Fri, 20 May 2022

10:30 - 12:00
L5

General Linear PDE with constant coefficients

Bogdan Raiță
(Scuola Normale Superiore di Pisa)
Further Information

Sessions will take place as follows:

17th May 14:00 -15:00

18th and 20th May 10:30 -12:00

Abstract

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

Wed, 18 May 2022

10:30 - 12:00
L5

General Linear PDE with constant coefficients

Bogdan Raiță
(Scuola Normale Superiore di Pisa)
Further Information

Sessions will take place as follows:

17th May 14:00 -15:00

18th and 20th May 10:30 -12:00

Abstract

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

Tue, 17 May 2022

14:00 - 15:00
L5

General Linear PDE with constant coefficients

Bogdan Raiță
(Scuola Normale Superiore di Pisa)
Further Information

Sessions will take place as follows:

17th May 14:00 -15:00

18th and 20th May 10:30 -12:00

Abstract

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

Thu, 24 Oct 2019

12:00 - 13:00
L4

Structure theory of RCD spaces up to codimension 1

Daniele Semola
(Scuola Normale Superiore di Pisa)
Abstract

The aim of this talk is to give an overview about the structure theory of finite dimensional RCD metric measure spaces. I will first focus on rectifiability, existence, uniqueness and constancy of the dimension of tangents up to negligible sets.
Then I will motivate why boundaries of sets of finite perimeter are natural codimension one objects to look at in this framework and present some recent structure results obtained in their study.
This is based on joint works with Luigi Ambrosio, Elia Bruè and Enrico Pasqualetto.
 

Fri, 01 Dec 2017

16:00 - 17:00
L1

New developments in the synthetic theory of metric measure spaces with Ricci curvature bounded from below

Luigi Ambrosio
(Scuola Normale Superiore di Pisa)
Abstract

The theory of metric measure spaces with Ricci curvature from below is growing very quickly, both in the "Riemannian" class RCD and the general  CD one. I will review some of the most recent results, by illustrating the key identification results and technical tools (at the level of calculus in metric measure spaces) underlying these results.
 

Thu, 08 Jun 2017
12:00
L4

DIVERGENCE-MEASURE FIELDS: GENERALIZATIONS OF GAUSS-GREEN FORMULA

GIOVANNI COMI
(Scuola Normale Superiore di Pisa)
Abstract

Divergence-measure fields are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose divergence is a Radon measure. Such vector fields form a new family of function spaces, which in a sense generalize the $BV$ fields, and were introduced at first by Anzellotti, before being rediscovered in the early 2000s by many authors for different purposes.
Chen and Frid were interested in the applications to the theory of systems of conservation laws with the Lax entropy condition and achieved a Gauss-Green formula for divergence-measure fields, for any $1 \le p \le \infty$, on open bounded sets with Lipschitz deformable boundary. We show in this talk that any Lipschitz domain is deformable.
Later, Chen, Torres and Ziemer extended this result to the sets of finite perimeter in the case $p = \infty$, showing in addition that the interior and exterior normal traces of the vector field are essentially bounded functions.
The Gauss-Green formula for $1 \le p \le \infty$ has been also studied by Silhavý on general open sets, and by Schuricht on compact sets. In such cases, the normal trace is not in general a summable function: it may even not be a measure, but just a distribution of order 1. However, we can show that such a trace is the limit of the integral of classical normal traces on (smooth) approximations of the integration domain.

Thu, 04 May 2017
16:00
L6

Joint Number Theory/Logic Seminar: On the Hilbert Property and the fundamental group of algebraic varieties

Umberto Zannier
(Scuola Normale Superiore di Pisa)
Abstract

This  concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem)  with the fundamental group of the variety.
 In particular, this leads to new examples (of surfaces) of  failure of the Hilbert Property. We also prove the Hilbert Property for a non-rational surface (whereas all previous examples involved rational varieties).

Thu, 04 May 2017
16:00
L6

Joint Number Theory/Logic Seminar: On he Hilbert Property and the fundamental groups of algebraic varieties

Umberto Zannier
(Scuola Normale Superiore di Pisa)
Abstract

This  concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem)  with the fundamental group of the variety.
 In particular, this leads to new examples (of surfaces) of  failure of the Hilbert Property. We also prove the Hilbert Property for a non-rational surface (whereas all previous examples involved rational varieties).

Mon, 26 May 2014

17:00 - 18:00
L6

A geometric approach to some overdetermined problems in potential theory

Lorenzo Mazzieri
(Scuola Normale Superiore di Pisa)
Abstract
We present a new method to establish the rotational symmetry

of solutions to overdetermined elliptic boundary value

problems. We illustrate this approach through a couple of

classical examples arising in potential theory, in both the

exterior and the interior punctured domain. We discuss how

some of the known results can be recovered and we introduce

some new geometric overdetermining conditions, involving the

mean curvature of the boundary and the Neumann data.

Thu, 01 Dec 2011
12:30
Gibson 1st Floor SR

Sobolev regularity for solutions of the Monge-Amp\`ere equation and application to the Semi-Geostrophic system

Guido De Philippis
(Scuola Normale Superiore di Pisa)
Abstract
I will talk about $W^{2,1}$ regularity for strictly convex Aleksandrov solutions to the Monge Amp\`ere equation

\[

\det D^2 u =f

\]

where $f$ satisfies $\log f\in L^{\infty} $. Under the previous assumptions in the 90's Caffarelli was able to prove that $u \in C^{1,\alpha}$ and that $u\in W^{2,p}$ if $|f-1|\leq \varepsilon(p)$. His results however left open the question of Sobolev regularity of $u$ in the general case in which $f$ is just bounded away from $0$ and infinity. In a joint work with Alessio Figalli we finally show that actually $|D^2u| \log^k |D^2 u| \in L^1$ for every positive $k$.

\\

If time will permit I will also discuss some question related to the $W^{2,1}$ stability of solutions of Monge-Amp\`ere equation and optimal transport maps and some applications of the regularity to the study of the semi-geostrophic system, a simple model of large scale atmosphere/ocean flows (joint works with Luigi Ambrosio, Maria Colombo and Alessio Figalli).

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