At Haverford, I have so far taught (proof-based) linear algebra, abstract algebra, topology, and a new course on the ethics of mathematical practice (Math 146) with an eye towards anti-racism. Here is a course syllabus for linear algebra.
Titles/descriptions of some senior theses I've advised:
Kolmogorov complexity and its application in inductive inference (2024-2025), by Angel Yang. Angel studied Kolmogorov complexity, useful for establishing a theory of randomness for individual data strings without the need of an ambient probability distribution. She then explored Solomonoff's theory of induction which attempts to formalize Occam's razor by combining the Bayesian method with (a computable approximation of) Kolmogorov complexity for choosing a universal prior.
Construction of unit entropy metrics on the lamppost graph that evenly distribute entropy between two subgraphs (2024-2025), by Peter LaRochelle. Peter studied metrics on a graph comprised of a figure 8 and a barbell, and studied the geometry of the hypersurface in the moduli space corresponding to metrics that assign each of these subgraphs the same value.
Entropy of metric graphs (2024-2025) by Al Farabie Akanda. Farabie explored metric graphs that took the form of a polygon with a self-loop at each vertex, and for some interesting cases computed the minimum possible entropy of a proper-subgraph of such a graph with unit entropy.
Cycles on metric graphs of given entropy (2024-2025) by Rowan Shigeno. Rowan studied the entropy function over the moduli space of metrics on a rose and collected evidence for some conjectures regarding the minimum possible entropy of a proper sub-rose within a unit entropy rose.
The group action of the free group on the Culler-Vogtmann Space (2023-2024) by Halley Kucirka. Halley studied the action of the outer automorphism group of the free group on the so-called Culler-Vogtmann outer space.
Entropy of metric graphs, by Peter Kulowiec (2023-2024). Peter studied the volume entropy of metric graphs and its relationship to the outer space.
Entropy and the outer space, by Atira Glenn-Keough (2023-2024). Atira studied the outer space for the outer automorphism group of a free group and its relationship to entropy of metric graphs. She in particular studied entropy minimizing, volume-1 graphs.
Computing the entropy metric on Culler-Vogtmann outer space, by Rachel Niebler (2023-2024). Rachel studied a metric on the outer space (the analog the Teichmuller space for the outer automorphism group of the free group) that resembles the Weil-Petersson metric on Teichmuller space. She in particular focused on formulae for the entropy-1 level set of graphs that come from the cycle complex.
SL(2,Z)-action on origamis associated to minimally intersecting filling pairs, by Jason Ma (2021-2022). Jason studied a family of origamis defined here and proved that they all have the alternating group as monodromy. This will feature in a forthcoming joint paper with Adam Friedman-Brown.
Constructing minimal origamis and minimal origami pairs, by Adam Friedman-Brown (2021-2022). Adam studied the same family of origamis mentioned above, and generated pictures that will be useful for computing Arf invariants for these origamis.
The axiom of choice and its equivalences, by Ben Haile (2021-2022). Ben studied various formulations of the Axiom of choice and proved their equivalencies. Along the way, he learned about transfinite induction/recursion.
The figure 8 knot complement: a topological and geometric perspective, by Ethan Flicker (2020-2021). Ethan studied the figure 8 knot complement; he proved the validity of the Wirtinger presentation and used it to give a presentation for its fundamental group, and also studied some of the mathematics behind showing that it admits a complete finite volume hyperbolic structure.
An exploration into fundamental groups, by Codie Collins (2020-2021). Codie studied the basics of fundamental groups and did an in-depth analysis of the fundamental group of a genus two surface from the point of view of both algebra and hyperbolic geometry.
A combinatorial identity for unicellular maps, by Matt Ludwig (2020-2021). Matt studied a formula for counting graphs on a surface of genus g with one face and a given number of edges. This relates to computing the Euler characteristic of Moduli space.
Simple closed curves on a torus intersecting at most k times, by Flame Ruethaimetapat (2020-2021). Flame studied the problem which asks for the largest collection of homotopically distinct simple closed curves on a torus so that no two curves cross more than k times.
Curves intersecting at most once, by Mujie Wang (2019-2020). Mujie studied the problem which asks for the largest collection of essential and homotopically distinct simple closed curves on a given surface so that no two curves cross more than once.
An exposition of Gromov's 1981 paper "Groups of polynomial growth and expanding maps", by Remy Erkel (2019-2020). Remy studied the celebrated theorem of Gromov equating polynomial growth to virtual nilpotence for finitely generated groups.