Title: Gradient activation problems ruled by the infinity Laplacian

Abstract: In this talk, we explore the interplay between congested traffic dynamics and the mathematical structure of the infinity Laplacian. We introduce the concept of comparison with k-cones and establish its equivalence to solving highly degenerate PDEs governed by the infinity Laplacian within the framework of viscosity solutions, where only test functions with gradients greater than k are considered. Building on this progress, we derive local Lipschitz estimates for functions that satisfy comparison with k-cones. Additionally, we derive monotonicity properties that, through a detailed analysis, reveal that blow-up solutions are linear.

Title: Symbolic powers in local rings of prime characteristic

Abstract: The celebrated theory of symbolic powers of ideals (in Noetherian rings) plays a significant role in commutative algebra and also in algebraic geometry. In this talk, after a brief general discussion, we will first show how symbolic powers can be eventually used in determining the Hilbert-Kunz multiplicity of the quotient by the "tight" closure of a given unmixed ideal in terms of the classical Hilbert-Samuel multiplicity.  Next, we will focus on one of the main questions in the theory, the so-called containment problem, which is particularly intriguing in regular local rings of prime characteristic. We will then present our central result, which improves a well-known theorem due to Hochster and Huneke, as well as explicit comparative examples. The results are part of the speaker's doctoral thesis and were obtained in collaboration with Cleto B. Miranda-Neto.

Title: On the lineability of the set of continuous functions that vanish at infinity and have a unique maximum

Abstract: Nesta apresentação, abordaremos o primeiro problema de lineabilidade formulado em 2004. Proposto por V. Gurariy, esse problema permaneceu sem solução por 15 anos, período em que diversas generalizações foram exploradas em diferentes direções. Sua resolução definitiva ocorreu apenas em 2020, utilizando ferramentas pouco convencionais da topologia, geometria e análise complexa. O objetivo da palestra é apresentar um caminho para uma generalização ainda parcialmente resolvida e incentivar o público a investigar esse novo problema, que parece apresentar uma ruptura na transição para dimensões superiores a 1.

Title: A Serrin's type Problem in Weighted Manifolds: Soap Bubble Results and Rigidity in Generalized Cones

Abstract: Inspired by a problem in fluid dynamics, Serrin [2] established a fundamental result on overdetermined problems: if a solution to a particular partial differential equation (PDE) satisfies both Dirichlet and Neumann boundary conditions, then the domain must be a ball, and the solution must be radially symmetric. More generally, an “overdetermined problem” refers to a partial differential equation (PDE) in which “too many” boundary conditions are assigned, such as both the Dirichlet and Neumann conditions. This means that the system of conditions is “overdetermined” relative to the degrees of freedom in the PDE. In this lecture we investigate a weighted overdetermined problem within the framework of Riemannian manifolds with density. Initially, by examining a Poisson problem associated with the drift Laplacian, we derive a Heintze-Karcher inequality and a soap bubble theorem that characterize geodesic balls in these spaces. Subsequently, by imposing both Dirichlet and Neumann boundary conditions, we establish a Serrin-type result in generalized cones, identifying metric balls as the unique solutions to this underlying overdetermined problem. This lecture is based in a joint work with A. Freitas and M. Santos ([1]).

References

[1] ARAÚJO, M.; FREITAS, A.; SANTOS, M. A Serrin’s type problem in weighted manifolds: Soap bubble results and rigidity in generalized cones. Potential Analysis, Springer, p. 1-16, 2024.

[2] SERRIN, J. A symmetry problem in potential theory. Archive for Rational Mechanics and Analysis, Springer, v. 43, p. 304-318, 1971.

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