Title: Um problemas do tipo Serrin na forma divergente em variedades riemannianas 

Abstract: Nesta palestra, iremos investigar um problema de valor de contorno sobredeterminado na forma divergente em domíninos limitados de variedades riemannianas com curvatura de Ricci não-negativa. Usando identidades integrais e estabelecendo condições naturais sobre a não linearidade, discutiremos desigualdades geométricas e teoremas de rigidez a partir do método de Weinberger (método da P-função).

Title: Critical metrics of the volume functional on complete manifolds

Abstract: A classical way to search for canonical metrics is to analyze critical points of geometric functionals. In that sense, V-static metrics appear as critical points of the volume functional when restricted to compact manifolds with constant scalar curvature and prescribed boundary metric. This variational problem can be translated into a prescribed metric equation problem, which allows us to study these metrics on complete manifolds without boundary. In this talk, we establish rigidity results for V-static metrics with curvature constraints. In particular, we classify the V-static metrics with parallel Ricci tensor and under the Bach-flat condition.

Title: Sot-large subspaces of non-cyclic operators

Abstract: In this lecture, we consider the strong operator topology (SOT) on the space L(X) of continuous linear operators on an infinite dimensional Fréchet space X. The existence of SOT-dense subspaces as well as of SOT-closed infinite dimensional subspaces inside the family of all non-cyclic operators is established. Other classes of operators, such as non-dense range or non-injective operators, are also studied to this respect.

Title: A characterization of monomial ideals via derivations

Abstract: In this talk, we present the module of derivations associated with the polynomial ring R=k[x1,...,xn], as well as recent developments concerning an important submodule: The logarithmic derivation module of an ideal I of R. In particular, we introduce the notion of log-decomposable ideals. Brumatti and Simis showed that every monomial ideal is log-decomposable. In this talk, we present a recent result showing that the converse also holds: log-decomposable ideals are precisely the monomial ideals.

Title: Expansivity for weighted composition operators on locally convex spaces of continuous functions

Abstract: We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces C0(X) and the locally convex spaces C(X)c.

Title: A tangential approach to Schauder estimates in elliptic PDEs with Dini regularity

Abstract: In this talk, we establish local Schauder estimates for flat viscosity solutions, that is, solutions with sufficiently small norms, to a class of fully nonlinear elliptic partial differential equations of the form

F(D²u, x) + ⟨B(x), Du⟩ = f(x), in B₁ ⊂ Rⁿ,

where the operator F is differentiable, though not necessarily convex or concave. In addition, we impose suitable Dini-type continuity assumptions on the data. Our methodology is based on geometric tangential techniques, combined with compactness and perturbative arguments.

Title: Optimal cost for the null controllability of the Stokes system with controls having n−1 components

Abstract: We establish a new spectral inequality for the low-frequency modes of the Stokes operator when only n−1 components of the solution are observed. As a consequence, we obtain the optimal cost for null controllability of the Stokes system with controls acting on only n−1 scalar components. In particular, the cost of controllability in this reduced-control setting is of the same order as in the fully controlled case with n components.

Title: On divide links and the B–S conjecture

Abstract: In 1968, Milnor proved, via the Fibration Theorem, that every (complex) algebraic link in S^(n+2) is fibered. In the particular case of S^3, there are well-known examples of fibered links that cannot be realized as the link of a complex singularity, such as the figure-eight knot. This observation motivated the introduction of the notion of real algebraic links. In 1998, Benedetti and Shiota formulated the B–S conjecture, which asserts that every fibered link in S^3 is real algebraic. Despite partial progress, a complete proof of this conjecture is still unknown. In this context, divide links stand out as fibered links exhibiting a distinctive type of symmetry, which may provide useful tools and techniques to shed light on the B–S conjecture.