The Infosys Prize 2022 in Mathematics is awarded to Prof. Mahesh Kakde of Indian Institute of Science, Bengaluru, for his outstanding contributions to algebraic number theory. Prof. Kakde’s deep work on the noncommutative Iwasawa main conjecture, his work on the Gross-Stark conjecture (with Samit Dasgupta and Kevin Ventullo), and his work on the Brumer-Stark conjecture (with Samit Dasgupta), resolves outstanding conjectures at the heart of modern number theory.

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### Towers of infinite possibility

### Scope and Impact of Work

Prof. Mahesh Kakde’s early work in his thesis and soon after made a decisive contribution to the main conjecture of non-commutative Iwasawa theory for the Tate motive. It used an insightful and ingenious combination of algebraic techniques and more analytic techniques drawn from the theory of modular forms. This has been a hallmark of his subsequent work as well.

Main conjectures in Iwasawa theory have played a huge role in modern number theory. Ideas from Iwasawa theory were a key inspiration for Wiles’s proof of Fermat’s Last Theorem. Prof. Kakde’s work since his thesis has been in the area of Iwasawa theory and related areas and thus at the center of current work in algebraic number theory.

Kakde’s work on the Gross-Stark conjecture (with Dasgupta and Ventullo) resolved an outstanding conjecture about leading terms of abelian p-adic L-functions and their relation to determinants made from units when the p-adic L-function vanished to order greater than one. This had been thought out of reach by experts in the area. A deepening of the ideas of this work, that again involved a deft combination of algebraic insights and a beautiful use of higher congruences between modular forms, led Prof. Kakde and Dasgupta to their landmark proof of the Brumer-Stark conjecture. This has led them to explicit p-adic constructions of units in abelian extensions of number fields, thus making a contribution to Hilbert’s 12th problem.

### Bio

Mahesh Kakde has been working at the Indian Institute of Science, Bengaluru since 2019 as Professor of Mathematics. Prior to that, he was at King's College London.

Born in Akola, Maharashtra in 1983, Prof. Mahesh Kakde did his B. Math. from Indian Statistical Institute, Bengaluru, and went on to complete his Ph.D. at the University of Cambridge in 2008, specializing in number theory. During his Ph.D., and shortly afterwards, he contributed essentially to Iwasawa theory. Prof. Kakde’s proof of the “Noncommutative Iwasawa Main Conjectures over Totally Real Fields” subsequently became the subject of conference proceedings dedicated to the topic.

Kakde was a Simons Visiting Professor, at The Centre de Recherches Mathématiques (CRM), Montreal, in 2020. Along with Samit Dasgupta, he delivered an invited talk titled "On the Brumer-Stark Conjecture and Refinements" at the International Congress of Mathematicians in 2022.

## Timeline

### Jury Citation

The relation between certain analytic functions called L-functions and prime numbers and other objects of interest in number theory emerged in the 19th century. Dirichlet’s proof of the infinitude of primes in arithmetic progression using Dirichlet L-functions is an early instance of this relation.

Beginning in the middle of the 20th century, p-adic analytic variants of these L-functions for prime numbers p have played a distinguished role in algebraic number theory. Iwasawa theory posits a deep relation between p-adic L-functions and arithmetic objects of interest to number theorists. Prof. Mahesh Kakde’s work on special values of p-adic L-functions and their relation to class groups and units of number fields resolves several outstanding conjectures in the subject. The methods used in his work (some of it joint with Dasgupta and Ventullo) are intricate and hinge on a deep study of congruences between modular forms.

Resolving Modern Number Theory's Outstanding Conjectures

“I would like to congratulate Mahesh for winning the Infosys Prize 2022. His work on the main conjecture of non-commutative Iwasawa theory, on the Gross-Stark conjecture and on the Brumer-Stark conjecture has had a big impact on the field of algebraic number theory. His work makes important progress towards a p-adic analytic analog of Hilbert’s 12th problem on construction of abelian extensions of number fields.”