6 Luglio, 2021 17:00

Seminario Matematico e Fisico di Milano

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

Jean Dolbeault, Université Paris Dauphine

abstract

abstract

Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various areas of Science. Research interests have evolved over the 80 years: while mathematicians were originally focussed on abstract properties (like notions of weak solutions and Cauchy problem in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. Entropy methods for nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as the optimal rate of decay of an entropy for an associated evolution equation. Much more can be learned on the way.
This lecture is intended to give an overview of various results on some Gagliardo-Nirenberg-Sobolev and Caffarelli-Kohn-Nirenberg inequalities obtained during the last decade. It will not be a global picture of an active area of research but more a series of snapshots aiming at the illustration of some emerging tools and new directions of research.

5 Luglio, 2021 15:00

Sezione di Fisica Matematica

Bosonic properties of fermionic systems

Christian Hainzl, LMU Munich, Germany

online

abstract

abstract

A pair of particles, one from inside, one from outside the Fermi sea, can be considered as a boson. Such a simplifying picture allows to obtain the correlation energy for both high density and low density Fermi systems, despite their completely different behavior.

28 Giugno, 2021 17:30 in punto

Sezione di Probabilità e Statistica Matematica

Regularization lemmas and convergence in total variation.

Lucia Caramellino, Università di Roma "Tor Vergata"

abstract

abstract

We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from a smooth (Wasserstein e.g.) distance to the total variation in a quantitative way. We provide a result removing the costly assumption that some non-degeneracy is required along the whole sequence, as we require only non-degeneracy at the limit. The price to pay is to control the smooth distance between the Malliavin matrix of the sequence and the Malliavin matrix of the limit which is particularly easy in the context of Gaussian limits as their Malliavin matrix is deterministic. We show some applications of this result.
From joint papers with Vlad Bally and Guillaume Poly.

21 Giugno, 2021 17:00

Seminario Matematico e Fisico di Milano

Dynamical rigidity of convex billiards

Jacopo De Simoi, University of Toronto

abstract

abstract

Convex billiards are a classical topic in conservative dynamics. Typically, their dynamics is qualitatively very intricate, since it showcases a coexistence of hyperbolic dynamics and KAM phenomena. Understanding long-term statistical properties of the dynamics with the current technology is essentially an intractable problem.

Here I venture in the opposite direction and I will discuss dynamical inverse problems: how much geometrical information can be extracted from the dynamics?

More precisely: what can be deduced about the billiard table if one knows the lengths of all periodic orbits? The quantum version of this question has been famously stated as "Can one hear the shape of a drum?"

In this talk I will review the latest results and describe the next steps in this direction. This is a joint project with Vadim Kaloshin.

16 Giugno, 2021 17:00

Sezione di Geometria, Algebra e loro applicazioni

Discrete analytic functions and Schur analysis

Daniel Alpay, Chapman University

On line

abstract

abstract

We first review both the theory of discrete analytic functions and the main features of Schur analysis (a collection of problems pertaining to functions analytic and contractive in the open unit disk, and with a wide range of applications). Then, we present new connections between the theory of discrete analytic functions and Schur analysis. This allows us to define a new class of problems pertaining to discrete analytic functions.

References:

D. Alpay, P. Jorgensen, R. Seager, and D. Volok. On discrete analytic functions: Products, Rational Functions, and Reproducing Kernels. Journal of Applied Mathematics and Computing. Volume 41, Issue 1 (2013), Page 393-426.

D. Alpay and D. Volok, Discrete analytic functions and Schur analysis. Preprint, 2021.

14 Giugno, 2021 17:30 in punto

Sezione di Probabilità e Statistica Matematica

On unique ergodicity, regularity, and mixing for the weakly-damped stochastic KdV equation.

Vincent Martinez, CUNY Hunter College

abstract

abstract

We discuss the existence, uniqueness, and regularity of invariant measures for the damped-driven stochastic Korteweg-de Vries equation, where the noise is additive and sufficiently non-degenerate. It is shown that a simple, but versatile control strategy, typically employed to establish exponential mixing for strongly dissipative systems such as the 2D Navier-Stokes equations, can nevertheless be applied in this weakly dissipative setting to establish elementary proofs of both unique ergodicity, albeit without mixing rates, as well as regularity of the support of the invariant measure. Under the assumption of large damping, however, a one-force, one-solution principle is established, from which we are able to deduce the existence of a spectral gap with respect to a Wasserstein distance-like function.

11 Giugno, 2021 14:00

MOX Seminar

La democratizzazione dell’Intelligenza Artificiale

Fabio Moioli, Head Consulting & Services @Microsoft Italia

Online seminar: mox.polimi.it/elenco-seminari/?id_evento=2053&t=763721

abstract

abstract

Le nuove tecnologie e l’Intelligenza Artificiale offrono oggi grandi opportunità di crescita economica e sociale, offrendo a tutti innovative soluzioni per migliorare le attività siano esse professionali o personali.

Di fronte a questi nuovi scenari si aprono importanti temi etici che vanno affrontati affinché l’impatto offerti da queste tecnologie possa portare i benefici attesi per migliorare la vita di tutti.

Con esempi e casi reali, durante l’intervento si valuteranno i molteplici ambiti di applicazione dell’AI, i progressi tecnologici fatti e i principi etici che guidano un’organizzazione come Microsoft nella definizione di sistemi di AI.

Sarà anche l’occasione per scoprire anche i vantaggi offerti dalle soluzioni di AI sui grandi urgenze mondiali, dalla crisi sanitaria a quelli legati ai temi della sostenibilità.

Contatto: paolo.zunino@polimi.it

10 Giugno, 2021 14:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Local Cox rings for klt singularities

Lukas Braun, Universität Freiburg, Germania

On line

abstract

abstract

In this talk, I will discuss notions of Cox rings for (klt) singularities. We investigate different local models and how the Cox rings behave when changing the model.
Finally, we investigate an iteration process for Cox rings and compare the associated covers with covers coming from the local fundamental group.


Cordiali saluti,

9 Giugno, 2021 15:15 in punto

Sezione di Analisi

Some adaptations of the moving planes method to singular elliptic problems

Francesco Esposito, Universita' della Calabria

abstract

abstract

In this talk we will consider positive singular solutions to some quasilinear elliptic problems under zero Dirichlet boundary conditions. We will study qualitative properties of the solutions via the moving plane procedure, that goes back to the celebrated papers of Alexandrov and Serrin. In particular, exploiting a fine adaptation of the moving plane procedure and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions in bounded smooth domains.

7 Giugno, 2021 17:00

Sezione di Fisica Matematica

Mathematics of magic angles in a model of twisted bilayer graphene

Macej Zworski, University of California, Berkeley CA, US

online

abstract

abstract

Magic angles are a hot topic in condensed matter physics:when
two sheets of graphene are twisted by those angles the resulting
material is superconducting. I will present a very simple operator whose
spectral properties are thought to determine which angles are magical.It
comes from a recent PR Letter by Tarnopolsky--Kruchkov--Vishwanath. The
mathematics behind this is an elementary blend of representation
theory(of the Heisenberg group in characteristic three), Jacobi theta
functions and spectral instability of non-self-adjoint operators
(involving Hörmander's bracket condition in a very simple setting).
Spectral characterization of magic angles also allows precise numerical
computations and I will discuss the error bound issues arising there
(joint work with S Becker, Membree and J Wittsten).