Mathematical Colloquia

[Tamara Grava]

Dear All,

I am sending the list of the Mathematical colloquia that will be held at SISSA till June 2017.

Monday the  3rd of April 16.00-17.00 room 128

Luigi Ambrosio, Scuola Normale Superiore

Title: New estimates on the matching problem.

Abstract: The matching problem consists in finding the optimal coupling between a random distribution of N points in a d-dimensional domain and another (possibly random) distribution. There is a large literature on the asymptotic behaviour as N tends to infinity of the expectation of the minimum cost, and the results depend on the dimension d and the choice of cost, in this random optimal transport problem, with challenging open problems. In a recent work, Caracciolo, Lucibello, Parisi and Sicuro proposed an ansatz for the expansion in N of the expectation. I will illustrate how a combination of semigroup smoothing techniques and Dacorogna-Moser interpolation provide first rigorous results for this ansatz.

Joint work with Federico Stra and Dario Trevisan, ArXiv:1611.04960

 Friday the 12th of May 16.00-17.00  room 005

Rita Pardini, Scuola Normale Superiore

 Title: Complex tori, abelian varieties and  irregular projective varieties.

 Abstract: A complex torus T is a complex variety that is the quotient of a complex vector space of dimension n by a discrete subgroup of rank 2n (a  "lattice"); if T can be realized as a closed subvariety of some complex projective space then it is called an abelian variety.

A complex torus/abelian variety  is  in some sense a linear object, since it has a natural group structure and its main  geometric invariants  can be explicitly described in terms of the lattice.

A  smooth complex projective variety  is called *irregular* if admits a non constant map  to a complex torus.  I will sketch  the construction of the Albanese map  of an irregular variety X, namely of  the  "maximal" map from X to a complex torus T, and  discuss some instances of how it  can be used to analyze geometrical properties of X.

 Monday 22nd of May, 4.00-5.00pm room 128

Vieri Mastropietro, Universita’ di Milano

Title: Localization of interacting quantum particles with quasi-random disorder

Abstract: It is well established at a mathematical level that disorder can produce Anderson localization of the eigenvectors of the single particle Schrödinger equation. Does localization survive in presence of many body interaction? A positive answer to such question would have important physical consequences, related to lack of thermalization in closed quantum systems. Mathematical results on such issue are still rare and a full understanding is a challenging problem. We present an example in which localization can be proved for the ground state of an interacting system of fermionic particles with quasi random Aubry-Andre' potential. The Hamiltonian is given by $N$ coupled almost-Mathieu Schrödinger operators. By assuming Diophantine conditions on the frequency and density, we can establish exponential decay of the ground state correlations. The proof combines methods coming from the direct proof of convergence of KAM Lindstedt series with Renormalization Group methods for many body systems. Small divisors appear in the expansions, whose convergence follows exploiting the Diophantine conditions and fermionic cancellations. The main difficulty comes from the presence of loop graphs, which are the signature of many body interaction and are absent in KAM series.

V.Mastropietro. Comm Math Phys 342, 217 (2016); Phys Rev Lett 115, 180401 (2015); Comm. Math. Phys .(2017)

 

Friday the 30th of May, 4.00-5.00pm, Room 128

Alessio Figalli, ETH Zurich

Title:TBA


Wednesday the 7th of June  5.00-6.00pm  room 128

Annalisa Buffa, CNR-IMATI, PAVIA

Title: TBA





 Friday 16th of June 2017   room 128

10:45 -11:45  Cinzia Casagrande (Dipartimento di Matematica, Università di Torino)

12:00 - 13:00  Isabelle Gallagher  (Université Paris-Diderot, IMJ-PRG)

 

Isabelle Gallagher

Title:  "On Hilbert's 6th problem: From particle systems to Fluid Mechanics"

Abstract. The question of deriving Fluid Mechanics equations from deterministic systems of interacting particles obeying Newton's laws, in the limit when the number of particles goes to infinity, is a longstanding open problem suggested by Hilbert in his 6th problem. In this talk we shall present a few attempts in this program, by explaining how to derive some linear models such as the Heat, acoustic and Stokes-Fourier equations. This corresponds to joint works with Thierry Bodineau and Laure Saint Raymond.

 

Cinzia Casagrande

Title: TBA




Best wishes,
T. Grava