The seminars will take place in room 1A150 of the Department of Mathematics "T. Levi-Civita".
9.00-9.50 A. Figalli
10.00-10.30 Coffee break
10.30-11.20 A. Skorobogatova
11.30-12.30 A. Carlotto
12.30 Lunch
Titles and abstracts:
A. Carlotto, Attaching faces of positive scalar curvature manifolds with corners
Abstract. Motivated by a conjecture due to Gromov, we prove a novel desingularization theorem that allows to smoothly attach two given manifolds with corners by suitably gluing a pair of isometric faces, with control on both the scalar curvature of the resulting space and the mean curvature of its boundary.
Various significant applications and related open problems will also be presented. This lecture is based on joint work with Chao Li (NYU).
A. Figalli, Complete classification of global solutions to the obstacle problem
Abstract. The characterization of global solutions to the obstacle problems in R^n, or equivalently of null quadrature domains, has been studied for more than 90 years. In this talk, I will discuss a recent result with Eberle and Weiss, where we give a conclusive answer to this problem by proving the following long-standing conjecture: The coincidence set of a global solution to the obstacle problem is either a half-space, an ellipsoid, a paraboloid, or a cylinder with an ellipsoid or a paraboloid as base.
A. Skorobogatova, Area-minimizing currents: structure of singularities and uniqueness of tangent cones
Abstract. The Plateau problem asks about the surfaces of least m-dimensional area spanning a given m-1 dimensional boundary. However, to guarantee existence of minimizers, one must relax to a weaker notion of “surface” and “boundary”, which means that “surfaces” have little a priori regularity. The problem of determining the size and structure of the interior singular set of area-minimizing surfaces has been studied thoroughly in a number of different frameworks, with many ground-breaking contributions in the last century. In the framework of integral currents, when the codimension of the surface is higher than 1, the presence of singular points with flat tangent cones creates an obstruction to easily understanding the size and structure of the interior singularities, as well as how the surface behaves at such singular points.
I will discuss joint works with Camillo De Lellis and Paul Minter, where we build on Almgren’s celebrated dimension estimate on the interior singular set, and establish (m-2)-rectifiability of the interior singular set of an m-dimensional area-minimizing integral current, as well as uniqueness and classification of the tangent cone at \mathcal{H}^{m-2}-a.e. interior point.