Professor of Mathematics

(Photo by Joe Rabinoff - 07/15.)

Department of Mathematics   Rice University MS 136   6100 S. Main St.   Houston, TX 77005, USA

Office: Herman Brown Hall, Room 222 Fax: (713)-348-5231  email: av15 **at** rice.edu

• Math 101: Single Variable Calculus I. (Fall 2024) Syllabus

• Locally recoverable algebro-geometric codes with multiple recovery sets from projective bundles (with K. Aguilar, A. Alvarez, R. Ardila, P.S. Ocal and C. Rodriguez Avila)
  Preprint.
• Reconstruction of generic anisotropic stiffness tensors from partial data around one polarization (with M. V. de Hoop, J. Ilmavirta and M. Lassas)
  Preprint.
• Reduction of Brauer Classes on K3 surfaces, Rationality, and Derived Equivalence (with S. Frei and B. Hassett)
  Journal für die reine und angewandte Mathematik (Crelle), 792 (2022), 289-305.
• Explicit Computation of Symmetric Differentials and its Application to Quasi-hyperbolicity (with N. Bruin and J. Thomas)
  Algebra and Number Theory, 16 (2022), 1377-1405.
• The Geometric Disposition of Diophantine Equations
  Notices of the American Mathematical Society, 68 (2021), 1291-1300.
• Locally Recoverable Codes on Surfaces (with C. Salgado and J. F. Voloch)
  IEEE Transactions on Information Theory, 69 (2021), 5765-5777.
• Campana points of bounded height on vector group compactifications (with M. Pieropan, A. Smeets, and S. Tanimoto)
  Proceedings of the London Mathematical Society (relaunched), 123 (2021), 57-101.
• Odd order obstructions to the Hasse principle on general K3 surfaces (with J. Berg)
  Mathematics of Computation, 89 (2020), 1395-1416.
• Cubic fourfolds fibered in sextic del Pezzo surfaces (with N. Addington, B. Hassett, and Yu. Tschinkel)
  American Journal of Mathematics, 141 (2019), 1479-1500.
• Kodaira dimension of moduli of special cubic fourfolds (with S. Tanimoto)
  Journal für die reine und angewandte Mathematik (Crelle), 752 (2019), 265-300.
• Campana Points, Vojta's conjecture, and level structures on semistable abelian varieties (with D. Abramovich)
  Journal de Théorie des Nombres de Bordeaux, 30 (2018), 525-532.
• Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture (with D. Abramovich)
  Advances in Mathematics, 329 (2018), 523-540.
• Abelian n-division fields of elliptic curves and Brauer groups of product Kummer and abelian surfaces (with B. Viray)
  Forum of Mathematics, Sigma, 5 doi:10.1017/fms.2017.16.
• Locally recoverable codes from algebraic curves and surfaces (with A. Barg, K. Haymaker, E. W. Howe, and G. Matthews)
  Algebraic Geometry for Coding Theory and Cryptography (E. Howe, K. Lauter, and J. Walker eds.), Association for Women in Mathematics 9 (2017), 95-127.
• Level structures on abelian varieties and Vojta's conjecture (with D. Abramovich; Appendix by Keerthi Madapusi Pera)
  Compositio Mathematica, 153 (2017), 373-394.
• Brauer groups on K3 surfaces and arithmetic applications (with K. McKinnie, J. Sawon and S. Tanimoto)
  Brauer groups and obstruction problems: moduli spaces and arithmetic (A. Auel, B. Hassett, A. Várilly-Alvarado, and B. Viray eds.), Progress in Mathematics 320 (2017), 177-218.
• Arithmetic of K3 surfaces (Notes for the 2015 Arizona Winter School)
  Geometry over nonclosed fields (F. Bogomolov, B. Hassett, and Yu. Tschinkel eds.), Simons Symposia 5 (2017), 197-248.
• Smooth compactifications of certain normic bundles (with B. Viray)
  European Journal of Mathematics, 1 (2015), 250-259.
• On the Unirationality of del Pezzo surfaces of degree two (with C. Salgado and D. Testa)
  Journal of the London Mathematical Society, 90 (2014), 121-139.
• Arithmetic of del Pezzo surfaces of degree 4 and vertical Brauer groups (with B. Viray)
  Advances in Mathematics, 255 (2014), 153-181.
• Cubic fourfolds containing a plane and a quintic del Pezzo surface (with A. Auel, M. Bernardara and M. Bolognesi)
  Algebraic Geometry, 1 (2014), 181-193.
• Failure of the Hasse principle on general K3 surfaces (with B. Hassett)
  Journal de l'Institut de Mathématiques de Jussieu 12 (2013), 853-877.
• Arithmetic of del Pezzo surfaces
  Birational geometry, rational curves, and arithmetic (F. Bogomolov, B. Hassett and Y. Tschinkel eds.) Simons Symposia 1 (2013), 293-319.
• Higher dimensional analogues of Châtelet surfaces (with B. Viray)
  Bulletin of the London Mathematical Society, 44 (2012) 125-135.
  Corrigendum 47 (2015), 217-218.
• Transcendental obstructions to weak approximation on general K3 surfaces (with B. Hassett and P. Varilly)
  Advances in Mathematics, 228 (2011), 1377-1404.
• Density of rational points on isotrivial rational elliptic surfaces
  Algebra and Number Theory, 5 (2011) 659-690.
• Failure of the Hasse principle for Enriques surfaces (with B. Viray)
  Advances in Mathematics, 226 (2011), 4884-4901.
• Laurent polynomials and Eulerian numbers (with D. Erman and G. Smith)
  Journal of Combinatorial Theory Series A, 118 (2011) 396-402.
• Big rational surfaces (with D. Testa and M. Velasco)
  Mathematische Annalen, 351 (2011) 95-107.
• Cox rings of degree one del Pezzo surfaces (with D. Testa and M. Velasco)
  Algebra and Number Theory 3 (2009) 729-761.
• Arithmetic E_8 lattices with maximal Galois action   (with D. Zywina)
  LMS Journal of Computation and Mathematics 12 (2009) 144-165.
• Arithmetic of del Pezzo surfaces of degree 1 Erratum
  Ph. D. Thesis, UC Berkeley (2009), 84 pages.
• Weak Approximation on del Pezzo surfaces of degree 1
  Advances in Mathematics 219 (2008), 2123-2145.   Magma scripts
• Location of Incenters and Fermat Points in Variable Triangles  
  Mathematics Magazine 74 (2001) 123-129.

I gave a mini-course on the arithmetic of del Pezzo surfaces at the conference Arithmetic of Surfaces in October 2010. I prepared some lecture notes to accompany the mini-course. The first two lectures were ''chalkboard talks''. For the last lecture I used slides for reasons that will be apparent if you look at the last few slides.  

• Latinx in the Mathematical Sciences 2021 IPAM (Co-organized with Selenne Bañuelos, Rodrigo Bañuelos, Pamela Harris, and Mariel Vázquez)
• Joint Mathematics Meetings AMS Special Session on Rational Points on Algebraic Varieties: Theory and Computation (Co-organized with Brendan Hassett and Drew Sutherland)
• Gulf Coast Undergraduate Research Symposium, Mathematics Session (Rice University)
• Birational Geometry and Arithmetic, ICERM (Co-organized with Asher Auel, Marta Pieropan, Sho Tanimoto and Yuri Tschinkel)
• SIAM Conference on Applied Algebraic Geometry Mini-symposium on Algebraic coding for storage applications (Co-organized with Alexander Barg)
• TAGS 2017 (Co-organized with John Calabrese)
• RTG Lectures in Arithmetic Geometry at Rice (Co-organized with Anastassia Etropolski and Jennifer Berg)
• Rational points / Pontos Racionais IMPA, Rio de Janeiro (Co-organized with Asher Auel, Oliver Lorscheid and Cecília Salgado)
• TAGS 2014 (Co-organized with Brendan Hassett).
• AIM Workshop: "Brauer groups and obstruction problems: moduli spaces and arithmetic" (Co-organized with Asher Auel, Brendan Hassett, and Bianca Viray).
• TAGS 2011 (Co-organized with Evan Bullock and Brendan Hassett).