Title: Symmetry, Rigidity, and Overdetermined Problems in Riemannian Manifolds

Abstract: Why are soap bubbles and black holes spherical? Why are pipes cylindrical? Many problems at the intersection of geometry, analysis, and physics can be understood through overdetermined problems, where the fundamental principle is that structures tend to be “as symmetric as possible.” In this lecture, we will explore how symmetric domains naturally arise as the unique solutions to such problems. Specifically, we will discuss contributions to Serrin’s problem, Alexandrov’s theorem, and the No-Hair Conjecture, primarily employing integral techniques. These methods have been particularly instrumental in extending classical results to Riemannian manifolds, and we will highlight some recent developments in this direction.

Title: ℓ1 Spreading Models and the Fixed Point Property 

Abstract:  Nesta plenária, iremos fazer uma digressão história acerca da chamada propriedade do ponto fixo (FPP), ilustrando as principais contribuições e alguns dos problemas que ainda estão em abertos. Falaremos também sobre as técnicas que foram desenvolvidas no trato dessa propriedade. Na segunda parte da exposição, daremos uma ênfase sobre a versão fraca dessa propriedade (weak-FPP), e na parte final apresentaremos uma contribuição publicada recentemente no Banach Journal of Mathematical Analysis.

Title: An overview on quasi-Einstein manifolds

Abstract: It is known by the classical book "Einstein Manifolds" (Besse, 1984) that quasi-Einstein manifolds correspond to a base of a warped product Einstein metric. Another interesting motivation to investigate quasi-Einstein manifolds derives from the study of diffusion operators by Bakry and Emery (1985), which is linked to the theories of smooth metric measure space, static spaces, and Ricci solitons.

In this presentation, we will explore the geometry of quasi-Einstein manifolds. We begin by reviewing the motivations behind this study, along with examples and classical results related to these manifolds. Following this, we will highlight recent classification results that incorporate contributions from researchers in Brazil. Finally, we will conclude with a discussion of several open problems in the field. 

Title: Large solutions for fractional elliptic equations

Abstract: In this talk, I will report some results about existence, uniqueness, multiplicity and regularity of large solutions for different types of fractional elliptic equations posed on bounded domains. Following the seminal (and independent) works of Keller and Osserman (1957), for large solutions, we mean functions that solve an equation in an open set and tend to infinity at points approaching its boundary. I will describe the motivations behind this type of solution, review part of the literature and discuss the structure of the solutions in the fractional setting. 

Title: An introduction to supergeometry

Abstract: I will give an introduction to the basic ideas of supergeometry, starting from its physical motivations, and highlighting the main research themes that are at present pursued.