Symplectic Dirac Operators on Hermitian Symmetric Spaces

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.Comment: 17 page

The adhesion molecule L1 regulates transendothelial migration and trafficking of dendritic cells

The adhesion molecule L1, which is extensively characterized in the nervous system, is also expressed in dendritic cells (DCs), but its function there has remained elusive. To address this issue, we ablated L1 expression in DCs of conditional knockout mice. L1-deficient DCs were impaired in adhesion to and transmigration through monolayers of either lymphatic or blood vessel endothelial cells, implicating L1 in transendothelial migration of DCs. In agreement with these findings, L1 was expressed in cutaneous DCs that migrated to draining lymph nodes, and its ablation reduced DC trafficking in vivo. Within the skin, L1 was found in Langerhans cells but not in dermal DCs, and L1 deficiency impaired Langerhans cell migration. Under inflammatory conditions, L1 also became expressed in vascular endothelium and enhanced transmigration of DCs, likely through L1 homophilic interactions. Our results implicate L1 in the regulation of DC trafficking and shed light on novel mechanisms underlying transendothelial migration of DCs. These observations might offer novel therapeutic perspectives for the treatment of certain immunological disorders

Introduction to symplectic Dirac operators

One of the basic ideas in differential geometry is that the study of analytic properties of certain differential operators acting on sections of vector bundles yields geometric and topological properties of the underlying base manifold. Symplectic spinor fields are sections in an L^2-Hilbert space bundle over a symplectic manifold and symplectic Dirac operators, acting on symplectic spinor fields, are associated to the symplectic manifold in a very natural way. Hence they may be expected to give interesting applications in symplectic geometry and symplectic topology. These symplectic Dirac operators are called Dirac operators, since they are defined in an analogous way as the classical Riemannian Dirac operator known from Riemannian spin geometry. They are called symplectic because they are constructed by use of the symplectic setting of the underlying symplectic manifold. This volume is the first one that gives a systematic and self-contained introduction to the theory of symplectic Dirac operators and reflects the current state of the subject. At the same time, it is intended to establish the idea that symplectic spin geometry and symplectic Dirac operators may give valuable tools in symplectic geometry and symplectic topology, which have become important fields and very active areas of mathematical research

Riemannian metrics of constant mass and moduli spaces of conformal structures

This monograph deals with recent questions of conformal geometry. It provides in detail an approach to studying moduli spaces of conformal structures, using a new canonical metric for conformal structures. This book is accessible to readers with basic knowledge in differential geometry and global analysis. It addresses graduates and researchers