- Diophantine approximations, equations, and problems
- Classical and adelic geometry of numbers
- Quadratic and linear forms; polynomials

- Theory of height functions, rational points on varieties, Mahler's measure, and connections with arithmetic geometry
- Geometric combinatorics, lattice point counting, Erhart polynomial
- Lattices, linear Diophantine problem of Frobenius

__Small zeros of quadratic forms with linear conditions__, Journal of Number Theory, vol. 108 no. 1 (2004), pg. 29-43 (pdf)- (with Matthias Beck,
Beifang Chen,
Christian
Haase, Allen
Knutson, Bruce
Reznick, Sinai
Robins, and Achill
Schuermann) Problems
from the Cottonwood Room
*, Contemporary Mathematics*374 (2005), 179-191 (Proceedings of the Summer AMS/MAA/SIAM Research Conference on Integer Points in Polyhedra, July 13 - July 17, 2003 in Snowbird, Utah) (pdf - my section of this paper: Lattice points in homogeneously expanding compact domains) - I have Erdös number 2 thanks to this paper!

- Integral points of small height outside of a hypersurface, Monatshefte für Mathematik, vol. 147 no. 1 (2006), pg. 25-41 (pdf)
- Siegel's lemma with additional conditions, Journal of Number Theory, vol. 120 no. 1 (2006), pg. 13-25 (pdf)
- (with Sinai Robins) Frobenius problem and the covering radius of a lattice, Discrete and Computational Geometry, vol. 37 no. 3 (2007), pg. 471-483 (.pdf)
- On effective Witt decomposition and Cartan-Dieudonné theorem, Canadian Journal of Mathematics, vol. 59 no. 6 (2007), pg. 1284-1300 (.pdf)
- Small zeros of
quadratic forms over the
algebraic closure of Q,
*International Journal of Number Theory*, vol. 4 no. 3 (2008), pg. 503-523 (.pdf) - On distribution of well-rounded sublattices of Z^2, Journal of Number Theory, vol. 128 no. 8 (2008), pg. 2359-2393 (.pdf)
- On similarity classes of well-rounded sublattices of Z^2, Journal of Number Theory, vol. 129 no. 10 (2009), pg. 2530-2556 (.pdf) -- see JNT Video Abstract
- Search bounds for zeros of polynomials over the algebraic closure of Q, Rocky Mountain Journal of Mathematics, vol. 39 no. 3 (2009), pg. 789-804 (.pdf)
- Effective structure theorems for symplectic spaces via height, in "Quadratic forms -- algebra, arithmetic, and geometry" (R. Baeza, W.K. Chan, D.W. Hoffmann, and R. Schulze-Pillot, eds.), Contemporary Mathematics 493 (2009), pg. 117--130 (.pdf)
- Integral orthogonal bases of small height for real polynomial spaces, Online Journal of Analytic Combinatorics, vol. 4 (2009), art. 7, 10 pp. (.pdf)
- (with Wai Kiu Chan) Small zeros of hermitian forms over quaternion algebras, Acta Arithmetica, vol. 142 no. 3 (2010), pg. 251--266 (.pdf)
- Algebraic points of small height missing a union of varieties, Journal of Number Theory, vol. 130 no. 10 (2010), pg. 2099--2118 (.pdf) -- see JNT Video Abstract; an earlier version of this paper also appears in the MPIM Preprint Series
- (with D. Moore, A. Ohana, W. Zeldow) On well-rounded sublattices of the hexagonal lattice, Discrete Mathematics, vol. 310 no. 23 (2010), pg. 3287--3302 (.pdf)
- Revisiting the hexagonal lattice: on optimal lattice circle packing, Elemente der Mathematik, vol. 66 no. 1 (2011) pg. 1--9 (.pdf)
- (with Sinai Robins) Bounds for solid angles of lattices of rank three, Journal of Combinatorial Theory A, vol. 118 no. 2 (2011) pg. 690--701 (.pdf)
- (with Achill Schuermann) Bounds on generalized Frobenius numbers, European Journal of Combinatorics, vol. 32 no. 3 (2011) pg. 361--368 (.pdf)
- (with Kate Petersen) On well-rounded ideal lattices, International Journal of Number Theory, vol. 8 no. 1 (2012) pg. 189--206 (.pdf)
- (with G. Henshaw, P. Liao, M. Prince, X. Sun, S. Whitehead) On integral well-rounded lattices in the plane, Discrete and Computational Geometry, vol. 48 no. 3 (2012), pg. 735--748 (.pdf)
- (with Iskander Aliev and Martin Henk) Generalized Frobenius numbers: bounds and average behavior, Acta Arithmetica, vol. 155 no. 1 (2012), 53-62 (.pdf)
- (with G. Henshaw, P. Liao, M. Prince, X. Sun, S. Whitehead) On well-rounded ideal lattices - II, International Journal of Number Theory, vol. 9 no. 1 (2013) pg. 139--154 (.pdf)
- Heights and quadratic forms: on Cassels' theorem and its generalizations, in "Diophantine methods, lattices, and arithmetic theory of quadratic forms" (W. K. Chan, L. Fukshansky, R. Schulze-Pillot, and J. D. Vaaler, eds.), Contemporary Mathematics, AMS vol. 587 (2013), pg. 77--94 (.pdf)
- (with Glenn Henshaw) Lattice point counting and height bounds over number fields and quaternion algebras, Online Journal of Analytic Combinatorics, vol. 8 (2013), art. 4, 20 pp. (.pdf)
- Well-rounded zeta-function of planar arithmetic lattices, Proceedings of the American Mathematical Society, vol. 142 no. 2 (2014), pg. 369--380 (.pdf)
- (with Wai Kiu Chan and Glenn Henshaw) Small zeros of quadratic forms missing a union of varieties, Transactions of the AMS, vol. 366 no. 10 (2014), pg. 5587--5612 (.pdf)
- (with Hiren Maharaj) Lattices from elliptic curves over finite fields, Finite Fields and Their Applications, vol. 28 (July 2014), pg. 67--78 (.pdf)
- (with Xun Sun) On the geometry of cyclic lattices, Discrete and Computational Geometry, vol. 52 no. 2 (2014), pg. 240--259 (.pdf)
- Stability of ideal lattices from quadratic number fields, The Ramanujan Journal, vol. 37 no. 2 (2015), pg. 243--256 (.pdf)
- (with Albrecht Boettcher, Stephan Garcia, and Hiren Maharaj) Toeplitz determinants with perturbations in the corners, Journal of Functional Analysis, vol. 268 no. 1 (2015), pg. 171--193 (.pdf)
- (with Albrecht Boettcher, Stephan Garcia, and Hiren Maharaj) On lattices generated by finite Abelian groups, SIAM Journal on Discrete Mathematics, vol. 29 no. 1 (2015), pg. 382--404 (.pdf)
- (with Stephan Garcia and Xun Sun) Permutation invariant lattices, Discrete Mathematics, vol. 338 no. 8 (2015), pg. 1536--1541 (.pdf)
- Height bounds on zeros of quadratic forms over Q-bar, Research in Mathematical Sciences, vol. 2 no. 1 (2015), art. 19, 26 pp. (.pdf)
- (with Wai Kiu Chan and Glenn Henshaw) Totally isotropic subspaces of small height in quadratic spaces, Advances in Geometry, vol. 16 no. 2 (2016), pg. 153--164 (.pdf)
- (with Albrecht
Boettcher, Stephan
Garcia, and Hiren
Maharaj) Lattices
from Hermitian function fields,
*Journal of Algebra*, vol. 447 (2016), pg. 560--579 (.pdf) - (with Albrecht
Boettcher, Stephan
Garcia, Hiren
Maharaj, and Deanna Needell)
Lattices from tight
equiangular frames,
*Linear Algebra and its Applications*, vol. 510 (2016), pg. 395--420 (.pdf) - (with Pavel
Guerzhoy and Florian
Luca)
__On arithmetic lattices in the plane__, Proceedings of the American Mathematical Society, vol. 145 no. 4 (2017), pg. 1453--1465 (.pdf) - (with Albrecht
Boettcher) Addendum
to: Lattices
from tight equiangular frames,
*Linear Algebra and its Applications*, vol. 531 (2017), pg. 592--601 (.pdf) - (with Albrecht
Boettcher, Stephan
Garcia, and Hiren
Maharaj)
__Lattice theory and Toeplitz determinants__,*Operator Theory: Advances and Applications*, vol. 262 (2018), pg. 117--138 (.pdf) - (with Albrecht Boettcher, Simon Eisenbarth, Stephan Garcia, and Hiren Maharaj) Spherical 2-designs and lattices from Abelian groups, to appear in Discrete and Computational Geometry (.pdf)
- (with Adib
Shaar)
__A new family of one-coincidence sets of sequences with dispersed elements for frequency hopping CDMA systems__,*Advances in Mathematics of Communications*, vol. 12 no. 1 (2018), pg. 181--188 - (with Nikolay
Moshchevitin)
__On an effective variation of Kronecker's approximation theorem avoiding algebraic sets__, Proceedings of the American Mathematical Society, vol. 146 no. 10 (2018), pg. 4151--4163 (.pdf) - (with Wai Kiu Chan) Small representations of integers by integral quadratic forms, preprint (.pdf)
- (with Deanna
Needell and Benny Sudakov)
An algebraic
perspective on integer sparse recovery,
*Applied Mathematics and Computation*, vol. 340 (2019), pg. 31--42 (.pdf) - (with Soren Aletheia-Zomlefer and Stephan Garcia)
__One conjecture to rule them all: Bateman-Horn__, preprint (.pdf) - (with Taoufiq Damir)
__Canonical basis twists of ideal lattices from real quadratic number fields__, preprint (.pdf)

Editorial:

- I am an editor for the Online Journal of
Analytic Combinatorics -- please submit papers through the
journal's online interface

- Diophantine methods, lattices, and arithmetic theory of quadratic forms -- proceedings of a BIRS workshop (Editors: W. K. Chan, L. Fukshansky, R. Schulze-Pillot, J. D. Vaaler), Contemporary Mathematics, AMS vol. 587 (2013)

- Lattices generated by equiangular tight frames, ICERM
workshop on ``Computational Challenges in the Theory of
Lattices", ICERM
- Brown, April 2018 (SLIDES
VIDEO)

- A difference in numbers, invited lecture at GEMS: Gateway to Exploring Mathematical Sciences, Claremont Colleges, November 2017 (TALK PROJECT SOLUTIONS)
__On two constructions of extremal lattices__, Vilnius Conference of Number Theory and Combinatorics, Vilnius, Lithuania (July 16 - 22, 2017) (.pdf)

__Generalized Frobenius numbers: bounds and average behavior via geometric techniques__, International Meeting on Numerical Semigroups with Applications, Levico Terme, Italy (July 4 - 8, 2016) (.pdf)

- Well-rounded lattices from algebraic constructions, Aalto University ANTA seminar, Helsinki, Finland (June 1, 2016) (.pdf)
__On arithmetic lattices in the plane__, Southern Illinois University Conference on Pure Mathematics, Carbondale, Illinois (May 15 - 16, 2016) (.pdf)__Lattices from Abelian groups__, Workshop on Lattices and Applications in Number Theory, Oberwolfach, Germany (January 17 - 23, 2016) (.pdf)

- Height bounds for
zeros of quadratic forms, Diophantine
Approximation and Related Topics, Aahrus, Denmark (July 13
- 17, 2015) (.pdf)

- Height bounds on
zeros of quadratic forms over Q-bar, AMS
Special
Session: Quadratic Forms in
Arithmetic and Geometry, AMS
Spring Western Section Meeting, Huntsville, AL, (March 27
- 29, 2015) (.pdf)

- Well-rounded
lattices from algebraic constructions, Erwin Schrödinger Institute for
Mathematical Physics, Program on "Minimal
Energy Point Sets, Lattices, and Designs", Workshop
on "Sphere Packings, Lattices, and Designs", Vienna,
Austria (October 27 - 31, 2014) (.pdf)

- On integral
well-rounded lattices, Karlsruhe Institute of
Technology, Mathematics
Colloquium, Karlsruhe, Germany, July 19, 2012 (.pdf)

- On the Frobenius
problem and its generalization, Universitaet des Saarlandes,
Number Theory Seminar, Saarbruecken, Germany, July 9, 2012 (.pdf)

- On distribution of integral well-rounded lattices in the plane, Special Session: Arithmetic of quadratic forms and integral lattices at the First Joint Meeting of AMS and Sociedad Mathematica de Chile, Pucon, Chile, December 2010 (.pdf)
- Heights and effective theory of quadratic forms over global fields, Workshop on the arithmetic of quadratic forms and integral lattices, Lake Ranco, Chile, December 2010 (.pdf) and University of Colorado at Boulder, Number Theory seminar, April 2014 (.pdf)
- Searching for
rational points on varieties over global fields, Colloquium,
Uppsala University,
November 2010 (.pdf) and Claremont
Colloquium, Claremont
Colleges, February 2014 (.pdf)

- On Siegel's lemma
outside of a union of varieties, Oberseminar
des Institutes für Algebra und Geometrie, University
of Magdeburg, November 2010 (.pdf)

- Small zeros of
hermitian forms over quaternion algebras, Number Theory
Seminar, Institut
de Mathématiques de Jussieu, Paris, France, October 2010 (.pdf)

- Orthogonal Siegel's
lemma, Illinois
Number Theory Conference, May 2010 (.pdf)

- Math in the grocery
aisle: from stacking oranges to constructing error-correcting
codes, invited lecture at GEMS: Gateway to
Exploring Mathematical Sciences, Claremont Colleges,
February 2009 (TALK PROJECT SOLUTIONS) - here is a VIDEO
clip about the GEMS program with bits of my lecture in it

- On distribution of
well-rounded sublattices of Z^2, Séminaire
de
Théorie des Nombres, Institut de
Mathématiques de Bordeaux, University of Bordeaux, France,
June 2008 (.pdf)

- Frobenius number,
covering radius, and well-rounded lattices, AMS
Special Session on the linear Diophantine problem of Frobenius,
Joint
Mathematics Meeting, San Diego, California, January 2008 (.pdf, data)

- Effective structure
theorems for quadratic spaces via height, Second International
Conference on The Algebraic and Arithmetic Theory of Quadratic
Forms 2007, Lake Llanquihue, Chile, December 2007 (.pdf)

- Sphere packing,
lattices, and Epstein zeta function, Claremont
Colloquium, Claremont
Colleges, November 2007 (.pdf)

- On distribution of
integral well-rounded lattices in dimension two, Illinois
Number Theory Fest, May 2007 (.pdf)

- Quadratic forms and
height functions, Colloquium
of
the Graduiertenkolleg, University of Göttingen,
June 2006 (.pdf)

- Simultaneous
Diophantine approximations and a conjecture of Minkowski
(expository talk), Introduction
to
Mathematical Research seminar series, University
of Texas at Austin, March 2006 (.pdf)

- Effective theorems
for quadratic spaces over the algebraic closure of Q, AMS
Special
Session on Mahler Measure and Heights, Joint
Mathematics Meeting, San Antonio, TX, January 2006 (.pdf)

- Frobenius problem
and the covering radius of a lattice, West
Coast
Number Theory, Pacific Grove, CA, December 2005 (.pdf, data)

- Some effective
Diophantine results over the algebraic closure
of Q,
XXIVth
Journées
Arithmétiques, Marseilles, France, July 2005 (.pdf)

- Effective
decompositions of quadratic spaces, ArithmeTexas,
College Station, TX, April 2005 (.pdf)

- Counting lattice
points in admissible adelic sets, MNTCG2,
Urbana-Champaign, IL, February 2005 (.pdf)

- Heights and Diophantine problems, September 2004, Colloquium, Rice University (.pdf)
- Siegel's lemma with additional conditions, June 2004, CNTA8 (.pdf)
- Integral points of small height outside of a hypersurface, May 2004, Illinois Number Theory Conference (.pdf)
- Small zeros of
quadratic forms with linear conditions, June 2003, Mahler's
Measure of Polynomials (.pdf)

- (with Oleg
German and Nikolay
Moshchevitin)
__On an effective variation of Kronecker's approximation theorem avoiding algebraic sets__, Oberwolfach preprint containing the material of 43 and an additional appendix on bad approximation and algebraic transference (.pdf) - (with Stefan
Kuehnlein and Rebecca
Schwerdt)
__Counting ideals in polynomial rings__, unpublished manuscript (.pdf) - (with Albrecht
Boettcher, Stephan
Garcia, and Hiren
Maharaj)
__Lattices from Abelian groups__, Oberwolfach report (2016) surveying the material of publications 27, 31, 35, 40 (.pdf)

- (with Wai Kiu Chan
and Glenn
Henshaw) Points
and subspaces of bounded height in quadratic and linear spaces,
unpublished manuscript containing preliminary version of results
of publication 26 and 34, as well as one of the results of 33,
with more details and some additional related results (.pdf)

- On integral
well-rounded lattices in the plane, unpublished
manuscript containing the results of publications 20 and 22
above with expanded details and arguments (.pdf)

- Simultaneous Diophantine approximation on a rational ellipse, unpublished manuscript (.pdf)
- Transcendental
Number Theory: lecture notes for the special topics
undergraduate course taught at Claremont McKenna, Spring 2015
semester (.pdf)

- Discrete geometry:
lecture notes for an undergraduate topics course I taught at
Claremont McKenna in the Fall 2008 & 2013 semesters (.pdf)

- Diophantine approximations and geometry of numbers: lecture notes for a graduate topics course I taught at Texas A&M University in the Spring 2006 semester (.pdf)

Algebraic points of small height with additional arithmetic conditions, Ph.D. Thesis, University of Texas at Austin, August 2004 (pdf)

Here are my Ph.D. students:

- Glenn Henshaw, (joint with Wai Kiu Chan) Ph.D. from Wesleyan University, May 2012
- Xun Sun, Ph.D. from Claremont Graduate
University, May 2015