2014 Fiscal Year Annual Research Report
| Project/Area Number |
14F04320
|
|---|
| Research Institution | Kyoto University |
|---|
Principal Investigator |
太田 慎一 京都大学, 理学(系)研究科(研究院), 准教授 (00372558)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
PALFIA Miklos 京都大学, 理学(系)研究科(研究院), 外国人特別研究員
|
|---|
| Project Period (FY) |
2014-04-25 – 2017-03-31
|
| Keywords | リーマン幾何 / 曲率 / 凸関数 / 勾配流 |
|---|
| Outline of Annual Research Achievements |
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.
|
| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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.
|
| Strategy for Future Research Activity |
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.
|
| Remarks |
代表者（太田）の研究についてのサイト。
|
