26 Maggio, 2021 12:15

Seminari di Cultura Matematica

Meccanica statistica e fenomeni sociali

Giuseppe Toscani,  Università di Pavia

online: tiny.cc/fdsCM2021

26 Maggio, 2021 15:15 in punto

Sezione di Analisi

Contact surface of Cheeger sets

Marco Caroccia, Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

Geometrical properties of Cheeger sets have been deeply studied by many authors since their introduction, as a way of bounding from below the first Dirichlet (p)-Laplacian eigenvualue. They represent, in some sense, the first eigenfunction of the Dirichlet (1)-Laplacian of a domain. In this talk we will introduce a recent property, studied in collaboration with Simone Ciani, concerning their contact surface with the ambient space. In particular we will show that the contact surface cannot be too small, with a lower bound on the (Hasudorff) dimension strictly related to the regularity of the ambient space. The talk will focus on the introduction of the problem and on the proof of the dimensional bounds. Functional to the whole argument is the notion of removable singularity, as a tool for extending solutions of pdes under some regularity constraint. Finally examples providing the sharpness of the bounds in the planar case are briefly treated.

24 Maggio, 2021 17:30 in punto

Sezione di Probabilità e Statistica Matematica

Limit Theorems for Exponential Random Graphs.

Alessandra Bianchi, Università di Padova

abstract

abstract

Exponential Random Graphs are defined through probabilistic ensembles with one or more adjustable parameters. They can be seen as a generalization of the classical Erdos Renyi random graph, obtained by defining a tilted probability measure that is proportional to the densities of certain given finite subgraphs. In this talk we will focus on the edge-triangle model, that is a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. Borrowing tools from statistical mechanics, together with large deviations techniques, we will characterize the limiting behavior of the edge density for all parameters in the so-called replica symmetric regime, where a complete characterization of the phase diagram of the model is accessible. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Joint work with Francesca Collet and Elena Magnanini.

19 Maggio, 2021 12:15

Seminari di Cultura Matematica

Design and Health

Stefano Capolongo, Politecnico di Milano

online: tiny.cc/fdsCM2021

Exponential Random Graphs are defined through probabilistic ensembles with one or more adjustable parameters. They can be seen as a generalization of the classical Erdos Renyi random graph, obtained by defining a tilted probability measure that is proportional to the densities of certain given finite subgraphs. In this talk we will focus on the edge-triangle model, that is a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. Borrowing tools from statistical mechanics, together with large deviations techniques, we will characterize the limiting behavior of the edge density for all parameters in the so-called replica symmetric regime, where a complete characterization of the phase diagram of the model is accessible. First, we determine the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases. Then we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Joint work with Francesca Collet and Elena Magnanini.

19 Maggio, 2021 17:00

Sezione di Geometria, Algebra e loro applicazioni

Fischer decomposition for massless field equations

Roman Lavicka, Charles University Prague

On line

abstract

abstract

Massless field equations are fundamental in particle physics. In Clifford analysis, a version of these equations in Euclidean space of dimension 4 have been studied. In this talk, we shall discuss a recent development on this topic. In particular, for fields with values in a general irreducible spin module, as an analogue of massless field equations we propose the so-called generalized Cauchy-Riemann equations introduced by E. Stein and G. Weiss. Then we describe Fischer decompositions of massless fields up to spin 3/2. This is a joint work with V. Soucek, W. Wang, F. Brackx and H. De Schepper.

18 Maggio, 2021 11:15 in punto

Sezione di Analisi

Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincaré inequalities

Giulia Meglioli, Dipartimento di Matematica Politecnico di Milano

abstract

abstract

The talk is concerned with the reaction-diffusion equation $u_t=\Delta(u^m)+u^p$, on a complete noncompact Riemannian manifold $M$. We consider the particularly delicate case when $p$ is less than $m$; moreover, we assume that the Poincaré and the Sobolev inequalities hold on $M$. We prove global existence in time of solutions for $L^m$ initial data. Furthermore, solutions are bounded for all positive times and their $L^\infty$ norm satisfy a certain quantitative bound. We also see that on a special class of Riemannian manifolds, solutions corresponding to sufficiently large $L^m$ data give rise to solutions that blow up in infinite time, a fact that cannot happen in the Euclidean setting.
The results have been recently obtained jointly with Gabriele Grillo and Fabio Punzo.

13 Maggio, 2021 14:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Cox rings and Algebraic stacks

Fabio Tonini, Università di Firenze

On line

abstract

abstract

We will discuss the notion of Cox ring for an algebraic stack, which extends the classical notion for varieties, via the language of torsors. We will then present some applications and possible direction of research.

12 Maggio, 2021 12:15

Seminari di Cultura Matematica

"L'amor che move il sole e l'altre stelle". Il cielo nella Divina Commedia

Fabio Peri, Civico Planetario U. Hoepli di Milano

online: tiny.cc/fdsCM2021

We will discuss the notion of Cox ring for an algebraic stack, which extends the classical notion for varieties, via the language of torsors. We will then present some applications and possible direction of research.

12 Maggio, 2021 15:15 in punto

Sezione di Analisi

Perturbation theory for Dirichlet eigenvalues in perforated domains

Laura Abatangelo, Dipartimento di Matematica, Politecnico di Milano

abstract

abstract

In this talk I will present some recent results on asymptotics of eigenvalues of the Dirichlet Laplacian when a small compact set is removed from the initial domain. If the small set is concentrating at a point in some sense, the eigenvalue variation is proved to be strictly related to the vanishing order of one of the relative eigenfunctions at that point. A good understanding of this asymptotics leads to new issues, for instance optimal location or optimal shape of the hole (open problem) as well as possible ramification of multiple eigenvalues.

10 Maggio, 2021 17:00

Seminario Matematico e Fisico di Milano

The DPG Method for Convection-Reaction Problems

Leszek Demkowicz, Oden Institute, The University of Texas at Austin

abstract

abstract

We present a progress report on the development of Discontinuous Petrov-Galerkin methods for the convection-reaction problem in context of time-stepping and space-time discretizations of Boltzmann equations [1].
The work includes a complete analysis for both conforming (DPGc) and non-nonconforming (DPGd) versions of the DPG method employing either globally continuous or discontinuous piece-wise polynomials to discretize the traces.
The results include construction of a local Fortin operator for the case of constant convection and a global discrete stability analysis for both DPGc and DPGd methods.
The theoretical findings are illustrated with numerous numerical experiments in two space dimensions.

This is a joint work with Nathan Roberts from Sandia National Laboratories.

Slides (PDF)

[1] L. Demkowicz, N. Roberts, "The DPG Method for the Convection–Reaction Problem Revisited", submitted.