out the main tools used to obtain them and the typical approach to deal with this kind of problems.

A problem in combinatorics: Simon's conjecture

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

algebra e geometria

Simon's conjecture is a long-standing open problem in Combinatorics. In the first part of the seminar, we provide its motivation and all the necessary definitions to formulate it. In the second part, we present some recent improvements on this topic, pointing out the main tools used to obtain them and the typical approach to deal with this kind of problems.

Multipenalty regularization for linear inverse problems

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

analisi numerica

Linear inverse problems are typically ill-posed in the sense of Hadamard and they require regularization strategies in order to compute meaningful approximation of the desired solution. Traditional regularization methods solve an optimization problem whose objective function consists of a data-fit term, which measures how well an image matches the observations, and one or more penalty terms, referred to as regularization terms, which promote some desirable properties of the sought-for solution. The quality of the computed solution strongly depends on the choice of the regularization parameters weighting the penalty terms. An advantage of multi-penalty regularization, when compared with one-penalty regularization, is that different features of the solution can be enhanced by using several regularization terms. However, a drawback of multi-penalty regularization is that one has to select the values of several regularization parameters. In this talk, we present spatially adapted multi-penalty regularization for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. In the presented approach, the function to be minimized contains one $\ell_1$ penalty term and several spatially adapted $\ell_2$ penalty terms. An iterative procedure is illustrated for the automatic computation of all the regularization parameters. As a case study, we present the application of adaptive multi-penalty regularization to the inversion of two-dimensional NMR data.

Luglio

08

Françoise Pène (Université de Bretagne Occidentale, Francia)

Invariance by induction of the asymptotic variance

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

It is well known that the integral of an observable is preserved by induction. We are interested here in extensions of this result to moments of order 2 and 3. We have two natural candidates for the second and third order moments: the classical asymptotic variance (given by the Green-Kubo formula) and an analogous quantity of the third order. This question arises from the proof of CLT. In some cases, the asymptotic variance in the CLT can be expressed on the one hand in terms of the classical Green-Kubo formula and on the other hand in terms of the Green-Kubo formula for the induced system. Under general assumptions (involving transfer operators), we prove that the asymptotic variance is preserved by induction and that the natural third order quantity is preserved up to an error term. This is joint work with Damien Thomine.

Luglio

01

Marta Maggioni (Universiteit Leiden, Netherlands)

Matching for random systems with an application to minimal weight expansions

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

We consider families of skew-product maps, representing systems evolving in discrete time in which, at each time step, one of a number of transformations is chosen according to an i.i.d process and applied. We extend the notion of matching for such dynamical systems and we show that, for a certain family of piecewise affine random maps of the interval, the property of random matching implies that any invariant density is piecewise constant. We give an application by introducing a one-parameter family of random maps generating signed binary expansions of numbers. This family has random matching for Lebesgue almost every parameter, producing matching intervals that are related to the ones obtained for the Nakada continued fraction transformations. We use this property to study the expansions with minimal weight. Joint with K. Dajani, and C. Kalle.

Giugno

24

Andreas Knauf (Universität Erlangen-Nürnberg, Germany)

Asymptotic velocity for scattering particles

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

fisica matematica

Partly with Jacques Fejoz, Richard Montgomery, Stefan Fleischer and Manuel Quaschner. The past and future of scattering particle systems is partly determined by their asymptotic velocity, that is, the Cesàro limit of the velocity. That this exists for bounded interactions and all initial conditions, is part of a statement sometimes called ‘asymptotic completeness’. The same statement does not apply to individual initial conditions in celestial mechanics. However, at least for up to four particles, nonexistence of asymptotic velocity is a measure zero phenomenon. We explain some new ideas connected with the proof (Poincaré section techniques for wandering sets, non-deterministic particle systems, and walks on a poset of set partitions).

Algebraic structures and representation stability in configuration spaces

algebra e geometria

Let FI be the category of finite sets and injections. For a fixed space X, Church-Ellenberg-Farb demonstrated that packaging the sequence of ordered configuration spaces {PConf_n(X)} as a single object, namely a contravariant functor FI -> Top is very convenient in analyzing the "stable behavior" of the sequence. After reviewing the general framework, I will present general elementary conditions on X (which need not be a manifold) which allows the action of FI to be extended to a larger category. The extra structure yields several improvements in the onset of stabilization.

Funzioni di Jack e moltiplicazione di funzioni sferiche su varietà simmetriche

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

algebra e geometria

Il seminario, che cercherà di essere introduttivo ed elementare quanto possibile, sarà dedicato a descrivere il problema (largamente aperto) della moltiplicazione nell'anello delle coordinate di una varietà simmetrica. Dopo aver introdotto gli oggetti necessari, descriverò una vecchia congettura di Stanley (1989) sulle funzioni simmetriche di Jack e spiegherò le sue conseguenze nel problema della moltiplicazione.

Giugno

17

Sunrose Shrestha (Tufts University, USA)

The topology and geometry of random square-tiled surfaces

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

SEMINARIO INTERDISCIPLINARE

A square-tiled surface (STS) is a branched cover of the standard square torus with branching over exactly one point. They are concrete examples of translation surfaces which are an important class of singular flat metrics on 2-manifolds with applications in Teichmüller theory and polygonal billiards. In this talk we will consider a randomizing model for STSs based on permutation pairs and use it to compute the genus distribution. We also study holonomy vectors (Euclidean displacement vectors between cone points) on a random STS. Holonomy vectors of translation surfaces provide coordinates on the space of translation surfaces and their enumeration up to a fixed length has been studied by various authors such as Eskin and Masur. In this talk, we obtain finer information about the set of holonomy vectors, Hol(S), of a random STS. In particular, we will see how often Hol(S) contains the set of primitive integer vectors and find how often these sets are exactly equal.

Giugno

04

Sandro Vaienti (Centre de Physique Théorique, Marsiglia, Francia)

Thermodynamic formalism for random weighted covering systems

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy a random covering condition. Joint with J. Atnip, G. Froyland and C. Gonzalez-Tokman

28

Francesco Cellarosi (Queen’s University, Canada)

Rational Horocycle lifts and the tails of Quadratic Weyl sums

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

SEMINARIO INTERDISCIPLINARE

Equidistribution of horocycles on hyperbolic surfaces has been used to dynamically answer several probabilistic questions about number-theoretical objects. In this talk we focus on horocycle lifts, i.e. curves on higher-dimensional manifolds whose projection to the hyperbolic surface is a classical horocycle, and their behaviour under the action of the geodesic flow. It is known that when such horocycle lifts are `generic’, then their push forward via the geodesic flow becomes equidistributed in the ambient manifold. We consider certain ‘non-generic’ (i.e. rational) horocycle lifts, in which case the equidistribution takes place on a sub-manifold. We then use this fact to study the tail distribution of quadratic Weyl sums when one of their arguments is random and the other is rational. In this case we obtain random variables with heavy tails, all of which only possess moments of order less than 4. Depending on the rational argument, we establish the exact tail decay, which can be described with the help of the Dedekind \psi-function. Joint work with Tariq Osman.

27

Cristina Di Girolami (Le Mans Université e Università di Chieti-Pescara)

Infinite dimensional stochastic calculus, an infinite dimensional PDE and some applications

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

probabilità

scarica allegato

This talk develops some aspects of stochastic calculus via regularization for processes with values in a general Banach space B. A new concept of quadratic variation which depends on a particular subspace is introduced. An Itô formula and stability results for processes admitting this kind of quadratic variation are presented. Particular interest is devoted to the case when B is the space of real continuous functions defined on [-T,0], T>0 and the process is the window process X(•) associated with a continuous real process X which, at time t, it takes into account the past of the process. If X is a finite quadratic variation process (for instance Dirichlet, weak Dirichlet), it is possible to represent a large class of path-dependent random variable h as a real number plus a forward integral which are explicitly given. This representation result of h makes use of a functional solving an infinite dimensional partial differential equation. This decomposition generalizes, in some cases, the Clark-Ocone formula which is true when X is the standard Brownian motion W. Some stability results will be given explicitly. This is a joint work with Francesco Russo (ENSTA ParisTech Paris).

21

Martin Leguil (Université Paris-Sud 11, Francia)

Some rigidity results for billiards and hyperbolic flows

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

sistemi dinamici

In a project with P. Bálint, J. De Simoi and V. Kaloshin, we have been studying the inverse problem for a class of open dispersing billiards obtained by removing from the plane a finite number of smooth strictly convex scatterers satisfying a non-eclipse condition. The dynamics of such billiards is hyperbolic (Axiom A), and there is a natural labeling of periodic orbits. We show that it is generically possible, in the analytic category and for billiard tables with two (partial) axial symmetries, to determine completely the geometry of those billiards from the purely dynamical data encoded in their Marked Length Spectrum (lengths of periodic orbits + marking). An important step is the obtention of asymptotic estimates for the Lyapunov exponents of certain periodic points accumulating a reference periodic point, which turn out to be useful in the study of other rigidity problems. In particular, I will explain the results obtained in a joint work with J. De Simoi, K. Vinhage and Y. Yang on the question of entropy rigidity for 3-dimensional Anosov flows and dispersing billiards.

Nuove tecnologie e didattica on line

didattica della matematica

La didattica a distanza in questo periodo emergenziale è stata l'unica modalità che ha permesso a scuole e università di proseguire le varie attività di formazione.Nel seminario si discuterà come le nuove tecnologie, quali ad esempio un Digital Learning Environment, possano facilitare la didattica online non solo in tempi di emergenza. Particolare attenzione sarà rivolta al loro ruolo chiave che possono svolgere nell'insegnamento e nell'apprendimento della matematica e più in generale delle discipline STEM.

Structure of Learning Tasks and the Information in the Weights of a Deep Network

nel ciclo di seminari: GEOMETRIA E DEEP LEARNING

algebra e geometria

Abstract: What are the fundamental quantities to understand the learning process of a deep neural network? Why are some datasets easier than other? What does it means for two tasks to have a similar structure? We argue that information theoretic quantities, and in particular the amount of information that SGD stores in the weights, can be used to characterize the training process of a deep network. In fact, we show that the information in the weights bounds the generalization error and the invariance of the learned representation. It also allows us to connect the learning dynamics with the so called "structure function" of the dataset, and to define a notion of distance between tasks, which relates to fine-tuning. The non-trivial dynamics of information during training give rise to phenomena, such as critical periods for learning, that closely mimics those observed in humans and may suggests that forgetting information about the training data is a necessary part of the learning process.

Limit theorems for Lévy flights on a 1D Lévy random medium

fisica matematica

The purpose of the presentation is to introduce a version of a stochastic process known in the physical literature as Lévy-Lorentz gas and derive laws of large numbers and functional limit theorems for it. The model can be described as follows: we consider a point process omega given by an ordered array of points on the real line. We call omega the random medium. The nearest-neighbour distances between the points are i.i.d variables in the domain of attraction of a beta-stable distribution with beta belonging to (0,1) U (1,2). A random walk Y takes place on the medium according to the following rule. Independently of omega there exists a random walk S on the integers with i.i.d increments in the normal domain of attraction of an alfa-stable distribution with alfa belonging to (0,1) U (1,2). The role of S is to drive the dynamics of Y on the point process omega. For example, if a realization of S is (0,3,-1,…), the process Y starts at the origin of the real line, then jumps to the third point of omega to the right of 0, then to the first point of omega to the left of 0, and so on. Our process of interest is Y. We may describe it as a Lévy flight on a one-dimensional Lévy random medium. For all combinations of the parameters alfa and beta, we prove the annealed functional limit theorem for the suitably rescaled process Y, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

14

Giulio Tiozzo, University of Toronto

Central limit theorems for counting measures in coarse negative curvature

nell'ambito della serie: DINAMICI: ANOTHER INTERNET SEMINAR

SEMINARIO INTERDISCIPLINARE

We establish general central limit theorems for an action of a group on a hyperbolic space with respect to counting for the word length in the group. In 2013, Chas, Li, and Maskit produced numerical experiments on random closed geodesics on a hyperbolic pair of pants. Namely, they drew uniformly at random conjugacy classes of a given word length, and considered the hyperbolic length of the corresponding closed geodesic on the pair of pants. Their experiments lead to the conjecture that the length of these closed geodesics satisfies a central limit theorem, and we proved this conjecture in 2018. In our new work, we remove the assumptions of properness and smoothness of the space, or cocompactness of the action, thus proving a general central limit theorem for group actions on hyperbolic spaces. We will see how our techniques replace the classical thermodynamic formalism and allow us to provide new applications, including to lengths of geodesics in geometrically finite manifolds and to intersection numbers with submanifolds. Joint work with I. Gekhtman and S. Taylor.

A classification of the performance of Gradient Descent for high dimensional statistical problems

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

I will report on recent work with Aukosh Jagannath and Reza Gheissari on the performance of gradient descent algorithms during the initial phase of the minimization of high dimensional loss functions, and try to understand how they manage to escape the region of maximal entropy.

Random deep neural networks are biased towards simple functions

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

We prove that the binary classifiers of bit strings generated by random wide deep neural networks with ReLU activation function are biased towards simple functions. The simplicity is captured by the following two properties. For any given input bit string, the average Hamming distance of the closest input bit string with a different classification is at least sqrt(n / (2π log n)), where n is the length of the string. Moreover, if the bits of the initial string are flipped randomly, the average number of flips required to change the classification grows linearly with n. These results are confirmed by numerical experiments on deep neural networks with two hidden layers, and settle the conjecture stating that random deep neural networks are biased towards simple functions. This conjecture was proposed and numerically explored in [Valle Pérez et al., ICLR 2019] to explain the unreasonably good generalization properties of deep learning algorithms. The probability distribution of the functions generated by random deep neural networks is a good choice for the prior probability distribution in the PAC-Bayesian generalization bounds. Our results constitute a fundamental step forward in the characterization of this distribution, therefore contributing to the understanding of the generalization properties of deep learning algorithms. Advances in Neural Information Processing Systems 32, 1964-1976 (2019)

The replica-symmetric prediction for random linear estimation with Gaussian matrices is exact

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

Scaling Optimal Transport for High dimensional Learning

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will review entropic regularization methods which define geometric loss functions approximating OT with a better sample complexity. More information and references can be found on the website of our book "Computational Optimal Transport " https://optimaltransport.github.io/

The interpolating replica trick for disordered models. Recent developments.

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

We introduce a new version of the replica trick for disordered models, based on the interpolation on the number of replicas, and not on analytic continuation. We show that this strategy allows to find the order parameter in the general case, and the associated variational principle. The method works also for multispecie models. We show some applications to neural networks, and related systems.

DIRECT/INVERSE HOPFIELD MODEL AND RESTRICTED BOLTZMANN MACHINES

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

The 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 for 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 techniques and standard approaches to the direct problem.

THE GENERALIZATION ERROR OF OVERPARAMETRIZED MODELS: INSIGHTS FROM EXACT ASYMPTOTICS

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

In a canonical supervised learning setting, we are given n data samples, each comprising a feature vector and a label, or response variable. We are asked to learn a function f that can predict the the label associated to a new --unseen-- feature vector. How is it possible that the model learnt from observed data generalizes to new points? Classical learning theory assumes that data points are drawn i.i.d. from a common distribution and argue that this phenomenon is a consequence of uniform convergence: the training error is close to its expectation uniformly over all models in a certain class. Modern deep learning systems appear to defy this viewpoint: they achieve training error that is significantly smaller than the test error, and yet generalize well to new data. I will present a sequence of high-dimensional examples in which this phenomenon can be understood in detail. [Based on joint work with Song Mei, Feng Ruan, Youngtak Sohn, Jun Yan]

A classification of loss landscapes for the difficulty of weak recovery via Stochastic Gradient Descent

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

We study the performance of stochastic gradient descent in high-dimensional inference tasks. Our focus is on the initial ``search'' phase where the algorithm is far from a trust region and the loss landscape is highly non-convex. We develop a classification of the difficulty of this problem, namely whether the problem requires linear, quasilinear, or polynomially many samples in the dimension to achieve weak recovery of the parameter. This classification depends on an intrinsic property of the population loss which we call the ``information exponent''. We illustrate our approach by applying it to a wide variety of estimation tasks such as parameter estimation for generalize linear models, two component Gaussian mixture models, phase retrieval, and spiked matrix and tensor models, as well as supervised learning for singe-layer networks with general activation functions. In this latter case, our results translate to the difficulty of this task for teacher-student networks in terms of the Hermite decomposition of the activation function.

Aprile

27

Nonconvex interactions in mean-field spin glasses

Jean Christophe Mourrat

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

I will describe a conjecture for the limit free energy of mean-field spin glasses with a bipartite structure, which I could prove to be an upper bound for the true limit. The conjectured limit is described in terms of the solution of an infinite-dimensional Hamilton-Jacobi equation. A fundamental difficulty of the problem is that the nonlinearity in this equation is not convex. I will also question the possibility to characterize this conjectured limit in terms of a saddle-point problem

Low-dimensional mutliplexed representations in large recurrent neural networks: illustrations in supervised and unsupervised learning.

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

Recurrent Neural Networks (RNN) are powerful tools to learn computational tasks from data. How the tasks to be performed are simultaneously encoded and the input/output data are represented in a high-dimensional network is an important question. I will consider two related problems, both inspired from computational neuroscience: (1) how multiple low-dimensional maps (environments) can be embedded in a RNN, (2) how multiplexed integrations of velocity signals can be carried out in the RNN to update positions in those maps. I will discuss the nature of the representations found in artificial RNN, and compare them to experimental recordings in the mammal brain.

The influence of data structure on learning in neural networks

nel ciclo di seminari: MATHEMATICAL METHODS AND MODELS IN MACHINE LEARNING

fisica matematica

In this talk will present a new generative model of structured data, the hidden manifold model, and analyse learning dynamics and phase diagrams for such structured data.

Double branched cover of knotoids and entanglement in proteins

algebra e geometria

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they’ve been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. We characterise forbidden moves between knotoids in terms of equivariant band attachments between strongly invertible knots, and in terms of crossing changes between theta-curves. Finally, we present some applications to the study of the topology of proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

On the geometry of learning

nel ciclo di seminari: GEOMETRIA E DEEP LEARNING

algebra e geometria

We will present and discuss the main geometric and topological approaches and results regarding Deep Neural Networks and Generative Adversarial Networks.

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

analisi matematica

TBA

Totally positive functions, sampling, and Gabor frames

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

analisi matematica

Totally positive functions play an important role in approximation theory and statistics. I will discuss some recent applications of totally positive functions in sampling theory and time-frequency analysis. At this time totally positive functions are the only functions for which optimal results for sampling in shift-invariant spaces and for Gabor frames have been proved.

Marzo

19

Giovanni Emanuele Corazza

Creatività nella matematica: la lezione di Henri Poincaré

nel ciclo di seminari: SEMINARI DI PROBABILITÀ

SEMINARIO INTERDISCIPLINARE

Il seminario si baserà anche sul contributo: Corazza, G.E. & Lubart, T.A. (2019). Science and Method: Henri Poincaré. In Glaveanu, V. P. (Ed.). (2019). The creativity reader. Oxford University Press.

Connectedness of drops in convex potentials

analisi matematica

An old conjecture of Almgren states that for every convex and coercive potential $g: \mathbb{R}^d\to \mathbb{R}$, every convex and one-homogeneous anisotropy $\Phi : \mathbb{R}^d\to \mathbb{R}^+$ and every volume $V>0$, the minimizers of \[ \min_{|E|=V} \int_{\partial E} \Phi(\nu) d\mathcal{H}^{d-1} + \int_{E} g dx \] are convex. I will review the known results on this problem and present recent progress obtained with G. De Philippis on the connectedness of the minimizers for smooth potentials and anisotropies. Our proof is based on the introduction of a new ``two-point function'' which measures the lack of convexity and which gives rise to a negative second variation of the energy.

Around Simon's Conjecture

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

algebra e geometria

Simon's conjecture is an interesting open problem in Combinatorics. In the first part of the seminar, we provide all the necessary notions to formulate it. Recently, several authors gave new improvements on this topic. In the second part, we present some of these results, pointing out the main ideas and giving some new questions arising from them.

Asynchronous Parallel Methods for nonlinear systems, theory and practice

nel ciclo di seminari: DANIEL SZYLD

analisi numerica

Asynchronous Parallel Methods for linear systems, theory and practice. Part II: two- level methods

nel ciclo di seminari: DANIEL SZYLD

analisi numerica

Asynchronous Parallel Methods for linear systems, theory and practice. Part I: one level methods

nel ciclo di seminari: DANIEL SZYLD

analisi numerica

A universal algebraic approach to Racks and quandles

algebra e geometria

Racks and quandles are binary algebraic structures arising in knot theory, representation theory of the braid groups and the study of Hopf algebras. The first part of the talk is an overview on quandle theory and the motivation behind it. The main tools used in the study of racks and quandles are group and module theory. In the second part of the talk we introduce some new ideas coming from universal algebra. In particular, we adapt the commutator theory developed by Freese and McKenzie for arbitrary algebraic structures to racks and quandles. The goal of such theory is to define the notions of abelian and central congruences, extending the familiar definitions in the setting of groups and other classical varieties in order to talk about solvable and nilpotent objects. For racks, the commutator theory has a nice and sharp interpretation in group theoretical language. We also provide some applications, both towards classification problems for finite quandles and knot theory.

SEMINARIO INTERDISCIPLINARE

We try to describe in mathematical terms a class of models M of languages and of (learning) transformations of M which, when applied to (a collection of texts from) a given library (corpus) L, somehow embedded into M, would result in a model of the language behind L.

Funzioni di Moebius, connessioni di Galois e criterio di omotopia di Quillen

Asynchronous Parallel Methods for linear systems, theory and practice. Part I: one level methods

nel ciclo di seminari: DANIEL SZYLD

analisi numerica

On some methods to build GENEOs in Topological Data Analysis

nel ciclo di seminari: GEOMETRIA E DEEP LEARNING

algebra e geometria

In this talk we will briefly introduce a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. Our focus will be on illustrating some methods to build Goup Equivariant Non-Expansive Operators (GENEOs), which are maps between function spaces associated with groups of transformations. The development of these techniques will give us the opportunity to obtain a better approximation of thetopological space of all GENEOs.

On some methods to build GENEOs in Topological Data Analysis

nel ciclo di seminari: GEOMETRIA E DEEP LEARNING

algebra e geometria

In this talk we will briefly introduce a general mathematical framework for group equivariance in the machine learning context. The framework builds on a synergy between persistent homology and the theory of group actions. Our focus will be on illustrating some methods to build Goup Equivariant Non-Expansive Operators (GENEOs), which are maps between function spaces associated with groups of transformations. The development of these techniques will give us the opportunity to obtain a better approximation of thetopological space of all GENEOs.

The Leray model of a matroid

algebra e geometria

We construct a combinatorial abstraction of the Leray spectral sequence associated with the De Concini—Procesi compactification of a complex hyperplane arrangement complement. We show that its homological properties generalise the ones that we expect from topology in the realizable case. Geometric and Hodge-theoretic arguments are replaced by combinatorial blowups, Groebner bases, and sheaves on posets.

Swapping 2× 2 blocks in the Schur and generalized Schur form

analisi numerica

Low-Rank Multigrid Methods for Stochastic Partial Differential Equations

analisi numerica

Approximating the Inverse Hessian in 4D-Var Data Assimilation

analisi numerica

Cohomological Hall algebras

nel ciclo di seminari: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

During the first part of the talk I will give a gentle introduction to the theory of cohomological Hall algebras and their relevance in the study of the topology of moduli spaces, such as the Hilbert schemes of points on a smooth surface. The second part of the talk is devoted to the definition of the 2-dimensional cohomological Hall algebras of curves and surfaces. If time permits, I will discuss the construction of a categorification of these algebras. This is based on joint works with Olivier Schiffmann and Mauro Porta.

18

Frank Sottile (Texas A&M)

Galois groups in Enumerative Geometry and applications

algebra e geometria

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

Group equivariant non-expansive operators and deep learning

nel ciclo di seminari: GEOMETRIA E DEEP LEARNING

algebra e geometria

In this talk we illustrate a new mathematical model for machine learning, which follows from the assumption that data cannot be studied directly, but only through the action of agents that transform them. In our framework each agent is represented by a group equivariant non-expansive operator acting on data. After endowing the space of agents with a suitable metric, we describe the main topological and geometrical properties of this space by means of methods developed for topological data analysis.

17

Monica Montardini (Università di Pavia)

Space-time isogeometric Galerkin method and efficient solver for the heat equation

analisi numerica

Isogeometric analysis is an evolution of the finite element method: it employs B-splines or their generalization both to represent the computational domain and to approximate the solution of the considered partial differential equation. The high-continuity of isogeometric basis functions leads to several advantages, e.g. higher accuracy per degree-of-freedom, but it introduces also challenging problems at the computational level: one of the major issues is the efficient solution of linear systems. In this talk, I will focus on the study of an efficient solver for a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, I propose a preconditioner that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. polynomial degree and the time required for the application is almost proportional to the number of degrees-of-freedom. This is based on a joint work with G. Sangalli, M. Tani and G. Loli.

Una breve introduzione alla teoria di Melnikov e al chaos in sistemi dinamici non-autonomi a tempo continuo

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

analisi matematica

In questo seminario inizierò parlando brevemente di alcuni meccanismi che stanno alla base di particolari fenomeni caotici, per poi concentrarmi sulla teoria di Melnikov. In particolare analizzerò il caso di un sistema dinamico a tempo continuo autonomo che presenta una traiettoria omoclina (che quindi converge ad un punto critico sia nel passato che nel futuro), soggetto ad una perturbazione non-autonoma. La teoria di Melnikov fornisce condizioni che garantiscono la persistenza dell’omoclina e la nascita di fenomeni caotici. Il modello più noto per questa tipologia di fenomeni è il pendolo (non-lineare) perturbato. Si vedranno brevemente estensioni al caso multidimensionale e a quello discontinuo (piecewise smooth) che trova applicazione nella modellizzazione dei rimbalzi o dell’attrito strisciante.

11

Parametrised chain complexes in persistence theory

Barbara Giunti (UNIMORE)

SEMINARIO INTERDISCIPLINARE

Topological Data Analysis (TDA) is a new and fast-growing branch of math, addressing the need for extracting information from big data sets. One of the methods of TDA is persistence theory, which has proven to be very informative. The standard way to approach persistence theory is using persistent homology. In this seminar, I present a different approach, based on the study of tame parametrised chain complexes. This choice is indeed useful, since, firstly, the study of tame parametrised chain complexes includes the one of persistent homology, and thus is more general. Secondly, there are many powerful theoretical tools available in this setting, and such tools allow us to retrieve invariants, and therefore information, in more general cases. I will provide a quick introduction to these theoretical tools, namely to model category theory, and then focus on some examples to explain in details the constructions and the invariants."

A Geometric Interpretation of Stochastic Gradient Descent

nel ciclo di seminari: GEOMETRIA E DEEP LEARNING

algebra e geometria

We start with a review of the main steps of the Deep Learning algorithm, together with some historical remarks. We then concentrate on the key ingredient, stochastic gradient descent (SGD), whose geometric significance appears elusive and was modelled using the SDE Fokker Planck by Chaudhari and Soatto. We then study a deterministic model in which the trajectories of our dynamical systems are described via geodesics of a family of metrics arising from the diffusion matrix (natural gradient method). These metrics encode information about the highly non-isotropic gradient noise in SGD. This is a joint work with S. Soatto (UCLA, Amazon) and P. Chaudhari (U. Penn.)

A rationale on artificial intelligence in shallow networks

fisica matematica

In this talk I will paint a self‐consistent scenario for information processing in shallow neural networks: I will present a minimal reference framework where what is learnt (e.g. via contrastive divergence on restricted Boltzmann machines) is then retrieved (e.g. via standard Hebbian mechanisms à la Hopfield). Then I will generalize this scheme by discussing three variations on theme: the tradeoff between dilution and multitasking capabilities, the tradeoff between storage and resolution and the mechanism of "sleeping and dreaming".

04

Alessandro Monguzzi (Università di Milano–Bicocca)

Spaces of Entire Functions in C^n+1

analisi matematica

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. After recalling some basic properties of P W A I will present a generalization of this space 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. Time permitting, I will provide a Paley–Wiener type characterization and a sampling result. This is a joint work in progress with Marco Peloso and Maura Salvatori.

Prima lezione corso su Gruppi di Lie

algebra e geometria

I

A Lanczos-like method for the time-ordered exponential

analisi numerica

The time-ordered exponential is defined as the function that solves a system of coupled first-order linear differential equations with generally non-constant coefficients. In spite of being at the heart of much system dynamics, control theory, and model reduction problems, the time-ordered exponential function remains elusively difficult to evaluate. In the talk, we present a Lanczos-like algorithm capable of evaluating it by producing a tridiagonalization of the original differential system. The algorithm is presented in a theoretical setting. Nevertheless, a strategy for its numerical implementation is also outlined and will be subject of future investigation.

A dynamical system approach to the scalar curvature equation: multiplicity results.

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

analisi matematica

In this talk we illustrate results concerning radial positive solutions of semi-linear elliptic equations such as $$\Delta u + k(|x|) u^{q-1}=0 \qquad \qquad \qquad (1)$$ where $x \in \mathbb{R}^n$, $n>2$, $k(|x|)>0$, and its generalization to the $p$-Laplace case. We focus in particular on the critical case $q=2^*=\frac{2n}{n-2}$. Our goal is to find conditions on $k$ ensuring existence and multiplicity of ground states with fast decay, i.e. solutions $u(x)$ defined and positive in the whole of $\mathbb{R}^n$ and decaying as $|x|^{-(n-2)}$ for $|x|$ large. Using Fowler transformation we pass from (1) to a two dimensional dynamical system so that we can apply phase plane techniques such as invariant manifold theory, shooting, Melnikov theory. In particular the search of ground states with fast decay is translated on the search of homoclinic trajectories.

Bernoulli type free-boundary problems: the vectorial and non-local cases

analisi matematica

Selection of real regular finite gap-solutions of soliton equations - II

Lecture 2: “Difficult case” 1. Focusing Nonlinear Schr¨odinger equation. Cherednik differential for the Nonlinear Schr¨odinger equation. Characterization of divisors for real solutions in terms of Cherednik differential. Elementary characterization of admissible divisors. Regularity of all real solutions. 2. Sine-Gordon equation. Cherednik differential for the Since-Gordon equation. Characterization of divisors for real solutions in terms of Cherednik differential. Elementary characterization of admissible divisors. Regularity of all real solutions. 3. Number of connected components in the space of solutions for a fixed spectral curve. 4. Real regular Kadomtsev-Perviashvili I solutions.

21

Alberto Viscardi (Università di Bologna)

Interpolating Curves via Subdivision

analisi numerica

Subdivision schemes are iterative methods for the construction of curves and surfaces from a set of initial control points. Here we focus on the univariate case (i.e. curves) and on interpolatory subdivision (i.e. schemes with a limit curve that interpolates the initial data). This family of schemes can be subdivided in two classes, primal and dual. An algebraic characterization of both classes in terms of certain Laurent polynomials (called symbols) will be provided, together with some examples.

Selection of real regular finite gap-solutions of soliton equations - I

Lecture 1: “Easy case” 1. The problem of selecting real and real regular solutions. Real algebraic curves. Real ovals. M-curves. 2. “Easy case” and “difficult case”. 3. Real solutions of the Korteweg–de Vries equation. Real regular solutions. Regular and singular real components. 4. Defocusing Nonlinear Schr¨odinger equation. Real regular solutions. Regular and singular real components. 5. Real regular Kadomtsev-Petviashvili II solutions and M-curves.

Exact solutions in mean-field disordered statistical mechanics

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

SEMINARIO INTERDISCIPLINARE

TBA

The strange case of 4D Wilson loops or a third way to TBA

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

I will show how the relativistic Thermodynamic Bethe Ansatz (TBA) by Al. Zamolodchikov, naturally attached to Dynkin diagrams, can be derived from an operator product expansion of null polygonal Wilson loops. This is a sort of third way by which TBA arises naturally, by counting as other means 1) the usual thermodynamics on the scattering theory and 2) its derivation from the functional equations of the so-called Ordinary Differential Equation/Integrable Models correspondence. I will illustrate both these ways.

Integrablity and Thermodynamic Bethe Ansatz in Generalised Quantum Hydrodynamics

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

SEMINARIO INTERDISCIPLINARE

I briefly review the topic of Generalised Hydrodynamics in the framework of Non Equilibrium Steady States (NESS) and how it is strongly connected to Integrability. As a result, the profiles of the steady currents of energy between two reservoirs at different temperatures can be expressed in terms of a Thermodynamic Bethe Ansatz (TBA) approach known as NESS-TBA. Various issues of this approach are presented.

From Nekrasov functions to theta functions

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

I'm going to demonstrate how to take an autonomous limit of the general solution of some (q-)isomonodromic system. Such solutions are usually given as Fourier transformations of Nekrasov functions. One can show using Seiberg-Witten equations that in the autonomous limit these Fourier transformations turn into Riemann theta functions, and thus satisfy Fay bilinear relations

Cluster varietes, integrable systems and supersymmetric gauge theories - III

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.

Dynamical systems solvable by algebraic operations

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

In this talk I shall focus on systems of nonlinear Ordinary Di§erential Equations, and introduce the notion of their solvability by algebraic operations: implying that their general solution, considered as a function of complex time, feature at most a Önite number of rational branch points, or equivalently deÖne a Riemann surface with a Önite number of sheets. Some properties of these systems shall be reviewed, including the subclasses of them featuring such remarkable properties as isochrony or asymptotic isochrony (as functions of real time). Techniques to identify such systems shall be reviewed, and several examples reported, including new classes of such systems. References: F. Calogero, Isochronous Systems, Oxford University Press, 2008 (264 pages, paperback 2012); Zeros of Polynomials and Solvable Nonlinear Evolution Equations, Cambridge University Press, 2018 (168 pages). F. Calogero and F. Payandeh, ìPolynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactionsî, J. Math. Phys. 60, 082701 (2019). F. Calogero, R. Conte and F. Leyvraz, "New solvable systems of two autonomous Örst-order ordinary di§erential equations with purely quadratic right-hand sides" (in preparation).

Cluster varietes, integrable systems and supersymmetric gauge theories - II

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.

Regularity of the free boundary for the two-phase Bernoulli problem.

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

analisi matematica

I will illustrate a recent result obtained in collaboration with B. Velichkov and L. Spolaor concerning the regularity of the free boundaries in the two phase Bernoulli problems. The new point is the analysis of the free boundary close to branch points, where we show that it is given by the union of two C^1 graphs. This complete the analysis started by Alt Caffarelli Friedman in the 80’s.

Cluster varietes, integrable systems and supersymmetric gauge theories - I

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

The minicourse is devoted to integrable systems on cluster varieties, their deautonomization and connection with supersymmetric gauge theories. We start with the cluster Poisson varieties and describe their main properties, keeping as a basic example the Fock-Goncharov construction of cluster co-ordinates on the (affine, co-extended) Lie groups. Then we discuss how this construction leads to appearence of a completely integrable system on their Poisson subvarieties, with the most well-known example given by relativistic Toda chains, while generally these integrable systems can be alternatively defined a la Goncharov and Kenyon. The whole picture allows natural deautonomization, still keeping traces of integrability in the (discrete, non-autonomous) equations of the Painleve type, whose solutions can be constructed in terms of supersymmetric gauge theories. To do that we remind the connection between Seiberg-Witten prepotentials and algebraic integrable systems, introduce Nekrasov functions and show, that their duals (just by Fourier transform) appear in this context as isomonodromic tau-functions, solving the Hirota equations for deautonomized cluster integrable systems.

On Painleve’/gauge theory correspondence

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

I will give an introduction to the correspondence between Painlevé equations, the associated isomonodromy deformation problems and supersymmetric gauge theories. The relation to Hitchin’s integrable system, (quantum) Toda chain and elliptic Calogero system will be highlighted.

Algebraic-geometrical integration theory of soliton equations - III

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Course description: A self-contained introduction to the theory of soliton equations with an emphasis on its applications to algebraic-geometry. Topics include: 1. General features of the soliton systems. Basic hierarchies of commuting flows: KP, 2D Toda, Bilinear Discrete Hirota hierarchy. Discrete and finite-dimensional integrable systems. 2. Algebraic-geometrical integration theory. Spectral curves. Baker-Akhiezer functions. 3. Hamiltonian theory of soliton equations.

Algebraic-geometrical integration theory of soliton equations - II

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

SEMINARIO INTERDISCIPLINARE

Course description: A self-contained introduction to the theory of soliton equations with an emphasis on its applications to algebraic-geometry. Topics include: 1. General features of the soliton systems. Basic hierarchies of commuting flows: KP, 2D Toda, Bilinear Discrete Hirota hierarchy. Discrete and finite-dimensional integrable systems. 2. Algebraic-geometrical integration theory. Spectral curves. Baker-Akhiezer functions. 3. Hamiltonian theory of soliton equations.

Integrable and superintegrable systems on moduli spaces of flat connections - III

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Let $G$ be a simple Lie group. For a compact topological surface the moduli space of $G$-flat connections has a natural symplectic structure when the surface is closed and a Poisson structure when the surface has boundary (Atyiah and Bott). Symplectic leaves of this Poisson variety are parametrized by conjugacy classes of monodromies along connected components of the boundary. The goal of these lectures is to explain that central functions on holonomies along any simple collection of curves on a surface define a superintegrable system and to describe these systems in some special cases. We will also see that these systems are natural generalizations of spin Calogero-Moser systems. The first lecture will focus on the definition and properties of superintegrable systems. In the second lecture spin Calogero-Moser systems will be introduced and it will be proven that they are superintegrable systems. After this superintegrable systems corresponding to simple curves on a surface will be introduced and the connection to Calogero-Moser type systems will be explained. The lectures are based on joint work with S. Artamonov and J. Stokman.

Total positivity, M-curves and real regular Kadomtsev-Petviashvili II solutions

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

This lecture is based on joint results with Simonetta Abenda, Bologna University. We establish a bridge between two approaches to constructing real regular solutions of the Kadomtsev-Petviashvili equation. Multiline soliton solutions are constructed in terms of totally non-negative Grassmannians, and real regular finite-gap solution correspond to spectral M-curves with divisors satisfying an extra condition. It is easy to construct soliton solutions by degenerating the spectral curves, but if we would like to stay in the real regular class, the problem becomes non-trivial. We present a construction associating a degenerate M-curve and a divisor on it with reality and regularity condition to a point of a totally non-negative Grassmannian. This construction essentially uses the parametrization of the totally non-negative Grassmannians in terms of the Le-networks from the Postnikov’s paper.

Hirota Difference Equation and Darboux System: Mutual Symmetry

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

We considered the relation between two famous integrable equations: The Hirota difference equation (HDE) and the Darboux system that describes conjugate curvilinear systems of coordinates in R3. We demonstrated that specific properties of solutions of the HDE with respect to independent variables enabled introduction of an infinite set of discrete symmetries. We showed that degeneracy of the HDE with respect to parameters of these discrete symmetries led to the introduction of continuous symmetries by means of a specific limiting procedure. This enabled consideration of these symmetries on equal terms with the original HDE independent variables

Mathematical Physics of Map Enumeration

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

SEMINARIO INTERDISCIPLINARE

Generating functions whose coefficients count maps (ribbon graphs) and hypermaps (Grothendieck's dessins d'enfant) satisfy several remarkable integrability properties. In particular, they obey Virasoro constraints, evolution equations, Kadomtsev-Petviashvili (KP) hierarchy and a topological recursion. (After a joint work with M. Kazarian)

Aspetti geometrici e astronomici nella Felsina fondata

SEMINARIO INTERDISCIPLINARE

Una città etrusca dipendeva da un sito esterno con una buona visione sulla città detto auguraculum (A). Nell’auguraculum l’augure dirigeva l’augurazione della città. La diagonale principale della città era fissata da un raggio di sole all’alba o al tramonto di un giorno dell’anno (Natale della città). Su questa diagonale si fissava un punto U (umbilicum=centro del mondo) a distanza R dall’auguraculum. In U si faceva l’inaugurazione della città e si tracciava la diagonale secondaria lungo un raggio di sole al tramonto dello stesso giorno o all’alba di quello successivo. Si fissava il raggio del pomerio, un cerchio di centro U che circondava l’urbe identificata con l’orbe dell’universo. Le intersezioni delle diagonali con il pomerio fissavano i quattro vertici di un quadrilatero che si dimostra essere un rettangolo per simmetria rispetto alle bisettrici degli angoli tra le diagonali. Le stesse bisettrici formano la croce sacrale che indica le strade principali: decumano massimo e cardio maximus. Le bisettrici si trovano tramite le semicorde: i semilati del rettangolo. L’asse dell’universo (Cardine) proveniente dal polo Nord celeste penetrava nel punto U della Terra e ne usciva nel punto S, estremo Sud del pomerio. La proiezione del Cardine sommerso sul piano della città era la parte meridionale del cardine. Si considerano due proiezioni del Cardine sulla verticale in U e in S, rappresentate da una coppia di colonne cosmiche. Le colonne cosmiche sono intese come sostegni del Cielo sulla Terra, o viceversa. A causa del diverso moto di rotazione del Cielo e della Terra queste colonne, intese come plastiche, appaiono a torciglione. Per gli Etruschi erano piuttosto dei perni rigidi detti cardini. A volte erano un sistema articolato di infiniti perni inseriti tra i primi vicini. Ovviamente questo sistema era totalmente instabile. Secondo gli Indiani attorno ad un perno si avvolgeva una corda che veniva tirata agli estremi in una specie di tiro alla fune che faceva girare il perno stesso. Vincenzo Grecchi: Bologna: il mistero delle quattro croci e il rito etrusco (Acacne Editrice, Bologna 2019)

Aspetti geometrici e astronomici nella Felsina fondata

SEMINARIO INTERDISCIPLINARE

Una città etrusca dipendeva da un sito esterno con una buona visione sulla città detto auguraculum (A). Nell’auguraculum l’augure dirigeva l’augurazione della città. La diagonale principale della città era fissata da un raggio di sole all’alba o al tramonto di un giorno dell’anno (Natale della città). Su questa diagonale si fissava un punto U (umbilicum=centro del mondo) a distanza R dall’auguraculum. In U si faceva l’inaugurazione della città e si tracciava la diagonale secondaria lungo un raggio di sole al tramonto dello stesso giorno o all’alba di quello successivo. Si fissava il raggio del pomerio, un cerchio di centro U che circondava l’urbe identificata con l’orbe dell’universo. Le intersezioni delle diagonali con il pomerio fissavano i quattro vertici di un quadrilatero che si dimostra essere un rettangolo per simmetria rispetto alle bisettrici degli angoli tra le diagonali. Le stesse bisettrici formano la croce sacrale che indica le strade principali: decumano massimo e cardio maximus. Le bisettrici si trovano tramite le semicorde: i semilati del rettangolo. L’asse dell’universo (Cardine) proveniente dal polo Nord celeste penetrava nel punto U della Terra e ne usciva nel punto S, estremo Sud del pomerio. La proiezione del Cardine sommerso sul piano della città era la parte meridionale del cardine. Si considerano due proiezioni del Cardine sulla verticale in U e in S, rappresentate da una coppia di colonne cosmiche. Le colonne cosmiche sono intese come sostegni del Cielo sulla Terra, o viceversa. A causa del diverso moto di rotazione del Cielo e della Terra queste colonne, intese come plastiche, appaiono a torciglione. Per gli Etruschi erano piuttosto dei perni rigidi detti cardini. A volte erano un sistema articolato di infiniti perni inseriti tra i primi vicini. Ovviamente questo sistema era totalmente instabile. Secondo gli Indiani attorno ad un perno si avvolgeva una corda che veniva tirata agli estremi in una specie di tiro alla fune che faceva girare il perno stesso. Vincenzo Grecchi: Bologna: il mistero delle quattro croci e il rito etrusco (Acacne Editrice, Bologna 2019)

15

Michael Semenov-Tian-Shansky

Quantum Toda Lattice and Representation Theory - III

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Quantum Toda Lattice is a famous model which displays deep connections both with the Representation Theory of semisimple Lie groups and with the Quantum Inverse Scatttering Method. Kostant's famous discovery linked Toda Lattice to the theory of generalized Whiitaker functions. New formulae, due mainly to Lebedev and his collaborators, are based on Sklyanin's ideology of quantum separation of variables. Interaction between the two approaches sheds a new light on classical constructions of Representation Theory.

The road closure conjecture

algebra e geometria

There is a new approach that has been proposed recently for the study of certain regular semigroups. This new approach involves the study of finite primitive groups. Via this method, many problems in the context of regular semigroups translate into natural problems in finite primitive groups. We give some details concerning the relation between regular semigroups and finite primitive groups. In particular, we discuss the Road Closure Conjecture on finite primitive groups, which is strictly related to a possible classification of certain idempotent generated regular semigroups.

Integrable and superintegrable systems on moduli spaces of flat connections - II

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Let $G$ be a simple Lie group. For a compact topological surface the moduli space of $G$-flat connections has a natural symplectic structure when the surface is closed and a Poisson structure when the surface has boundary (Atyiah and Bott). Symplectic leaves of this Poisson variety are parametrized by conjugacy classes of monodromies along connected components of the boundary. The goal of these lectures is to explain that central functions on holonomies along any simple collection of curves on a surface define a superintegrable system and to describe these systems in some special cases. We will also see that these systems are natural generalizations of spin Calogero-Moser systems. The first lecture will focus on the definition and properties of superintegrable systems. In the second lecture spin Calogero-Moser systems will be introduced and it will be proven that they are superintegrable systems. After this superintegrable systems corresponding to simple curves on a surface will be introduced and the connection to Calogero-Moser type systems will be explained. The lectures are based on joint work with S. Artamonov and J. Stokman.

Integrable and superintegrable systems on moduli spaces of flat connections - I

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Let $G$ be a simple Lie group. For a compact topological surface the moduli space of $G$-flat connections has a natural symplectic structure when the surface is closed and a Poisson structure when the surface has boundary (Atyiah and Bott). Symplectic leaves of this Poisson variety are parametrized by conjugacy classes of monodromies along connected components of the boundary. The goal of these lectures is to explain that central functions on holonomies along any simple collection of curves on a surface define a superintegrable system and to describe these systems in some special cases. We will also see that these systems are natural generalizations of spin Calogero-Moser systems. The first lecture will focus on the definition and properties of superintegrable systems. In the second lecture spin Calogero-Moser systems will be introduced and it will be proven that they are superintegrable systems. After this superintegrable systems corresponding to simple curves on a surface will be introduced and the connection to Calogero-Moser type systems will be explained. The lectures are based on joint work with S. Artamonov and J. Stokman.

Algebraic-geometrical integration theory of soliton equations - I

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

SEMINARIO INTERDISCIPLINARE

Course description: A self-contained introduction to the theory of soliton equations with an emphasis on its applications to algebraic-geometry. Topics include: 1. General features of the soliton systems. Basic hierarchies of commuting flows: KP, 2D Toda, Bilinear Discrete Hirota hierarchy. Discrete and finite-dimensional integrable systems. 2. Algebraic-geometrical integration theory. Spectral curves. Baker-Akhiezer functions. 3. Hamiltonian theory of soliton equations.

14

Michael Semenov-Tian-Shansky

Quantum Toda Lattice and Representation Theory - II

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Quantum Toda Lattice is a famous model which displays deep connections both with the Representation Theory of semisimple Lie groups and with the Quantum Inverse Scatttering Method. Kostant's famous discovery linked Toda Lattice to the theory of generalized Whiitaker functions. New formulae, due mainly to Lebedev and his collaborators, are based on Sklyanin's ideology of quantum separation of variables. Interaction between the two approaches sheds a new light on classical constructions of Representation Theory.

14

Michael Semenov-Tian-Shansky

Quantum Toda Lattice and Representation Theory - I

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

Quantum Toda Lattice is a famous model which displays deep connections both with the Representation Theory of semisimple Lie groups and with the Quantum Inverse Scatttering Method. Kostant's famous discovery linked Toda Lattice to the theory of generalized Whiitaker functions. New formulae, due mainly to Lebedev and his collaborators, are based on Sklyanin's ideology of quantum separation of variables. Interaction between the two approaches sheds a new light on classical constructions of Representation Theory.

Shuffle algebras: main examples

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

My talk is supposed to be an addition to B. Feigin's lecture course. I plan to discuss in detail the connection between shuffle algebras and simplest quantum affine and toroidal algebras, (mainly following A. Negut.)

Cohomology of algebraic structures and integrability

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

will explain an operadic approach to cohomology theory, which allows one to develop cohomology for Poisson vertex algebras and vertex algebras. This is applied to the proof of integrability for classical and quantum Hamiltonian PDE.

Combinatorial integrability -III

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects. A preliminary layout includes - Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation) Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements) The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions) Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)

Affine and toroidal algebras and integrable systems -III

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

I plan to discuss the constructions of different quantum groups . Our main tool - shuffle algebras . Definition of such algebras is rather elementary and simple.In shuffle terms it is possible to get the simple construction of affine and toroidal quantum groups and also in some cases deformed W-algebras. Representations of quantum groups and R -matrices are also naturally appear. Shuffle algebras are good for the constructions of coordinate rings of the quantizations of the different moduli spaces -for example instanton manifolds.I plan to explain how to do it.Actually such coordinate rings can be realised as quotients or subalgebras of the shuffle algebras. In some special case subalgebras in the suffle algebras become commutative. By this way we get "big" commutative subalgebras which in representations become the quantum hamiltonians of interesting integrable systems.

Combinatorial integrability -II

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects. A preliminary layout includes - Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation) Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements) The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions) Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)

Combinatorial integrability -I

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

The minicourse will be devoted to a description of combinatorial solutions to integrable hierarchies of Kadomtsev--Petviashvili type that arise naturally in enumeration of various topological and algebro-geometric objects. A preliminary layout includes - Permutations and their decompositions into products of transpositions (simple Hurwitz numbers, Okounkov's theorem, Hurwitz formula, Bousquet-M\'elou--Schaeffer formula, cut-and-join equation) Symmetric group representations (diagonilizability of the cut-and-join operator, the group algebra of the symmetric group, Schur polynomial, Jucys--Murphy elements) The semiinfinite Grassmannian and the Kadomtsev--Petviashvili hierarchy (Pl\"ucker embeddings, semiinfinite planes in the space of Laurent series, Orlov--Shcherbin family of solutions) Ramified coverings of the 2-sphere (coverings and ramified coverings, Hurwitz numbers and ramified coverings, Caley formula, genus expansion)

The Hitchin fibration as an integrable system

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

SEMINARIO INTERDISCIPLINARE

TBA

Affine and toroidal algebras and integrable systems - II

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

I plan to discuss the constructions of different quantum groups . Our main tool - shuffle algebras . Definition of such algebras is rather elementary and simple.In shuffle terms it is possible to get the simple construction of affine and toroidal quantum groups and also in some cases deformed W-algebras. Representations of quantum groups and R -matrices are also naturally appear. Shuffle algebras are good for the constructions of coordinate rings of the quantizations of the different moduli spaces -for example instanton manifolds.I plan to explain how to do it.Actually such coordinate rings can be realised as quotients or subalgebras of the shuffle algebras. In some special case subalgebras in the suffle algebras become commutative. By this way we get "big" commutative subalgebras which in representations become the quantum hamiltonians of interesting integrable systems.

Affine and toroidal algebras and integrable systems

nel ciclo di seminari: WINTER SCHOOL ON INTEGRABLE SYSTEMS AND REPRESENTATION THEORY

I plan to discuss the constructions of different quantum groups . Our main tool - shuffle algebras . Definition of such algebras is rather elementary and simple.In shuffle terms it is possible to get the simple construction of affine and toroidal quantum groups and also in some cases deformed W-algebras. Representations of quantum groups and R -matrices are also naturally appear. Shuffle algebras are good for the constructions of coordinate rings of the quantizations of the different moduli spaces -for example instanton manifolds.I plan to explain how to do it.Actually such coordinate rings can be realised as quotients or subalgebras of the shuffle algebras. In some special case subalgebras in the suffle algebras become commutative. By this way we get "big" commutative subalgebras which in representations become the quantum hamiltonians of interesting integrable systems.

09