Research

My areas of research are Number Theory and Discrete Geometry, more specifically my interests are in

Here are my publications (also see my papers and reviews on MathSciNet, as well as my papers and reviews on Zentralblatt Math):

  1. Small zeros of quadratic forms with linear conditions, Journal of Number Theory, vol. 108 no. 1 (2004), pg. 29-43 (pdf)
  2. (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!
  3. Integral points of small height outside of a hypersurface, Monatshefte für Mathematik, vol. 147 no. 1 (2006), pg. 25-41 (pdf)
  4. Siegel's lemma with additional conditions, Journal of Number Theory, vol. 120 no. 1 (2006), pg. 13-25 (pdf)
  5. (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)
  6. On effective Witt decomposition and Cartan-Dieudonné theorem, Canadian Journal of Mathematics, vol. 59 no. 6 (2007), pg. 1284-1300 (.pdf)
  7. 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)
  8. On distribution of well-rounded sublattices of Z^2, Journal of Number Theory, vol. 128 no. 8 (2008), pg. 2359-2393 (.pdf)
  9. 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
  10. 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)
  11. 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)
  12. Integral orthogonal bases of small height for real polynomial spaces, Online Journal of Analytic Combinatorics, vol. 4 (2009), art. 7, 10 pp. (.pdf)
  13. (with Wai Kiu Chan) Small zeros of hermitian forms over quaternion algebras, Acta Arithmetica, vol. 142 no. 3 (2010), pg. 251--266 (.pdf)
  14. 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
  15. (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)
  16. Revisiting the hexagonal lattice: on optimal lattice circle packing, Elemente der Mathematik, vol. 66 no. 1 (2011) pg. 1--9 (.pdf)
  17. (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)
  18. (with Achill Schuermann) Bounds on generalized Frobenius numbers, European Journal of Combinatorics, vol. 32 no. 3 (2011) pg. 361--368 (.pdf)
  19. (with Kate Petersen) On well-rounded ideal lattices, International Journal of Number Theory, vol. 8 no. 1 (2012) pg. 189--206 (.pdf)
  20. (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)
  21. (with Iskander Aliev and Martin Henk) Generalized Frobenius numbers: bounds and average behavior, Acta Arithmetica, vol. 155 no. 1 (2012), 53-62 (.pdf)
  22. (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)
  23. 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)
  24. (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)
  25. Well-rounded zeta-function of planar arithmetic lattices, Proceedings of the American Mathematical Society, vol. 142 no. 2 (2014), pg. 369--380 (.pdf)
  26. (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)
  27. (with Hiren Maharaj) Lattices from elliptic curves over finite fields, Finite Fields and Their Applications, vol. 28 (July 2014), pg. 67--78 (.pdf)
  28. (with Xun Sun) On the geometry of cyclic lattices, Discrete and Computational Geometry, vol. 52 no. 2 (2014), pg. 240--259 (.pdf)
  29. Stability of ideal lattices from quadratic number fields, The Ramanujan Journal, vol. 37 no. 2 (2015), pg. 243--256 (.pdf)
  30. (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)
  31. (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)
  32. (with Stephan Garcia and Xun Sun) Permutation invariant lattices, Discrete Mathematics, vol. 338 no. 8 (2015), pg. 1536--1541 (.pdf)
  33. Height bounds on zeros of quadratic forms over Q-bar, Research in Mathematical Sciences, vol. 2 no. 1 (2015), art. 19, 26 pp. (.pdf)
  34. (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)
  35. (with Albrecht Boettcher, Stephan Garcia, and Hiren Maharaj) Lattices from Hermitian function fields, Journal of Algebra, vol. 447 (2016), pg. 560--579 (.pdf)
  36. (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)
  37. (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)
  38. (with Albrecht Boettcher) Addendum to: Lattices from tight equiangular frames, Linear Algebra and its Applications, vol. 531 (2017), pg. 592--601 (.pdf)
  39. (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)
  40. (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)
  41. (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)
  42. (with Deanna Needell and Benny Sudakov) An algebraic perspective on integer sparse recovery, Applied Mathematics and Computation, vol. 340 (2019), pg. 31--42 (.pdf)
  43. (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)
  44. (with Wai Kiu Chan) Small representations of integers by integral quadratic forms, Journal of Number Theory, vol. 201 (2019) pg. 40--52 (.pdf)
  45. (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)
  46. (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)
  47. (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)
  48. (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)
  49. (with Albrecht Boettcher) Representing integers by multilinear polynomials, Research in Number Theory, vol. 6 no. 4 (2020), Article #38 (.pdf)
  50. (with David Kogan) On the geometry of nearly orthogonal lattices, Linear Algebra and its Applications, vol. 629 (2021), pg. 112--137 (.pdf)
  51. (with Pavel Guerzhoy and Stefan Kuehnlein) On sparse geometry of numbers, Research in the Mathematical Sciences, vol. 8 no. 1 (2021), Article #2 (.pdf)
  52. (with David Kogan) Cyclic and well-rounded lattices, Moscow Journal of Combinatorics and Number Theory, vol. 11 no. 1 (2022), pg. 79--96 (.pdf)
  53. (with Maxwell Forst) Counting basis extensions in a lattice, Proceedings of the American Mathematical Society, vol. 150 no. 8 (2022), pg. 3199--3213 (.pdf)
  54. (with Siki Wang) Positive semigroups in lattices and totally real number fields, Advances in Geometry, vol. 22 no. 4 (2022), pg. 503--512 (.pdf)
  55. (with David Kogan) On average coherence of cyclotomic lattices, Communications in Mathematics, vol. 31 no. 1 (2023), pg. 57--72 (.pdf)
  56. (with Maxwell Forst) On zeros of multilinear polynomials, Journal of Number Theory, vol. 245 (2023), pg. 169--186 (.pdf)
  57. (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)
  58. (with Camilla Hollanti) Euclidean lattices: theory and applications, Communications in Mathematics, vol. 31 no. 2 (2023) pg. 251--263 (.pdf)
  59. (with Maxwell Forst) On lattice extensions, Monatshefte fur Mathematik, vol. 203 no. 3 (2024) pg. 613--634 (.pdf)
  60. (with Maxwell Forst) On a new absolute version of Siegel's lemma, Research in the Mathematical Sciences, vol. 11 no.1 (2024) Article #10, 16 pp. (.pdf)
  61. (with Pavel Guerzhoy and Tanis Nielsen) Deep hole lattices and isogenies of elliptic curves, Research in Number Theory, vol. 10 no. 2 (2024) Article #33, 12 pp. (.pdf)
  62. (with Sehun Jeong) Diophantine avoidance and small-height primitive elements in ideals of number fields, Combinatorics and Number Theory, vol. 13 no. 4 (2024), pg. 333-350 (.pdf)
  63. (with Sehun Jeong) Integral zeros of quadratic polynomials avoiding sublattices, The Ramanujan Journal, vol 66 no. 2 (2025), Paper no. 26, 10 pp.  (.pdf)
  64. On lattice illumination of smooth convex bodies, to appear in Archiv der Mathematik (.pdf)

 

Most of my papers can also be found on the arXiv by clicking here.

Editorial:

Here are some of my talks:

  1. On lattice extensions, Conference on Modern Problems of Number Theory, Sirius University of Science and Technology, Sochi, Russia (online), July 9, 2024 (.pdf)
  2. Deep hole lattices and isogenies of elliptic curves, Southern Illinois University Conference on Pure Mathematics, Carbondale, Illinois (May 16-17, 2024) (.pdf)
  3. Quadratic forms and lattices, Diophantine Approximation online seminar, Independent University of Moscow, May 11, 2024 (.pdf -- in Russian)
  4. On a new version of Siegel's lemma, Diophantine Approximation, Dynamics and Related Topics, Yau Center for Mathematical Sciences, Hainan, China, January 29 - February 2, 2024 (.pdf)
  5. 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)
  6. Geometric constructions for sparse integer signal recovery, Combinatorics Seminar, UC Irvine, February 2022 (.pdf)
  7. On frame coherence and nearly orthogonal lattices, Algebra / Number Theory / Combinatorics seminar, Claremont Colleges, February 2022 (.pdf)
  8. On sparse geometry of numbers, Diophantine Analysis and Related Topics, online conference at Moscow State University (Moscow, Russia), May 31 - June 5, 2021 (.pdf)
  9. Representing integers by multilinear polynomials, Diophantine Analysis Online Seminar, Moscow State University (Moscow, Russia), April 8, 2021 (.pdf)
  10. Lattices from group frames and vertex transitive graphs, Algebra Seminar, Florida State University, Tallahassee, Florida, November 20, 2019 (.pdf)
  11. Heights and quadratic forms, Colloquium, University of Texas Rio Grande Valley, Edinburg, Texas, November 19, 2019 (.pdf)
  12. 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)
  13. An algebraic perspective on integer sparse recovery, USC Combinatorics Seminar, October 31, 2018 (.pdf)
  14. Lattices generated by equiangular tight frames, ICERM workshop on ``Computational Challenges in the Theory of Lattices", ICERM - Brown, April 2018 (SLIDES VIDEO)
  15. A difference in numbers, invited lecture at GEMS: Gateway to Exploring Mathematical Sciences, Claremont Colleges, November 2017 (TALK  PROJECT  SOLUTIONS)
  16. On two constructions of extremal lattices, Vilnius Conference of Number Theory and Combinatorics, Vilnius, Lithuania (July 16 - 22, 2017) (.pdf)
  17. 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)
  18. Well-rounded lattices from algebraic constructions, Aalto University ANTA seminar, Helsinki, Finland (June 1, 2016) (.pdf)
  19. On arithmetic lattices in the plane, Southern Illinois University Conference on Pure Mathematics, Carbondale, Illinois (May 15 - 16, 2016) (.pdf)
  20. Lattices from Abelian groups, Workshop on Lattices and Applications in Number Theory, Oberwolfach, Germany (January 17 - 23, 2016) (.pdf)
  21. Height bounds for zeros of quadratic forms, Diophantine Approximation and Related Topics, Aahrus, Denmark (July 13 - 17, 2015) (.pdf)
  22. 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)
  23. 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)
  24. On integral well-rounded lattices, Karlsruhe Institute of Technology, Mathematics Colloquium, Karlsruhe, Germany, July 19, 2012 (.pdf)
  25. On the Frobenius problem and its generalization, Universitaet des Saarlandes, Number Theory Seminar, Saarbruecken, Germany, July 9, 2012 (.pdf)
  26. 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)
  27. 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) 
  28. Searching for rational points on varieties over global fields, Colloquium, Uppsala University, November 2010 (.pdf) and Claremont Colloquium, Claremont Colleges, February 2014 (.pdf)
  29. On Siegel's lemma outside of a union of varieties, Oberseminar des Institutes für Algebra und Geometrie, University of Magdeburg, November 2010 (.pdf)
  30. Small zeros of hermitian forms over quaternion algebras, Number Theory Seminar, Institut de Mathématiques de Jussieu, Paris, France, October 2010 (.pdf)
  31. Orthogonal Siegel's lemma, Illinois Number Theory Conference, May 2010 (.pdf)
  32. 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
  33. 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)
  34. 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)
  35. 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)
  36. Sphere packing, lattices, and Epstein zeta function, Claremont Colloquium, Claremont Colleges, November 2007 (.pdf)
  37. On distribution of integral well-rounded lattices in dimension two, Illinois Number Theory Fest, May 2007 (.pdf)
  38. Quadratic forms and height functions, Colloquium of the Graduiertenkolleg, University of Göttingen, June 2006 (.pdf)
  39. Simultaneous Diophantine approximations and a conjecture of Minkowski (expository talk), Introduction to Mathematical Research seminar series, University of Texas at Austin, March 2006 (.pdf)
  40. 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)
  41. Frobenius problem and the covering radius of a lattice, West Coast Number Theory, Pacific Grove, CA, December 2005 (.pdf, data)
  42. Some effective Diophantine results over the algebraic closure of Q, XXIVth Journées Arithmétiques, Marseilles, France, July 2005 (.pdf)
  43. Effective decompositions of quadratic spaces, ArithmeTexas, College Station, TX, April 2005 (.pdf)
  44. Counting lattice points in admissible adelic sets, MNTCG2, Urbana-Champaign, IL, February 2005 (.pdf)
  45. Heights and Diophantine problems, September 2004, Colloquium, Rice University  (.pdf)
  46. Siegel's lemma with additional conditions, June 2004, CNTA8 (.pdf)
  47. Integral points of small height outside of a hypersurface, May 2004, Illinois Number Theory Conference (.pdf)
  48. 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:

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:

Here is my mathematical genealogy page. Here is my Google Scholar profile. Here is my ResearchGate profile.


 

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