The isoperimetric problem on smooth Riemannian manifolds aims at minimizing the perimeter among sets having a fixed volume, where the perimeter measures the area of the boundary of a set. Minimizers of the problem are called isoperimetric sets. The isoperimetric profile is the function assigning to any volume the infimum of the problem.

In this mini course, we consider the problem on Riemannian manifolds with Ricci curvature bounded below, and we discuss some selected topics.

• In the first lecture, we introduce the notion of perimeter and we define the isoperimetric problem, discussing examples and related functional analytic aspects.

• In the second lecture, we present the second order differential inequalities satisfied by the isoperimetric profile on manifolds with Ricci lower bounds.

• In the third lecture, we present some isoperimetric inequalities on manifolds with nonnegative Ricci curvature.

We shall also mention some modern developments of the theory, as well as important open problems. 

Prerequisites: it is assumed a basic knowledge of Riemannian Geometry, Functional Analysis and Measure Theory.

In this talk, we present local and global higher integrability properties for quasiminimizers of a class of double phase integrals characterized by non-standard growth conditions. We work purely on a variational level in the setting of a doubling metric measure space supporting a Poincaré inequality. The main novelty is the use of an intrinsic approach, based on a double phase Sobolev-Poincaré inequality.

Positive intermediate Ricci curvature is a family of interpolating curvature conditions between positive sectional and positive Ricci curvature. While many results that hold for positive sectional or positive Ricci curvature have been extended to these intermediate conditions, only relatively few examples are known so far. In this talk I will present several extensions of construction techniques from positive Ricci curvature to these curvature conditions, such as surgery and bundle techniques. As an application we obtain a large class of new examples of manifolds with a metric of positive intermediate Ricci curvature, including all homotopy spheres that bound a parallelizable manifold, and show that Gromov's Betti number bound for manifolds of non-negative sectional curvature does not hold from positive Ricci curvature up to roughly halfway towards positive sectional curvature. This is joint work with David Wraith.

We provide some background on the construction of harmonic maps and generalizations. Afterwards we show that there exists infinitely p-harmonic maps of S^m provided that p<m<p+2+2sqrt{p}. 

En esta plática, hablaremos del perfil isoperimétrico de variedades MxR^k, con M una variedad cerrada. En particular, hablaremos de algunos resultados recientes de cotas inferiores explícitas para el perfil isoperimétrico de T^mxR^k (2021), donde T^m es el producto de m copias del círculo unitario, y de

S^m x R^k (2023), con S^m  esfera de dimensión baja. Este es un trabajo en colaboración con el Dr. Miguel Ruiz. 

En esta plática hablaré sobre un proceso de regularización y aproximación de corrientes de Georges De Rham y como Igor Nikolaev adaptó dicho proceso en el caso de métricas riemannianas en espacios con curvatura acotada. Además, en la segunda parte de la charla, expondré una parte de mi trabajo doctoral, donde establecí una versión de dichos procesos cuando tenemos una acción de un grupo de Lie en una variedad o en un espacio con curvatura acotada, respectivamente.

La presencia de campos espinoriales paralelos no triviales caracteriza la existencia de métricas de holonomía especial sobre una n-variedad 

Riemanniana M. En particular, el grupo estructural de cualquier haz de espinores sobre M admite una reducción al subgrupo de isotropía de un

espinor unitario en la representación espinorial de Spin(n), que además induce una reducción del grupo estructural del haz tangente al grupo de holonomía. En esta plática revisaremos la reciente clasificación homotópica para Spin(7)-estructuras en variedades de dimensión ocho. Como una aplicación, mostraremos que una Spin(7)-variedad compacta y conexa admite exactamente dos Spin(7)-estructuras compatibles con una orientación dada.

The class of RCD(K,N)-spaces for a given real number K, and non.negative real number N, consists of proper metric measure spaces that satisfy a synthetic notion of having Ricci curvature bounded below by K and Hausdorff dimension bounded above by N. Most notably this class is known to be stable under measured Gromov-Hausdorff convergence and examples of such spaces include Riemannian manifolds and Alexandrov spaces.

A very classical problem in geometry is to study the relation between curvature and topology. So, one can then for instance consider studying properties of the fundamental group of these spaces. However, since RCD(K,N)-spaces are not necessarily topological manifolds some work was needed before one could talk about their Universal covers, this was done by Mondino and Wei. More recently, Wang proved that the Universal cover is indeed simply connected.

In this talk we will focus first on giving a motivation for studying synthetic notions of curvature bounds. Then we will define and give examples of RCD-spaces as well as describe some of their properties. Finally, we will present some of the results obtained regarding their fundamental groups.

This is based on joint work with Sergio Zamora-Barrera.