Ivan Fesenko

Fesenko was educated at St. Petersburg State University where he was awarded a PhD in 1987.

After continuing at St. Petersburg State University as assistant and associate professor, he became professor in pure mathematics at the University of Nottingham in the UK. He moved to Westlake University in China as a distinguished professor of mathematics in 2023.{{cite web|url=https://en.westlake.edu.cn/news_events/westlakenews/Faculty_Staff/202401/t20240118_36677.html|title=Former Nottingham University Professor Ivan Fesenko Joins Westlake University Full-time|publisher=Westlake University|date=December 2023|access-date=2024-07-27}}

Fesenko was awarded the Prize of the Petersburg Mathematical Society{{cite web|url=http://www.mathsoc.spb.ru/mol_mat.html|title=Prize of the Petersburg Mathematical Society}} in 1992.

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 symbol on local fields and higher local field,{{Cite book |last1=Fesenko |first1=I. B.|last2=Vostokov |first2=S. V. |title=Local Fields and Their Extensions, Second Revised Edition, American Mathematical Society |year=2002 |publisher=American Mathematical Soc. |isbn=978-0-8218-3259-2 |url=https://books.google.com/books?id=CQXTAQAAQBAJ}} higher class field theory,{{Cite journal |title=Class field theory of multidimensional local fields of characteristic 0, with the residue field of positive characteristic |first=I. |last=Fesenko |journal=St. Petersburg Mathematical Journal |volume=3 |year=1992 |pages=649–678}}{{Cite journal |last=Fesenko |first=I. |title=Abelian local p-class field theory|journal=Math. Ann. |volume=301 |year=1995 |pages=561–586 |doi=10.1007/bf01446646|s2cid=124638476 }} p-class field theory,{{cite journal |title=Local class field theory: perfect residue field case |first=I.|last=Fesenko | publisher=Russian Academy of Sciences |journal=Izvestiya: Mathematics |volume=43 |number=1 |year=1994 |pages=65–81|doi=10.1070/IM1994v043n01ABEH001559|bibcode=1994IzMat..43...65F}}{{Cite journal |last=Fesenko |first=I. |title=On general local reciprocity maps|journal=Journal für die reine und angewandte Mathematik |volume=473 |year=1996 |pages=207–222 }} arithmetic noncommutative local class field theory.{{Cite book |last=Fesenko |first=I. |chapter=Nonabelian local reciprocity maps |title=Class Field Theory – Its Centenary and Prospect, Advanced Studies in Pure Math |year=2001 |pages=63–78 | isbn = 4-931469-11-6}}

He coauthored a textbook on local fields{{Cite book |last1=Fesenko |first1=I. B.|last2=Vostokov |first2=S. V. |title=Local Fields and Their Extensions, Second Revised Edition, American Mathematical Society |year=2002 |isbn=978-0-8218-3259-2 |url=https://www.maths.nottingham.ac.uk/personal/ibf/book/book.html}} and a volume on higher local fields.{{Cite journal|last1=Fesenko |first1=I.|last2=Kurihara |first2=M. | title=Invitation to higher local fields, Geometry and Topology Monographs|journal=Geometry and Topology Monographs|url=http://www.msp.warwick.ac.uk/gtm/2000/03/ |issn=1464-8997 |publisher=Geometry and Topology Publications|year=2000 |arxiv=math/0012131}}

Fesenko discovered a higher Haar measure and integration on various higher local and adelic objects.{{cite journal |url=http://www.mathematik.uni-bielefeld.de/documenta/vol-kato/vol-kato.html |title=Analysis on arithmetic schemes. I|first=I.|last= Fesenko|journal=Documenta Mathematica|year=2003 |pages=261–284 |isbn=978-3-936609-21-9}}{{Cite journal |last=Fesenko |first=I. |title=Adelic study of the zeta function of arithmetic schemes in dimension two|journal=Moscow Mathematical Journal |volume=8|year=2008 |pages=273–317 |doi=10.17323/1609-4514-2008-8-2-273-317 }} He pioneered the study of zeta functions 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 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, where L-functions and replaced by zeta functions and automorphicity is replaced by mean-periodicity.{{cite journal |url=https://www.maths.nottingham.ac.uk/personal/ibf/a2.pdf|title=Analysis on arithmetic schemes. II|first=I.|last=Fesenko| journal=Journal of K-theory |volume=5|year=2010 |issue=3|pages=437–557|doi=10.1017/is010004028jkt103}} This work was followed by a joint work with Suzuki and Ricotta.{{Cite journal |last1=Fesenko |first1=I. |last2=Ricotta|first2=G.|last3=Suzuki|first3=M.|title=Mean-periodicity and zeta functions|journal=Annales de l'Institut Fourier|volume=62|year=2012 |issue=5 |pages=1819–1887 |doi= 10.5802/aif.2737|arxiv=0803.2821|s2cid=14781708 }}

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.{{Cite journal |last=Fesenko |first=I. |title=Adelic study of the zeta function of arithmetic schemes in dimension two|journal=Moscow Mathematical Journal |volume=8|year=2008 |pages=273–317|doi=10.17323/1609-4514-2008-8-2-273-317 }}{{cite journal |url=https://www.maths.nottingham.ac.uk/personal/ibf/a2.pdf|

title=Analysis on arithmetic schemes. II|first=I.|last=Fesenko| journal=Journal of K-theory |volume=5|year=2010 |

issue=3|pages=437–557|

doi=10.1017/is010004028jkt103}}

{{Cite journal |last=Suzuki |first=M.|title=Positivity of certain functions associated with analysis on elliptic surfaces|journal=J. Number Theory |volume=131 |year=2011|issue=10|pages= 1770–1796|doi=10.1016/j.jnt.2011.03.007|s2cid=14225498|doi-access=free}}

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 for the zeta function of elliptic surfaces.{{Cite journal |last=Fesenko |first=I. |title=Adelic study of the zeta function of arithmetic schemes in dimension two|journal=Moscow Mathematical Journal |volume=8|year=2008 |pages=273–317 |doi=10.17323/1609-4514-2008-8-2-273-317 }}{{cite journal |url=https://www.maths.nottingham.ac.uk/personal/ibf/a2.pdf|

title=Analysis on arithmetic schemes. II|first=I.|last=Fesenko| journal=Journal of K-theory |volume=5 |year=2010 |

issue=3|pages=437–557|

doi=10.1017/is010004028jkt103}} 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 of Mochizuki.{{cite journal |last=Fesenko |first=I. |title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki |journal=Europ. J. Math. |volume=1 |year=2015 |issue=3 |pages=405–440 |doi=10.1007/s40879-015-0066-0 |s2cid=52085917 |doi-access=free }}

His contributions include his analysis of class field theories and their main generalizations.{{cite web|url=https://www.maths.nottingham.ac.uk/personal/ibf/232.pdf|last=Fesenko |first=I. |title=Class field theory guidance and three fundamental developments in arithmetic of elliptic curves}}

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 inter-universal Teichmüller theory of Shinichi Mochizuki. He is the author of a survey{{cite journal |last=Fesenko |first=I. |title=Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki |journal=Europ. J. Math. |volume=1 |year=2015 |issue=3 |pages=405–440 |doi=10.1007/s40879-015-0066-0 |s2cid=52085917 |doi-access=free }} and a general article{{cite journal |url=http://inference-review.com/article/fukugen|last=Fesenko |first=I.|title=Fukugen|journal=Inference: International Review of Science|volume=2|year=2016|issue=3 |doi=10.37282/991819.16.25 }} on this theory. He co-organized two international workshops on IUT.{{cite journal|url=https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html|title=Oxford Workshop on IUT theory of Shinichi Mochizuki |date=December 2015}}{{cite web|url=https://www.maths.nottingham.ac.uk/personal/ibf/files/kyoto.iut.html|title=Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop), July 18-27 2016}}

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Category:1962 births

Category:20th-century Russian mathematicians

Category:21st-century Russian mathematicians

Category:Living people

Category:Number theorists

Category:Academics of the University of Nottingham