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Schedule and abstracts

IMPORTANT : 

On Monday 8 and Friday 12 the seminars will take place at the Scuola Galileiana di Studi Superiori, Aula Magna, via Marzolo 6, Padova : Google Maps link

On Tuesday 9, Wednesday 10, Thursday 11 the seminars will take place at the Dipartimento di Matematica "T. Levi-Civita", Torre Archimede, Aula 1AD100, via Trieste 63, Padova : Google Maps link

 

Schedule (tentative):

  Mo 8 Tu 9 We 10 Th 11 Fr 12
8.30-9.00 Registration        
9.00-9.45 Tolomeo Tolomeo Pozzetti Santambrogio Santambrogio
9.55-10.40 Tolomeo Tolomeo Pozzetti Santambrogio 9.55-10.20 break
          10.20-11.05 Pozzetti
11.15-12.00 Caporaso Pozzetti Caporaso Caporaso 11.10-11.55 Pozzetti
12.10-12.55 Caporaso Pozzetti Caporaso Caporaso 12.10-12.55 Pozzetti
          13.00 Final apero
14.30-15.15 Tolomeo Santambrogio   Caporaso  
15.25-16.10 Tolomeo Santambrogio   15.20-15.50 break  
        15.50-16.35 Santambrogio  
16.45-17.30   Tolomeo   Santambrogio  
           

The final seminars are highlighted in bold.

The social dinner will take place on Tuesday 9th.

Titles and abstracts:

Lucia Caporaso

Course title: Moduli spaces of algebraic and tropical curves

The goal of the course is the study of the moduli space of algebraic curves of fixed genus, including the recent approach via tropical curves and their moduli spaces. The course will cover the basic theory of curves (both algebraic and tropical), construct the moduli spaces of (stable) algebraic curves and of tropical curves, and describe the connection between them.

Seminar title:  Recent applications of moduli spaces in various areas of mathematics

The final seminar will present some  applications of the theory of moduli spaces in various areas of mathematics and ongoing research directions.

Maria Beatrice Pozzetti

Course title: An introduction to geometric structures

A (G,X) structure on a manifold M is the datum of an atlas on M with charts in X whose change of charts are restrictions of elements of G. While (Diffeo(R^n), R^n)-structures are simply smooth structures, the theory becomes richer when G is a finite dimensional Lie group, such as SL(2,R), and X is a homogeneous G-space. We will provide an introduction to this theory focusing on hyperbolic and convex projective structures, mostly on topological surfaces.

Seminar title: Higher rank Teichmüller spaces

An higher rank Teichmüller space is a connected component of the variety of homomorphisms from the fundamental group of a surface of genus at least two to a semisimple Lie groups. I will discuss joint work with Beyrer-Guichard-Laburie-Wienhard where we prove that these arise precisely when the group G satisfies an algebraic notion known as  Theta-positivity. In the proof we show that these homomorphisms have a lot in common with the action of the fundamental group of a hyperbolic surface on its universal covering by deck transformations.

Filippo Santambrogio

Course title: From Convex Analysis to Optimal Transport

The course will provide students with an introduction to the main notions of convex analysis in finite and infinite dimensions, focusing on subgradients, Fenchel–Legendre transforms, and duality. These tools will then be applied within the framework of the Monge–Kantorovich theory of optimal transport, which will occupy most of the course. Topics will include the existence of optimal transport plans, the dual formulation, the existence of optimal maps in the sense of Brenier, as well as Wasserstein distances, geodesics, and the dynamic formulation of Benamou–Brenier.

An approximate program would be:

Convex functions, representation as sup of affine functions, subdifferentials, continuity and differentiability properties of convex functions. Fenchel–Legendre transforms, Fenchel–Rockafellar duality, examples of dual problems, discrete transport. The Monge–Kantorovich problem and its dual, conditions for solving the Monge problem. Brenier’s theorem, the Monge–Ampère equation, monotone maps, application to the isoperimetric inequality. Wasserstein distances: triangle inequality, induced topology, geodesics. Curves in Wasserstein space, the dynamic formulation of Benamou–Brenier.

Seminar title: The sliced Wasserstein distance and the sliced Wasserstein flow

Leonardo Tolomeo

Course title: Transport of measures under the flow of partial differential equations

In this course, we explore how the flow of Hamiltonian PDEs transports Gaussian measures, and how information about the measure can be exploited to deduce information about the flow itself. We start by recalling several concepts regarding finite-dimensional Hamiltonian systems, the definition of the Gibbs measure, and how the Gibbs measure can be used to deduce information about the flow for almost every initial data. We then continue by showing how these concepts can be extended to the infinite-dimensional setting, focusing on specific PDEs, such as the nonlinear Schrödinger equation. Finally, we focus on non-invariant measures and show how one can use information about the flow to deduce information about the evolution of the measure, and in turn how this can provide more detailed information about the flow itself.

Seminar title: Transport of measures under the flow of SPDEs

We describe how the flow of stochastic PDEs transports measures, highlighting the differences from the PDE case. We show that, under very general assumptions, the flow of SPDEs admits a unique invariant measure, and that the techniques from the course can be exploited to deduce fine properties of the invariant measure itself. Finally, we present a recent result on the invariant measure for the stochastic Navier–Stokes equation in two dimensions. This is based on joint work with J. Coe (Edinburgh) and M. Hairer (EPFL and Imperial College London).