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2014 年度 実績報告書

距離空間内の勾配流とその応用

研究課題

研究課題/領域番号 14F04320
研究機関京都大学

研究代表者

太田 慎一  京都大学, 理学(系)研究科(研究院), 准教授 (00372558)

研究分担者 PALFIA Miklos  京都大学, 理学(系)研究科(研究院), 外国人特別研究員
研究期間 (年度) 2014-04-25 – 2017-03-31
キーワードリーマン幾何 / 曲率 / 凸関数 / 勾配流
研究実績の概要

In this academic year, we have continued working on gradient flows in Alexandrov spaces. Last year in the joint work of Ohta-Palfia we established the convergence of discrete time gradient flows on Alexandrov spaces with upper or lower curvature bounds. We also proved an abstract law of large numbers extending the one proved by Sturm for CAT(0)-spaces (non-positively curved metric spaces). The paper is to appear in the journal "Calc. Var. PDE". Based on new ideas from this paper we extended the results to the continuous case, generalizing the theory of gradient flows developed by Ambrosio-Gigli-Savare in CAT(0)-spaces to CAT(1)-spaces (metric spaces of curvature bounded above by 1). Our results are somewhat more general, a space with tangent cones (possessing angles) and a semi-convex squared distance function is sufficient for our analysis.

Also Palfia continued working on a possible extension of Loewner’s theorem to several noncommuting variables. A previous preprint about a possible extension of the theorem contained a serious flaw, but eventually this gap seems to be removable. The missing piece appears to be a generalized C*-algebraic notion of convexity, namely the matrix convexity of Effros and Winkler. With this notion at hand to a certain extent the original argumentation of the preprint seems to work, but leads to a more general and abstract representation formula for operator monotone functions in several noncommuting variables.

現在までの達成度 (区分)
現在までの達成度 (区分)

2: おおむね順調に進展している

理由

We have originally considered a generalization of the Ambrosio-Gigli-Savare theory to gradient flows in Finsler manifolds or normed spaces. Then we found that possessing “angles” is an essential condition to follow their theory and established the above mentioned results in CAT(1)-spaces. Our analysis is based on the very essential property of a space being “Riemannian” (having angles). Thus it is natural to proceed to Finsler manifolds as a next step.

An extension of Loewner’s theorem to several noncommuting variables is a challenging problem. Although the original argument contained a gap, we believe that this work will be an important contribution.

今後の研究の推進方策

The goal is to continue working on gradient flows in various geometric settings. Establishing a further generalization of the Ambrosio-Gigli-Savare theory to the Finsler setting is one of the problems to be pursued. Also now with a continuous CAT(1) theory at hand we can investigate a continuous time version of the law of large numbers, possibly relating it to the heat flow on the space. This will enable us to consider a large deviation theory and possibly an ergodic theory on such spaces. If an extension of the theory of gradient flows to a particular geometric object is established then one can consider these further problems on this geometric object.

Our other set of goals is to rewrite the preprint on Loewner’s theorem and completely fill in the gaps of the original argument establishing a representation formula for operator monotone functions in several noncommuting variables.

備考

代表者(太田)の研究についてのサイト。

  • 研究成果

    (4件)

すべて 2014 その他

すべて 雑誌論文 (2件) (うち査読あり 2件) 学会発表 (1件) (うち招待講演 1件) 備考 (1件)

  • [雑誌論文] Weighted inductive means2014

    • 著者名/発表者名
      Yongdo Lim, Miklos Palfia
    • 雑誌名

      Linear Algebra and its Applications

      巻: 453 ページ: 59-83

    • DOI

      10.1016/j.laa.2014.04.002

    • 査読あり
  • [雑誌論文] Weighted multivariable operator means of positive definite operators2014

    • 著者名/発表者名
      Miklos Palfia, Denes Petz
    • 雑誌名

      Linear Algebra and its Applications

      巻: 463 ページ: 134-153

    • DOI

      10.1016/j.laa.2014.08.025

    • 査読あり
  • [学会発表] Gradient flows and law of large numbers in Alexandrov spaces2014

    • 著者名/発表者名
      Miklos Palfia
    • 学会等名
      Geometry and Probability
    • 発表場所
      東京工業大学(東京都)
    • 年月日
      2014-09-23 – 2014-09-23
    • 招待講演
  • [備考] Web Page of Shin-ichi OHTA

    • URL

      https://www.math.kyoto-u.ac.jp/~sohta/

URL: 

公開日: 2016-06-01  

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