Title: Some research topics and recent results in Commutative Algebra

Abstract: Algebra is one of the oldest and most fundamental areas of Mathematics. Commutative Algebra lies within its domain. This lecture will present some of the most active lines of research associated with this field, focusing on its fascinating intersection with Homological Algebra, which, in turn, offers a wide range of applications in other areas of Mathematics (e.g., Geometry, Topology, PDE Theory) and even outside of it (e.g., Neuroscience). Some technical results obtained in recent years will be presented, particularly regarding several long-standing homological and differential conjectures, as well as certain applications in Algebraic Geometry.

Title: Soon.

Abstract: Soon.

Title: Recent Contributions to Linear Dynamics

Abstract:  Systematic investigations on the dynamics of linear operators began in the 1990s with seminal works by R. Gethner, G. Godefroy, D. Herrero, C. Kitai and J. Shapiro, among others. For a long time the focus was on the study of various concepts of chaotic behavior, including hypercyclicity, frequent hypercyclicity, Devaney chaos, Li-Yorke chaos and distributional chaos. More recently, some researchers have revisited some fundamental concepts in dynamical systems that are not concepts of chaos, including expansivity, hyperbolicity, structural stability and the shadowing property, and have begun to develop a systematic study of these concepts in the setting of linear dynamics. This opened a new branch of research in the area of linear dynamics. Many exciting achievements have already been made, including the solution of a long-standing open problem. Below, we list some of these achievements:

• The existence of operators that have the shadowing property but are not hyperbolic.

• The existence of structurally stable operators that are not hyperbolic.

• The equality “hyperbolicity = expansivity + shadowing” for operators on Banach spaces.

• A characterization of the shadowing property for weighted shifts on classical Banach sequence spaces.

• The emergence of the concept of generalized hyperbolicity.

• The structural stability of every generalized hyperbolic operator.

• The generalized Grobman-Hartman theorem.

• The equivalence between shadowing and finite shadowing for operators on Banach spaces and the failure of this equivalence for operators on Fréchet spaces.

In our talk, we will provide an overview of this subject. We will review relevant concepts, present recent solutions to several basic problems, and discuss some basic questions that remain open.

Our exposition will be based on some recent papers of the speaker with various collaborators, namely, B. Caraballo, P. Cirilo, U. Darji, V. Fávaro, A. Messaoudi, A. Peris and E. Pujals.

Title: Soon

Abstract: Soon.

Title: A functional Central Limit Theorem for the most general BM on the half-line

Abstract: We prove, in this joint work with E. Erhard, M. Jara and E. Pimenta, a functional central limit theorem for a random walk on the non negative integers. Depending on the chosen parameters, it will converge in distribution to each one of the possible Brownian motion (BM) on the half-line, which can be: the absorbed BM, the reflected BM, the elastic BM,  the sticky BM, exponential holding BM, or mixed BM. Furthermore, with a small modification on the topology,  we can also obtain the Killed BM as the limiting process, and rate of convergence are provided in most  cases. 

Title: Asymptotic behaviour of viscoelastic models

Abstract: We are concerned with the global existence as well as the asymptotic behaviour of some viscoelastic evolution models. In the presence of viscoelastic effects given by a memory term with past history, we establish the global existence of solutions and the uniform decay rates of those solutions.