Derived geometry and duality

• Project/Area Number: 25800001
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tohoku University
• Principal Investigator: Iwanari Isamu 東北大学, 理学研究科, 准教授 (70532547)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 淡中双対 / 高次圏 / モチーフ / DG代数 / 変形理論 / 有理ホモトピー / 基本群 / 安定∞圏 / 周期写像 / 導来代数幾何学 / 混合モチーフ / オペラッド / ガロア群

Outline of Final Research Achievements

The principal purpose of this program is to study a duality of tannakian type for higher categories and to apply it to various theory such as mixed motives. I proved a tannakian characterization theorem for symmetric monoidal stable infinity-categories that satisfy a certain simple condition (so-called fine tannakian infinity-categories). I applied this theory to mixed motives to obtain motivic Galois stacks and associated motivic Galois group. I also define a motivic rational homotopy type and its relation with motivic Galois actions and related notions. I applied the tannaka duality theory to motivic rational homotopy types.

Research Products

• [Journal Article] Tannakization in derived algebraic geometry (2015)
  • Author(s): Isamu Iwanari
  • Journal Title: Journal of K-Theory Volume: Volume 14 Issue: 3 Pages: 642-700
  • DOI: 10.1017/is014008019jkt278
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bar construction and tannakization (2014)
  • Author(s): Isamu Iwanari
  • Journal Title: Publication of Research Institute of Mathematical Sciences Volume: Volume 50 Issue: 3 Pages: 515-568
  • DOI: 10.4171/prims/143
  • NAID: 110009851501
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Monoidal infinity category of complexes from tannakian viewpoint (2012)
  • Author(s): Hiroshi Fukuyama and Isamu Iwanari
  • Journal Title: Mathematiche Annalen Volume: Online first Issue: 2 Pages: 519-553
  • DOI: 10.1007/s00208-012-0843-8
  • Peer Reviewed
• [Presentation] Period maps in noncommutative geometry (2017)
  • Author(s): 岩成勇
  • Organizer: Conference in homotopy and category theory
  • Place of Presentation: Max Planck Institute for Mathematics （ドイツ）
  • Int'l Joint Research / Invited
• [Presentation] Period mapping for noncommutative algebras I,II,III (2016)
  • Author(s): 岩成勇
  • Organizer: 非可換代数多様体に対するホッジ理論と周期写像
  • Place of Presentation: 大阪大学理学研究科 (日本）
  • Year and Date: 2016-08-01
  • Invited
• [Presentation] Tannaka duality, stable infinity-categories and motivic Galois groups (2016)
  • Author(s): 岩成勇
  • Organizer: Seminaires Geometrie Algebriques, Champs et Homotopie
  • Place of Presentation: Institut de Math’ematique de Toulouse （フランス）
  • Int'l Joint Research / Invited
• [Presentation] Period mapping for noncommutative algebras (2016)
  • Author(s): 岩成勇
  • Organizer: Seminaires Geometrie Algebriques, Champs et Homotopie
  • Place of Presentation: Institut de Math'ematique de Toulouse（フランス）
  • Int'l Joint Research / Invited
• [Presentation] Tannaka duality, stable infinity-categories and motivic Galois groups (2016)
  • Author(s): 岩成勇
  • Organizer: Seminaires Analyse Algebriques,
  • Place of Presentation: Institut de Math'ematique de Jussieu (フランス)
  • Int'l Joint Research / Invited
• [Presentation] Reconstructing schemes and Deligne-Mumford stacks from derived infinity-categories (2015)
  • Author(s): Isamu Iwanari
  • Organizer: Algebra seminar
  • Place of Presentation: Fudan University, 中国
  • Year and Date: 2015-11-06
  • Int'l Joint Research / Invited
• [Presentation] Derived algebraic geometry and Tannaka duality (2015)
  • Author(s): Isamu Iwanari
  • Organizer: Colloquium
  • Place of Presentation: Fudan University, 中国
  • Year and Date: 2015-11-05
  • Int'l Joint Research / Invited
• [Presentation] From Bousfield localizations to dualities of tannakian type (2015)
  • Author(s): Isamu Iwanari
  • Organizer: Bousfield localizations form a set: a workshop in memory of Tetsusuke Ohkawa
  • Place of Presentation: Nagoya University
  • Year and Date: 2015-08-30
  • Int'l Joint Research / Invited
• [Presentation] ∞圏についてのレクチャー (2015)
  • Author(s): 岩成勇
  • Organizer: 第三回代数トポロジー信州春の学校
  • Place of Presentation: 信州大学理学部 （松本市）
  • Year and Date: 2015-03-03 – 2015-03-06
  • Invited
• [Presentation] 導来淡中双対性とモチーフ的ガロア群 (2015)
  • Author(s): 岩成勇
  • Organizer: 名古屋大学代数幾何学セミナー
  • Place of Presentation: 名古屋大学理学部 （名古屋市）
  • Year and Date: 2015-01-19
  • Invited
• [Presentation] Derived Tannaka dulaity and motivic Galois groups (2014)
  • Author(s): Isamu Iwanari
  • Organizer: International workshop on motives in Tokyo 2014
  • Place of Presentation: 東京大学数理 （東京都目黒区）
  • Year and Date: 2014-12-16
  • Invited
• [Presentation] 導来淡中双対性について (2014)
  • Author(s): 岩成勇
  • Organizer: 日本数学会2014 年秋季総合分科会 代数学分科会特別講演
  • Place of Presentation: 広島大学 （東広島市）
  • Year and Date: 2014-09-28
  • Invited
• [Presentation] Tannaka duality in derived algebraic geometry and applications (2014)
  • Author(s): Isamu Iwanari
  • Organizer: Homological methods in algebra, geometry and physics
  • Place of Presentation: City University London (イギリス）
  • Year and Date: 2014-07-26
  • Invited
• [Presentation] Kimura finiteness and derived Tannaka duality theorems (2014)
  • Author(s): Isamu Iwanari
  • Organizer: 函館数論幾何研究集会
  • Place of Presentation: 函館市中央図書館
  • Year and Date: 2014-05-26
  • Invited
• [Presentation] Categorified ring theory (2014)
  • Author(s): 岩成勇
  • Organizer: Workshop on noncommutative motives and derived algebraic geometry
  • Place of Presentation: 中央大学理工学部
  • Invited
• [Presentation] 楕円曲線の混合モチーフのガロア群 (2013)
  • Author(s): 岩成勇
  • Organizer: 混合モチーフ・K理論セミナー
  • Place of Presentation: 中央大学理工学部
  • Invited
• [Presentation] アーベル多様体の混合モチーフのモチーフ的ガロア群 (2013)
  • Author(s): 岩成勇
  • Organizer: 大阪大学整数論＆保型形式セミナー
  • Place of Presentation: 大阪大学理学部
  • Invited
• [Presentation] アーベル多様体の混合モチーフのモチーフ的ガロア群 (2013)
  • Author(s): 岩成勇
  • Organizer: 東北大代数幾何セミナー
  • Place of Presentation: 東北大学理学部
  • Invited