Research Projects

(Most recent projects listed first)

Determinantal Varieties Modulo Hyperplanes
Preprint coming soon! In the meantime, here's a presentation that I gave in Spain: Resolutions of Determinantal Ideals and Their Initial Ideals
Eisenbud-Mazur Conjecture
My senior thesis at the University of Notre Dame. It begins with a pretty thorough introduction to the module of differentials and evolutions and ends by studying the paper of David Eisenbud and Barry Mazur on symbolic squares. Thesis

Completions of Excellent Rings
In the summer of 2007 I studied completions of excellent rings at the Williams College REU. Below are the two papers we wrote as well as a poster we presented at the 2008 Joint Math Meetings and a Beamer presentation outlining the results.
- Dimensions of Formal Fibers of Height One Prime Ideals (With Michael Daub and S. Loepp, to appear in Communications in Algebra)
- Unique Factorization Domains and Geometrically Regular Formal Fibers (With Michael Daub, Ryan Johnson, H. Lindo, S. Loepp and Paul Woodard, to appear in the Canadian Journal of Mathematics
- Presentation
- Poster
An Excellent Poem: I wrote a short poem about the methods we used in our proofs for constructing these excellent rings. It's mathematically accurate and meant to be humorous, but I'm not sure if the humor extends much past those in our group... Enjoy!
- Poem
Lissajous Knots
In Summer 2006 I was at the Claremont Colleges REU where I worked under the supervision of Prof. Jim Hoste (Pitzer College) with Jay Daigle (Pomona College) and Wenjing Zheng (U. California Berkeley) on the subject of Lissajous knots. We were able to find bounds for the number of knot types of Lissajous knots of certain frequencies. For an expository look at our research, the following links are for the poster we presented at the Joint Math Meetings in Janurary 2007, and the presentation we gave at the Claremont Colleges in August. This is really cool stuff!
- Preprint (to appear in J. Experimental Math)
- Poster - "Sampling of Lissajous and Fourier Knots"; our poster from the Joint Math Meetings 2007.
- Presentation - A beamer presentation given July 28, 2006 at the Claremont Colleges
- Notre Dame Talk - A shorter presentation given September 30, 2007 at Notre Dame
Totally Positive Matrices
In Summer 2005 I participated in the Notre Dame REU. Working with Prof. Misha Gekhtman, my partner Brad Froehle and I studied the topic of Totally Positive Matrices
- On generators of bounded ratios of totally positive matrices (Linear Algebra Appl. 428 (2008), no. 7)

Expository Papers

Algebraic Geometry:
In the Spring of 2007 I organized an undergraduate seminar in algebraic geometry aimed at sophomore level honors students. I have prepared a set of lecture notes to accompany the seminar.
- Lecture Notes
- Seminar Syllabus
- Free Resolutions This was meant to be a later chapter in the algebraic geometry notes, but I haven't gotten around to combining them yet. These may be a bit rough but carry one to the start of free resolutions and computing Hilbert functions of monomial ideals.
Miscellaneous
- Koszul Complexes - A short expository paper on Koszul Complexes to accompany a series of talks I gave at the Commutative Algebra seminar.
- Tietze Extension Theorem - An expository paper on the Tietze Extension Theorem and the entrance paper to the Notre Dame SUMR program.
- Commutative Algebra Notes A start to some typed notes on introductory commutative algebra. These mainly follow Atiyah & MacDonald and contain some solutions to the homework problems.