Study of foundation and applications of cluster algebras

• Project/Area Number: 23540012
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: Nakanishi Tomoki 名古屋大学, 多元数理科学研究科, 教授 (80227842)
• Project Period (FY): 2011-04-28 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 団代数 / 完全WKB解析 / 二重対数関数 / ダイログ関数 / 曲面の三角分割 / Y-system

Outline of Final Research Achievements

We studied the foundation and applications of cluster algebras, and we obtained the following results as a part of the research result.
A. Applications to dilogarithm identity and quantum dilogarithm identity, B. Classification of c-vectors and d-vectors of cluster algebras of finite type, C. Applications to exact WKB analysis, D. Study of fundamental properties of generalized cluster algebras and formulation of quantum generalized cluster algebras.
These results are based on joint works with Rinat Kashaev, Salvatore Stella, and Kohei Iwaki.

Research Products

• [Int'l Joint Research] Northeastern University(米国)
• [Journal Article] Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras (2015)
  • Author(s): Kohei Iwaki and Tomoki Nakanishi
  • Journal Title: International Mathematics Research Notices Volume: 印刷中 Issue: 14 Pages: 4375-4417
  • DOI: 10.1093/imrn/rnv270
  • Peer Reviewed / Open Access
• [Journal Article] Structure of seeds in generalized cluster algebras (2015)
  • Author(s): Tomoki Nakanishi
  • Journal Title: Pacific Journal of Mathematics Volume: 277 Issue: 1 Pages: 201-202
  • DOI: 10.2140/pjm.2015.277.201
  • Peer Reviewed / Open Access
• [Journal Article] Quantum generalized cluster algebras and quantum dilogarithms of higher degrees (2015)
  • Author(s): Tomoki Nakanishi
  • Journal Title: Theoretical and Mathematical Physics Volume: 185 Issue: 3 Pages: 1759-1768
  • DOI: 10.1007/s11232-015-0377-9
  • Peer Reviewed / Open Access
• [Journal Article] Exact WKB analysis and cluster algebras (2014)
  • Author(s): Kohei Iwaki and Tomoki Nakanishi
  • Journal Title: Journal of Physics A: Mathematical and Theoretical Volume: 47 Issue: 47 Pages: 474009-474009
  • DOI: 10.1088/1751-8113/47/47/474009
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type (2014)
  • Author(s): Tomoki Nakanishi, Salvatore Stella
  • Journal Title: Electron. J. Combin. Volume: 21
  • Peer Reviewed
• [Journal Article] Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r (2013)
  • Author(s): Rei Inoue 他
  • Journal Title: Publ. RIMS Volume: 49 Issue: 1 Pages: 1-42
  • DOI: 10.4171/prims/95
  • Peer Reviewed
• [Journal Article] Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2 (2013)
  • Author(s): Rei Inoue 他
  • Journal Title: Publ. RIMS Volume: 49 Issue: 1 Pages: 43-85
  • DOI: 10.4171/prims/96
  • Peer Reviewed
• [Journal Article] Tropicalization method in cluster algebras (2012)
  • Author(s): Tomoki Nakanishi
  • Journal Title: Contemporary Mathematics Volume: 580 Pages: 95-115
  • DOI: 10.1090/conm/580/11486
  • Peer Reviewed
• [Journal Article] On tropical dualities in cluster algbras (2012)
  • Author(s): Tomoki Nakanishi, Andrei Zelevinsky
  • Journal Title: Contemp. Math. Volume: 565 Pages: 217-226
  • Peer Reviewed
• [Journal Article] Classical and quantum dilogarithm identities (2011)
  • Author(s): Rinat M. Kashaev, Tomoki Nakanishi
  • Journal Title: SIGMA Volume: 7
  • Peer Reviewed
• [Presentation] Introduction to cluster algebras および Tropical variables in cluster algebras （計２講演） (2016)
  • Author(s): Tomoki Nakanishi
  • Organizer: Tropical Geometry and Related Topic
  • Place of Presentation: 京都大
  • Year and Date: 2016-03-07
  • Int'l Joint Research / Invited
• [Presentation] Introduction to cluster algebras (2015)
  • Author(s): Tomoki Nakanishi
  • Organizer: Symposium on Representation Theory 2015
  • Place of Presentation: 伊豆長岡
  • Year and Date: 2015-11-17
  • Int'l Joint Research / Invited
• [Presentation] Riemann surface and cluster algebra (2015)
  • Author(s): Tomoki Nakanishi
  • Organizer: Quantization of Spectral Curves
  • Place of Presentation: 大阪市大
  • Year and Date: 2015-11-02
  • Int'l Joint Research / Invited
• [Presentation] Cluster algebras and applications (2015)
  • Author(s): Tomoki Nakanishi
  • Organizer: School and International Conference on Geometry and Quantization GEOQUANT 2015
  • Place of Presentation: ICMAT, Madrid, Spain
  • Year and Date: 2015-09-07
  • Int'l Joint Research / Invited
• [Presentation] On generalized cluster algebras (2015)
  • Author(s): Tomoki Nakanishi
  • Organizer: The Ninth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
  • Place of Presentation: University of Georgia, Atlanta, USA
  • Year and Date: 2015-04-01
  • Int'l Joint Research / Invited
• [Presentation] On generalized cluster algebras (2014)
  • Author(s): Tomoki Nakanishi
  • Organizer: Conference on strings, quivers and cluster algebras in mathematical physics
  • Place of Presentation: KIAS, Seoul
  • Year and Date: 2014-12-18 – 2014-12-22
  • Invited
• [Presentation] Cluster algebras and Y-systems (2014)
  • Author(s): Tomoki Nakanishi
  • Organizer: Summer School on Quantum Groups and Integrability - Algebraic, Analytic and Geometric Aspects
  • Place of Presentation: Universtat Hamburg
  • Year and Date: 2014-07-21 – 2014-07-25
  • Invited
• [Presentation] Exact WKB analysis and cluster algebras (2014)
  • Author(s): Tomoki Nakanishi
  • Organizer: 4TH Workshop on Cominatorics of Moduli Spaces, Cluster Algebras, and Topological Recursion
  • Place of Presentation: Laboratoire J.-V. Poncelet, Steklov Mathematical Institute, and the Higher School of Economics, Moscow
  • Year and Date: 2014-05-26 – 2014-05-31
  • Invited
• [Presentation] Exact WKB analysis and cluster algebras (2014)
  • Author(s): Tomoki Nakanishi
  • Organizer: Combinatorial Representation theory
  • Place of Presentation: Centre de recherches mathematiques - CRM - Universite de Montreal
  • Year and Date: 2014-04-21 – 2014-04-25
  • Invited
• [Presentation] Exact WKB analysis and cluster algebras (2014)
  • Author(s): Tomoki Nakanishi
  • Organizer: Geometry, physics and representation theory
  • Place of Presentation: 名古屋大学
  • Invited
• [Presentation] Wonder of sine-Gordon Y-system (2013)
  • Author(s): Tomoki Nakanishi
  • Organizer: Algebra, Combinatorics and Representation Theory: in memory of Andrei Zelevinsky
  • Place of Presentation: Northeastern University, Boston, USA
  • Invited
• [Presentation] Diagrammatic description of c-vectors and d-vectors in cluster algebras of finite type Wonder of sine-Gordon Y-system (2013)
  • Author(s): 中西知樹
  • Organizer: 日本数学会2013年度秋期総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] Wonder of sine-Gordon Y-system (2013)
  • Author(s): 中西知樹
  • Organizer: 日本数学会2013年度秋期総合分科会
  • Place of Presentation: 愛媛大学
• [Presentation] Exact WKB analysis and cluster algebras (2013)
  • Author(s): Tomoki Nakanishi
  • Organizer: Cluster algebras and related topics
  • Place of Presentation: MFO, Oberwolfach, Germany
  • Invited
• [Presentation] Wonder of sine-Gordon Y-system (2013)
  • Author(s): Tomoki Nakanishi
  • Organizer: Infinite Analysis 13: Bethe ansatz, quantum groups and beyond
  • Place of Presentation: 京大
  • Invited
• [Presentation] Diagrammatic description of c-vectors and d-vectors of cluster algebras of finite type (2012)
  • Author(s): Tomoki Nakanishi
  • Organizer: Cluster Algebras in Combinatorics, Algebra, and Geometry
  • Place of Presentation: MSRI, Berkeley, USA
  • Invited
• [Presentation] Classical and quantum dilogarithm identities (2012)
  • Author(s): Tomoki Nakanishi
  • Organizer: Workshop: Representation theory of Chevalley groups and related topics(招待講演)
  • Place of Presentation: 名大
• [Presentation] T-systems, Y-systems, and cluster algebras (2011)
  • Author(s): Tomoki Nakanishi
  • Organizer: Workshop: ISLAND 4 Tropical geometry and integrable systems(招待講演)
  • Place of Presentation: グラスゴー大
• [Presentation] T-systems, Y-systems, and cluster algebras (2011)
  • Author(s): Tomoki Nakanishi
  • Organizer: Workshop: Cluster algebras and statistical physics(招待講演)
  • Place of Presentation: ブラウン大
• [Presentation] T-systems, Y-systems, and cluster algebras (2011)
  • Author(s): Tomoki Nakanishi
  • Organizer: Worskhop: Cluster algebras, representation theory, and Poisson geometry(招待講演)
  • Place of Presentation: バンフ数学研究所
• [Presentation] 団代数とその応用 (2011)
  • Author(s): 中西 知樹
  • Organizer: 日本数学会(招待講演)
  • Place of Presentation: 信州大
• [Presentation] Classical and quantum dilogarithm identities (2011)
  • Author(s): Tomoki Nakanishi
  • Organizer: Workshop: Inifnite Analysis 11 Winter School on Quantum cluster algebras(招待講演)
  • Place of Presentation: 阪大
• [Remarks]
  • URL: http://www.math.nagoya-u.ac.jp/~nakanisi/index.html