Research
My areas of research are Number Theory and Discrete Geometry, more specifically my
interests are in
- 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
Here are my publications (also see my papers
and reviews
on MathSciNet,
as well as my papers
and reviews on Zentralblatt
Math):
- 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 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 (.pdf)
- (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 Deanna
Needell and Benny Sudakov)
An algebraic
perspective on integer sparse recovery, Applied
Mathematics and Computation, vol. 340 (2019), pg. 31--42 (.pdf)
- (with Albrecht
Boettcher, Simon Eisenbarth, Stephan Garcia,
and Hiren
Maharaj) Spherical
2-designs and lattices from Abelian groups, Discrete and Computational
Geometry, vol. 61 no. 1 (2019) pg. 123--135 (.pdf)
- (with Wai Kiu Chan)
Small representations
of integers by integral quadratic forms, Journal of Number Theory,
vol. 201 (2019) pg. 40--52 (.pdf)
- (with Taoufiq Damir) Canonical basis twists of ideal
lattices from real quadratic number fields, Houston Journal of Mathematics,
vol. 45 no. 4 (2019) pg. 999--1019 (.pdf)
- (with Deanna
Needell, Josiah
Park and Jessie Xin) Lattices from tight frames
and vertex transitive graphs, Electronic Journal of Combinatorics, vol. 26
no. 3 (2019), P3.49, 30 pp. (.pdf)
- (with Ed Shi) Positive
semigroups and generalized Frobenius numbers over totally real
number fields, Moscow
Journal of Combinatorics and Number Theory, vol. 9 no.
1 (2020) pg. 29--41 (.pdf)
- (with Soren Aletheia-Zomlefer and Stephan Garcia)
The Bateman-Horn conjecture: Heuristic, history, and
applications, Expositiones
Mathematicae, vol. 38 no. 4 (2020), pg. 430--479 (.pdf)
- (with Albrecht
Boettcher) Representing
integers by multilinear polynomials, Research in Number Theory,
vol. 6 no. 4 (2020), Article #38 (.pdf)
- (with David Kogan) On the geometry of nearly orthogonal
lattices, Linear Algebra and its Applications,
vol. 629 (2021), pg. 112--137 (.pdf)
- (with Pavel
Guerzhoy and Stefan
Kuehnlein) On sparse geometry of numbers, Research
in the Mathematical Sciences, vol. 8 no. 1 (2021), Article
#2 (.pdf)
- (with David Kogan) Cyclic and well-rounded lattices, Moscow Journal of Combinatorics
and Number Theory, vol. 11 no. 1 (2022), pg. 79--96 (.pdf)
- (with Maxwell Forst) Counting basis extensions in a
lattice, Proceedings of
the American Mathematical Society, vol. 150 no. 8
(2022), pg. 3199--3213 (.pdf)
- (with Siki Wang) Positive semigroups in lattices and
totally real number fields, Advances in Geometry,
vol. 22 no. 4 (2022), pg. 503--512 (.pdf)
- (with David Kogan) On average coherence of cyclotomic
lattices, Communications in Mathematics, vol. 31
no. 1 (2023), pg. 57--72 (.pdf)
- (with Maxwell Forst) On zeros of multilinear polynomials,
Journal of Number Theory, vol. 245 (2023), pg. 169--186 (.pdf)
- (with Alex Hsu) Covering
point-sets with parallel hyperplanes and sparse signal
recovery, Discrete and Computational Geometry,
vol. 69 no. 3 (2023) pg. 919--930 (.pdf)
- (with Maxwell Forst) On lattice extensions, preprint (.pdf)
Most of my papers can also be found on the arXiv by clicking here.
Editorial:
Here are some of my talks:
- Positive semigroups in lattices and totally real number
fields, AMS
Special Session on "Number Theory at Non-PhD Granting
Institutions", JMM
2023, Boston, MA, January 4-7, 2023 (.pdf)
- Geometric constructions for sparse integer signal recovery,
Combinatorics
Seminar, UC Irvine, February 2022
(.pdf)
- On frame coherence and nearly orthogonal lattices, Algebra
/ Number Theory / Combinatorics seminar, Claremont Colleges,
February 2022 (.pdf)
- On sparse geometry of numbers, Diophantine
Analysis and Related Topics, online conference at Moscow
State University (Moscow, Russia), May 31 - June 5, 2021 (.pdf)
- Representing integers by multilinear polynomials, Diophantine
Analysis Online Seminar, Moscow State University (Moscow,
Russia), April 8, 2021 (.pdf)
-
Lattices from group frames
and vertex transitive graphs, Algebra Seminar, Florida State University,
Tallahassee, Florida, November 20, 2019 (.pdf)
- Heights and
quadratic forms, Colloquium, University of Texas Rio
Grande Valley, Edinburg, Texas, November 19, 2019 (.pdf)
- Bounding linear
forms on a lattice away from zero, Conference on
the Arithmetic Theory of Quadratic Forms, Seoul National
University, January 7 - 11, 2019 (.pdf)
- An algebraic
perspective on integer sparse recovery, USC
Combinatorics Seminar, October 31, 2018 (.pdf)
- 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)
Here are some additional manuscripts and lecture notes that I have
written:
- Topics in Discrete Optimization: set of lecture notes
used for a special topics course I taught at Claremont Graduate
University, Fall 2022 (.pdf)
- Geometric Number Theory (GNT): extended set of lecture
notes, different parts of which were used for different special
topics courses I taught at Claremont McKenna, including a GNT
course in the Spring 2019 semester (.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)
Here is my Ph.D. Thesis:
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:
Here are my undergraduate senior thesis students:
- Tanis Nielsen (Harvey Mudd College,
B.A. May 2023) Tanis's
thesis
- Carl Bell (Pomona
College, B.A. May 2023)
- Siki Wang (Claremont McKenna
College, B.A. May 2022) Siki's thesis
- Yingqi (Ed) Shi (Claremont
McKenna College, B.A. May 2020) Ed's
thesis
- Alexander Hsu (Claremont
McKenna College, B.A. December 2019) Alex's
thesis
- Yuxin (Jessie) Xin (Claremont
McKenna College, B.A. May 2019) Jessie's
thesis
- Scott Maislin (Claremont
McKenna College, B.A. May 2017) Scott's
thesis
- Mia Siracusa (Scripps
College, B.A. May 2017) Mia's
thesis
- Michelle Goodwin (Claremont
McKenna College, B.A. May 2016) Michelle's
thesis
- Emma Cranston (Scripps
College, B.A. May 2016)
- Tatiana Bradley (Scripps
College, B.A. May 2015) Tatiana's
thesis
- John Shaughnessy (Claremont
McKenna College, B.A. May 2014) John's
thesis
- Michael Mei (Pomona
College, B.A. May 2014)
- Jonathan Raiman (Pomona
College, B.A. May 2013)
- Amanda Clemm (Scripps
College, B.A. May 2012)
Here is my mathematical
genealogy page. Here is my Google
Scholar profile. Here is my ResearchGate
profile.