Abstracts
for the Seminar

Fall 2022

**Speaker: **Florian Frick, Carnegie Mellon University

**Title: **The Borsuk-Ulam theorem in combinatorics and geometry

**Time:** 2:30 PM, Monday, October 17, 2022

**Place:** Malott 206

**Abstract:** The classical Borsuk--Ulam theorem has numerous consequences across mathematics. It states that any continuous map from an $n$-sphere to $\mathbb{R}^n$ must identify antipodal points. I will present some new applications of this result, such as:

- Codes in projective spaces through the Borsuk-Ulam theorem control structural results for zeros of raked trigonometric polynomials and more general maps.
- For which pairs $(d,n)$ does a continuous fibration of a region in $\mathbb{R}^n$ by unit $d$-spheres exist?
- Generalizations of Lovász' lower bounds for chromatic numbers of graphs to obstructions for chromatic mixing.

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