Ivan Fesenko

Ivan Borisovich Fesenko (Russian: Иван Борисович Фесенко; born 1962) is a mathematician working in number theory, higher arithmetic geometry and its relations with geometry, algebraic K-theory, harmonic analysis, representation theory and mathematical logic.

Education and first professional years

Fesenko got his master's degree and his PhD in St Petersburg University and worked in St Petersburg University since 1986. He was awarded a number of prizes including the Prize of the Petersburg Mathematical Society in 1992 for his work on explicit higher class field theory.^[1] Since 1995 he holds the chair in pure mathematics in University of Nottingham.

Research work

Fesenko has contributed to number theory, higher reciprocity formulas, higher class field theory, higher arithmetic geometry, higher adeles, higher zeta integrals, arithmetic zeta functions, harmonic analysis on non-locally compact groups, infinite group theory.

Fesenko discovered several new explicit formulas for the generalized Hilbert symbol on local fields and higher local fields,^[2] which belong to the branch of Vostokov's explicit formulas. He developed new class field theories which generalize class field theory to their higher-dimensional versions. He extended the explicit method of Jürgen Neukirch in class field theory to higher dimensions to deal with Galois modules which do not satisfy the property of Galois descent. Using his method he constructed an explicit class field theory for complete objects associated to arithmetic schemes such as higher local fields with last perfect residue field. ^[3][4] In this theory Milnor K-groups play a central role. He developed an explicit class field theory for local fields with perfect and imperfect residue field.^[5][6] In 2000 Fesenko initiated another branch of class field theories, a "noncommutative local class field theory" for arithmetically profinite Galois extensions of local fields^[7] which relates quotients of the field of norms with the Galois group via a 1-cocycle and provides a certain alternative to representation theoretical local Langlands correspondence.

Fesenko contributed to infinite and higher ramification theory, while in pro-p-group theory he studied subgroups of the Nottingham group and relation to infinite ramification theory; one of them, the Fesenko group, is called after him. Fesenko is a coauthor of a textbook on local fields^[8] and a coeditor of a volume on higher local fields.^[9]

Fesenko contributed to the theory of higher adeles associated to arithmetic schemes and their dualities. He pioneered the theory of analytic adelic structure associated to rank two integral structures on relative surfaces and studied its relation to the geometric adelic structure on the surface.

In harmonic analysis he developed a higher version (Fesenko measure) of the translation invariant Haar measure, integration and harmonic analysis for non locally compact objects associated to arithmetic schemes such as higher local fields and higher analytic adeles.^[10][11]

Fesenko was the first mathematician to systematically study zeta functions in higher dimensions using higher adelic zeta integrals which involve the higher Haar measure. He introduced zeta integrals on arithmetic schemes of dimension two, such as proper regular models of elliptic curves over global fields. He extended and generalized the unramified part of Tate's thesis in its two-dimensional version. The theory reduces the study of the zeta function of the surface to the study of a boundary term and its geometric and analytic properties.^[12] His work relates two adelic dualities on surfaces and measure theoretical and topological properties of subquotients of adelic spaces to fundamental properties of the arithmetic zeta functions.

Further developments of his higher adelic work led to 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 replaces by zeta functions and automorphicity is replaced by mean-periodicity. Unlike the latter, the mean-periodicity correspondence is of commutative nature.^[13][14]

Another application of his work on higher zeta integrals provides a new approach to the generalized Riemann hypothesis for the zeta function of elliptic surfaces.^[15] The higher zeta integral and explicit higher class field theory, as well as two dualities on elliptic surfaces are applied to provide a direct higher adelic relation between the analytic and arithmetic-geometric ranks of the surfaces, with key applications to the study of the Birch and Swinnerton-Dyer conjecture.^[16]

Fesenko is the principal investigator of a research team of Universities of Oxford and Nottingham supported by EPSRC Programme Grant on Symmetries and Correspondences and intra-disciplinary developments.^[17] In 2015 Fesenko published notes^[18] on inter-universal Teichmüller theory of Shinichi Mochizuki and organised the first international workshop on the IUT theory of Shinichi Mochizuki.^[19]

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