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#### Research interests:

My thesis is on maximal subgroup growth, but I am interested in subgroup growth more generally (and also probabilistic group theory and profinite groups). Here is a preliminary version of a paper:

Maximal subgroup growth of abelian by cyclic groups. Note: The November 1 version of this paper should have stated at the beginning of section 2.2 that the rings are commutative (and unital).

In my spare time, I study Computational Complexity and hope to do research in this area too.

Colleague: Rachel Skipper

##### What is subgroup growth?

One way to try to understand some infinite groups is to study their subgroups of finite index. Given a finitely generated group $G$, we have a function $s_G(n): \mathbb{N} \to \mathbb{N}$, where $s_G(n)$ is the number of subgroups of $G$ of index at most $n$. How fast does this function grow? And how does this growth rate relate to structural properties of $G$? Sometimes, we can say quite a bit.

Subgroup growth also includes similar functions. For example, maximal subgroup growth is about counting the number maximal subgroups of $G$ of index $n$ (or at most $n$). In fact, the prime number theorem gives an asymptotic formula for the number of maximal subgroups of $\mathbb{Z}$ of index at most $n$.

One place to learn more is the book Subgroup Growth by Lubotzky and Segal.

##### What is a profinite group?

From calculus, we are familiar with taking limits of sequences of numbers, but how can we take limits of sequences of algebraic objects such as sets, groups or rings?

Let $G_1, G_2, \ldots$ be a sequence of groups. To take the limit, we need to somehow relate these groups to each other; we want the groups to be successive “approximations” of the limit object.

One possibility is that each group be a subgroup of the one that follows it; we then get the direct limit. The projective limit however, is different:

Let each group be a quotient of the next one. Then the limit is called the projective limit. If the groups in the sequence are finite, then the limit object is a profinite group (being the projective limit of finite groups). An example of this is approximating the integers by a sequence of cyclic groups: let $G_k =$ the cycle group of order $2^k$. Then addition mod $2^k$ does in fact approximate addition in $\mathbb{Z}$; in fact, this is how computers do arithmetic (the 2's complement method). And for this example, the limit object is the 2-adic integers.

Some aspects of profinite group theory, by Segal, is an excellent survey.

##### Talks given:
• Maximal subgroup growth of some groups*, Atlanta Georgia, January 2017
• (*A longer version of this talk was also given at CSU Fort Collins in December 2016.)
• Subgroup growth: a brief survey, UCCS (alma mater), December 2016
• Barriers to proving $P \neq NP$, November 2016
• Maximal subgroup growth: current progress and open questions, November 2016
• The maximal subgroup growth of some metabelian groups, March 2016
• Exact counting of maximal subgroups, March 2016
• Counting subgroups according to their index, May 2015
• Polynomial subgroup growth: the pro-p case and more, May 2015
• Counting subgroups via split extensions, April 2015
• Profinite groups: Random generation and maximal subgroup growth, December 2014
• Introduction to Profinite Groups and p-adic Numbers, December 2014
• Random Groups, (20 min) Ohio State-Denison Conference, May 2014
• Random Groups, (1 hour) April 2014
• The Banach-Tarski Paradox, March 2013

(Most of the above talks were given at Binghamton University. Some were given in the algebra seminar, and four were given in the graduate student seminar. The ones given in May of 2015 were the two parts of my admission to candidacy exam.)

##### Conferences attended:
• 2017: Joint Math Meetings - Atlanta, Georgia
• 2016: A Celebration of Mathematics and Computer Science - in honor of Avi Wigderson's 60th Birthday - IAS
• 2015: Geometric and Probabilistic Methods in Group Theory and Dynamical Systems, Texas A&M
• 2015: Combinatorics and Computer Algebra, Fort Collins, CO
• 2014: Ohio State-Denison Mathematics Conference
• 2014: Young Geometric Group Theory Conference in Luminy, France
• 2014: Geometric Group Theory introductory school in Luminy, France
• 2012: Binghamton University Graduate Conference in Algebra and Topology
• 2012: Cornell Probability Summer School (a non-registered participant)
• 2011: Binghamton University Graduate Conference in Algebra and Topology