Topics

Each of the three instructors will focus on a different topic. See titles and descriptions below.

Instructor: Florian Frick

Title: The topology of chromatic numbers  

Description: This lecture series explores the rich interplay between topology and the theory of chromatic numbers of (hyper-)graphs. We will trace key developments from classical results, such as chromatic numbers of Kneser graphs, to recent advances, such as applications in Ramsey theory. Emphasis will be placed on both foundational methods and current frontiers in the area.

Slides Lecture 1

Slides Lecture 2

Slides Lecture 3

Slides Lecture 4

Slides Lecture 5

Instructor: Andreas Holmsen

Title: Topological Helly theorems and combinatorial convexity

Description: 

In this mini-course we will investigate combinatorial and topological generalizations of Helly-type problems from discrete geometry, such as the colorful and fractional versions of Helly’s theorem and the celebrated (p,q)-theorem due to Alon and Kleitman. We will mainly focus on the use of techniques from extremal and topological combinatorics.

Instructor: Pablo Soberón

Title: Topological methods in fair partitions and allocations

Description: In this course, we will explore the connections between fairness in resource allocation and topology.  Participants will learn how topological tools - such as the Borsuk-Ulam theorem, Sperner's lemma, and equivariant topology - can be applied to problems of fair division, resource allocation, and geometric partitioning. Key techniques include the configuration space/test map scheme and fixed-point theorems, which provide powerful frameworks for proving the existence of fair or balanced solutions under symmetry and continuity assumptions.