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. Search bounds for zeros of polynomials over the algebraic closure of Q, to appear in Rocky Mountain Journal of Mathematics (.pdf)
10. On similarity classes of well-rounded sublattices of Z^2, preprint (.pdf)
11. Effective structure theorems for symplectic spaces via height, preprint (.pdf)
12. Algebraic points of small height missing a union of varieties, preprint (.pdf)
    

1. 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)
   
2. 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)
   
3. 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)
   
4. Sphere packing, lattices, and Epstein zeta function, Claremont Colloquium, Claremont Colleges, November 2007 (.pdf)
   
5. On distribution of integral well-rounded lattices in dimension two, Illinois Number Theory Fest, May 2007 (.pdf)
   
6. Quadratic forms and height functions, Colloquium of the Graduiertenkolleg, University of Göttingen, June 2006 (.pdf)
   
7. Simultaneous Diophantine approximations and a conjecture of Minkowski (expository talk), Introduction to Mathematical Research seminar series, University of Texas at Austin, March 2006 (.pdf)
   
8. 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)
   
9. Frobenius problem and the covering radius of a lattice, West Coast Number Theory, Pacific Grove, CA, December 2005 (.pdf, data)
   
10. Some effective Diophantine results over the algebraic closure of Q, XXIVth Journées Arithmétiques, Marseilles, France, July 2005 (.pdf)
    
11. Effective decompositions of quadratic spaces, ArithmeTexas, College Station, TX, April 2005 (.pdf)
    
12. Counting lattice points in admissible adelic sets, MNTCG2, Urbana-Champaign, IL, February 2005 (.pdf)
    
13. Heights and Diophantine problems, September 2004, Colloquium, Rice University  (.pdf)
14. Siegel's lemma with additional conditions, June 2004, CNTA8 (.pdf)
15. Integral points of small height outside of a hypersurface, May 2004, Illinois Number Theory Conference (.pdf)
16. Small zeros of quadratic forms with linear conditions, June 2003, Mahler's Measure of Polynomials (.pdf)
    

• Simultaneous Diophantine approximation on a rational ellipse, unpublished manuscript  (.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)