Ivan Fesenko works in number theory and higher number theory, and intra-disciplinary interaction with geometry and analysis.
He was awarded grants from Germany, Japan, France, Russia, Canada, the UK and the USA. He has organized 35 international conferences and symposia.
Among his former 50 PhD students and postdoctoral visitors several hold permanent positions in leading universities in the USA, UK, Japan, Germany, France, Lithuania, Russia, Turkey. One of his former PhD students, C. Birkar is awarded the 2018 Fields Medal.
During the period of 2015-2021 he is the principal investigator of Nottingham-Oxford EPSRC Programme Grant on Symmetries and Correspondences: https://www.maths.nottingham.ac.uk/plp/pmzibf/symcor.html
Class field theory, explicit class field theory, higher number theory and K-theory, higher class field theory, translation invariant measure and integration on higher non-locally compact objects, zeta integrals and their applications, two adelic structures on arithmetic surfaces and their interaction, applications to the BSD conjecture, anabelian geometry, inter-universal Teichmueller theory and applications
Number theory including local number theory, higher dimensional number theory and arithmetic geometry, commutative and non-commutative class field theories, zeta functions studied via zeta integrals,… read more
FESENKO, I., 2010. Analysis on arithmetic schemes. II Journal of K-theory. 5(3), 437-557
FESENKO, I., RICOTTA, G. and SUZUKI, M., 2012. Mean-periodicity and zeta functions Annales de l'Institut Fourier. 62(5), 1819-1887
Number theory including local number theory, higher dimensional number theory and arithmetic geometry, commutative and non-commutative class field theories, zeta functions studied via zeta integrals, arithmetic measure and integration theory on higher dimensional objects.
Interaction of number theory with representation theory, algebraic geometry, functional analysis, measure and integration theory, infinite group theory, mathematical logic, quantum physics.
Interaction of higher number theory with anabelian geometry and IUT. Applications of two-dimensional adelic analysis and geometry to the study of zeta functions of elliptic curves over global fields and the three main conjectures about them.
Local fields, higher local fields, local class field theory, higher local class field theory, infinite ramification theory, higher ramification theory, arithmetic non-commutative local class field theory, subgroups of the Nottingham group and number theory, representation theory for algebraic groups over local fields, model theoretic approach to zeta and L functions, measure and integration on large non locally compact spaces originating in number theory, including arithmetic loop spaces, new adelic structures on arithmetic surfaces, 2d local and adelic zeta integrals, 2d adelic analysis and its applications to the study of fundamental properties of zeta functions of arithmetic surfaces.
Further developments related to two-dimensional adelic analysis, i.e. the study of zeta functions of arithmetic surfaces using two-dimensional zeta integrals and higher class field theory.
Applications of two-dimensional adelic analysis to meromorphic continuation and functional equation of the zeta functions via mean-periodicity correspondence, to the generalized Riemann Hypothesis for the zeta functions via positivity of the log derivative of the boundary term, to the Birch and Swinnerton-Dyer conjecture via the boundary term and underlying adelic geometry and arithmetic structures.
Analysis of modern developments in algebraic number theory from the standpoint view of class field theory.
New relations between two-dimensional adelic analysis and geometry and the inter-universal Teichmueller theory of Shinichi Mochizuki.