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The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems…

With any convex function F on a finite-dimensional linear space X such that F goes to infinity at infinity, we associate a Borel measure on the dual space X*. This measure is obtained by pushing forward the measure exp(-F(x))dx under the…

In this article we consider the length functional defined on the space of immersed planar curves. The $L^2(ds)$ Riemannian metric gives rise to the curve shortening flow as the gradient flow of the length functional. Motivated by the…

In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the K\"ahler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow…

We investigate aspects of the metric bubble tree for non-collapsing degenerations of (log) K\"ahler-Einstein metrics in complex dimensions one and two, and further describe a conjectural higher dimensional picture.

This is a double exposure of the probabilistic construction of Kahler-Einstein metrics on a complex projective algebraic variety X - where the Kahler-Einstein metric emerges from a canonical random point process on X - and the variational…

This book gives an introduction to fundamental aspects of generalized Riemannian, complex, and K\"ahler geometry. This leads to an extension of the classical Einstein-Hilbert action, which yields natural extensions of Einstein and…

In this article, we consider linearly convex complex cones in complex Banach spaces and we define a new projective metric on these cones. Compared to the hyperbolic gauge of Rugh, it has the advantage of being explicit, and easier to…

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces…

In this article we study convexity properties of distance functions in infinite dimensional Finsler unitary groups, such as the full unitary group, the unitary Schatten perturbations of the identity and unitary groups of finite von Neumann…

In this paper we investigate the possibility to obtain locally new Sasaki-Einstein metrics on the space $T^{1,1}$ considering a deformation of the standard metric tensor field. We show that from the geometric point of view this deformation…

We develop a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. We model the intrinsically non-linear structure…

We introduce a class of flows on the Wasserstein space of probability measures with finite first moment on the Cartan-Hadamard Riemannian manifold of positive definite matrices, and consider the problem of differentiability of the…

A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally…

We define a functional for Hermitian metrics using the curvature of the Chern connection. The Euler-Lagrange equation for this functional is an elliptic equation for Hermitian metrics. Solutions to this equation are related to…

We define the notion of a moment map and reduction in both generalized complex geometry and generalized K\"{a}hler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\C\P^n$, Hirzebruch…

In this paper, we investigate set-valued maps of strongly and approximately Jensen convex and Jensen concave type. We present counterparts of the Bernstein--Doetsch Theorem with Tabor type error terms.

Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the…

We describe a construction of Stein kernels using moment maps, which are solutions to a variant of the Monge-Amp\`ere equation. As a consequence, we show how regularity bounds on these maps control the rate of convergence in the classical…

We prove a compactness theorem for K\"ahler metrics with various bounds on Ricci curvature and the $\mathcal I$ functional. We explore applications of our result to the continuity method and the Calabi flow.