Alison Miller

Alison Beth Miller

abm (at) ams (dot) org
alimil (at) umich (dot) edu

I am a mathematician and an Associate Editor at Mathematical Reviews.

I am interested in number theory, broadly speaking, especially when there is interesting algebraic or geometric structure involved. I currently work primarily in arithmetic invariant theory, arithmetic statistics, and arithmetic dynamics. I have an ongoing interest in exploring connections with topology and knot theory, specifically related to the Alexander module and Blanchfield pairing.

I completed my PhD at Princeton in 2014 under Manjul Bhargava. From 2014-2019 I was an NSF and Benjamin Peirce postdoctoral fellow at Harvard University.

Publications and preprints

Seifert surfaces in the four-ball and composition of binary quadratic forms, with Menny Aka, Peter Feller, and Andreas Wieser (arxiv preprint)

Dynamical Mahler Measure: A survey and some recent results, with Annie Carter, Matilde Lalín, and Michelle Manes (arxiv preprint), Proceedings of WIN5 (Women in Numbers 5), to appear

Two-variable polynomials with dynamical Mahler measure zero, with Annie Carter, Matilde Lalín, Michelle Manes, and Lucia Mocz, Research in Number Theory 8, 25 (2022) (arxiv preprint).

On the number of binary quadratic forms having discriminant 1−4p, p prime, with Stanley Yao Xiao (arxiv preprint)

Asymptotics for the number of Simple (4a+1)-Knots of Genus 1, International Mathematics Research Notices Volume 2022, Issue 9, May 2022, Pages 6742–6769. (arxiv preprint)

PhD Thesis: Counting Simple Knots via Arithmetic Invariants.

See here for my older papers, mostly dating from my undergraduate years.


Arithmetic statistics for knot invariants: PDF slides from my talk at the Boston University/Keio University 2019 Workshop on Number Theory.

Old course websites and notes

Notes may contain typos: please contact me with any questions or corrections.

Math 223a: Local Class Field Theory notes (pdf), 2018 website, 2017 website
Math 223b: Global Class Field Theory notes (pdf), 2019 website, 2018 website
Math 221: Graduate Algebra notes (pdf)
Math 222: Lie Groups and Lie Algebras notes (pdf)

People who might be confused with me

Allison N. Miller is a low-dimensional topologist and professor at Swarthmore College.

Alison L. Miller is a developmental psychologist at the University of Michigan.