2020 (Aug)--2023(Aug)-Director-Kerala School of Mathematics, India
Research Interests
1. Class number of number fields: (i) Class numbers of number fields are mysterious objects and not much is known about these numbers. One of the many questions is to show the existence of infinitely many number fields (for quadratic field cases many results are known!) whose class number is divisible by a given integer. Quantifying these fields is also an interesting problem. The Cohen-Lenstra heuristics related to these questions is yet to be established. In the direction of indivisibility not much is known though. It's worth looking into these directions relating the arithmetic of some kind of modular forms and that of elliptic curves. (ii) There are few well known conjectures (e.g. Vandiver conjecture) relating to the class number of maximal real subfields of cyclotomic fields. The above problems of divisibility and indivisibility are worth exploring here in this set up. This will give us information about the class numbers of the corresponding cyclotomic fields too. These numbers are not easy to calculate by using the class number formula.
2. Modular relations: (i) Consider applications of the theory of zeta-functions to ‘Theoretical Computer Science’. We begin by studying the work of Anshell and Goldfeld in which they add one more condition on the coefficients of the Euler product, i.e. polynomial time evaluation algorithm for the values at prime powers. Then we study the work of Kirschenhoffer and Prodinger about the trie algorithm. In contrast to Knuth’s work, they deduce the exact formula for the expectation and for the variance they used Ramanujan’s formulas for the Riemann zeta-values involving Lambert series. We believe there must be a disguised form of the modular relation. (ii) Study Mertens’ formula for a class of zeta-functions as we have studied in one of our previous work. We also reveal the hidden structure of Vinogradov’s work on a general Mertens’ formula in number fields.
3. Chow groups, class groups and Tate-Shafarevich groups: Aimed at studying the inter-relations between these three important groups. In particular it aims at transferring certain properties of one group to the other.
Awards & Fellowships
2021- Fellow of West Bengal Academy of Science and Technology- West Bengal Academy of Science and Technology
2024-Ganesh Prasad Memorial Award-Indian Mathematical Society
Memberships
Working as a vice-president of Society for Special Functions and Applications (SSFA)
Set up the Mathematics Department at Tribhuvan University, Nepal
Organized some INSPIRE camp.
Organized 6 international conferences under the name ‘ICCGNFRT’.
