Ivan Fesenko

Ivan Borisovich Fesenko (Russian: Иван Борисович Фесенко; born 1962) is a mathematician working in number theory.

Education and professional years

In 1979 Ivan Fesenko was the winner of All-Russian mathematical olympiad. He got his PhD in St Petersburg University and worked at St Petersburg University since 1986. He was awarded a number of prizes including the Prize of the Petersburg Mathematical Society.^[1] Since 1995 he is professor in pure mathematics at University of Nottingham. Since 2015 he is the principal investigator of a research team of Universities of Nottingham and Oxford supported by EPSRC Programme Grant on Symmetries and Correspondences and intra-disciplinary developments.^[2]

Contributions

Ivan Fesenko has contributed to number theory, algebraic K-theory, infinite group theory, measure and integration theory. In number theory he played a fundamental role in explicit reciprocity formulas, class field theories, higher local fields, higher class field theories, higher adelic structures, higher zeta integrals, arithmetic zeta functions, and other areas. He is a coauthor of a textbook on local fields^[3] and a coeditor of a volume on higher local fields.^[4]

Work in local number theory and class field theory

Fesenko discovered several types of explicit formulas for the generalized Hilbert symbol on local fields and higher local fields,^[5] which belong to the branch of Vostokov's explicit formulas. He developed several generalizations of class field theory. He extended the explicit method of Jürgen Neukirch in class field theory in various directions, to deal with generalized class formations which do not satisfy the property of Galois descent. Together with Kazuya Kato, he is the main contributor to higher local class field theory.^[6][7] This theory uses Milnor K-groups instead of the multiplicative group of a usual local field with finite residue field. He constructred explicit p-class field theory for local fields with perfect and imperfect residue field where indices of norm groups can be infinite.^[8][9] In 2000 he initiated "noncommutative local class field theory" for arithmetically profinite Galois extensions of local fields.^[10] This arithmetic theory can be viewed as an alternative to the representation theoretical local Langlands correspondence.

Work on higher Haar measure and harmonic analysis on higher local fields and other non locally compact groups

Solving a long-standing problem, Fesenko invented a "higher Haar measure" on various local and adelic objects, such as higher local fields, associated to arithmetic schemes of arbitrary dimension.^[11][12] This measure takes values in the formal power series over complex numbers, while in dimension one the theory is the usual Haar measure. This measure is associated to integral structures of corank n of n-dimensional local fields and other objects. The measure is translation invariant, finitely additive and countably additive in some refined sense. In the case of formal power series over reals or complex numbers, the associated integral has various links with Feynman path integral.^[13]

Work in higher adelic geometry and zeta functions

Fesenko discovered a new adelic structure on relative surfaces, called an analytic adelic structure. He studied zeta functions in higher dimensions using higher adelic zeta integrals which he defined using his higher measure. The zeta integral of a surface is closely related to the square of the zeta function of the surface. Using the higher adelic zeta integral Fesenko generalized Iwasawa-Tate theory and Tate's thesis from 1-dimensional global fields to 2-dimensional objects such as proper regular models of elliptic curves over global fields. Using higher adelic duality and higher harmonic analysis, this theory reduces the study of the zeta function of the surface to the study of a boundary integral and its analytic properties.^[14]

Three associated developments have started with the proof of the two-dimensional version of Iwasawa-Tate theory. The first development is the study of functional equation and meromorphic continuation of the zeta function of a proper regular model of an elliptic curve over a global field, and hence the study of the same aspects of the L-function of the curve. This development s led to the emergence and statement of a new mean-periodicity correspondence between the arithmetic zeta functions and mean-periodic elements of the space of smooth functions on the real line of not more than exponential growth at infinity. It can be viewed as a weaker version of Langlands correspondence where L-functions and replaced by zeta functions and automorphicity is replaced by mean-periodicity.^[15][16] The second development is an application to the generalized Riemann hypothesis, which in this higher theory is reduced to a certain positivity property of small derivatives of the boundary function and to the properties of the spectrum of the Laplace transform of the boundary function.^[17][18] The third application provides a new higher adelic method to relate the arithmetic and analytic ranks in the Birch and Swinnerton-Dyer conjecture for the zeta function of elliptic surfaces.^[19][20] This method uses two higher adelic structures on arithmetic surfaces: a geometric additive structure and an arithmetic multiplicative structure and an interplay between them via the reciprocity map in class field theory.^[21]

Other contributions

Ivan Fesenko played a leading role in organizing the study of inter-universal Teichmüller theory (IUT) of Shinichi Mochizuki. He published notes^[22] on this theory and co-organized two international workshops on IUT.^{[23] [24]}

References

External links

Wikimedia Commons has media related to Ivan Fesenko.

• Ivan Fesenko's webpage
• Research papers of Ivan Fesenko
• Ivan Fesenko at the Mathematics Genealogy Project