The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates ...

The authors investigate composition operators on Hardy-Orlicz spaces when the Orlicz function \Psi grows rapidly: compactness, weak compactness, to be p-summing, order bounded, \ldots, and show how these notions behave according to the growth of \Psi. They introduce an adapted version of Carleson measure. They construct various examples showing that their results are essentially sharp. In the last part, they study the case of Bergman...

The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. The discussion includes: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does n...

"Volume 209, number 983 (second of 5 numbers)."

The goal of the Entropy and the Quantum schools has been to introduce young researchers to some of the exciting current topics in mathematical physics. These topics often involve analytic techniques that can easily be understood with a dose of physical intuition. In March of 2010, four beautiful lectures were delivered on the campus of the University of Arizona. They included Isoperimetric Inequalities for Eigenvalues of the Laplacia...

This volume contains the proceedings of the NIMS Thematic Workshop on Mathematical and Statistical Methods for Imaging, which was held from August 10-13, 2010, at Inha University, Incheon, Korea. The goal of this volume is to give the reader a deep and unified understanding of the field of imaging and of the analytical and statistical tools used in imaging. It offers a good overview of the current status of the field and of direction...

This paper deals with elliptic equations in the plane with degeneracies. The equations are generated by a complex vector field that is elliptic everywhere except along a simple closed curve. Kernels for these equations are constructed. Properties of solutions, in a neighborhood of the degeneracy curve, are obtained through integral and series representations. An application to a second order elliptic equation with a punctual singular...

Let G=G(K) be a simple algebraic group defined over an algebraically closed field K of characteristic p=0. A subgroup X of G is said to be G-completely reducible if, whenever it is contained in a parabolic subgroup of G, it is contained in a Levi subgroup of that parabolic. A subgroup X of G is said to be G-irreducible if X is in no proper parabolic subgroup of G; and G-reducible if it is in some proper parabolic of G. In this paper,...

The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems $u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,$ under the main assumption of polynomial growth at rate $p$ i.e. $|a(x,t,u,Du)|\leq L(1+|Du|^{p-1}), p \geq 2.$ They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the firs...

The notion of a fusion system was first defined and explored by Puig, in the context of modular representation theory. Later, Broto, Levi, and Oliver extended the theory and used it as a tool in homotopy theory. The author seeks to build a local theory of fusion systems, analogous to the local theory of finite groups, involving normal subsystems and factor systems. Among other results, he defines the notion of a simple system, the ge...