Studies of cohomology groups in Poisson geometry

• Project/Area Number: 23540067
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Akita University
• Principal Investigator: MIKAMI KENTARO 秋田大学, 工学(系)研究科(研究院), 名誉教授 (70006592)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2011: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
• Keywords: ポアソン構造 / シンプレクティック構造 / 形式的ハミルトン・ベクトル場 / 運動量写像 / ゲルファント・フックス・コホモロジー群 / 既約分解 / 結晶基底(Crystal Basis) / グレブナー基底(Groebner Basis) / シンプレクティック多様体 / ゲルファント・フックス・コホモロジー / 既約表現 / 結晶基底 / グレブナー基底 / シンプレクティック / ポアソン / コホモロジー群 / ゲフフアント・フックス / 同変運動量写像 / 表現論 / クリスタルベース / ゲフファント・フックス

Research Abstract

I studied about Sp(2n) relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on the 2n-dimensional symplectic vector space ant got the results below. 1) In 2-dim case, we determined the GKF cohomology groups for weight 20. 2) In the case of 4-dim, using Littlewood-Richardson rule, we determined the GKF cohomology groups for weight 2,4 and 6. This was published in J.Math.Sci.Univ.Tokyo 19(2012)1-18, by the title Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R4. 3) By using crystal base theory, in 6-dimensional case, we studied the GKF cohomology groups for weight 2,4 and 6. The result is uploaded on http://arxiv.org by The relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on 6-dimensional plane, arXiv:1402.6834. 4) I gave an affirmative answer to a conjecture, and submitted to some Journal by the title, ``An affirmative answer to a conjecture for Metoki class''.

Research Products

• [Journal Article] Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R4 (2012)
  • Author(s): K. Mikami and Y. Nakae
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 19 Pages: 1-18
  • NAID: 120006906096
  • Peer Reviewed
• [Journal Article] Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4 (2012)
  • Author(s): Kentaro Mikami, Yasuharu Nakae
  • Journal Title: J. Math. Sci. Univ. Tokyo Volume: 19
  • NAID: 120006906096
  • Peer Reviewed
• [Presentation] Morita conjecture about Metoki class (2013)
  • Author(s): K. Mikami
  • Organizer: 慶應義塾大学幾何セミナー
  • Place of Presentation: 慶應義塾大学矢上キャンパス
  • Year and Date: 2013-03-09
• [Presentation] Conjecture about Metoki class (2013)
  • Author(s): K. Mikami
  • Organizer: 「接触構造, 特異点, 微分方程式及びその周辺」研究集会
  • Place of Presentation: 秋田カレッジプラザ
  • Year and Date: 2013-01-29
• [Presentation] Lower weight Gel'fand- Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4 (2012)
  • Author(s): K. Mikami
  • Organizer: Foliations and Diffeomorphism Groups 2012
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2012-11-01
• [Presentation] Gel'fand-Fuks cohomology groups of Hamiltonian vector fields (2011)
  • Author(s): K. Mikami
  • Organizer: Exchange Program of Japan and Belgium by JSPS
  • Place of Presentation: Universite Catholique de Louvain (Belgium)
  • Year and Date: 2011-12-15
• [Presentation] Gel'fand-Fuks cohomology groups of Hamiltonian vector fields (2011)
  • Author(s): Kentaro Mikami
  • Organizer: Exchange Program of Japan and Belgium by JSPS(招待講演)
  • Place of Presentation: Universite Catholique de Louvain (at Belgium)
• [Presentation] Lower weight Gel'fand-Kalinin-Fuks cohomology groups of formal Hamiltonian vector fields of R^4
  • Author(s): Kentaro Mikami
  • Organizer: Foliations and Diffeomorphism Grooups 2012
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Invited
• [Presentation] Conjecture about Metoki class
  • Author(s): Kentaro Mikami
  • Organizer: 「接触構造、特異点、微分方程式及びその周辺」研究集会
  • Place of Presentation: 秋田カレッジプラザ
  • Invited
• [Presentation] Morita conjecture about Metoki class
  • Author(s): Kentaro Mikami
  • Organizer: 慶應義塾大学幾何セミナー
  • Place of Presentation: 慶應義塾大学
  • Invited
• [Remarks] ホームページ等(1)
  • URL: http://www.math.akita-u.ac.jp/~mikami/Conj4MetokiClass/
• [Remarks] ホームページ等(2)
  • URL: http://www.math.akita-u.ac.jp/~mikami/GKF_R6/
• [Remarks] 1402.6834
  • URL: http://www.arxiv.org/
• [Remarks] A proof to Kotschick-Morita theorem of G-K-F class
  • URL: http://www.math.akita-u.ac.jp/~mikami/Conj4MetokiClass
• [Remarks] A proof to a Kotschick-Morita theorem by Risa/Asir
  • URL: http://www.math.akita-u.ac.jp/~mikami/Conj4MetokiClass
• [Remarks] Proof by RisaAsir to a conjecture for Metoki class
  • URL: http://www.math.akita-u.ac.jp/~mikami/Conj4MetokiClass