# A5 Seminars

### IME - USP

Português

Welcome to the A5 Group Seminar page! This is a general-audience colloquium series for all members of the IME math community, our aim is to introduce research level math for undergraduate and gradute students at IME.

The lectures this term will take place every two weeks, on Tuesdays at 2 pm. A hybrid model will be adopted: when they are virtual, seminars will be broadcasted in some classroom at IME-USP.

To see the next lecture click here.

## Talks this term

March 29
Speaker
Pietro Mesquita Piccione
Title
A counterexample for the periodic orbit conjeture
Abstract
In this presentation, I will give a counterexample to the periodic orbit conjecture, which is: "There is no flow in a compact variety whose orbits are all closed and such that the length of the orbits is unbounded." The counterexample will be the flow of a vector field, in the product of the Heisenberg manifold by a 2-torus, which will result in a (real) analytic manifold of dimension 5. This presentation will be based on the article "A counterexample to the periodic orbit conjecture", by Dennis Sullivan, the most recent winner of the of the Abel Prize.
Time
2 pm
Model
Online lecture with broadcast
Recording
April 19th
Speaker
Claudio Gorodski
Title
The diameter of quotients of the sphere
Abstract
We consider an arbitrary quotient $$X = S^n(1)/G$$ of the unit sphere $$S^n(1)$$ by a group of isometries $$G$$ and show that the diameter of $$X$$ is zero or greater than a universal constant $$\epsilon \gt 0$$. What is new is the independence of $$\epsilon$$ from $$n$$. The classification of finite simple groups is used in this proof. (Joint work with C. Lange, A. Lytchak, and R. A. E. Mendes).
Time
2 pm
Model
On-site
Room
B-6
May 3rd
Speaker
Severino Toscano
Title
The Toeplitz Operator Index
Abstract

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.

Time
2 pm
Model
On-site
Room
B-9
May 17th
Speaker
João Ruiz
Title
Modular Forms: why should I care?
Abstract

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"!

Time
2 pm
Model
Online lecture with broadcast
Recording