Representation theory, mathematical physics, combinatorics, number theory, algebraic geometry, low-dimensional topology, mathematical biology, quantitative finance
My PhD was in arithmetic geometry. I obtained the p-adic uniformization of Shimura curves associated with quaternion algebras, which was later used in the proof of Fermat's Last Theorem. Then, I mostly switched to representation theory and integrable models in physics, which resulted in the definition of Double Affine Hecke Algebras, proof of the Macdonald conjectures in q-combinatorics, new theory of hypergeometric functions, theory of DAHA invariants of algebraic knots and links, new theory of Rogers-Ramanujan identities and other developments in algebra-geometry-topology and mathematical physics. My latest papers are focused on applications of DAHA knot invariants and their generalizations in number theory, including classical zeta and L-functions. Also, I recently wrote papers in financial mathematics, related to general-purpose AI, and on modeling the spread of COVID-19 via Bessel functions.
Momentum Managing Epidemic Spread and Bessel Functions
Chaos, Solitons & Fractals, 139, 110234, 2020
Jones Polynomials of Torus Knots Via DAHA
International Mathematics Research Notices , https://doi.org/10.1093/imrn/rns202, 2013
Double Affine Hecke Algebras
Cambridge University Press, https://doi:10.1017/CBO9780511546501, 2005