Research

• 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
  

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, no. 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. Well-rounded zeta-function of planar arithmetic lattices, to appear in the Proceedings of the AMS (.pdf)
25. (with Wai Kiu Chan and Glenn Henshaw) Small zeros of quadratic forms missing a union of varieties, preprint (.pdf)
26. (with Glenn Henshaw) Lattice point counting and height bounds over number fields and quaternion algebras, preprint (.pdf)
27. (with Xun Sun) On the geometry of cyclic lattices, preprint (.pdf)
    

Editorial:

• 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)

1. On integral well-rounded lattices, Karlsruhe Institute of Technology, Mathematics Colloquium, Karlsruhe, Germany, July 19, 2012 (.pdf)
   
2. On the Frobenius problem and its generalization, Universitaet des Saarlandes, Number Theory Seminar, Saarbruecken, Germany, July 9, 2012 (.pdf)
   
3. 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)
4. 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)
   
5. Searching for rational points on varieties over global fields, Colloquium, Uppsala University, November 2010 (.pdf)
   
6. On Siegel's lemma outside of a union of varieties, Oberseminar des Institutes für Algebra und Geometrie, University of Magdeburg, November 2010 (.pdf)
   
7. Small zeros of hermitian forms over quaternion algebras, Number Theory Seminar, Institut de Mathématiques de Jussieu, Paris, France, October 2010 (.pdf)
   
8. Orthogonal Siegel's lemma, Illinois Number Theory Conference, May 2010 (.pdf)
   
9. 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
   
10. 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)
    
11. 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)
    
12. 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)
    
13. Sphere packing, lattices, and Epstein zeta function, Claremont Colloquium, Claremont Colleges, November 2007 (.pdf)
    
14. On distribution of integral well-rounded lattices in dimension two, Illinois Number Theory Fest, May 2007 (.pdf)
    
15. Quadratic forms and height functions, Colloquium of the Graduiertenkolleg, University of Göttingen, June 2006 (.pdf)
    
16. Simultaneous Diophantine approximations and a conjecture of Minkowski (expository talk), Introduction to Mathematical Research seminar series, University of Texas at Austin, March 2006 (.pdf)
    
17. 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)
    
18. Frobenius problem and the covering radius of a lattice, West Coast Number Theory, Pacific Grove, CA, December 2005 (.pdf, data)
    
19. Some effective Diophantine results over the algebraic closure of Q, XXIVth Journées Arithmétiques, Marseilles, France, July 2005 (.pdf)
    
20. Effective decompositions of quadratic spaces, ArithmeTexas, College Station, TX, April 2005 (.pdf)
    
21. Counting lattice points in admissible adelic sets, MNTCG2, Urbana-Champaign, IL, February 2005 (.pdf)
    
22. Heights and Diophantine problems, September 2004, Colloquium, Rice University  (.pdf)
23. Siegel's lemma with additional conditions, June 2004, CNTA8 (.pdf)
24. Integral points of small height outside of a hypersurface, May 2004, Illinois Number Theory Conference (.pdf)
25. Small zeros of quadratic forms with linear conditions, June 2003, Mahler's Measure of Polynomials (.pdf)
    

• (with Wai Kiu Chan and Glenn Henshaw) Points and subspaces of bounded height in quadratic and linear spaces, unpublished manuscript containing the results of publication 25 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)
  
• Discrete geometry: lecture notes for an undergraduate topics course I taught at Claremont McKenna in the Fall 2008 semester (.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)

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