Graduate Projects & Papers



Papers

Applying Distributional Compositional Categorical Models of Meaning to Language Translation, 2018.
A New Universal Definition of F_q [t] in F_q (t), 2019.  
A Note on Hilbert's "Geometric" Tenth Problem, 2020.


Theses

Efforts in the Direction of Hilbert's Tenth Problem, 2018.

This was my master's thesis, completed under the supervision of Professor Damian Roessler. The thesis had three objectives: motivate the connection between the integers Z and F_q[t] from a decidability and definability standpoint, excurse through the definability results of Koenigsmann and subsequent authors in global fields, and provide a shorter and simpler universal definition of F_q[t] in F_q(t) than existed at the time.

An Analysis of Tame Topology using O-Minimality, 2017,
with accompanying poster.

This was my bachelor's thesis, completed under the supervision of Professor Andreea Nicoara. Here I explored some 'tame' properties of o-minimal structures, in comparison to the corresponding properties for semialgebraic, semianalytic, and subanalytic sets, and also proved some algebraic results using quantifier elimination in ACF and RCF.


Oxford Broadening Essays

To be added: Sums of Squares (And Higher Powers).
Flatness in Algebraic Geometry.
Higher Category Theory.
Applying Distributional Compositional Categorical Models of Meaning to Language Translation.

These are essays written to fulfill in part the broadening requirement for Oxford DPhil students. (I know of at least one mistake present in the first, which I may get around to fixing.)
The last is an essay I wrote for the "Distributional Models of Meaning" reading course for MFOCS students at Oxford. It has since been sharpened, condensed and its Irish corrected, and is published in Electronic Proceedings in Theoretical Computer Science. This is the original version (with many mistakes present). 

Undergraduate Projects


Models Wanted: A Talk in Mathematical Logic.

On Wednesday February 15th I presented a talk on my summer research at the University of Notre Dame to the Trinity College Mathematical Society.


Getting strung up & Dithering around (link upon request).

This is a paper I wrote a previous summer while exploring some ideas in the intersection of Computer Science and Art. I was initially inspired by the work of Petros Vrellis, a brilliant artist who creates artwork using computer programming and algorithms. I ended up tackling three problems; recreating Vrellis' work, recreating the work of Yumi Yamashita, another artist who also designs continuous line illustrations, and Kel Cruz, an artist who makes art from squares of coloured tape. I am currently in the process of physically making the art inspired by these artists.


Homological Algebra.

This was a report written for my sophister algebra seminar, MA341E. I enjoyed learning about this new area of mathematics and I'm glad I got the chance to present a lecture on this subject, as part of my assessment. The talk notes can be found here.


Scott Sentences in Uncountable Structures
with accompanying abstract.

I completed this work during the summer of 2016 at the University of Notre Dame under Prof. Julia Knight. This research allowed me to explore model theory more from when I first came across it at UChicago, and I'm very grateful to have been given that chance.
This paper was published by the RHIT Undergraduate Math Journal in the Spring issue.


An investigation into the use of a genetic algorithm to obtain numerical solutions to the Moving Sofa problem.

I completed this fun little project during the summer of 2015 at Trinity under Prof. Mike Peardon. This project (which I'm currently shortening and rewriting) gave me my first taste of evolutionary algorithms, and is also a great icebreaker at parties.


Using numerical methods to solve the gravitational n-body problem.

This was my first internship, completed at the end of my first year in 2014 at Trinity, under Prof. Mike Peardon. Here to demonstrate the leapfrog algorithm I simulated the solar system, and constructed a Barnes-Hut algorithm to deal with larger numbers of bodies. At the end, I tried my hand at animating the results in OpenGL.