Ivan Fesenko
Encyclopedia
Ivan Fesenko is a mathematician
working in number theory
and other areas of mathematics. In 1992 Fesenko won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on class field theory.
to their higher-dimensional version for arithmetic schemes, inventing new mathematical objects and theories.
In class field theory
Fesenko constructed an explicit class field theory
for complete objects associated to arithmetic schemes, which is part of higher class field theory where Milnor K-groups of the fields play a central role. He developed an explicit class field theory theory for local fields with perfect and imperfect residue field. Fesenko initiated a "noncommutative local class field theory" for arithmetically profinite Galois extensions of local fields which relates quotients of the field of norms with the Galois group via a 1-cocycle. He is a coauthor of a popular textbook on local field
s and a coeditor of a volume on higher local field
s.
Generalizing the Haar measure
and integration to non locally compact objects associated to arithmetic schemes, Fesenko invented a translation invariant measure, integration and Fourier transform
on higher-dimensional local fields. Extending the theory of geometric adele ring
s associated to arithmetic surfaces he introduced analytic adelic objects associated to rank two integral structures and developed the theory of measure and integration on them.
Fesenko pioneered the study of zeta functions of arithmetic surfaces using zeta integrals. He introduced zeta integrals on arithmetic schemes of dimension two, generalized Tate's thesis
and proved a two-dimensional version which reduces the study of the zeta function to the study of geometric and analytic properties of adelic spaces. His theory relates adelic dualities and measure theoretical and topological properties of quotients of adelic spaces with fundamental properties of the zeta functions.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
working in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and other areas of mathematics. In 1992 Fesenko won the Young Mathematician Prize of the Petersburg Mathematical Society for his work on class field theory.
Work
Ivan Fesenko extended and generalized several core theories for one-dimensional objects in algebraic number theoryAlgebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
to their higher-dimensional version for arithmetic schemes, inventing new mathematical objects and theories.
In class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
Fesenko constructed an explicit class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
for complete objects associated to arithmetic schemes, which is part of higher class field theory where Milnor K-groups of the fields play a central role. He developed an explicit class field theory theory for local fields with perfect and imperfect residue field. Fesenko initiated a "noncommutative local class field theory" for arithmetically profinite Galois extensions of local fields which relates quotients of the field of norms with the Galois group via a 1-cocycle. He is a coauthor of a popular textbook on local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
s and a coeditor of a volume on higher local field
Higher local field
In mathematics, a higher local field is an important example of a complete discrete valuation field. The usual local fields have associated to them a unique surjective discrete valuation , this is by virtue of simply being a complete discrete...
s.
Generalizing the Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
and integration to non locally compact objects associated to arithmetic schemes, Fesenko invented a translation invariant measure, integration and Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
on higher-dimensional local fields. Extending the theory of geometric adele ring
Adele ring
In algebraic number theory and topological algebra, the adele ring is a topological ring which is built on the field of rational numbers . It involves all the completions of the field....
s associated to arithmetic surfaces he introduced analytic adelic objects associated to rank two integral structures and developed the theory of measure and integration on them.
Fesenko pioneered the study of zeta functions of arithmetic surfaces using zeta integrals. He introduced zeta integrals on arithmetic schemes of dimension two, generalized Tate's thesis
Tate's thesis
In number theory, Tate's thesis is the thesis of where he reformulated the work of Erich Hecke on L-series in terms of Fourier analysis on adelic groups....
and proved a two-dimensional version which reduces the study of the zeta function to the study of geometric and analytic properties of adelic spaces. His theory relates adelic dualities and measure theoretical and topological properties of quotients of adelic spaces with fundamental properties of the zeta functions.