Blättern Fakultät für Mathematik und Informatik (inkl. GAUSS) nach "Schick, Thomas Prof. Dr." Betreuer
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The Spectra on Lie Groups and Its Application to twisted L2-Invariants
(2024-02-29)We computed spectra of various $G$-invariant differential operators on the universal cover of 2x2 unimodular groups. This was achieved by applying tools from harmonic analysis/representation theory to corresponding groups. ... -
Twisted equivariant K-theory and equivariant T-duality
(2023-09-18)This thesis is about two things: twisted equivariant K-theory and equivariant topological T-duality. First, we prove a fixed point decomposition theorem for twisted equivariant K-theory, generalising a result of Atiyah and ... -
Novikov-Shubin Invariants of Nilpotent Lie Groups
(2023-05-25)Novikov-Shubin invariants are so-called L2-invariants of non-compact manifolds. They are defined using the Laplace operators and measure the density of their spectra near zero. This near-zero part of the spectrum is ... -
Concordances in Positive Scalar Curvature and Index Theory
(2023-02-24)We apply the strategy to study of diffeomorphisms via block diffeomorphisms to the world of positive scalar curvature (psc) metrics. For each closed psc manifold, we construct the cubical set of all psc block metrics, the ... -
Foliated Positive Scalar Curvature
(2021-11-11)In this thesis we study different questions on scalar curvatures. The first part is devoted to obstructions against existence of a (Riemannian) metric with positive scalar curvature on a closed manifold. The second ... -
L²-Invariants for Self-Similar CW-Complexes
(2020-11-11)L²-invariants are commonly defined only for periodic spaces, that is, spaces with a cocompact action by some discrete group. This thesis develops a theory of L²-invariants for quasi-periodic spaces instead, where the group ... -
Index Theory and Positive Scalar Curvature
(2020-07-24)The aim of this dissertation is to use relative higher index theory to study questions of existence and classification of positive scalar curvature metrics on manifolds with boundary. First we prove a theorem relating ... -
Benjamini-Schramm Convergence of Normalized Characteristic Numbers of Riemannian Manifolds
(2020-04-30)We study a weak form of Gromov-Hausdorff convergence of Riemannian manifolds, also known as Benjamini-Schramm convergence. This concept is also applicable to other areas and has widely been studied in the context of ... -
A Mayer-Vietoris Spectral Sequence for C*-Algebras and Coarse Geometry
(2019-04-10)Let $A$ be a C*-algebra that is the norm closure $A = \overline{\sum_{\beta \in \alpha} I_\beta}$ of an arbitrary sum of C*-ideals $I_\beta \subseteq A$. We construct a homological spectral sequence that takes as input the ... -
Shape space in terms of Wasserstein geometry and application to quantum physics
(2018-11-26)This thesis offers a mathematical framework to treat quantum dynamics without reference to a background structure, but rather by means of the change of the shape of the state. For this, Wasserstein geometry is used. The ... -
New topological and index theoretical methods to study the geometry of manifolds
(2018-07-31)For a $\mathit{Spin}$ manifold $M$ the Rosenberg index $\alpha([M])$ is an obstruction against positive scalar curvature metrics. When $M$ is non-$\mathit{Spin}$ but $\mathit{Spin}^c$, Bolotov and Dranishnikov suggested ... -
Characters on infinite groups and rigidity
(2018-05-02)We show that for a strong extension of discrete measured groupoids $1\to\mathcal{S}\to\mathcal{G}\to\mathcal{Q}\to 1$ with $L\mathcal{G}$ a finite factor, $\mathcal{Q}$ has poperty (T) if and only if the inclusion of ... -
Secondary large-scale index theory and positive scalar curvature
(2016-09-06)We develop a theory of secondary invariants associated to complete Riemannian metrics of uniformly positive scalar curvature outside a prescribed subset on a spin manifold. We work in the context of large-scale (or "coarse") ... -
On an analogue of L2-Betti numbers for finite field coefficients and a question of Atiyah
(2016-07-12)We construct a dimension function for modules over the group ring of an amenable group. This may replace the von Neumann dimension in the definition of L2-Betti numbers and thus allows an analogue definition for finite ... -
Geometric twisted K-homology, T-duality isomorphism and T-duality for circle actions
(2015-11-16)We discuss topological T-duality and the associated geometric or topological objects in this thesis. Concretely, it consists of three parts. In this first part we prove two versions of geometric twisted K-homology are ... -
Algebraic Structure and Integration in Generalized Differential Cohomology
(2014-01-22)The present thesis deals with the construction of algebraic structure, particularly products, on generalized differential cohomology from an abstract homotopy-theoretic point of view. Beginning with a multiplicative ... -
Categorification and applications in topology and representation theory
(2013-08-28)This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's ... -
L2-invariants of nonuniform lattices in semisimple Lie groups
(2013-05-03)We compute L²-invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel-Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for ... -
Index Theory and Positive Scalar Curvature
(2012-06-22)Obstructions to the existence of positive scalar curvature metrics are considered. This thesis has two fairly independent parts. The first part of this thesis proves that the Roe (or coarse) index of the Dirac-Rosenberg ... -
Higher Lefschetz invariants for foliated manifolds
(2012-05-22)Let $(M, F)$ be a compact foliated manifold. Suppose that a compact Lie group $Γ$ acts on $(M, F)$ by diffeomorphisms of Mthat map each leaf onto itself. We call such a structure a foliated $Γ$ -manifold and denote it ...