Problems on geometric asymptotic analysis over convex polytopes

• Project/Area Number: 21740117
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Global analysis
• Research Institution: Nagoya University
• Principal Investigator: TATE Tatsuya 名古屋大学, 大学院・多元数理科学研究科, 准教授 (00317299)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
• Keywords: 幾何学的漸近解析 / 凸多面体 / リーマン和 / トーリック多様体 / ランダム行列理論 / 量子ウォーク / エータ関数 / Euler-Maclaurin公式 / 量子酔歩 / 格子凸多面体 / 離散幾何解析学

Research Abstract

A convex polytope is said to be a lattice polytope if each vertex is a lattice point. An asymptotic expansion formula for the Riemann sums of general smooth functions over lattice polytopes was obtained. This formula is regarded as a generalization of the so-called local Euler-Maclaurin formula due to Berline-Vergne. Furthermore, an explicit formula for the third term in the expansion was obtained in case where the polytope is Delzant.

Research Products

• [Journal Article] Asymptotic behavior of quantum walks on the line (2012)
  • Author(s): Toshikazu Sunada and Tatsuya Tate
  • Journal Title: Journal of Functional Analysis Volume: 262 Pages: 2608-2645
  • Peer Reviewed
• [Journal Article] Problems on asymptotic analysis over convex polytopes (2011)
  • Author(s): Tatsuya Tate
  • Journal Title: Travaux Mathematique Volume: 19 Pages: 65-96
  • Peer Reviewed
• [Journal Article] Asymptotic Euler-Maclaurin formula over lattice polytopes (2011)
  • Author(s): Tatsuya Tate
  • Journal Title: Journal of Functional Analysis Volume: 260 no.2 Pages: 501-540
  • Peer Reviewed
• [Journal Article] A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions (2010)
  • Author(s): Tatsuya Tate
  • Journal Title: Asymptotic Analysis Volume: 67 no.1-2 Pages: 101-123
  • Peer Reviewed
• [Journal Article] Bernstein measures on convex polytopes, in Spectral analysis in geometry and number theory
  • Author(s): Tatsuya Tate
  • Journal Title: Contemporary Mathematics Volume: 484 Pages: 295-319
  • Peer Reviewed
• [Journal Article] A spectral analogue of the Meinardus theorem on asymptotics of the numer of partitions
  • Author(s): Tatsuya Tate
  • Journal Title: Asymptotic Analysis (未定)(掲載決定済)
  • Peer Reviewed
• [Presentation] One-dimensional quantum walks (2013)
  • Author(s): 楯辰哉
  • Organizer: 第8回代数・解析・幾何学セミナー
  • Place of Presentation: 鹿児島大学
  • Year and Date: 2013-02-18
• [Presentation] One-dimensional quantum walks (2013)
  • Author(s): 楯辰哉
  • Organizer: 第8回代数・解析・幾何学セミナー
  • Place of Presentation: 鹿児島大学理学部
  • Invited
• [Presentation] Asymptotics of quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: Workshop of "Quantum Dynamics and Quantum Walks
  • Place of Presentation: 自然科学研究機構岡崎コンファレンスセンター
  • Year and Date: 2012-11-24
• [Presentation] 一次元量子ウォークに対する弱収束定理の簡単な証明 (2012)
  • Author(s): 楯辰哉
  • Organizer: 福岡大学微分幾何研究会「Geometry and Something」
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2012-11-03
• [Presentation] Asymptotics of quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「確率解析とその周辺」
  • Place of Presentation: 名古屋大学
  • Year and Date: 2012-10-25
• [Presentation] Asymptotic behavior of quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「確率論と幾何学」
  • Place of Presentation: 大学コンソーシアムやまがたゆうキャンパス
  • Year and Date: 2012-10-20
• [Presentation] 量子ウォークの数理 (2012)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「物質科学の数学的手法と数理物理」
  • Place of Presentation: 理化学研究所
  • Year and Date: 2012-06-17
• [Presentation] 一次元量子酔歩に対するPlancherel-Rotach 型の漸近公式 (2012)
  • Author(s): 楯辰哉
  • Organizer: 応用数学連携フォーラム第27回ワークショップ
  • Place of Presentation: 東北大学情報科学研究科
  • Year and Date: 2012-04-20
• [Presentation] A formula of Plancherel-Rotach type for quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 第11回名古屋国際数学コンファレンス "Topology and Analysis on Foliations"
  • Place of Presentation: 名古屋大学
  • Year and Date: 2012-03-24
• [Presentation] A formula of Plancherel-Rotach type for quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 第11回名古屋国際数学コンファレンス"Topology and Analysis on Foliations"
  • Place of Presentation: 名古屋大学(招待講演)
  • Year and Date: 2012-03-24
• [Presentation] A formula of Plancherel-Rotach type for quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 研究集会 "Mathematical approach to emerging topics in material science 2012"
  • Place of Presentation: 東北大学
  • Year and Date: 2012-02-18
• [Presentation] A formula of Plancherel-Rotach type for quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 研究集会"Mathematical approach to emerging topics in material science 2012"
  • Place of Presentation: 東北大学(WPI-AIMR)(招待講演)
  • Year and Date: 2012-02-18
• [Presentation] Asymptotics of quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「確率解析とその周辺」
  • Place of Presentation: 名古屋大学大学院多元数理科学研究科
  • Invited
• [Presentation] Asymptotics of quantum walks on the line (2012)
  • Author(s): 楯辰哉
  • Organizer: Workshop of "Quantum Dynamics and Quantum Walks"
  • Place of Presentation: 自然科学研究機構岡崎コンファレンスセンター
• [Presentation] 量子酔歩に対するPlancherel-Rotach 型公式 (2011)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「量子化の幾何学2011」
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-11-19
• [Presentation] 量子酔歩に対するPlancherel-Rotach型公式 (2011)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「量子化の幾何学2011」
  • Place of Presentation: 早稲田大学(招待講演)
  • Year and Date: 2011-11-19
• [Presentation] A formula of Plancherel-Rotach type for quantum walks on the line (2011)
  • Author(s): 楯辰哉
  • Organizer: 福岡大学微分幾何研究会「Geometry and Something」
  • Place of Presentation: 福岡大学セミナーハウス
  • Year and Date: 2011-11-04
• [Presentation] A formula of Plancherel-Rotach type for quantum walks on the line (2011)
  • Author(s): 楯辰哉
  • Organizer: 福岡大学微分幾何研究会「Geometry and (Something)
  • Place of Presentation: 福岡大学セミナーハウス(招待講演)
  • Year and Date: 2011-11-04
• [Presentation] Asymptotic Euler-Maclaurin formula over lattice polytopes (2011)
  • Author(s): 楯辰哉
  • Organizer: DFG-JSPS Seminar, "Lie Groups: Geometry and Analysis"
  • Place of Presentation: Universitat Paderborn, Germany
  • Year and Date: 2011-09-09
• [Presentation] Asymptotic Euler-Maclaurin formula over lattice polytopes (2011)
  • Author(s): 楯辰哉
  • Organizer: DFG-JSPS Seminar, "Lie Groups : Geometry and Analysis"
  • Place of Presentation: Universitat Paderborn(招待講演)
  • Year and Date: 2011-09-09
• [Presentation] Asymptotic Euler-Maclaurin formula over lattice polytopes (2011)
  • Author(s): 楯辰哉
  • Organizer: Long Term Workshop on"Geometry and Analysis"
  • Place of Presentation: 京都大学.
  • Year and Date: 2011-03-17
• [Presentation] Asymptotic Euler-Maclaurin formula over lattice polytopes (2011)
  • Author(s): Tatsuya Tate
  • Organizer: Long Term Workshop on "Geometry and Analysis"
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2011-03-17
• [Presentation] Euler-Maclaurin expansion over Delzant polytopes (2010)
  • Author(s): 楯辰哉
  • Organizer: 福岡大学微分幾何研究会「Geometry and Something」
  • Place of Presentation: 福岡大学セミナーハウス.
  • Year and Date: 2010-10-08
• [Presentation] Asymptotic Euler-Maclaurin expansion over Delzant polytopes (2010)
  • Author(s): 楯辰哉
  • Organizer: AMS-SMM 8th International Meeting at Berkeley, Special Session "Toeplitz Operators and Discrete Quantum Models"
  • Place of Presentation: University of California, Berkeley, USA
  • Year and Date: 2010-06-04
• [Presentation] Asymptotic Euler-Maclaurin expansion over Delzant polytopes (2010)
  • Author(s): Tatsuya Tate
  • Organizer: AMS-SMM 8^<th> International Meeting at Berkeley Special Session "Toeplitz Operators and Discrete Quantum Models"
  • Place of Presentation: カリフォルニア大学 バークレー校(アメリカ)
  • Year and Date: 2010-06-04
• [Presentation] Asymptotic Euler-Maclaurin formula over lattice polytopes (2010)
  • Author(s): 楯辰哉
  • Organizer: International Conference "Functions in Number Theory and Their Probabilistic Aspects"
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2010-02-17
• [Presentation] Asymptotic Euler-Maclaurin formula over Delzant polytopes (2009)
  • Author(s): 楯辰哉
  • Organizer: 研究集会「ポテンシャル論とベルグマン核」
  • Place of Presentation: 京都大学数理解析研究所.
  • Year and Date: 2009-12-03
• [Presentation] Asymptotic Euler-Maclaurin formula over Delzant polytopes (2009)
  • Author(s): 楯辰哉
  • Organizer: ポテンシャ論とベルグマン核
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-12-03
• [Presentation] An asymptotic Euler-Maclaurin formula for Delzant polytopes (2009)
  • Author(s): 楯辰哉
  • Organizer: Third International Conference on Geometry and Quantization
  • Place of Presentation: Universite de Luxembourg, Luxembourg
  • Year and Date: 2009-09-10
• [Presentation] An asymptotic Euler-Maclaurin formula for Delzant polytopes (2009)
  • Author(s): Tatsuya Tate
  • Organizer: Third International Conference on Geometry and Quantization
  • Place of Presentation: University of Luxembourg
  • Year and Date: 2009-09-10
• [Presentation] Problems on asymptotic analysis over convex polytopes (2009)
  • Author(s): 楯辰哉
  • Organizer: International School on Geometry and Quantization
  • Place of Presentation: University of Luxembourg, Luxembourg.
• [Presentation] Problems on asymptotic analysis over convex polytopes (2009)
  • Author(s): Tatsuya Tate
  • Organizer: International School on Geometry and Quantization
  • Place of Presentation: University of Luxembourg
• [Presentation] 一次元量子酔歩に対する Plancherel-Rotach 型の漸近公式
  • Author(s): 楯辰哉
  • Organizer: 応用数学連携フォーラム第 27 回ワークショップ
  • Place of Presentation: 東北大学大学院情報科学研究科
  • Invited
• [Book] "Spectral analysis in geometry and number theory", Contemporary Mathematics 484, American Mathematical Society, Providence, RI (2009)
  • Author(s): M.Kotani,H.Naito and T.Tate