
The TUM School of Computation, Information and Technology brings together programs from the former Departments of Mathematics, Informatics, and Electrical and Computer Engineering. The school focuses on providing top-notch instruction and supporting students to prepare them for future professional careers. By consolidating synergies and integrating innovative technologies, the school aims to offer interdisciplinary education that equips students for a changing global job market.
This dissertation explores advancements in Riemannian geometry, specifically focusing on how scalar curvature influences the shape and size of mathematical spaces. It introduces new examples related to stability conjectures concerning the characterization of spheres by scalar curvature. The work addresses the necessity of conditions that prevent 'bubbling' in sequences of manifolds and constructs examples that converge in a specific sense (volume-preserving intrinsic flat sense) but not in another (Gromov–Hausdorff sense). This is achieved by refining a technique called the 'tunnel construction' to allow for the attachment of 'tunnels' or 'wells' to manifolds while minimally affecting scalar curvature. The research also builds upon and extends previous work on constructing manifolds with specific properties, including limits with no geodesics and counterexamples to existing conjectures. The study delves into the theoretical aspects of scalar curvature and its implications for classifying geometric structures. It presents novel constructions that challenge existing conjectures and expand the understanding of manifold properties. The dissertation aims to provide a deeper insight into the relationship between curvature, convergence of manifolds, and their topological characteristics, offering non-perturbative counterexamples with complex topologies.