Takashi Shioya – författare

This is a self-contained account of how some modern ideas in differential geometry can be used to tackle and extend classical results in integral geometry. The authors investigate the influence of total curvature on the metric structure of complete, non-compact Riemannian 2-manifolds, though their work, much of which has never appeared in book form before, can be extended to more general spaces. Many classical results are introduced and then extended by the authors. The compactification of complete open surfaces is discussed, as are Busemann functions for rays. Open problems are provided in each chapter, and the text is richly illustrated with figures designed to help the reader understand the subject matter and get intuitive ideas about the subject. The treatment is self-contained, assuming only a basic knowledge of manifold theory, so is suitable for graduate students and non-specialists who seek an introduction to this modern area of differential geometry.

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.