This book develops a general theory of Steenrod operations in spectral sequences. It gives special attention to the change-of-rings spectral sequence for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces and homotopy orbit spaces. In treating the change-of-rings spectral sequence, the book develops from scratch the necessary properties of extensions o...

The authors consider the Hodge Laplacian DELTA on the Heisenberg group H n , endowed with a left-invariant and U(n) -invariant Riemannian metric. For 0<=k<=2n 1 , let DELTA k denote the Hodge Laplacian restricted to k -forms. In this paper they address three main, related questions:*(1) whether the L 2 and L p -Hodge decompositions, 1 , hold on H n;*(2) whether the Riesz transforms dDELTA -12 k are L p -bounded, for 1" ; *(3) h...

The classical Grothendieck inequality is viewed as a statement about representations of functions of two variables over discrete domains by integrals of two-fold products of functions of one variable. An analogous statement is proved, concerning continuous functions of two variables over general topological domains. The main result is the construction of a continuous map f from l2 (A) into L2 (O A, PA), where A is a set, OA = {-1,1}A...

This book is based on lectures given at Stanford University in 2009. The purpose of the lectures and of the book is to give an introductory overview of how to use Ricci flow and Ricci flow with surgery to establish the Poincare Conjecture and the more general Geometrization Conjecture for 3-dimensional manifolds. Most of the material is geometric and analytic in nature; a crucial ingredient is understanding singularity development fo...

The result of lectures given by the authors at New York University, the University of Utah, and Michigan State University, the material is written for students who have had only one term of calculus, but it contains material that can be used in modeling courses in applied mathematics at all levels through early graduate courses. Numerous exercises are given as well as solutions to selected exercises, so as to lead readers to discover...

This book gives a presentation of topics in Hamilton's Ricci flow for graduate students and mathematicians interested in working in the subject. The authors have aimed at presenting technical material in a clear and detailed manner. In this volume, geometric aspects of the theory have been emphasized. The book presents the theory of Ricci solitons, Kähler-Ricci flow, compactness theorems, Perelman's entropy monotonicity and no local ...

The authors study the following singularly perturbed problem: - 2 ?u V(x)u=f(u) in R N . Their main result is the existence of a family of solutions with peaks that cluster near a local maximum of V(x) . A local variational and deformation argument in an infinite dimensional space is developed to establish the existence of such a family for a general class of nonlinearities f .

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, the authors prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problem...

Volume 211, number 991 (first of 5 numbers).

Make learning math fun by sharing these hands-on labs with your child. Math Lab for Kids presents more than 50 activities that incorporate coloring, drawing, games, and items like prisms to make math more than just numbers. With Math Lab for Kids, kids can: - Explore geometry and topology with hands-on examples like prisms, antiprisms, Platonic solids, and Möbius strips. - Build logic skills by playing and strategizing through tangra...