All talks will be held on the University of Arkansas campus in the Science and Engineering Building (SCEN) Room 408.
| Time | Title | Speaker |
| 8:30-9:15a | Coffee and bagels | |
| 9:15-10:05a | Cusped hyperbolic 3-manifolds with a compact totally geodesic boundary | Anuradha Ekanayake |
| 10:30-11:20a | Tools for Mathematical Illustration | Edmund O. Harriss |
| 12:00-2:00p | Lunch | |
| 2:00-2:50p | Torsion-Free Uniform Lattices in Aut(X m,n) | Maya Verma |
| 3:05-3:55p | Avoiding inessential edges | Henry Segerman |
| 4:10-5:00p | Geometric and topological data analysis in biology applications | Jiahui Chen |
Abstracts
Maya Verma
Torsion-Free Uniform Lattices in Aut(X m,n)
For the Cayley 2-complex Xm,n of the Baumslag-Solitar group BS(m,n), one can consider the group of combinatorial automorphisms of Xm,n, denoted by Aut(Xm,n). In this talk, I will answer a question posed and partially answered by Forester: For which pairs of nonzero integers (m,n), does Aut(Xm,n) contain incommensurable lattices?
Anuradha Ekanayake
Cusped hyperbolic 3-manifolds with a compact totally geodesic boundary
Determining the lowest volume examples in various classes of hyperbolic 3-manifolds is a part of an extensive program focused on understanding the hyperbolic volume structure. Its roots can be tracked down at least to the notes written by William Thurston in 1979. He observed that the volume is a good measurement of topological complexity. Since then, many authors have worked on identifying smallest-volume examples in different classes of hyperbolic manifolds.
We denote by Nc,c the class of orientable cusped hyperbolic 3-manifolds with a compact totally geodesic boundary. A notion of particular importance associated with these manifolds is return paths – geodesic arcs perpendicular to the boundary of the manifold at both end points. The volume of a manifold with a geodesic boundary depends on the length of its shortest return path in several ways. I will discuss this dependence and how it can be used to identify the lowest volume manifold in Nc,c.
This talk is based on my thesis work, supervised by Jason DeBlois.
Edmund O. Harriss
Tools for Mathematical Illustration
How can we use the physical world and our intuitions about it to challenge mathematical understanding and lead to new research questions? In this talk I will look at some of the ways I connect mathematical ideas to physical geometry and illustrations and how this can both lead to research questions and new applications of mathematical ideas, such as the Gearhart Curvahedra sculpture in the courtyard of Gearhart Hall.
Jiahui Chen
Geometric and topological data analysis in biology applications
This talk will discuss geometric and topological data analysis in biology applications and focus on an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale analysis of evolving manifolds constructed from filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exact topological invariants. To demonstrate the utility of the proposed method, the application is considered for the predictions of binding free energy (BFE) changes of protein-protein interactions (PPIs) induced by mutations with machine learning modeling. It has a great application in studying the SARS-CoV-2 virus’ infectivity, antibody resistance, and vaccine breakthrough, which will be presented in this talk.
Henry Segerman
Avoiding inessential edges
Results of Matveev, Piergallini, and Amendola show that any two triangulations of a three-manifold with the same number of vertices are related to each other by a sequence of local combinatorial moves (namely, 2-3 and 3-2 moves). For some applications however, we need our triangulations to have certain properties, for example that all edges are essential. (An edge is inessential if both ends are incident to a single vertex, into which the edge can be homotoped.)
We show that any two triangulations with all edges essential can be related to each other by a sequence of 2-3 and 3-2 moves, keeping all edges essential as we go.
This is joint work with Tejas Kalelkar and Saul Schleimer.