# Akshay Venkatesh

Akshay Venkatesh (born November 21, 1981 in New Delhi ) is an Indo-Australian mathematician who deals with number theory , ergodic theory and automorphic forms. In 2018 he received the Fields Medal .

## Life

Venkatesh grew up in Perth , Australia. In 1994 he received a bronze medal at the International Mathematical Olympiad . From 1995 he studied mathematics at the University of Western Australia (Bachelor 1997 with first class honors ). From 1998 he was at Princeton University with Peter Sarnak , where he received his doctorate in 2002 (Limiting forms of the trace formula). As a post-doc , he was a Moore Instructor at the Massachusetts Institute of Technology . From 2004 he was Associate Professor at the Courant Institute of Mathematical Sciences of New York University and from 2008 Professor at Stanford University . Since 2018 he has been Professor at the Institute for Advanced Study (IAS) in Princeton, New Jersey , where he was previously Distinguished Visiting Professor in the 2017/18 academic year.

From 2004 to 2006 he was a Clay Research Fellow. In 2007 he was a Packard Fellow and received the Salem Prize . In 2008 he won the SASTRA Ramanujan Prize and in 2016 the Infosys Prize. For 2017 he was awarded the Ostrowski Prize . Venkatesh has been a member of the Royal Society since 2019 .

In 2006 he gave a lecture at the International Congress of Mathematicians in Madrid (Equidistribution, L-functions and ergodic theory: on some problems of Juri Linnik , with Philippe Michel ) and in 2010 he was an Invited Speaker at the ICM in Hyderabad ( Statistics of number fields and function fields with Jordan S. Ellenberg ). At the ICM 2018 in Rio de Janeiro he received the Fields Medal "for his synthesis of analytical number theory, homogeneous dynamics, topology and representation theory, which solved long open assumptions about the equal distribution of arithmetic objects" (laudation).

## plant

With Jordan S. Ellenberg, he applied methods of ergodic theory to the question of the representation of integer quadratic forms by those with fewer variables and demonstrated the validity of a local-global principle (in the sense of Helmut Hasse ).

Partly with Elon Lindenstrauss , Manfred Einsiedler and Grigori Margulis , he dealt with questions of equal distribution in homogeneous spaces . He proved the equal distribution of the orbits of many semi-simple groups with Einsiedler, Margulis and Amir Mohammadi and with Einsiedler, Lindenstrauss and Michel the equal distribution of periodic orbits in the locally symmetrical space , which is related to the distribution of ideal classes of totally real cubic number fields in the limit of infinite discriminants. ${\ displaystyle SL (3, \ mathbb {Z}) / SL (3, \ mathbb {R})}$

With Lindenstrauss he proved Sarnak's conjecture about the validity of Hermann Weyl's law for tip shapes as eigenfunctions of the Laplace operator in locally symmetric spaces . In its original Weyl form, this law establishes a connection between the number of eigenvalues ​​of the Laplace operator and the volume of the manifold. Locally symmetrical spaces are given by the formation of quotients according to a discrete subgroup in a large class of algebraic groups . With Lior Silberman he also made progress on another hypothesis of Sarnak, the QUE conjecture ( quantum unique ergodicity, with Zeev Rudnick ).

Also with Ellenberg, he improved the upper bound (asymptotically for large degrees) of the number of number fields of fixed degree with limited discriminants . Manjul Bhargava had previously dealt with the special case of number fields with degrees less than 5. The work was, as he said in an interview (Quanta Magazine 2018), a psychological breakthrough for Venkatesh, as it showed him in his post-doctoral time that he could discover new things in fields of his own choosing (for his dissertation, his doctoral supervisor Sarnak had the Question still suggested).

In the analytical theory of automorphic forms he made progress (partly with Philippe Michel ) in the question of sub-convexity bounds for L-functions of automorphic representations on the critical line. The problem also has applications in questions of uniformity in the geometry of numbers. The method proposed by Venkatesh in 2004 from the theory of dynamic systems (ergodic theory) enabled a completely new, more general approach in this area. In particular, he was able to deal with all sub-convexity issues for group GL (2).

With Harald Helfgott , he gave new limits for the number of integer points on elliptic curves .

With Craig Westerland and Ellenberg he proved special cases of the Cohen - Lenstra conjectures about class groups in the function body case.

In the 2010s he deals with the role of torsion (in the homology of arithmetic groups) in the Langlands program , in part with Nicolas Bergeron and Frank Calegari . He made a number of conjectures, for example with Kartik Prasanna about the coherence of the cohomology of arithmetic groups with motivic cohomology in the context of the Beilinson conjectures about special values ​​of L-functions.

In 2012 he found a mistake with Vesselin Dimitrov in the attempt to prove the abc conjecture by Shin'ichi Mochizuki (part 3, 4 of his preprint series). He admitted the mistake, but felt that it should be corrected, and subsequently published revisions of his work.

In 2018 he and Brian Lawrence gave a new proof of Faltings ' theorem , which still follows the basic structure of Faltings, but uses the analysis of the variation of p-adic Galois representations instead of Abelian varieties.

## Fonts (selection)

• With Michel: Equidistribution, L-functions and ergodic theory: on some problems of Yu. Linnik. International Congress of Mathematicians. Vol. II, pp. 421-457, Eur. Math. Soc., Zurich 2006.
• With Michel: The subconvexity problem for GL 2 . In: Publ. Math. Inst. Hautes Études Sci. , No. 111, 2010, pp. 171-271.
• Sparse equidistribution problems, period bounds and subconvexity. In: Ann. of Math. , (2) 172, 2010, No. 2, pp. 989-1094.
• With Ellenberg, Westerland: Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields . In: Ann. of Math. , (2) 183, 2016, No. 3, pp. 729-786.

## Individual evidence

1. ^ Ostrowski Prize 2017.
2. For his synthesis of analytic number theory, homogeneous dynamics, topology, and representation theory, which has resolved long-standing problems in areas such as the equidistribution of arithmetic objects. " Official website of the IMU for the Fields Medal.
3. From p-adic groups, Marina Ratner's theorem .
4. Local global principles for representations of quadratic forms . In: Inventiones Mathematicae , Volume 171, 2008, p. 257, arxiv : math / 0604232 .
5. Einsiedler, Lindenstrauss, Margulis, Venkatesh: Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces . 2007, arxiv : 0708.4040 .
6. Einsiedler, Margulis, Mohammadi, Venkatesh: Effective equidistribution and property tau. 2015, arxiv : 1503.05884 , will appear in: J. Am. Math. Soc.
7. Manfred Einsiedler, Elon Lindenstrauss, Philippe Michel, Akshay Venkatesh: Distribution of periodic torus orbits and Duke's theorem for cubic fields. In: Annals of Mathematics , Volume 173, 2010, pp. 815-885, 2007 arxiv : 0708.1113 .
8. Lindenstrauss, Venkatesh: Existence and Weyl's law for spherical cusp forms. In: Geom. Funct. Anal. , Volume 17, 2007, pp. 220-251, arxiv : math / 0503724 .
9. ^ Silberman, Venkatesh: On Quantum unique ergodicity for locally symmetric spaces I. In: Geom. Funct. Anal. , Volume 17, 2007, pp. 960-998, arxiv : math / 0407413 .
10. Regarding the limit given by Wolfgang Schmidt , Asterisque, Volume 228, 1995, p. 189.
11. ^ Ellenberg, Venkatesh: The number of extensions of a number field with fixed degree and bounded discriminant. In: Annals of Mathematics , Volume 163, 2006, pp. 723-741, arxiv : math / 0309153 .
12. ^ Michel Venkatesh: Equidistribution, L-Functions and Ergodic theory: on some problems of Juri Linnik. Lecture, International Congress of Mathematicians 2006.
13. ^ Venkatesh: Sparse equidistribution problems, period bounds and subconvexity . In: Annals of Mathematics , Volume 172, 2010, pp. 989-1094, 2005 arxiv : math / 0506224 .
14. Michel, Venkatesh: Subconvexity Problem for GL (2) . In: Pub. Math. IHES , Volume 111, 2010, pp. 171-280, arxiv : 0903.3591 .
15. Integral points on elliptic curves and 3-torsion in class groups. In: American J. Math. , Volume 19, 2006, p. 527, arxiv : math / 0405180 .
16. Ellenberg, Venkatesh, Westerland: Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields. In: Annals of Mathematics , Volume 183, 2016, pp. 729-786, 2009 arxiv : 0912.0325 .
17. ^ Calegari, Venkatesh: A torsion Jacquet-Langlands correspondence. arxiv : 1212.3847 Arxiv 2012.
18. Prasanna, Venkatesh: Automorphic cohomology, motivic cohomology, and the adjoint L-function . 2016, arxiv : 1609.06370 .
19. Kevin Hartnett: An abc proof too tough even for mathematicians. In: Mircea Pitici (ed.): The best writings in mathematics 2013. Princeton UP, 2014, p. 228, originally Boston Globe , November 4, 2012.
20. Lawrence, Venkatesh: Diophantine problems and p-adic period mappings. 2018, arxiv : 1807.02721 .