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Ivan Fesenko

Professor in Pure Mathematics, Faculty of Science

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Biography

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

Expertise Summary

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

Teaching Summary

Mathematics

Research Summary

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

Selected Publications

Current Research

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.

Past Research

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.

Future Research

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.

  • 2019. On asymptotic equivalence of classes of elliptic curves over Q Available at: <https://www.maths.nottingham.ac.uk/plp/pmzibf/asym2.pdf>
  • 2019. Class field theory, its three main generalisations, and applications Available at: <https://www.maths.nottingham.ac.uk/plp/pmzibf/232.pdf> (In Press.)
  • FESENKO I.B., VOSTOKOV S.V. and YOON S.H., 2018. Generalised Kawada--Satake method for Mackey functors in class field theory, Europ. J. Math.. 4, 953-987 (In Press.)
  • FESENKO, I., 2016. Fukugen Inference: International Review of Science. 2(3),
  • FESENKO, I., 2015. Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces Moscow Mathematical Journal. 15(3), 435-453
  • FESENKO, I., RICOTTA, G. and SUZUKI, M., 2012. Mean-periodicity and zeta functions Annales de l'Institut Fourier. 62(5), 1819-1887
  • FESENKO, I., 2010. Analysis on arithmetic schemes. II Journal of K-theory. 5(3), 437-557
  • FESENKO, I., FRIEDLANDER, E., MERKURIEV, A. and REHMANN, U., eds., 2010. Documenta Mathematica Volume dedicated to A.A. Suslin Bielefeld : Deutsche Mathematiker-Vereinigung. (In Press.)
  • FESENKO, I., 2008. Adelic approach to the zeta function of arithmetic schemes in dimension two Moscow Mathematical Journal. 8(2), 273-317
  • FESENKO, I., 2008. Model theory guidance in number theory?. In: CHATZIDAKIS, Z., MACPHERSON, D., PILLAY, A. and WILKIE, A., eds., Model theory with applications to algebra and analysis Cambridge University Press. 327-334
  • I. FESENKO, 2006. In: V. KAIMANOVICH, A. LODKIN, ed., Representation Theory, Dynamical Systems, and Asymptotic Combinatorics: Advances in the Mathematical Sciences 217. American Mathematical Society. 37-50
  • FESENKO, I., 2005. Measure, integration and elements of harmonic analysis on generalized loop spaces. In: URALTSEVA, N.N., ed., Proceedings of St Petersburg Mathematical Society: Dedicated to Sergey Vladimirovich Vostokov 12. Providence, RI: American Mathematical Society. 149-165
  • FESENKO, I., 2005. On the image of noncommutative local reciprocity map Homology, Homotopy and Applications. VOL 7(NUMB 3), 53-62
  • FESENKO, I., 2003. Analysis on arithmetic schemes. I. In: BLOCH, S., FESENKO, I., ILLUSIE, L., KURIHARA, M., SAITO, S., SAITO, T. and SCHNEIDER, P., eds., A collection of manuscripts written in honour of Kazuya Kato on the occasion of his fiftieth birthday Bielefeld : Deutsche Mathematiker-Vereinigung. 261-284
  • FESENKO, I.B and VOSTOKOV, S.V., 2002. Local fields and their extensions 2nd ed. American Mathematical Society, Providence, Rhode Island.
  • FESENKO, I., 2002. Sequential topologies and quotients of Milnor K-groups of multidimensional local fields St. Petersburg Mathematical Journal. VOL 13(PART 3), 485-502
  • FESENKO, I., 2001. Nonabelian local reciprocity maps. In: MIYAKE, K., ed., Class Field Theory - Its Centenary and Prospect: Advanced Studies in Pure Mathematics 30. Mathematical Society of Japan, Tokyo, Japan. 63-78
  • DU SAUTOY, M. and FESENKO, I., 2000. Where the wild things are: ramification groups and the Nottingham group. In: New horizons in pro-p-groups Birkhäuser Publishing Ltd, Basel, CHE. 287-328
  • FESENKO, I., 2000. Topological Milnor <em>K</em>-groups of higher local fields. In: Invitation to higher local fields 3. Geometry & Topology Publications, University of Warwick, Mathematics Institute. 61-74
  • FESENKO, I., 2000. Local reciprocity cycles. In: Invitation to higher local fields 3. Geometry & Topology Publications, University of Warwick, Mathematics Institute. 293-298
  • FESENKO, I., 2000. Parshin's higher local class field theory in characteristic <em>p</em>. In: Invitation to higher local fields 3. Geometry & Topology Publications, University of Warwick, Mathematics Institute. 75-79
  • FESENKO, I., 2000. Explicit higher local class field theory. In: Invitation to higher local fields 3. Geometry & Topology Publications, University of Warwick, Mathematics Institute. 95-101
  • FESENKO, I., 2000. Higher class field theory without using <em>K</em>-groups. In: Invitation to higher local fields 3. Geometry & Topology Publications, University of Warwick, Mathematics Institute. 137-142
  • FESENKO, I. and KURIHARA, M., eds., 2000. Invitation to higher local fields, Geometry and Topology Monographs Geometry & Topology Publications, University of Warwick, Mathematics Institute.
  • EFRAT, I. and FESENKO, I., 1999. Fields Galois-equivalent to a local field of positive characteristic Mathematical Research Letters. VOL 6(NUMBER 3/4), 345-356
  • FESENKO, I., 1999. On just infinite pro-p-groups and arithmetically profinite extensions of local fields Journal für die reine und angewandte Mathematik. ISSU 517, 61-80
  • FESENKO,I.B., 1998. On deeply ramified extensions Journal of the London Mathematical Society. 57, 325-335
  • FESENKO, I. B., 1996. On general local reciprocity maps Journal für die reine und angewandte Mathematik. ISSUE 473, 207-222
  • FESENKO, I., 1996. Complete discrete valuation fields. Abelian local class field theories. In: Handbook of Algebra 1. Elsevier Science BV, Amsterdam. 221-268
  • FESENKO, I., 1996. Abelian extensions of complete descrete valuation fields In: Number Theory. 47-74

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