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.
Action of motivic cohomology on cohomology of arithmetic groups:
 The behavior of random reduced bases, with Seungki Kim, submitted.
 Effective equidistribution and property (tau), with Manfred Einsiedler, Gregory Margulis, Amir Mohammadi, submitted.
 Functoriality, Smith theory, and the Brauer homomorphism, with David Treumann. To appear
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 JacquetLanglands correspondence,
F. Calegari, submitted. Here is a short version which focusses on the JacquetLanglands 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, submitted.
 Homological stability for Hurwitz spaces and the CohenLenstra 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 CohenLenstra 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 CohenLenstra 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 padic 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 illdistributed 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).

Localglobal 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 3torsion in class
groups, with H. Helfgott, JAMS (2006).

Equidistribution, Lfunctions 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 RankinSelberg Lfunctions,
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 BurgerSarnak method and operations on the unitary dual of GL(n),
Representation Theory (2005).

On uniform bounds for rational points on nonrational 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 2dimensional 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.