## Research

There is a generalization of the Birkhoff ergodic theorem for certain Fuchsian groups.
Right now I am working on expanding the list of groups known to enjoy this ergodic
property beyond the Fuchsian examples. The essential property required of the groups,
in order for the proof to go through, is that their action on hyperbolic space be
controlled by some combinatorial data. This combinatorial data manifests in one of (at
least) two distinct forms. It may present as a special kind of partial order, or perhaps
dually as a generalized notion of `median' i.e., algebraic structure. Presently, I am
taking a closer look at the categorical aspects of this duality. In particular, there
is a notion of ultrafilter on these partial orders which, in a lot of ways, mimics the
behavior of ultrafilters on boolean algebras. On the other side, the ideals (in the sense
of universal algebra) of the median operation behave like convex sets. In either case
there are traces of codensity. For instance, the codensity monad of the inclusion of
finite sets into sets is the ultrafilter monad - the proof I know invokes the language
of measure and integration. Similarly, the inclusion of convex subsets into measure
all of these situations together. In the process of proving that the half-spaces of
ultrafilters functor or the ultrafilters of half-spaces functor admit monad structures and
moreover, arise as the codensity monad of some functor, one expects to learn about integration
operations on the pertinent combinatorial objects. Another example from the literature uses
the concept of codensity to understand the possible (group) actions on a set. This approach
should allow us to push the current techniques (close to) as far as they go in this direction.

## Teaching

Term Course Position
Fall 2018 Math 221: Calculus and Analytic Geometry 1 Teaching Assistant
Spring 2019 Math 221 TA coordinator
Fall 2019 Math 234: Calculus -- Functions of Several Variables TA coordinator
Spring 2020 Math 222: Calculus and Analytic Geometry 2 TA coordinator
Summer 2020 Math 221 Teaching Assistant
Fall 2020 Math 234 TA coordinator
Spring 2021 Math 340: Elementary Matrix and Linear Algebra Teaching Assistant
Summer 2021 Math 240: Introduction to Discrete Mathematics Insructor of Record
Fall 2021 Math 112: Algebra Instructor of Record
Spring 2022 Math 112 Instructor of Record
Summer 2022 Math 222 Instructor of Record
Fall 2022 Math 221 TA coordinator
Spring 2023 Math 222 Teaching Assistant
Summer 2023 Math 240 Instructor of Record

## Discrete Math Lecture Notes

If you choose to view or download these notes, keep in mind the fact that
I have noticed certian browsers tend to omit some pen strokes (especially
single points/dots). This does not seem to be a problem for me when browsing
mobile Safari, for example.
Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5
Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10
Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15
Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20
Lecture 21 Lecture 22 Lecture 23 Lecture 24

## Calc 2 Lecture Notes

FToC U-Sub By Parts Trig
Partial Fractions Linear ODEs

## Higgins: Categories & Groupoids

Lastly, to cite the sources used to create the documents linked in the research section.
Everything in the document about hyperbolic space lives in the canonical refrence for
the subject, "Metric Spaces of Nonpositive Curvature" by Bridson and Haefliger.
The introduction to ultrafilters follows the book by J.L. Bell and A.B. Slomson titled
"Models and Ultraproducts an Introduction" just far enought to see the Ultrafilter
lemma. Along the way, I illustrate examples and solve all of the exercises. This
document should be understandable to anyone who has read through lesson 8 say,
of the discrete math notes above. The main ingredient in the famous proof of Arrow's
Impossibility Theorem is the ultrafilter lemma. The remainder of the document treats
this theorem following the presentation in
this video.