SISSA mathematical Colloquia

[Emanuele Tuillier Illingworth]

*Monday the  3^rd 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 12^th of May 16.00-17.00 room 005*

*Rita Pardini, Università di Pisa*

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 22^nd of May, 4.00-5.00pm room 128*

*Vieri Mastropietro, Università 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 7^th 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 
deterministicsystems 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