Research

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The arithmetic of thin orbits and Kleinian groups

Most recently, I’ve been interested in reciprocity obstructions in thin orbits.  In analogy to the study of rational or integer points on varieties, one might consider the arithmetic of a group or semigroup orbit.  If the group is an algebraic group, we are studying points on a variety.  If we choose a thin subgroup of an algebraic group, we obtain a thin orbit, and we don’t have access to the same tools.  Some well-known classical problems can be rephrased in this way.  For example, Zaremba conjectured that every natural number appears as the denominator of some rational number with continued fraction expansion involving only the coefficients 1,2,3,4,5.  This can be rephrased as a statement about a thin orbit of a semigroup lying inside SL(2,Z).  The study of the curvatures of Apollonian circle packings is very analogous (in fact, work of Asmus Schmidt shows how close this analogy is; Apollonian packings arose in his work on complex continued fractions), and can be phrased as the study of the orbit of the Apollonian group.  For two decades, it was conjectured that the Apollonian orbit had a local-to-global property, meaning that besides some local (congruence) obstructions, all sufficiently large integers should appear.  In 2023, as part of an REU/G project, we (Haag, Kertzer, Rickards and myself) found counterexamples:  there are certain quadratic and quartic infinite families of disallowed curvatures that arise from quadratic and quartic reciprocity.  We have been referring to these as reciprocity obstructions.

My interest was originally motivated by a connection between Apollonian circle packings and the structure of imaginary quadratic fields (in turn coming out of a connection to abelian sandpiles!). A Schmidt arrangement (named for Asmus Schmidt) is the orbit of the extended real line in the extended complex plane under the Mobius action of the Bianchi group PSL2(OK) (here, OK is the ring of integers of the imaginary quadratic field). The resulting fractal of nested circles is an analogue to the Farey subdivision of the real line into nested intervals. It has interesting connections to the arithmetic of the field: for example, it is connected if and only if OK is Euclidean. It turns out the Schmidt arrangement of the Gaussian integers is a union of all primitive integral Apollonian circle packings. In other fields, there are other new packings — and associated thin groups — that govern the picture. In this age of computer visualization, there are still new things to discover about topics as classical as imaginary quadratic fields, quadratic forms, hermitian forms, etc.

Students

Postdocs

  • James Rickards

Collaborators

Summer/Lab Projects

The arithmetic of Abelian varieties

The Fibonacci numbers are governed by a twisted multiplicative group — for example, primes p appear in the (p2-1)-th term because this is the cardinality of the multiplicative group of Fp2. If you replace the multiplicative group with an elliptic curve, you obtain a recurrence sequence called an elliptic divisibility sequence (this is because the division polynomials satisfy a recurrence coming from the group law). Much of my interest in arithmetic geometry is connected to algebraic divisibility sequences such as these. The sequences carry all sorts of arithmetic information: for example, using a multi-dimensional generalization of elliptic divisibility sequences called elliptic nets, you can compute the Tate-Lichtenbaum pairing as a quotient of certain terms. 

Cryptography and algorithmic/computational number theory

I learned from my mentor Kristin Lauter that cryptographic applications can give rise to interesting and totally new number theory problems. I’ve worked on elliptic curve cryptography and lattice-based cryptography, the latter of which provides the promise of secure post-quantum crypto (quantum computers will be able to solve the elliptic curve discrete logarithm problem). Lately my focus has been on isogeny-based cryptography.  My interest has often been in the security of hard problems.  I find it fascinating to see the interplay between theory and practice, and to work across disciplines.

I am always interested in number theory algorithms of all kinds, and I enjoy computer experimentation.

Miscellaneous number theory

The core of all three topics above is, of course, algebraic number theory. I’m interested in quadratic forms, Hermitian forms, explicit class field theory and complex multiplication, continued fractions and elementary number theory, as well as connections to dynamics.

Students

  • Amy Feaver (Ph.D. 2014)
  • Ryan Ibarra (Honors thesis 2019)
  • Annie S. Chen (Boulder High School)

Postdocs

Collaborators

Edmund Harriss and Steve Trettel. Ryan Ibarra, Henry Lembeck, Mohammad Ozaslan, and Hanson Smith. Annie S. Chen, T. Alden Gassert

Summer/Lab Projects

  • 2018: Ryan Ibarra, Henry Lembeck, Mohammad Ozaslan