- What: "Three Days between Analysis and Geometry in Trento"
- When: 2023/08/28 - 2023/08/30
- Where: University of Trento, Italy
- Who: Organised by the Department of Mathematics, University of Trento
Elia Bruè (Bocconi University)
The fundamental groups of manifolds with nonnegative Ricci curvature.
In 1968, Milnor proposed a conjecture stating that Riemannian manifolds with nonnegative Ricci curvature have finitely generated fundamental groups.
The first part of the course will focus on comprehending this conjecture and discussing the progress made in addressing it throughout the years.
In the subsequent part, we will discuss recent work with Naber and Semola, where we provide a counterexample to Milnor's conjecture. More specifically, we construct a seven-dimensional manifold with nonnegative Ricci curvature whose fundamental group is isomorphic to
Adam Parusinski (Côte d'Azur University)
Introduction to semialgebraic and subanalytic geometry.
The goal of this course is to give an introduction to semialgebraic and subanalytic sets and mappings with emphasis on the properties used in various areas of differential geometry and analysis. In particular we cover such subjects as Łojasiewicz Inequalities, decompositions and stratifications. One lecture will be devoted to the metric properties of these sets including the L-regular decomposition, subanalytic preparation theorem and Lipschitz stratification. We also discuss the integration on subanalytic sets, local densities, and the dependence of integrals on parameters in families.
Luigi De Rosa (University of Basel)
Intermittency and Minkowski content in turbulence.
In 1941 Kolmogorov theorized that all p-th moments of increments of the velocity in a turbulent flow have 1/3 has a universal regularity exponent. However, downward deviations from K41 prediction are experimentally observed. This phenomenon is nowadays known as "intermittency", which theoretical physicists, starting from Landau, linked to the spottiness of the region where the dissipation is supported. We propose a couple of Minkowski-type notions of dimensions, one Eulerian and one Lagrangian, which lay down a setup to make Landau's objection quantitative. The approach is quite geometrical and it is in fact part of a more general picture in which most of the PDEs in fluid dynamics fall.
Stefano Spirito (University of L'Aquila)
On the inviscid limit for 2D incompressible fluid.
We review some recent results concerning the inviscid limit for the 2D Euler equations with irregular vorticity. In particular, by using techniques from the theory of transport equations with non smooth vector fields, we show that solutions of the incompressible 2D Euler equations obtained from the ones of the 2D incompressible Navier-Stokes equations via the vanishing viscosity limit satisfy a representation formula in terms of the flow of the velocity and that the strong convergence of the vorticity holds. Moreover, we also prove a rate of convergence. The talk is based on results obtained in collaboration with Gianluca Crippa (Univ. Basel) and Gennaro Ciampa (Università Milano Statale).
Barbara Nelli (University of L'Aquila)
The Jenkins-Serrin Theorem reloaded.
Jenkins and Serrin in the sixties proved a famous theorem about minimal graphs in the Euclidean 3-space with infinite boundary values. After reviewing the classical results, we show how to solve the Jenkins-Serrin problem in a 3-manifold with a Killing vector field. This is a joint work with A. Del Prete and J. M. Manzano.
Emanuela Radici (University of L'Aquila)
On curvature and five gradients inequality on manifolds.
Introduced almost ten years ago, the five gradients inequality has been used to provide estimates on Sobolev norms of minimizers involving the Wasserstein distance. In conjunction with the JKO scheme, this inequality can grant compactness for the minimizing movement scheme. We investigate the geometric and functional meaning of the five gradients inequality in two generalizations. In the setting of Lie groups the proof naturally suggests that it is a second order optimality condition for the Kantorovich potentials, while in general compact Riemannian manifolds the curvature plays a role. This is a joint work with Simone Di Marino and Simone Murro.
Emanuele Tasso (TU Wien)
Rectifiability of a class of integralgeometric measures and applications
In his textbook 'Geometric Measure Theory', Federer proposed the following problem: (Q) Is the restriction of the m-dimensional integralgeometric measure to a finite set a m-rectifiable measure? With this motivation, after a brief introduction to the integral geometric measure, I will discuss how rectifiability issues in the spirit of (Q) play an important role in some variational problems. As a sort of unifying theory, I will then introduce a novel class of measures in the euclidean space based upon the idea of slicing. The central result of this talk will follow, which is a sufficient condition for rectifiability in the above class. Two main applications will be shown: the solution to Federer's problem, as well as a novel rectifiability criterion for Radon measures via slicing, the latter being reminiscent of White's rectifiable slices theorem for flat chains. If time permits, I will discuss how to extend the main result to the Riemannian case by means of the notion of transversal family of maps. In the very last part of the talk I will propose some related open problems.
Enrico Savi (University of Trento)
Embedded
In 2020 Parusiński and Rond proved that every algebraic set