Associate Head and Associate Professor
My Contact Info
Office: MSCS 532
Office hours: Office hours for each of my courses is listed on the Canvas page.
E-mail: paul.fili (at) okstate.edu
In the Spring of 2023, I am teaching Math 4753/5753: Introduction to Cryptography. Course pages are available on Canvas.
My research interests are in number theory and analysis. I am primarily interested in topics relating to the distribution of algebraic numbers and points of small height, potential theory, and arithmetic dynamics. I am interested in Bogomolov and Lehmer type problems for heights, unlikely intersections, and equidistribution for dynamical systems. Lately I've become interested in stochastic dynamics.
- Dynatomic polynomials, necklace operators, and universal relations for dynamical units (with J.R. Doyle and T. Hyde), New York J. Math. 28 (2022), 534–556.
- Wandering points for the Mahler measure (with L. Pottmeyer and M. Zhang), Acta Arith. 204 (2022), no. 3, 225-252.
- On the behavior of Mahler's measure under iteration (with L. Pottmeyer and M. Zhang), Monatshefte für Mathematik 193 (2020), 61–86.
- Quantitative height bounds under
splitting conditions (with L. Pottmeyer), Trans. of the Amer.
Math. Society, 372 (2019), no. 7, 4605-4626.
- Energy integrals and small points for the Arakelov
height (with C. Petsche and I. Pritsker), Archiv der Mathematik, 109
(2017), no. 5, 441-454.
- Height bounds for algebraic numbers
satisfying splitting conditions (with I. Pritsker), J. Number Theory,
175 (2017), 250-264.
- Equidistribution and the heights of totally real and totally p-adic numbers (with Z. Miner), Acta Arith., vol. 170, no. 1 (2015), 15-25.
- Energy integrals over local fields and global height bounds (with C. Petsche), Int. Math. Res. Not. 2015(5): 1278-1294, 2015.
- On the heights of totally p-adic numbers, J. Théor. Nombres Bordeaux 26(1): 103-109, 2014.
- A generalization of Dirichlet's unit theorem (with Z. Miner), Acta Arith., 162(4): 355-368, 2014.
- Norms extremal with respect to the Mahler measure (with Z. Miner), J. Number Theory, 132(1): 275-300, 2012.
- Orthogonal decomposition of the space of algebraic numbers and Lehmer's problem (with Z. Miner), J. Number Theory, 133(11), 3941-3981, 2013.
On the non-Archimedean metric Mahler measure (with C.L. Samuels), J. Number Theory, 129: 1698--1708, 2009.
Here is my doctoral thesis, Orthogonal decomposition of the space of algebraic numbers modulo torsion from the University of Texas at Austin (May 2010).
I currently have one Ph.D. student, Preston Kelley.
My Curriculum Vitae (updated Fall 2022).
I received my bachelor's degree in Mathematics and Physics with a language citation in Classical Greek from Harvard University in 2004. I wrote my senior honors thesis on Lenstra's Elliptic Curve method of factorization under the direction of Frank
Calegari. I received my Ph.D in Mathematics in May 2010 at the University of Texas at Austin studying under the supervision of Jeffrey Vaaler.
Last updated Feb. 1, 2023.