My research interests include arithmetic dynamics, arithmetic statistics, geometry of configuration spaces, Galois theory, and algebraic combinatorics.
Publications
- The arithmetic of semirings part I: ideals
with Jay Chen, Dorien Laurens, Jasper Piermarini, and Harriet Simons,
preprint (2026). - Rational orbits under correspondences,
preprint (submitted) (2026),
arXiv preprint 2605.04391. - Profinite iterated monodromy groups of unicritical polynomials,
with Ophelia Adams,
to appear in Algebra and Number Theory (2026).
arXiv preprint 2504.13028. - Polynomials with many rational preperiodic points,
with John R. Doyle,
Transactions of the AMS, Vol. 378, No. 08 (2025), 5981-6011.
arXiv preprint 2201.11707. - Cyclotomic factors of necklace polynomials,
Acta Arithmetica, Vol. 204 (2022), 287-316. - Dynatomic polynomials, necklace operators, and universal relations for dynamical units,
with John R. Doyle and Paul Fili,
New York Journal of Mathematics, Vol. 28 (2022), 534-556. - Dynamical moduli spaces and polynomial endomorphisms of configurations,
with Talia Blum, John R. Doyle, Colby Kelln, Henry Talbott, and Max Weinreich,
Arnold Mathematical Journal, Vol. 8 (2022), 285-317, DOI: 10.1007/s40598-022-00197-z - Euler characteristic of the space of real multivariate irreducible polynomials,
Proceedings of the AMS, Vol. 150 (2022), 2331-2343. - Density of periodic points for Lattès maps over finite fields,
with Zoë Bell, Jasmine Camero, Karina Cho, Chieh-Mi Lu, Rebecca Miller, Bianca Thompson, and Eric Zhu,
Journal of Number Theory, Vol. 238 (2022), 951-966, DOI:10.1016/j.jnt.2021.10.009 - Polynomial factorization statistics and point configurations in R^3,
International Mathematical Research Notices (2018), DOI: 10.1093/imrn/rny271. - Liminal reciprocity and factorization statistics,
Algebraic Combinatorics, Vol. 2, No. 4 (2019), 521-539. - Factorization statistics and the twisted Grothendieck-Lefschetz formula,
extended abstract, Proceedings of 30th Conference on Formal Power Series and Algebraic Combinatorics (2018). - Polynomial splitting measures and cohomology of the pure braid group,
with Jeffrey C. Lagarias.
Arnold Mathematical Journal, Vol. 3, No. 2 (2017), 219-249, DOI: 10.1007/s40598-017-0064-z. - Gauss’ hidden menagerie: from cyclotomy to supercharacters,
with Stephan Ramon Garcia and Bob Lutz.
Notices Amer. Math. Soc., Vol. 62, No. 8 (2015), 878-888. - Galois theory of the lemniscate,
with David A. Cox,
The Journal of Number Theory, Vol. 135 (2014), 43-59. - Small dynamical heights for quadratic polynomials and rational functions,
with Rob Benedetto, Ruqian Chen, Yordanka Kovacheva, and Colin White.
Experimental Mathematics, Vol. 23 (2014), 433-447. - A Wallis product on clovers,
The American Mathematical Monthly, Vol. 121, No. 3 (2014), 237-243. - Pre-images in quadratic dynamical systems,
with Ben Hutz and Ben Krause.
Involve, Vol. 4, No. 4 (2011), 343-363. - On the number of rational iterated pre-images of -1 under quadratic dynamical systems,
American Journal of Undergraduate Research, Vol. 9, No. 1 (2010), 19-26.
Doctoral Thesis
- Polynomial statistics, necklace polynomials, and the arithmetic dynamical Mordell-Lang conjecture
(Complete thesis.) - Chapters 5 and 6 on the arithmetic dynamical Mordell-Lang conjecture
(Joint work with Mike Zieve.) - Slides from the defense.
Awards and Grants
- In 2022 I received the Glen and Claire Swogger Award for Exemplary Classroom Teaching from the University of Chicago.
- In 2020 I received the NSF Mathematical Sciences Postdoctoral Research Fellowship.
- In 2019 I received the Wirt and Mary Cornwell Prize in mathematics from the University of Michigan.
- In 2018 I received the Rackham Predoctoral Fellowship from the University of Michigan.
Invited Talks
- Academia Sinica, Taiwan
Algebra/Number Theory Seminar, May 2026 - National Central University, Taiwan
Algebra/Number Theory Seminar, May 2026 - Boston College
AMS Sectional Meeting, March 2026 - Amherst College
Colloquium, March 2026 - AG@PUI Seminar
March 2026 - American Institute of Mathematics
Dynamics of Multiple Maps Workshop, November 2025 - University of Rochester
Algebra/Number Theory Seminar, March 2025 - Florida State University
Algebra Seminar, February 2025 - Joint Math Meetings
Arithmetic Dynamics Special Session, January 2025 - Penn State University
Algebra Seminar, November 2024 - Oklahoma State University
Number Theory Seminar, April 2023 - Oregon State University
Number Theory Seminar, April 2023 - Vassar College
Colloquium, February 2023 - Baruch College
Colloquium, February 2023 - Joint Math Meetings
Arithmetic Dynamics Special Session, January 2023 - AMS Sectional Meeting
Arithmetic Dynamics Special Session, October 2022 - Arithmetic Dynamics International Online Seminar (ADIOS)
February 2022 - Harvard University
Algebraic Dynamics Seminar, February 2022 - Oklahoma State University
Number Theory Seminar, March 2021
Video - University of Washington
Algebraic Geometry Seminar, February 2021 - University of Connecticut
Math Club, October 2020 - Joint Math Meetings
Arithmetic Dynamics Special Session, January 2020 - CUNY Graduate Center
Combinatorial and Additive Number Theory Workshop, May 2019 - University of Chicago
Midwest Representation Stability Research Meeting, April 2019 - AMS Western Sectional Meeting
Emerging Connections in Number Theory, March 2019 - University of Michigan
Graduate Recruitment Symposium, March 2019 - University of Michigan
Complex Analysis, Dynamics, and Geometry Seminar, March 2019 - University of Michigan
Combinatorics Seminar, February 2019 - Wayne State University
Algebra Seminar, January 2019 - University of California, Berkeley
Combinatorics Seminar, November 2018 - University of Chicago
Geometry/Topology Seminar, November 2018 - Dartmouth College
Formal Power Series and Algebraic Combinatorics Conference, July 2018 - CUNY Graduate Center
Combinatorial and Additive Number Theory Workshop, May 2018 - AMS Eastern Sectional Meeting
Arithmetic Dynamics Special Session, April 2018 - Joint Math Meetings
Arithmetic Dynamics Special Session, January 2018 - Fields Institute
Workshop on Interactions between Model Theory and Arithmetic Dynamics, July 2016 - Amherst College
Colloquium, April 2013 - University of Connecticut
Pi Day Colloquium, March 2013