Bannai, Kenichi

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Professor

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Career 【 Display / hide 】

  • 2001.04
    -
    2001.08

    JSPS Post Doctoral Fellow, 特別研究員・PD

  • 2001.09
    -
    2007.03

    Assistant Professor, Graduate School of Mathematics, Nagoya University, 多元数理科学研究科, 助手

  • 2005.04
    -
    2007.03

    JSPS Fellowship Abroad (ENS, Paris), 海外特別研究員（派遣先：パリ高等師範学校）

  • 2007.04
    -
    2008.03

    Nagoya University, Graduate School of Mathematics,, Assistant Professor

  • 2008.04
    -
    2012.03

    University of Keio, Department of Mathematics, Senior Assistant Professor

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Academic Background 【 Display / hide 】

  • 1995.03

    The University of Tokyo, Faculty of Science, Department of Mathematics

    University, Graduated

  • 1997.03

    The University of Tokyo, Graduate School, Division of Mathematical Sciences

    Graduate School, Completed, Master's course

  • 2000.03

    The University of Tokyo, Graduate School, Division of Mathematical Sciences

    Graduate School, Completed, Doctoral course

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Academic Degrees 【 Display / hide 】

  • 博士（数理科学）, 東京大学, Coursework, 2000.03

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Research Areas 【 Display / hide 】

  • Natural Science / Algebra

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Research Keywords 【 Display / hide 】

  • Arithmetic Geometry

  • Number Theory

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Research Themes 【 Display / hide 】

  • 代数多様体の数論幾何的予想の解決に向けた戦略的研究, 

    2009
    -
    2013

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Papers 【 Display / hide 】

  • Topological structures of large-scale interacting systems via uniform functions and forms

    Bannai K., Kametani Y., Sasada M.

    Forum of Mathematics Sigma 12 2024.11

    　View Summary

    In this article, we investigate the topological structure of large-scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large-scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large-scale interacting systems.

  • The Hodge realization of the polylogarithm and the Shintani generating class for totally real fields

    Bannai K., Bekki H., Hagihara K., Ohshita T., Yamada K., Yamamoto S.

    Advances in Mathematics 448 2024.06

    ISSN  00018708

    　View Summary

    In this article, we construct the Hodge realization of the polylogarithm class in the equivariant Deligne–Beilinson cohomology of a certain algebraic torus associated to a totally real field. We then prove that the de Rham realization of this polylogarithm gives the Shintani generating class, a cohomology class generating the values of the Lerch zeta functions of the totally real field at nonpositive integers. Inspired by this result, we give a conjecture concerning the specialization of this polylogarithm class at torsion points, and discuss its relation to the Beilinson conjecture for Hecke characters of totally real fields.

  • CANONICAL EQUIVARIANT COHOMOLOGY CLASSES GENERATING ZETA VALUES OF TOTALLY REAL FIELDS

    Bannai K., Hagihara K., Yamada K., Yamamoto S.

    Transactions of the American Mathematical Society Series B 10   613 - 635 2023

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    It is known that the special values at nonpositive integers of a Dirichlet L-function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L- functions of totally real fields. Hecke L-functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393—417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shin- tani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.References

  • p-adic polylogarithms and p-adic Hecke L-functions for totally real fields

    Bannai K., Hagihara K., Yamada K., Yamamoto S.

    Journal fur die Reine und Angewandte Mathematik (Journal fur die Reine und Angewandte Mathematik)  2022 ( 791 ) 53 - 87 2022.10

    ISSN  00754102

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    The purpose of this article is to newly define the p-adic polylogarithm as an equivariant class in the cohomology of a certain infinite disjoint union of algebraic tori associated to a totally real field. We will then express the special values of p-adic L-functions interpolating nonpositive values of Hecke L-functions of the totally real field in terms of special values of these p-adic polylogarithms.

  • Category of mixed plectic Hodge structures

    Kenichi Bannai, Kei Hagihara, Shinichi Kobayashi, Kazuki Yamada, Shuji Yamamoto, and Seidai Yasuda

    Asian J. Math. (International Press)  24 ( no.1 ) 31 - 76 2020

    Research paper (scientific journal), Joint Work, Accepted

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Papers, etc., Registered in KOARA 【 Display / hide 】

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Presentations 【 Display / hide 】

  • Shintani Revisited

    Kenichi Bannai

    Boston University/Keio University Workshop 2019 (Boston) , 

    2019.06
    , 

    Boston University

  • On the p-adic polylogarithm function for totally real fields

    Kenichi Bannai

    p-adic Cohomology and Arithmetic Geometry 2019 (東北大学片平キャンパス) , 

    2019.11

  • 総実代数体に付随する代数トーラスの新谷生成類

    坂内健一

    RIMS研究集会（公開型）「代数的整数論とその周辺」 (京都大学数理解析研究所) , 

    2019.12

  • 総実代数体に付随する代数トーラスのポリログについて

    坂内健一

    談話会 (大阪大学) , 

    2019.12

  • 総実代数体のp進ポリログとp進L関数

    坂内健一

    整数論・保形型式セミナー (大阪大学) , 

    2019.12

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • Construction of a universal theory of hydrodynamic limits by developing a new geometry on infinite product spaces

    2024.06
    -
    2027.03

    Grants-in-Aid for Scientific Research, Grant-in-Aid for Challenging Research (Exploratory), No Setting

    　View Summary

    流体力学極限は、膨大な自由度を持ち確率的に振る舞う相互作用粒子系から、時空間変数に関する適切なスケール極限を用いて、その保存量が従う決定論的な偏微分方程式を導出する手法である。非平衡統計力学を基礎付ける方法として長年盛んに研究され続け、個々のモデルに対する定理の積み重ねとして発展してきたが、既存の流体力学極限の証明は、個々のモデルの詳細に依存しており、物理的に期待されるようなロバストで普遍的な理論には程遠い。本研究は、非平衡統計力学を基礎づける重要な手法である流体力学極限の普遍的理論体系の構築を、無限直積空間上の新しい幾何学の創出によって、実現することを目指すものである。

  • Strategic research to construct motivic units using new symmetry

    2018.06
    -
    2023.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Bannai Kenichi, Grant-in-Aid for Scientific Research (S), Principal investigator

    　View Summary

    Our aim was to prospect the construction of motivic units applicable to the proof of conjectures in arithmetic geometry via a motivic object called the polylogarithm. As a concrete objective, we studied the polyogarithm on an algebraic torus associated to a totally real field with equivariant action of the unit group. We discovered the Shintani generating class which universally generates the special balues of the Hecke L-functions of the totally real field. Using a conjectural structure called a plectic structure, we formulate a precise conjecture concerning the equivariant polylogarithm and its relation to the Beilinson conjecture for the Hecke L-function of totally real fields.

  • Strategic Reseach using Eisensterin classes to prove Conjectures in Arithmetic Geometry

    2014.04
    -
    2019.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Bannai Kenichi, TAKAI Yuuki, OTA Kazuto, ONO Masataka, KIRAL Erin Mehmet, Grant-in-Aid for Scientific Research (A) , Principal investigator

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    Our original goal was to study the polylogarithm in the case of totally real fields. Our original goal was to study the polylogarithm via the Eisenstein class, but in course of our research, we realized the importance of a certain algebraic torus associated to a totally real field, and using the ideas from plectic structures proposed by Nekovar and Scholl, we succeeded in proving that the Shintani generating function which generates special values of Shintani zeta functions, defines a canonical class on the algebraic torus.

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Courses Taught 【 Display / hide 】

  • TOPICS IN ALGEBRA C

    2025

  • RESEARCH SKILLS

    2025

  • MATHEMATICS 1B

    2025

  • MATHEMATICS 1A

    2025

  • INTRODUCTION TO INTERDISCIPLINARY SCIENCE AND TECHNOLOGY

    2025

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