A5 Seminars

The final goal of the seminar is to give an outline of the proof of a formula for the Fredholm index of a Toeplitz operator on the circle with continuous symbol. I hope to be able to explain all the definitions and give a good idea of why the formula is true assuming only concepts of Measure Theory and Functional Analysis studied in the courses of our Bachelor's degree in mathematics. This "index theorem" for Toeplitz operators is the simplest of a series of results that express the Fredholm index of an operator (an analytic datum) in terms of some topological information carried by the operator. In this case, the Fredholm index is equal to minus the number of rotations of a continuous function wich doesn't vanish, used to define the Toeplitz operator in question.

This lecture will be a condensed version of two lectures I I gave in the course Panoramas da Matemática (MAT 554) in 2020.

In this talk we will explore a little bit of the theory of modular forms: how they arise in nature, how they can be used to solve Number Theory problems, and how we can place them in a more general context, with deep connections context, with deep connections to other areas of mathematics.

Highlights of the lecture include: "With how many squares can an integer be made?" and "A meeting of Number Theory and Algebraic Geometry"!

The concepts of connection and parallel transport are fundamental in Riemannian geometry and afford, among other things, interpretations of curvature.

This talk is aimed at a broad undergraduate mathematics audience and will be roughly divided into three parts. First we will introduce the concepts of connection, parallel transport, Holonomy groups, geodesics, and curvature. We will then cover Ambrose Singer's theorem.

Finally we will comment on relations between parallel transport and singular foliations, presenting results on recent research.

In the article "Opinion Dynamics on Discourse Sheaves", published in 2021, Jakob Hansen and Robert Ghrist present how to use cellular sheaves and their respective cohomology theory to model the evolution of people's opinions/preferences over time. The main idea of the paper is to generalize the Laplacian matrix via the Laplacian sheaf in the context of graphs.

In this lecture I will introduce basic concepts of category theory, necessary for understanding the article, and discuss the main advantages of this approach such as the ability to identify the presence of people who lie.

In the area of Representation of Algebras, current research topics, such as tau-tilting theory and subcategory \(n\)-cluster-tilting, are inspired by tilting theory. This theory is based on the presence of a special module \(T\) called the tilting module.

In this lecture, an example in the category of finitely generated modules over a \(C\)-algebra \(A\) of finite dimension will be presented. The tilting theory shows an intimate connection between the algebras \(A\) and \(B = \operatorname{End} T\).