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Mathematics & Statistics
Colloquium Archive

# Colloquia — Fall 2020

## October 30, 2020

Title
Speaker

Time
Place

Surgery theory and cell-like maps
Michelle Daher
University of Florida
3:00pm–4:00pm
Location

Abstract

Cell-like maps play an important role in the topology of manifolds since they appear as limits of homeomorphisms. Typically, the image of a cell-like map of a manifold is a manifold, but generally it is a manifold with singularities (homology manifold).

In the 60s, Lacher asked the question whether two closed manifolds that can be mapped by cell-like maps onto the same space $$X$$ must be homeomorphic. In the 70s, Quinn proved that if such an $$X$$ exists it has to be infinite dimensional. Since it was known that cell-like maps cannot raise the dimension by a finite number, the chances for a positive answer to Lacher's question became slim.

Nevertheless, in a paper published in 2020, Dranishnikov, Ferry, and Weinberger gave an example of two closed non-homeomorphic 6-manifolds that can be mapped by cell-like maps onto the same space. In this talk, we show that for any $$n$$, we can find n non-homeomorphic manifolds that can be mapped by cell-like maps onto the same space $$X$$. These examples are closely related to Surgery theory, the main tool in the classification of higher dimensional manifolds.

## Friday, September 25, 2020

Title
Speaker

Time
Place
The round sphere provides the least-perimeter way to enclose prescribed volume in $$R^m$$. The $$n$$-bubble problem seeks the least-perimeter way to enclose and separate $$n$$ prescribed volumes in $$R^m$$. The solution is also known only for $$n = 2$$ in $$R^m$$ (the standard double bubble) and $$n = 3$$ in $$R^2$$ (the standard triple bubble). If you give $$R^m$$ Gaussian density, the solution was recently proved by Milman and Neeman for $$n \le m$$. There is further news for other densities.