Ivan Fesenko works in number theory and its intra-disciplinary interaction with other branches of mathematics.
He was awarded grants from Germany, France, Russia, Canada, the UK and the USA. He has led various research projects achieving significant outcomes. He organized 30 international conferences and symposia.
Since May 2015 he is the principal investigator of Nottingham-Oxford EPSRC Programme Grant on Symmetries and Correspondences: https://www.maths.nottingham.ac.uk/personal/ibf/symcor.html
See e.g. the wiki page
Number theory including local number theory, higher dimensional number theory and arithmetic geometry, commutative and noncommutative 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 noncommutative 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.
Local fields, higher local fields, local class field theory, higher local class field theory, infinite ramification theory, higher ramification theory, arithmetic noncommutative local class field theory, subgroups of the Nottingham group and number theory, representation theory for algebraic groups over local fields, model theoretical approach to the Manin real multiplication programme, 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, local and adelic zeta integrals, two-dimensional adelic analysis and its applications.
Two-dimensional adelic analysis, i.e. the study of zeta functions of arithmetic surfaces using two-dimensional zeta integrals.
Applications of two-dimensional adelic analysis to meromorphic continuation and functional equation of the zeta functions via mean-periodicity correspondence.
Applications of two-dimensional adelic analysis to the generalized Riemann Hypothesis for the zeta functions via positivity of the log derivative of the boundary term.
Applications of two-dimensional adelic analysis to the Birch and Swinnerton-Dyer conjecture via the boundary term and underlying adelic geometry and arithmetic structures.
Other two-dimensional arithmetic geometry theories such as inter-universal Teichmueller theory of Shinichi Mochizuki