| 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.
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