Ph.D., Brown University (1998)
Sc.M., Brown University (1996)
A.B., Harvard University (1993)
I am a pure mathematician working in the field of arithmetic dynamics, meaning that I study problems at the interface between number theory (the study of rational numbers, integers, and the primes) and dynamical systems (chaos theory, loosely speaking). One of the central problems in my field is to understand, given a polynomial or rational function f(x) with rational numbers as coefficients, the set Per(f) of rational numbers that happen to be periodic under the iteration (i.e., repeated application) of f(x). It is known that if f(x) has degree at least two, then the set Per(f) is always finite; but otherwise, very little is known about how big this finite set can be, let alone what sorts of dynamical or number-theoretic properties it can have. The questions that arise from this setup are notoriously difficult to solve. To attack them, I spend most of my time working with more technical machinery: p-adic numbers and p-adic dynamics, Julia sets and Fatou sets, Berkovich spaces, arithmetic height functions, and arboreal Galois representations.
I enjoy teaching at all levels of the undergraduate mathematics curriculum, from introductory calculus through honors topics like measure theory and basic algebraic topology. My courses feature a heavy dose of rigor, especially at the upper levels, to get my students thinking like mathematicians. In other words, I want my students not only to know how to design a logical argument over a grand scale to prove a theorem or analyze a complicated sum or integral, but also to be fluent enough with the fundamentals to check all the nuts and bolts along the way. As a pure mathematician, I aim to instill in my students an appreciation for the intrinsic beauty of the field. I want them to question why various mathematical facts are true, rather than simply taking them for granted as notes to be memorized on flash cards. My favorite part is seeing the involuntary grin or occasional dropped jaw when a student is enlightened by an especially elegant mathematical gem.
NSF Mathematical Sciences Grant, 2015-2018, extended to 2019.
NSF Mathematical Sciences Grant, 2012-2015
NSF Mathematical Sciences Grant, 2009-2012
NSF Mathematical Sciences Grant, 2006-2009
NSA Young Investigator's Grant, 2004-2006
NSF Mathematical Sciences Research Postdoctoral Fellowship (Boston University), 2000-2002
Organizing Committee Member for the ``Ideals, Varieties, Applications'' Conference in honor of David Cox, at Amherst College, June 10-14, 2019.
Organizing Committee Member for the ``Silvermania'' Arithmetic Conference in honor of Joseph H. Silverman, at Brown University, August 11-15, 2015.
Co-organizer of the Five-College Number Theory Seminar, 2003-present.