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Title: Surgery theory and cell-like maps Speaker: Michelle Daher, University of Florida Time: 2:30pm–3:30pm Place: MS Teams Sponsor: Mohamed Elhamdadi

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.

Title: Double Bubbles and Densities Speaker: Frank Morgan, Atwell Professor of Mathematics, Emeritus, Williams College Time: 2:30pm–3:30pm Place: MS Teams Sponsor: Mohamed Elhamdadi

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.

In 2000 Hales proved that regular hexagons provide a least-perimeter way to partition the plane into unit areas. Undergraduates recently obtained a partial extension to closed hyperbolic manifolds. The 3D Euclidean case remains open. The best tetrahedral tile was proved recently. (Despite what Aristotle said, the regular tetrahedron does not tile.) We'll describe many such results and open questions. Undergraduates welcome.