Sequences of metric spaces, MS presentation, Oregon State University, March 16th, 2026. 

Abstract: I discuss the basic definitions and examples for the theory of Gromov-Hausdorff convergence of metric spaces, in both the compact and non-compact setting. Along the way, we prove a classical algebraio-topological result known as  lower-semi-continuity of the fundamental group. We conclude by noting a couple of future lines of inquiry for my PhD research.

Approximately 49 minutes to talk about topological groups, chalk talk for Geometry and Topology Seminar, Oregon State University, October 1st, 2025. 

Abstract: We discuss the basic theory for topological groups, including (1) the structure of their open and closed subgroups, and (2) the fact their fundamental groups are abelian. We then discuss the resolution of Hilbert's Fifth Problem, which asked when topological groups graduate to Lie groups: one answer is that every locally-Euclidean group is isomorphic to a Lie group. This discussion includes a theorem due to Gleason and Yamabe, which concludes that locally-compact, connected groups are either discrete or else are "almost" Lie groups, in the sense that for arbitrarily small compact normal subgroups, K \leq G, the quotient G/K is a Lie group. In particular, locally compact groups are Lie groups if and only if they satisfy the no-small-subgroups property. 

Gromov-Hausdorff convergence and lower semi-continuity of the fundamental group, talk for Graduate Student Topology & Geometry Conference (GSTGC), Indiana University Bloomington, April 13, 2025. (Received funding, but was unable to attend, due to sickness.) 

Abstract: The Gromov-Hausdorff distance allows us to distinguish metric spaces by how far they are from being isometric. Among compact and proper metric spaces, we may talk about sequences of metric spaces and the limits of these sequences with respect to this distance. Lower semi-continuity of the fundamental group is an algebraic result relating elements of such a sequence (of metric spaces) with the limit of this sequence. In this talk, I cover the basic definitions and background for Gromov-Hausdorff convergence and lower semi-continuity of the fundamental group, and discuss conditions under which lower semi-continuity does (and does not) hold.

Decreasing paths of polygons, talk for MathForAll Conference on Math Education and Research, Oregon State University, Spring 2024.

Abstract: We call a continuous path of polygons “decreasing” if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by a decreasing path (attainability problem). In particular, this characterization is such that there is a finite procedure for checking attainability.