Ivan Fesenko is a

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

working in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

and its interaction with other areas of modern mathematics.

Education

Fesenko was educated at

St. Petersburg State University Saint Petersburg State University (SPBU; russian: Санкт-Петербургский государственный университет) is a public research university in Saint Petersburg, Russia. Founded in 1724 by a decree of Peter the G ...

where he was awarded a

PhD PHD or PhD may refer to: * Doctor of Philosophy (PhD), an academic qualification Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * ''Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group ** Ph.D. (Ph.D. albu ...

in 1987.

Career and research

Fesenko was awarded the Prize of the

Petersburg Mathematical Society The Saint Petersburg Mathematical Society (russian: Санкт-Петербургское математическое общество) is a mathematical society run by Saint Petersburg mathematicians. Historical notes The St. Petersburg Mathe ...

in 1992. Since 1995, he is professor in pure mathematics at University of Nottingham. He contributed to several areas of number theory such as class field theory and its generalizations, as well as to various related developments in pure mathematics. Fesenko contributed to explicit formulas for the generalized

Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...

symbol on

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...

s and

higher local field In mathematics, a higher (-dimensional) local field is an important example of a complete discrete valuation field. Such fields are also sometimes called multi-dimensional local fields. On the usual local fields (typically completions of number f ...

, higher

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

, p-class field theory, arithmetic noncommutative local class field theory. He coauthored a textbook on

s and a volume on

s. Fesenko discovered a higher Haar measure and integration on various higher local and adelic objects. He pioneered the study of

zeta functions In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * Ar ...

in higher dimensions by developing his theory of higher adelic zeta integrals. These integrals are defined using the higher Haar measure and objects from higher class field theory. Fesenko generalized the Iwasawa-Tate theory from 1-dimensional global fields to 2-dimensional arithmetic surfaces such as proper regular models of

elliptic curves In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

over global fields. His theory led to three further developments. The first development is the study of functional equation and meromorphic continuation of the Hasse zeta function of a proper regular model of an elliptic curve over a global field. This study led Fesenko to introduce 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. This correspondence can be viewed as a weaker version of the

Langlands correspondence In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

, where L-functions and replaced by zeta functions and automorphicity is replaced by mean-periodicity. This work was followed by a joint work with Suzuki and Ricotta. The second development is an application to the

generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...

, 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. The third development is a higher adelic study of relations between the arithmetic and analytic ranks of an elliptic curve over a global field, which in conjectural form are stated in the

Birch and Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory an ...

for the zeta function of elliptic surfaces. This new method uses FIT theory, two adelic structures: the geometric additive adelic structure and the arithmetic multiplicative adelic structure and an interplay between them motivated by higher class field theory. These two adelic structures have some similarity to two symmetries in

inter-universal Teichmüller theory Inter-universal Teichmüller theory (abbreviated as IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arith ...

of Mochizuki. His contributions include his analysis of class field theories and their main generalizations.

Other contributions

In his study of infinite ramification theory, Fesenko introduced a torsion free hereditarily just infinite closed subgroup of the Nottingham group. Fesenko played an active role in organizing the study of

Shinichi Mochizuki is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperboli ...

. He is the author of a survey and a general article on this theory. He co-organized two international workshops on IUT.

Selected publications

References

{{DEFAULTSORT:Fesenko, Ivan 1962 births 20th-century Russian mathematicians 21st-century Russian mathematicians Living people Number theorists