Department of Mathematics

The 20th century saw both the birth and rapid, widespread adoption of one of the most ubiquitous organizational tools available in modern mathematics: category theory. Category theory provides a unifying language in which one may formally express relationships and analogies within and between very different classes of objects, whether algebraic, geometric, analytic, or otherwise; it is structured enough to be meaningful, and it is flexible enough to be expressive. However, while already a major success story, the categorical phenomenon has continued to expand into higher-dimensional variants– so-called higher categories. These 21st-century generalizations come in many different forms and often serve vastly different purposes. In this talk, we will take a peek at three kinds of higher categories: double categories, \((\infty, 1)\)-categories, and \(A_\)-categories. As demonstrations of the utilities of these gadgets, we will briefly discuss applications in dynamical systems theory, derived algebraic geometry, and homological mirror symmetry.

Abstract: Nematic liquid crystals are classified as partially-ordered materials due to the tendency of their molecules to align along preferred directions. This partial alignment, combined with fluid-like properties, render these materials useful in a range of applications, including their widespread use in optical displays. When confined to thin channels, the fluid flow extends the applications of nematics further to optofluidic devices and guided micro-cargo transport. We examine the intrinsic coupling between the fluid motion and the orientation of nematic molecules by considering steady unidirectional flows in thin channels. We employ a reduced Beris-Edwards framework which leads to a system of nonlinear and coupled differential equations. Our analytical and numerical results highlight the universality of order reconstruction solutions, characterized by distinct orientational sub-domains that are separated by domain walls.

The math club is hosting a panel discussion on undergraduate mathematics research opportunities. The panelists will discuss how to become involved in mathematics research as an undergraduate and what the research process is like.

In this talk, we will survey progress on Ricci solitons in dimension 4. Ricci solitons, which are self-similar solutions to the Ricci flow, play a central role in singularity analysis for Ricci flow. Singularity analysis is crucial for understanding how to extend Ricci flow past singularities and for potential topological applications. In Bamler’s theory, singularity models are generically shrinking solitons, though steady solitons can also appear as singularity models. Additionally, we will discuss both recent and older works by Bamler, Chan, Deng, Freedman, Lu, Ma, Shin, Yang, and Zhang.