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, to appear in the Ramanujan Journal (.pdf)
  30. (with Wai Kiu Chan and Glenn Henshaw) Totally isotropic subspaces of small height in quadratic spaces, preprint (.pdf)
  31. (with Albrecht Boettcher, Stephan Garcia, and Hiren Maharaj) On lattices generated by finite Abelian groups, preprint (.pdf)
  32. (with Albrecht Boettcher, Stephan Garcia, and Hiren Maharaj) Toeplitz determinants with perturbations in the corners, preprint (.pdf)
  33. (with Stephan Garcia and Xun Sun) Permutation invariant lattices, preprint (.pdf)
Most of my papers can also be found on the arXiv by clicking here.

Editorial:

Here are some of my talks:
  1. Computational complexity of lattice problems and cyclic lattices, 2014 Undergraduate Summer Research Program Polygons and Polynomials, ICERM - Brown University, July 28, 2014 (.pdf)
  2. Well-rounded lattices from algebraic constructions, International Conference on The Algebraic and Arithmetic Theory of Quadratic Forms 2013, Puerto Natales, Patagonia, Chile (December 16-21, 2013) (.pdf)
  3. On integral well-rounded lattices, Karlsruhe Institute of Technology, Mathematics Colloquium, Karlsruhe, Germany, July 19, 2012 (.pdf)
  4. On the Frobenius problem and its generalization, Universitaet des Saarlandes, Number Theory Seminar, Saarbruecken, Germany, July 9, 2012 (.pdf)
  5. 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)
  6. 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
  7. Searching for rational points on varieties over global fieldsColloquium, Uppsala University, November 2010 (.pdf) and Claremont Colloquium, Claremont Colleges, February 2014 (.pdf)
  8. On Siegel's lemma outside of a union of varieties, Oberseminar des Institutes für Algebra und Geometrie, University of Magdeburg, November 2010 (.pdf)
  9. Small zeros of hermitian forms over quaternion algebras, Number Theory Seminar, Institut de Mathématiques de Jussieu, Paris, France, October 2010 (.pdf)
  10. Orthogonal Siegel's lemma, Illinois Number Theory Conference, May 2010 (.pdf)
  11. Math in the grocery aisle: from stacking oranges to constructing error-correcting codes, invited lecture at GEMS: Gateway to Exploring Mathematical Sceinces, Claremont Colleges, February 2009 (TALK  PROJECT  SOLUTIONS) - here is a VIDEO clip about the GEMS program with bits of my lecture in it
  12. 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)
  13. 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)
  14. 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)
  15. Sphere packing, lattices, and Epstein zeta function, Claremont Colloquium, Claremont Colleges, November 2007 (.pdf)
  16. On distribution of integral well-rounded lattices in dimension two, Illinois Number Theory Fest, May 2007 (.pdf)
  17. Quadratic forms and height functions, Colloquium of the Graduiertenkolleg, University of Göttingen, June 2006 (.pdf)
  18. Simultaneous Diophantine approximations and a conjecture of Minkowski (expository talk), Introduction to Mathematical Research seminar series, University of Texas at Austin, March 2006 (.pdf)
  19. 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)
  20. Frobenius problem and the covering radius of a lattice, West Coast Number Theory, Pacific Grove, CA, December 2005 (.pdf, data)
  21. Some effective Diophantine results over the algebraic closure of Q, XXIVth Journées Arithmétiques, Marseilles, France, July 2005 (.pdf)
  22. Effective decompositions of quadratic spaces, ArithmeTexas, College Station, TX, April 2005 (.pdf)
  23. Counting lattice points in admissible adelic sets, MNTCG2, Urbana-Champaign, IL, February 2005 (.pdf)
  24. Heights and Diophantine problems, September 2004, Colloquium, Rice University  (.pdf)
  25. Siegel's lemma with additional conditions, June 2004, CNTA8 (.pdf)
  26. Integral points of small height outside of a hypersurface, May 2004, Illinois Number Theory Conference (.pdf)
  27. 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 is my mathematical genealogy page. Here is my google scholar profile.

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