My research interests are centered around effective methods in arithmetic geometry, arithmetic statistics, and number theory.
Ongoing Projects
 Lfunctions and Modular Forms Database (LMFDB)
 A census of zeta functions of quartic K3 surfaces over $\mathbf{F}_3$ (with David Harvey)
 Effective computations of Bernoulli numbers (with David Harvey)
 Counting points on singular plane curves (with David Harvey)
 Computation of zeta functions of nondegenerate projective/toric hypersurfaces via controlled reduction in $p$adic cohomology (with David Harvey and Kiran Kedlaya) see also the following associated software packages:
Publications

Computing Zeta Functions of Cyclic Covers in Large Characteristic
(with
Vishal Arul,
Alex Best,
Richard Magner,
and
Nicholas Triantafillou
)
presented at ANTS (Algorithmic Number Theory Symposium)
arXiv:1806.02262, SageMath implementation 
Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$adic cohomology
(with David Harvey and Kiran Kedlaya)
presented at ANTS (Algorithmic Number Theory Symposium)
arXiv:1806.00368 
Rigorous computation of the endomorphism ring of a Jacobian
(with
Nicolas Mascot,
Jeroen Sijsling, and
John Voight
)
to appear in Mathematics of Computation.
arXiv:1705.09248, SageMath/Magma implementation 
On the arithmetic of a family of degreetwo K3 surfaces
(with
Florian Bouyer,
Dino Festi,
Chris Nicholls, and
Mckenzie West
)
Mathematical Proceedings (2018).
DOI, arXiv:1703.02127 
On the distribution of the Picard ranks of the reductions of a
K3 surface
(with
AndreasStephan Elsenhans and
Jörg Jahnel
)
submitted
arXiv:1610.07823. 
Traces, High powers and One level density for families of curves over finite fields
(with
Alina Bucur,
Chantal David,
João Guerreiro,
David LowryDuda
)
Mathematical Proceedings 162(2) (2018).
DOI, arXiv:1610.00164. 
Effective computations of HasseWeil zeta functions
Ph.D. thesis pdf 
Variation of NéronSeveri ranks of reductions of K3 surfaces (with Yuri Tschinkel)
Experimental Mathematics 23 (2014), 475481.
DOI, arXiv:1405.2265, raw data. 
Asymptotic expansions, $L$values and a new Quantum Modular Form (with Korneel Debaene and João Guerreiro)
The Ramanujan Journal 35 (2014), 141148.
DOI, arXiv:1311.3469. 
A search for Wilson primes (with Robert Gerbicz
and David Harvey)
Mathematics of Computation 82 (2014), 30713091.
DOI, arXiv:1209.3436. 
Faster deterministic integer factorization (with David Harvey)
Mathematics of Computation 83 (2014), 339345.
DOI, arXiv:1201.2116.
Software
 Lfunctions and Modular Forms Database (LMFDB)
 Controlled reduction: Computation of zeta functions of projective hypersurfaces (C++) and its SageMath wrapper
 Endomorphisms: Rigorous computation of the endomorphism ring of a Jacobian (SageMath/Magma)
Talks

Frobenius Distributions
(slides)
Department of Mathematics, University of Washington, April 2019. 
Upper bounds for the endomorphism algebra of an abelian variety
(slides)
JMM, AMS Special Session on Number Theory, Arithmetic Geometry, and Computation, January 2019 
Randomness in number theory
(slides)
Department of Mathematics, Colorado Sate University, November 2018 
Frobenius Distributions
(slides)
Colóquio de Geometria e Aritmética Rio de Janeiro, October 2018. 
Frobenius Distributions
(slides)
Department of Mathematics, Massachusetts Institute of Technology, September 2018. 
Computing zeta functions of nondegenerate hypersurfaces in toric varieties
(slides)
Algorithmic Number Theory Symposium XIII, University of Wisconsin, Madison, July 2018. 
Distributions in arithmetic geometry
(slides)
Communicating Mathematics Effectively, University of Washington, June 2018. 
Computing zeta functions of nondegenerate hypersurfaces in toric varieties
(slides)
Birational Geometry and Arithmetic, ICERM, Brown University, May 2018. 
Computing zeta functions of nondegenerate toric hypersurfaces
(slides)
Department of Mathematics & Statistics, Boston University, December 2017. 
Computing zeta functions of nondegenerate toric hypersurfaces
(slides)
MaineQuébec Number Theory Conference, October 2017 
Computing zeta functions of nondegenerate toric hypersurfaces
(slides, video)
Banff International Research Station, $p$adic Cohomology and Arithmetic Applications, October 2017 
Toric point counting and applications (slides, video)
Banff International Research Station, New Trends in Arithmetic and Geometry of Algebraic Surfaces, March 2017 
Toric point counting and applications
Department of Mathematics & Statistics, University of Vermont, February 2017 
Counting points on curves (slides)
Université Laval, Québec, QuébecMaine Number Theory Conference, October 2016 
Counting points on curves (slides)
University of New South Wales, School of Mathematics and Statistics, July 2016 
Zeta functions of quartic K3 surfaces over $\mathbf{F}_3$
(slides)
Sage Days 71: Explicit padic methods in number theory, Oxford University, March 2016 
Equidistributions in arithmetic geometry
(slides)
Department of Mathematics, Dartmouth College, Jan 2016 
Equidistributions in arithmetic geometry
(slides)
Departamento de Matemática, Instituto Superior Técnico, Dec 2015 
Effective computations of HasseWeil zeta functions
(slides)
Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives, ICERM, Brown University, Oct 2015 
Effective computations of HasseWeil zeta functions
Courant Institute of Mathematical Sciences, NYU, May 2015 
Variation of NéronSeveri ranks of reductions of algebraic surfaces
Department of Mathematics, Rice University, Feb 2015 
Variation of NéronSeveri ranks of reductions of algebraic surfaces
School of Mathematics and Statistics, UNSW, Oct 2014 
Variation of NéronSeveri ranks of reductions of algebraic surfaces
Courant Institute of Mathematical Sciences, NYU, Oct 2014 
The point counting problem
Courant Institute of Mathematical Sciences, NYU, Dec 2012 
Faster deterministic integer factorization
Courant Institute of Mathematical Sciences, NYU, Jan 2012 
Étale Cohomology
Courant Institute of Mathematical Sciences, NYU, Mar 2011 
Étale Cohomology
Departamento de Matemática, Instituto Superior Técnico, July 2010