Degrees

Ph.D., University of California, Berkeley (2010)
B.S., The University of Arizona (2003)

Post Doctoral Fellow
University of Wisconsin-Madison
Van Vleck Assistant Professor and RTG postdoctoral fellow (2010)

Research Interests

I am broadly interested in number theory, arithmetic geometry, and algebraic geometry, with some side interests in combinatorics and commutative algebra, and with a focused interest on the arithmetic of varieties, stacks, arithmetic statistics, modular curves, and Galois representations. I am especially interested in making connections between different subareas of mathematics.

These are ancient areas of inquiry that began with very basic questions about solving polynomial equations and were motivated by concrete applications; for example, the Greek astronomer Apollonius of Perga (240-190BC) developed his theory of conics and ellipses to facilitate the study of astronomy. Questions about numbers and shapes still remain central to the frontier of mathematical research, and my research has a particular emphasis on using modern technical tools to study classical problems.

Teaching Interests

Mathematical study is a powerful means of communing with the intellectual heritage of our predecessors, bridging the gap between generations and connecting us to the rich tapestry of mathematical discovery throughout history. Mathematics also teaches abstraction, precision, and structured thinking, and is a great centerpiece for a liberal arts education.

Mathematics instruction is often egalitarian, in that faculty typically teach a wide range of courses at various levels and across the curriculum. I especially enjoy teaching "pedagogy" courses focused on foundations and proof-writing, and emphasize problem-solving, intuition, and interesting cross-topic connections in my courses.

Awards and Honors

(2016-2022) NSF Career (DMS-1555048)
(2011-2013) NSA Young Investigator Grant

Selected Publications

Angle Ranks of Abelian Varieties, with Taylor Dupuy and Kiran S. Kedlaya, Mathematische Annalen, 2023

Heights on stacks and a generalized Batyrev--Manin--Malle conjecture, with Jordan S. Ellenberg and Matthew
Satriano, Forum of Mathematics, Sigma, Volume 11, 2023

ell-adic images of Galois for elliptic curves over Q, with Jeremy Rouse and Andrew Sutherland and an appendix with John Voight; Forum of Mathematics, Sigma, Volume 10, 2022

Sporadic Cubic Torsion, with Maarten Derickx, Anastassia Etropolski, Mark van Hoeij, and Jackson S. Morrow; Algebra & Number Theory 15-7 (2021), 1837--1864

The canonical ring of a stacky curve, with John Voight, Mem. Amer. Math. Soc. 277 (2022), no. 1362

Uniform bounds for the number of rational points on curves of small Mordell--Weil rank, with Eric Katz and Joe Rabinoff; Duke Math J. 165 (2016), no. 16, 3189--3240

Personal Website

https://dmzb.github.io/