Paul Fili

Associate Head and Associate Professor

Department of Mathematics

My Contact Info

Office: MSCS 402A (administrative), MSCS 532
Office hours: Office hours for each of my courses is listed on the Canvas page.
E-mail: paul.fili (at) okstate.edu

Teaching

In the Fall of 2024, I am teaching Math 5143: Real Analysis and Math 2890: Uncomputability in Mathematics (Honors). Course pages are available on Canvas.

Research

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.

Preprints

Stochastic Equidistribution and Generalized Adelic Measures (with J. Doyle and B. Tobin), preprint. A revision is coming soon! You can find pictures of Stochastic Julia sets and more here and slides from a recent talk here.
A metric of mutual energy and unlikely intersections for dynamical systems, preprint. Revision in progress.

Publications

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.

Unpublished notes

On totally real numbers and equidistribution (with Z. Miner).

Here is my doctoral thesis, Orthogonal decomposition of the space of algebraic numbers modulo torsion from the University of Texas at Austin (May 2010).

Graduate students

I currently have one Ph.D. student, Preston Kelley.

About me

My Curriculum Vitae (updated Fall 2024).

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 Aug. 27, 2024.