### Bi-College Math Colloquium

Monday, April 8, 2024 - 4:15pm

#### Dr. Bülent Tosun

Institute for Advanced Study (Princeton) and the University of Alabama

It is a fundamental result in geometric topology that every closed 3-dimensional manifold can be realized as the boundary of a 4-dimensional manifold, and many 4-dimensional manifolds with different topology can have the same boundary. A difficult problem, for a given closed 3-dimensional manifold, is to find a 4-dimensional manifold with the smallest possible topology (e.g. contractible) that the 3-manifold bounds. This fits under the problem of embedding 3-dimensional manifolds into 4-dimensional Euclidean space which has a rich history and proved to be tremendously important for the development of geometric topology since the 1950s. In this talk, I will provide further context and motivations for the problem above. Next, I will introduce a seemingly different problem where we will try to understand curves on Seifert surfaces of knots in three dimensional sphere (lots of pictures, lots of fun). Finally, we will see how the progress in the previous step will help us to construct many 3-manifolds that bound contractible 4-manifolds. The talk will feature some recent results which were obtained with REU students at the speaker's home institution.