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.
Seminar Discrete analytic functions and Schur analysis Daniel Alpay, Chapman University mercoledì 16 giugno 2021 alle ore 17:00 On line
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.
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.
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.
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.
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.
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.
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).
Seminar Meccanica statistica e fenomeni sociali Giuseppe Toscani , Università di Pavia mercoledì 26 maggio 2021 alle ore 12:15 online: tiny.cc/fdsCM2021
Seminar Contact surface of Cheeger sets Marco Caroccia, Dipartimento di Matematica, Politecnico di Milano mercoledì 26 maggio 2021 alle ore 15:15 precise polimi-it.zoom.us/j/89745265407
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.
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.
Seminar Design and Health Stefano Capolongo , Politecnico di Milano mercoledì 19 maggio 2021 alle ore 12:15 online: tiny.cc/fdsCM2021