 # Courses

Here you can find the abstracts of the courses that will be taught in the school. In the Spanish version you can find academic resources for them.

## “Computer Driven Questions and Theorems in Geometry”Dr. Moira Chas, Stony Brook University

Three numbers can be associated to the free homotopy class of a closed curve in a surface with boundary and negative Euler characteristic:

• The number of self-intersections,
• The number of letters in the word associated to the class, where the letters are generators of the fundamental group of the surface,
• The length of the geodesic curve in that homotopy class, if the surface has a metric of constant negative curvature.

These numbers can be explicitly determined or approximated by nontrivial algorithmic means and the use of a computer. These calculations exhibit various patterns in the relationships between the three mentioned numbers. In this course it will be discussed how these calculations can lead to obtaining counterexamples of existing conjectures, as well as discovering new conjectures and subsequent theorems in certain cases.

#### Resources:

See Spanish version

## “An invitation to homotopy theory”Dr. Angélica Osorno, Reed College

This course will begin by discussing the notions of homotopy and deformation of a topological space. The fundamental group of a topological space will be defined and it will be shown that it is invariant under the notion of homotopy. Then the notion of a covering space will be studied, and the relationship between covering spaces of a topological space and subgroups of its fundamental group will be presented. With these tools the fundamental group of the circle will be calculated. The course will end with the Seifert–van Kampen theorem and various examples. In particular, the result that any discrete group is the fundamental group of a topological space that can be explicitly constructed will be discussed.

#### Resources:

See Spanish version

## “Introduction to noncommutative topology”Dr. John Skukalek, Universidad San Francisco de Quito

This mini-course is an introduction to C*-algebras accessible to advanced undergraduate students. The goal will be to explain why C*-algebras are referred to as noncommutative topological spaces. We will introduce commutative C*-algebras as a natural algebraic structure arising from the study of ordinary, "commutative" topological spaces before moving on to general, abstract C*-algebras. Our focal point will be the Gelfand-Naimark Theorem, which establishes the precise link between commutative C*-algebras and topological spaces.

#### Resources:

See Spanish version