# About Me

I am a commutative algebraist from Severn, Maryland. My graduate degree is from Cornell University in 2011, and my undergraduate degree is from St. Mary's College of Maryland in 2004.

For information about Hood College Mathematics
Department's upcoming *Celebration of Mind* event in honor of
Martin Gardner or to RSVP see:

**Tea with Martin: An Afternoon of Mathematical Diversions**

I am also a recent Project NExT Fellow (Silver Dot, 2012-2013) and a Section NExT Fellow for the MD-DC-VA section(2011-2012).

## Primary Research Interests

My research primarily focused on computing the projective dimension, regularity, and Betti numbers of special classes of ideals over polynomial rings. In particular, my thesis results were focused on:

- Stabilization of Betti diagrams of powers of equigenerated ideals \(I\).
- Constructing linear quotient orderings on special classes of monomial ideals.
- Bounding the degrees of syzygies of r-generated edge ideals in degree 2 in terms of smallest induced cycle \(C_k\).
- Construction and resolution of nerve complexes related to edge ideals, using them to study properties of graphs on subgraphs rather than induced subgraphs.
- Bounding the regularity or projective dimension of an ideal \(I = (f_1,...,f_k)\subseteq\Bbbk[x_1,..,x_n]\) in terms of the degree of its generators, independent of \(n\), constructing ideals with a high projective dimension relative to the fixed numbers and degrees of generators.

My current research has moved into producing explicit resolutions of residue fields \(\Bbbk\) over quotient rings \(S=\Bbbk[x_1,..,x_n]/M\), where \(M=(m_1,..,m_r)\) is a monomial ideal. This extends results of Berglund, Blasiak and Hersh on the Poincaré series of such rings \(S\).

Link to abstract for my talk at AMS Session on Commutative Algebra, JMM San
Diego: *Betti
Numbers of Infinite Free Resolutions*.

## Undergraduate Teaching and Research

### Hood College 2011-present

- Fall 2014
- Math 111G: Mathematics of Games and Sports
- Math 201: Calculus I
- Math 339: Linear Algebra

- Spring 2014
- Math 112W: Workshop Statistics
- Math 440: Introduction to Abstract Algebra
- Math 502: Explorations in Algebra

- Fall 2013
- Calculus I
- Probability and Statistics
- "Workshop Calculus"
- Contest Problem Solving (seminar)

- Spring 2013
- Calculus II
- "Workshop Calculus"
- Mathematics of Games and Sports

- Fall 2012
- Calculus I
- "Workshop Calculus"
- Contest Problem Solving (seminar)

- Spring 2012
- Calculus II
- Abstract Algebra
- Econometrics (independent study)

- Fall 2011
- Calculus I
- Probability and Statistics

### Undergraduate Research Group

Currently, I have two undergraduate research students (Brian
Penko '15 and Megan Rodriguez '15) working on a project in
*Buchberger Graphs of Trivariate Monomial Ideals*. Our focus is
on enumerating minimal (strongly generic) monomial ideals with a
Buchberger graph realizing a fixed planar graph, and providing
minimal bounds on degrees of generators for such ideals.

In Summer 2012, I had two undergraduate research students working on
a project on the *Combinatorics of the Mandelbrot Set*. In
particular, they focused on identifying points in the Mandelbrot set
with orbit patterns corresponding to \(H\)-compositions of numbers
\(n\in{\mathbb N}\), classifying and counting points \(x\in{\mathbb R}\) with
period \(n\) by their combinatorial type, and producing explicit
coefficients of infinitely iterated Multibrot polynomials.

Our paper from that summer research has been submitted, and
both students attended *MathFest 2012* in Madison, WI to
present their work.

## Section NExT Link:

## Technology Resources

# News

### Upcoming Math Department Events:

- Math Lecture:
*Title:*A Combinatorial Gaming Zoo*Date:*October 20th*Time:*3-4pm*Speaker:*Gwyn Whieldon*Location:*HT 316

- Tea with Martin: A Celebration of Mind Event
*Date:*October 27th*Time:*3-5pm*Event:*Math Tea*Location:*HT 316*RSVP:*link

### Past Math Department Events:

- Math Lecture:
*Date:*February 5th*Speaker:*James Tanton*Location:*HT 316

- Math Lecture:
*Date:*March 5th*Speaker:*Jill Tysse*Location:*HT 316