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Noncommutative geometry and global analysis : conference in honor of Henri Moscovici, June 29-July 4, 2009, Bonn, Germany
This volume represents the proceedings of the conference on Noncommutative Geometric Methods in Global Analysis, held in honor of Henri Moscovici, from June 29-July 4, 2009, in Bonn, Germany. Henri Moscovici has made a number of major contributions to noncommutative geometry, global analysis, and representation theory. This volume, which includes articles by some of the leading experts in these fields, provides a panoramic view of the interactions of noncommutative geometry with a variety of areas of mathematics. It focuses on geometry, analysis and topology of manifolds and singular spaces, index theory, group representation theory, connections of noncommutative geometry with number theory and arithmetic geometry, Hopf algebras and their cyclic cohomology.
eBook
Cornered Heegaard Floer Homology
Bordered Floer homology assigns invariants to 3-manifolds with boundary, such that the Heegaard Floer homology of a closed
3-manifold, split into two pieces, can be recovered as a tensor product of the bordered invariants of the pieces. We construct cornered
Floer homology invariants of 3-manifolds with codimension-2 corners, and prove that the bordered Floer homology of a 3-manifold with
boundary, split into two pieces with corners, can be recovered as a tensor product of the cornered invariants of the pieces.
eBook
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,\\mathsf d,\\mathfrak m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong \\mathrm {CD}^{*}(K,N) condition of Bacher-Sturm.
eBook
Degree Theory of Immersed Hypersurfaces
The authors develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature, extrinsic curvature and special Lagrangian curvature and show that in all these cases, this number is equal to -\\chi(M), where \\chi(M) is the Euler characteristic of the ambient manifold M.
eBook
Spaces of Curves with Constrained Curvature on Hyperbolic Surfaces
Let S be a hyperbolic surface. We investigate the topology of the space of all curves on S which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval (κ1, κ2). Such a space falls into one of four qualitatively distinct classes, according to whether (κ1, κ2) contains, overlaps, is disjoint from, or contained in the interval [−1, 1]. Its homotopy type is computed in the latter two cases. We also study the behavior of these spaces under covering maps when S is arbitrary (not necessarily hyperbolic nor orientable), and show that if S is compact, then they are always nonempty.
Journal Article