Extremal Spectral Dynamics and a Fractal Theory for Simplicial Complexes
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...
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Format:  Doctoral Thesis 
Language:  English 
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PhilippsUniversität Marburg
2023

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Summary:  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 nontrivially 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 nontrivially 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 graphdirected
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 higherdimensional 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. 

DOI:  10.17192/z2023.0540 