As an undergraduate, I attended Carnegie Mellon University. I was accepted into and completed the Carnegie Mellon Honors Program in Mathematics. This program allows a student to attain a Masters and a Bachelors degree in Mathematics within the standard four-year period. My research was guided by Rami Grossberg and Andrés Villaveces.
I graduated in May 2009 with my PhD in Mathematics from the University of California, Berkeley. My advisor was Thomas Scanlon. My dissertation research focused on an open conjecture in the model theory of groups and rings. Copies of the dissertation are available through ProQuest; a reformatted web version is linked on my publications page. As part of the requirements of my PhD degree, I took a qualifying examination on December 17th, 2004. The syllabus is availabe as a Word document here.
While a graduate student, I TA'ed for two courses; their descriptions and course materials are here. I also became heavily involved in running the NSF Research Experience for Undergraduates (REU) run by Scott Chapman at Trinity University. A description of the student projects and my involvement can be found here.
In July 2009, I began a two-year postdoc at Université Claude Bernard Lyon 1. I was funded through a grant awarded by the National Science Foundation International Research Fellowship Program. I worked with Tuna Altınel and Frank Wagner on problems relating to groups with chain conditions on their centralizers.
In the summer of 2011, I returned to the United States to teach as a visiting assistant professor at Smith College. I am currently teaching two sections of second semester calculus and one of probability. Syllabi can be found here. In the summer of 2012, I will help Vadim Ponomarenko mentor the undergraduate projects at the San Diego State University REU.
I work in two fields of mathematics: non-unique factorization theory (commutative algebra) and the model theory of groups (logic and group theory). My publications.
Factorization Theory: This field studies commutative domains -- and more generally atomic, cancellative monoids -- whose elements do not have a unique factorization into irreducible elements. Classically, the theory of non-unique factorizations was developed with a view towards algebraic number rings and, more generally, Dedekind and Krull domains. However, it has also been discovered that these problems have applications to Ramsey theory, module theory, and control theory, where one must study the factorization properties of monoids quite different from those in number theory.
Two main ideas have driven much of the research in this field: 1. To understand the factorization properties of complex structures (e.g. algebraic number rings) in terms of simpler, combinatorial structures (block monoids over the class group). 2. To study invariants that measure how "far" a ring or monoid is from having unique factorization. My research has concentrated on certain subjects:
block monoids over abelian groups, particularly, finite groups and groups of torsion-free rank ≤ 1. When M is a Krull monoid with class group G, then the block monoid over the group G captures all the factorization properties of M, encapsulating the mantra "the class group controls the factorization".
characterizations of half-factoriality in Krull domains and Krull monoids;
arithmetic congruence monoids, which are arithmetic progressions closed under multiplication;
semigroup rings and generalized polynomial rings;
elasticity invariants of these monoids.
Due to the role of zero-sum sequences and sumsets in the study of block monoids, my research in this area has also ventured into additive number theory and Erdős-Ginzburg-Ziv problems in Ramsey theory.
Model Theory: Through model theory, logic has been successfully applied to many algebraic contexts. My own interests have centered around the use of model theory in the theory of groups.
My PhD dissertation concerned groups and rings which possess two strong model theoretic properties: stability and aleph0-categoricity. It is conjectured that algebraically these properties force very simple structures: up to a finite extension, they are both just abelian groups of finite exponent (and so the ring is essentially only a ring in name, since multiplication is trivial). A more detailed description of the dissertation is given here.
Recently, I have been studying groups with a purely group-theoretic property: finite chains of centralizers. Many classic groups in group theory possess this property, as do stable groups, a particularly important class of groups in model theory. Wagner and others have demonstrated that several fundamental group-theoretic facts about stable groups, such as the relationship between the Fitting subgroup and bounded left Engel elements, are provable simply from this basic chain property. Tuna Altinel and I have shown that a logical property of stable groups, namely the existence of definable envelopes of nilpotent subgroups, is also true in this purely group-theoretic class. Since this class of groups is not first-order definable, such a logical result is a bit surprising. It is also useful, since groups with such chain conditions are appearing in other investigations of the model theory of groups, such as rosy groups with NIP and pseudofinite groups.
In the past, I have also had an interest in the model theory of differential fields, generalizations of Ax's Theorem, and model theoretic constructions of universal graphs omitting families of subgraphs.
"Model Theory and Nilpotence in Groups With Bounded Chains of Centralizers" Antalya Algebra Days XIV, Çeşme, Turkey. May 2012.
"Model Theoretic Advances for Groups With Bounded Chains of Centralizers" Connecticut Logic Seminar. Wesleyan University, Middletown, Connecticut. January 2012.
"Nilpotence in Groups with Bounded Chains of Centralizers". AMS Session on Mathematical Logic, and Ordered and General Algebraic Structures, AMS/MAA Joint Meetings, Boston, Massachusetts. January 2012.
"Factoring Within Arithmetic Progressions" Plenary Lecture, Undergraduate Research Symposium. University of North Texas, Denton, Texas. August 2011.
"Factoring Within Arithmetic Progressions" Combinatorics and Additive Number Theory (CANT), City University of New York, New York City, NY. May 2011.
"Baer-Suzuki Theorems for Infinite Groups" British Postgraduate Model Theory Conference, Leeds, England. January 2011.
"Factorization Techniques for Numerical Semigroup Rings" AMS Session on Commutative Rings and Fields, AMS/MAA Joint Meetings, New Orleans, Louisiana. January 2011.
"Baer-Suzuki Theorems for Infinite Groups" AMS Special Session on Model Theory of Fields and Applications (Mathematics Research Communities session), AMS/MAA Joint Meetings, New Orleans, Louisiana. January 2011.
"Tautness: Stretched Between Half-Factoriality and Bounded Factorization" Karl-Franzens Universität Graz, Graz, Austria. December 2010.
"Lengths of Factorizations in Numerical Semigroup Rings" Third International Meeting on Integer Valued Polynomials and Problems in Commutative Algebra, CIRM, Luminy, France. November 2010.
"Factorization in Numerical Semigroup Rings" University of Stockholm. September 2010.
"Elimination of Imaginaries for the p-adics" Model Theory of Fields, AMS Mathematics Research Community, Snowbird Resort, Utah. June 2010.
"Factorization in Numerical Semigroup Rings" Combinatorics and Additive Number Theory (CANT), City University of New York, New York City, NY. May 2010.
"Factorization in Semigroup Rings" Second Annual Iberian Meeting on Numerical Semigroups, Granada, Spain. February 2010.
"Model Theory of Stable Groups" AMS Session on Algebras, AMS/MAA Joint Meetings, San Francisco, California. January 2010.
"Equivalence Relations and Factorization" Karl-Franzens Universität Graz, Graz, Austria. November 2009.
"Stable, Countably Categorical Algebraic Structures" Institut Camille Jordan, Université Lyon 1, Lyon, France. October 2009.
"Stable Aleph0-categorical Algebraic Structures" Special Session on Model Theory and Its Applications, AMS Meeting Fall Eastern Section, Wesleyan University, Middletown, CT. October 2008.
"Stable, Aleph0--categorical Algebraic Structures" Fourth MATHLOGAPS training workshop, University of Manchester, England, July 2008.
"Stable Aleph0-Categorical Algebraic Structures" Model Theory Working Seminar, McMaster University, Hamilton, Canada, October 2007.
"A Case Study in Factorization Theory: Arithmetic Congruence Monoids" The Art of Factorization in Multiplicative Structures (MAA PREP Workshop), Trinity University, San Antonio, TX, May 2007.
"Monoids and Combinatorics I, II, and III" The Art of Factorization in Multiplicative Structures (MAA PREP Workshop), Trinity University, San Antonio, TX, May 2007.
"Elasticity in Polynomial Rings and Block Monoids" Karl-Franzens Universität Graz, Graz, Austria. December 2006.
"Elasticity in Polynomial and Power Series Rings" AMS/MAA Joint Meetings; San Antonio, TX, January 2006.