title: Extremal Spectral Dynamics and a Fractal Theory for Simplicial Complexes
title: Extremale spektrale Dynamik und eine Fraktaltheorie für Simplizialkomplexe
titleType: AlternativeTitle
Creators: name: Märte, Julian
Publisher: name: Philipps-Universität Marburg
Year: 2023
Subjects: subject: Perturbationstheorie
subject: Spektrale Graphentheorie
subject: Zentraler Grenzwertsatz
subject: Unterteilung
subject: Perturbation theory
subject: Laplace Operator
subject: Sobolev theory
subject: Simplicial geometry
subject: Unterteilungen,Graphkonvergenz
subject: Spectral graph theory
subject: Simpliziale Geometrie
subject: Geometrie
subject: Kombinatorischer Laplace-Operator
subject: Graph convergence
subject: Mathematics
subject: FOS: Mathematics
subject: Mathematik
subject: Sobolevtheorie
subject: subdivision
subject: Sobolev
subject: Combinatorial Laplacian

description: The aim of this work is the exploration of spectral asymptotics of certain geometries associated to simplicial complexes. We will state how combinatorial and differential Laplacians can be associated to a simplicial complex and describe certain asymptotics linked to their spectra. First we take under consideration the change of spectrum for the combinatorial Laplacian under a certain class of subdivision procedures and show a universal limit theorem regarding the sequence arising from this construction. Universality in this case means that the limit spectrum carries no spectral information related to the input complex. It is thus only dependent on the dimension of the complex and the subdivision procedure used. We will carry out the explicit calculation of such a limit for one particular example of a subdivision related to barycentric subdivision. Next we point out obstructions to the application of the same procedure to the full barycentric subdivision. It will turn out that the procedure is not favorable if the given subdivision procedure is acting non-trivially on lower dimensional faces. Lastly we give an example of a subdivision procedure of high symmetry, i.e. edgewise subdivision, for which we can determine the spectrum by a group action argument even though it acts non-trivially on lower dimensional faces. Furthermore dual relations to fractal theory are examined and the particular class of fractals arising from subdivision of a complex in the sense of a graph-directed limit construction is formalized. In the end open question regarding the nature of the limit are formulated and initiating thoughts are presented. Secondly we associate to a simplicial complex a geometry (not necessarily embeddable in euclidean space) and show that there exists a natural differential Laplacian on this geometry. These complexes can be used to model thin structures around their geometry. As this modelling procedure is a higher-dimensional generalization of quantum graphs we will call a complex equipped with this differential structure a quantum complex. Thin structures over such a complex allow for modelling of systems with a larger number of dimensions not constraint by a small diameter. We show generalizations of estimates used in the proof of the spectral asymptotic of these thin structures for the graph case indicating that a general spectral asymptotic might be possible. We formulate open questions on spectral asymptotics and the relation of the combinatorial and differential Laplacian associated to the complex.