### Research Interests

- Geometric and combinatorial group theory, especially problems related to Grigorchuk's group, iterated monodromy groups of complex dynamical systems, and other self-similar groups.
- The scholarship of teaching and learning mathematics at the undergraduate level, especially problems related to higher-order thinking skills, knowledge retention, and motivation.

### Research Statement

I find that mathematics expresses itself most beautifully in the interplay between different branches of the field. By this, I mean that I find most aesthetically pleasing any arguments that leverage multiple interpretations of particular mathematical objects to prove deep conclusions. To me, high school students receive their best taste of pure mathematics when they employ both algebraic and geometric properties to solve problems (with linear functions, for example). As a geometric group theorist, I study groups by considering their actions on nice geometric spaces; this approach gives me two (or sometimes more) ways of learning about the group, and they usually complement each other well. ... Full document

If you are looking for a more specific example of what I do (with an interactive JavaScript!), you may find this interesting.

### Current Projects

- Topological mating is a procedure by which two topological polynomials are used to produce a Thurston map. Sometimes, mating different pairs of polynomials will result in equivalent Thurston maps. The reason for this phenomenon remains unclear. I am investigating this problem by studying the associated iterated monodromy groups and the inclusions induced by the mating.
- In the Fall 2010 and Spring 2011 semesters, I will investigate the effectiveness of using wikis to facilitate class projects. I will consider not only the effect wiki technology has on students' mastery of mathematical concepts, but also its effect on affective learning goals, such as students' interest in mathematics and views on mathematics' importance.

### Papers

- "Mega-bimodules of topological polynomials:
Sub-hyperbolicity and Thurston obstructions"

My Thesis

Abstract: In 2006, Bartholdi and Nekrashevych solved a decade-old problem in holomorphic dynamics by creatively applying the theory of self-similar groups. Nekrashevych expanded this work in 2009 to define what we refer to as mega-bimodules which capture the topological data of Hurwitz classes of topological polynomials. He also showed that proving that these mega-bimodules are sub-hyperbolic will have two important implications: that all iterated monodromy groups of topological polynomials are contracting and that the Hubbard-Schleicher spider algorithm for complex polynomials generalizes to topological polynomials. We prove sub-hyperbolicity in the simplest non-trivial case and apply these mega-bimodules to holomorphic dynamics to prove a partial converse to the Berstein-Levy Theorem proved in 1985. - "Mapping
schemes realizable by obstructed topological polynomials"

Preprint

Abstract: In 1985, Levy used a theorem of Berstein to prove that all hyperbolic topological polynomials are equivalent to complex polynomials. We prove a partial converse to the Berstein-Levy Theorem: given post-critical dynamics that are, in a sense, strongly non-hyperbolic, we prove the existence of topological polynomials realizing these post-critical dynamics whiich are not equivalent to any complex polynomial. This proof employs the theory of self-similar groups to demonstrate that a topological polynomial admits an obstruction, and produces a wealth of examples of obstructed topological polynomials. - "Group work and self-efficacy in a business calculus class"

Preprint

Abstract: In a business calculus course with lecture and discussion sections, TAs ran half the discussion sections in the standard format and half in a group work format. At the beginning and end of the semester, students completed brief surveys indicating their confidence in their ability to perform mathematical procedures and their ability to learn new mathematics. The changes in the reported learning self-efficacy of the students in the group work sections correlated strongly with their final course grades, unlike in the standard sections. This suggests that while group work may not increase students learning self-efficacy, it might improve the accuracy with which students perceive their own abilities. - "Wordlength in Alternative
Finite Presentations of Thompsons' Group F"

Manuscript

This was my honors project in mathematics at Bowdoin College, done under the supervision of Jen Taback. I prove that the wordlength of an element of Thompson's group F with respect to the standard finite generating set (which we can compute by a result of Fordham) provides a linear bound on the wordlength of that element with respect to alternative two element generating sets. I then use this result to show that most dead end words with respect to the standard generating set are not dead end words with respect to the "next" natural generating set. - "Muddy Children, Temporary
Deafness, and the 10,000th Digit of the Square Root of 2: Modeling
Knowledge"

Manuscript

This was my honors project in economics at Bowdoin College, done under the supervision of Dorothea Herreiner. I review some concepts from model theory and logic and discuss how they can be applied to easing the troublesome assumption of complete knowledge that crops up so frequently in game theory. - "Mapping Schemes Realizable by Obstructed Topological Polynomials"

AMS Session on Dynamical Systems, and Topics in Analysis

2011 Joint Mathematical Meetings (January 2011)

Slides - "Group Work and Self-Efficacy in a Business Calculus Class"

MAA Session on the Scholarship of Teaching and Learning in Collegiate Mathematics

2011 Joint Mathematical Meetings (January 2011)

Slides - "An Introduction to Iterated Monodromy Groups of Complex Rational Maps"

Graduate Geometry and Topology Seminar

UIUC Department of Mathematics (October 2010)

Slides - "Using Self-Similar Groups to Find Thurston Obstructions and Determine
Thurston Equivalence"

Workshop on Holomorphic Dynamics around Thurston's Theorem

Roskilde University (October 2010)

Slides - "Using Self-Similar Groups to
Find Thurston Obstructions"

Contributed Talk

Spring Topology and Dynamics Conference (March 2010)

Slides - "Using Self-Similar Groups to
Find Thurston Obstructions"

Group Theory Seminar

UIUC Department of Mathematics (February 2010)

Slides - "Obstucted Topological
Polynomials Realizing Particular Ramification Graphs"

Groups and Dynamics Seminar

Texas A&M Department of Mathematics (October 2009) - "Combinatorics of Polynomial
Iterations"

Group Theory Seminar

UIUC Department of Mathematics (March 2009) - "Amenable Actions of
Nonamenable Groups"

Group Theory Seminar

UIUC Department of Mathematics (September 2007) - "A Virtually Free Automaton
Group"

Group Theory Seminar

UIUC Department of Mathematics (May 2007) - Hudson River Valley Undergraduate Mathematics Conference: "Wordlength in Alternative Finite Presentations of Thompson's Group F" (April 2005)

### Talks and Presentations

### Conferences

2011 Joint Mathematical Meetings (January 2011)

Workshop on Holomorphic Dynamics around Thurston's Theorem (September 2010)

Spring Topology and Dynamics Conference (March 2010)

Conference on Research in Undergraduate Mathematics Education (February 2010)

Geometry and Topology Conference at Munster (June 2009)

AMS Central Sectional Meeting (March 2009)

Examples of Groups Summer Program (Summer 2008)

Graduate Student Topology Conference (March 2008)

University of Nebraska's IMMERSE Program (Summer 2005)

Hudson River Valley Undergraduate Mathematics Conference (April 2005)

MathFest 2004 (August 2004)