Wilderich Tuschmann – författare

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kähler and Sasaki geometry.

This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

This proceedings volume presents a collection of fifteen peer-reviewed and expanded articles based on expert talks delivered at two workshops held in September 2023: the RIMS Symposium on Analysis, Geometry and Stochastics on Metric Spaces at the Research Institute of Mathematical Sciences (Kyoto), organized by S. Shimizu and W. Tuschmann, and Metrics and Measures 2023 at the Mathematical Institute of Tohoku University (Sendai), organized by T. Shioya. These events were part of the 2023 activities of the HeKKSaGOn Mathematics Group, a collaborative network founded in 2010 that brings together six leading universities: Heidelberg, Karlsruhe (KIT), Kyoto, Sendai (Tohoku), Göttingen, and Osaka.The volume reflects recent developments in analysis, geometry, non-Euclidean statistics, and stochastics, as well as their applications. The articles offer a rich and diverse cross section of the themes explored during the workshops. Topics include long-time limits of diffusion means, Helmholtz–Weyl decompositions in exterior domains, Green functions in nonnegative Ricci curvature, locally homogeneous RCD spaces, Wald spaces for phylogenetic trees, Korevaar–Schoen energy forms on fractals, gluing constructions under lower curvature bounds, combinatorial Riemannian manifolds, the string topology coproduct and intersection multiplicity, probabilistic morphisms and geometric methods in statistical, manifold, and machine learning, gradient flows of convex functions, open problems in metric measure geometry, dynamical aspects of Hadamard spaces, obstacle problems for mean curvature flows, and moduli spaces of nonnegatively curved metrics on Riemannian manifolds and RCD spaces.Together, these contributions aim to inspire both experts and newcomers in the field, offering insights into cutting-edge research and fostering further exploration across disciplines.