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.

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.

### 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

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.

# News

### Upcoming Math Department Events:

• To be announced.

### Past Math Department Events:

• Math Lecture:
• Date: November 6th
• Topic: Math for America
• Location: HT 316
• Math Lecture:
• Date: November 13th
• Speaker: Edwin O'Shea
• Location: HT 316
• Math Lecture:
• Date: November 27th
• Speaker: Michael Dorff
• Location: HT316
• Math Lecture:
• Date: February 5th
• Speaker: James Tanton
• Location: HT 316
• Math Lecture:
• Date: March 5th
• Speaker: Jill Tysse
• Location: HT 316