Speaker
Description
The Borsuk--Ulam theorem is one of the most widely used and influential results in algebraic topology, with applications in mathematics, physics, computer science, robotics, economics, and many other fields. In dimension two, it states that if one continuously flattens a beach ball without tearing or puncturing it, then there exists a point on the surface that lies directly above its antipodal point.
In these lectures, we introduce two of the most fundamental invariants in algebraic topology, homology and homotopy groups. Using these invariants, we establish a generalization of the Borsuk--Ulam theorem. We then apply this generalized theorem to prove the topological Tverberg theorem, a topological extension of Tverberg's theorem on the combinatorics of points in Euclidean space.