| Date: | February 27, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 624 |
| Speaker: | Maggie Miller, University of Texas at Austin |
| Title: | Branched covers of twist-roll spun knots |
| Abstract: | Twist-roll spun knots are a family of 2-spheres that are smoothly knotted in the 4-sphere. Many of these 2-spheres are known to be branch sets of cyclic covers of the 4-sphere over itself (maybe counterintuitively to 3-dimensional topologists, since this never happens for nontrivial knots in the 3-sphere). It’s very difficult to come up with interesting examples of 2-spheres in the 4-sphere, so this family typically serves as the examples in any theorem about surfaces in the 4-sphere. I’ll discuss a few different versions of their construction and prove a relationship between some of their branched coverings. As a corollary, we’ll conclude that some interesting families of manifolds known to be homeomorphic are actually diffeomorphic. This is joint with Mark Hughes and Seungwon Kim.
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| Date: | March 19, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Seokbeom Yoon, Southern University of Science and Technology |
| Title: | The (twisted/L^2)-Alexander polynomial of an ideally triangulated 3-manifold. |
| Abstract: | Ideal triangulation is a useful tool for studying 3-manifolds. It allows us to efficiently compute certain 3-manifold invariants (e.g. volume). In this talk, I would like to explain how the Alexander polynomial, as well as twisted/L^2 ones, are related to ideal triangulations. This work is joint with Stavros Garoufalidis.
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| Date: | April 16, 2024 |
| Time: | 4:00pm |
| Location: | BLOC 302 |
| Speaker: | Sara Maloni, University of Virginia |
| Title: | Geometric Structures associated to Higher Teichmüller Theory |
| Abstract: | The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, this space can also be seen as a connected component of representations from the fundamental group of S into Isom(H^2). Generalizing this point of view, Higher Teichmüller Theory studies connected components of representations from the fundamental group of S into Lie groups of rank greater than 1.
We will discuss parts of the classical theory of deformations of geometric structures, Higher Teichmüller Theory and the notion of Anosov representation. We will then describe how Anosov representations correspond to deformation of certain geometric structures, and a joint work with Alessandrini, Tholozan and Wienhard about their topology. |
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