Akshay Venkatesh -- Research Interests

My research is in number theory and various related topics. I like problems where there is interesting interaction between analysis and algebra.

Cohomology of arithmetic groups and periods of automorphic forms Lecture notes for Takagi lectures given November 2014. Japanese Journal of Math (2016)
Derived Galois deformation rings, with Soren Galatius, to appear, Advances in Math.
Derived Hecke algebra and cohomology of arithmetic groups, submitted.
Automorphic cohomology, motivic cohomology, and the adjoint L-function, with Kartik Prasanna, submitted.
Derived Hecke algebra for weight one forms with Michael Harris, to appear, Experimental Mathematics.

The behavior of random reduced bases, with Seungki Kim, to appear, IMRN (2017).
Effective equidistribution and property (tau), with Manfred Einsiedler, Gregory Margulis, Amir Mohammadi, submitted.
Functoriality, Smith theory, and the Brauer homomorphism, with David Treumann. Annals (2016).
Torsion homology growth and cycle complexity of arithmetic manifolds, with N. Bergeron and M.H. Sengun, Duke (2016)
Hurwitz monodromy and full number fields, with D. Roberts. Algebra and number theory (2015).
A torsion Jacquet-Langlands correspondence, F. Calegari, submitted. Here is a short version which focusses on the Jacquet-Langlands correspondence, but omits many of the experiments and other material. Use the short version where possible, since it corrects a few minor errors in the long version.
Periods and harmonic analysis on spherical varieties , with Yiannis Sakellaridis, to appear Asterisque.
Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields , with Jordan Ellenberg and Craig Westerland, Annals (2016).
[The sequel to the above paper, available here, had a gap affecting the results on Cohen-Lenstra heuristics, i.e. parts of the intro, Section 6, and Section 12. Please note that the other sections are not affected, and nor is the preceding paper. See Jordan's blog post for a description of the gap. We can recover some of the original applications to Cohen-Lenstra heuristics under extra restrictions on target group and boundary monodromy; unfortunately, we have not had time to write a revision, despite my original optimistic claim that we would do so "shortly".]
A note on sphere packings in high dimension , IMRN. (2012)
Asymptotic growth of torsion homology for arithmetic groups, with N. Bergeron, JIMJ (2012).
Linnik's ergodic method and the distribution of integer points on spheres, with Jordan Ellenberg and Philippe Michel, proceedings, International Colloquium at the Tata Institute (2012).
Modelling lambda invariants by p-adic random matrices, Jordan Ellenberg and Sonal Jain, Communications in Pure and Applied Math (2011).
Statistics of number fields and function fields, with Jordan Ellenberg, Proceedings of the 2010 ICM (2010).
How small must ill-distributed sets be?, with Harald Helfgott. Analytic number theory: essays in honor of Klaus Roth (2009).
Some effective results for xaxb, with Jean Bourgain, Elon Lindenstrauss, Philippe Michel, Ergodic theory and dynamical systems (2009).
Distribution of periodic torus orbits:
- The distribution of periodic torus orbits on homogeneous spaces, with Manfred Einsiedler, Elon Lindenstrauss and Philippe Michel, DMJ (2009).
- The distribution of periodic torus orbits on homogeneous spaces II: Duke's theorem for quadratic fields. , with Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, l'Enseignment Math.
- The distribution of periodic torus orbits and Duke's theorem for cubic fields, with Manfred Einsiedler, Elon Lindenstrauss and Philippe Michel, Annals of Mathematics (2011).
The subconvexity problem for GL(2), with Philippe Michel, journal Publications Math IHES (2010).
Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces,, Manfred Einsiedler and Gregory Margulis, Inventiones Math. (2009).
Local-global principles for representations of quadratic forms, with Jordan Ellenberg, Inventiones, (2008)
Sparse equidistribution problems, period bounds, and subconvexity, Annals (2010).
Entropy bounds for Hecke eigenfunctions on division algebras , with Lior Silberman.
Appendix to "Counting maximal arithmetic subgroup", by M. Belolipetsky, with Jordan Ellenberg. Duke (2007).
The work of Einsiedler, Katok and Lindenstrauss on the Littlewood conjecture , Bulletin AMS (2007).
Quantum unique ergodicity on locally symmetric spaces, with L. Silberman. GAFA (2007).
Existence and Weyl's law for spherical cusp forms, with E. Lindenstrauss. GAFA (2007).
Reflection principles and bounds for class group torsion, with Jordan Ellenberg, IMRN (2007).
Effective computation of Maass forms, with A. Booker and A. Strombergsson journal, IMRN (2007).
The number of extensions of number field with fixed degree and bounded discriminant, with J. Ellenberg, Annals (2006).
Integral points on elliptic curves and 3-torsion in class groups, with H. Helfgott, JAMS (2006).
Equidistribution, L-functions and ergodic theory: on some problems of Yu. V. Linnik, with Ph. Michel, Proceedings of the 2006 ICM. This is a longer version than the published one.
Heegner points and nonvanishing of Rankin-Selberg L-functions, with P. Michel Analytic number theory: a tribute to Gauss and Dirichlet (2006).
Large sieve inequalities for GL(n) forms in the conductor aspect, Advances in Math, 2006.
The Burger-Sarnak method and operations on the unitary dual of GL(n), Representation Theory (2005).
On uniform bounds for rational points on non-rational curves, with J. Ellenberg, IMRN (2005).
Counting extensions of function fields with specified Galois group and bounded discriminant, with J. Ellenberg, Geometric methods in algebra and number theory (2005).
Small solutions to linear congruences and Hecke equidistribution, with A. Strombergsson, Acta Arithmetica, (2005).
Beyond endoscopy and the classification of special forms on GL(2), Crelle (2004).
Dimension of the space of cusp forms associated to 2-dimensional Galois representations, with P. Michel, IMRN (2002).
Limiting forms of the trace formula , my PhD thesis (2002).
Locally quasiprimitive graphs, with C. Prager, C.H. Li, S. Zhou, Discrete Math (2002).

I would like to acknowledge the support of the National Science Foundation, the Alfred P. Sloan foundation, and the Packard foundation. Any opinions expressed in the above papers do not necessarily reflect the views of any sponsoring agency.