Abstract: Networks provide a mathematical framework for studying complex systems across biology, physics, ecology, neuroscience, and many other fields. In this talk, I will present two projects from my thesis that use spectral graph-theoretic methods to understand both the structure and dynamics of complex networks. One part of the talk focuses on community detection, where communities represent groups of nodes that are more strongly connected to each other than to the rest of the network. I will present a new analytical approach for detecting community structure in directed weighted networks. Unlike many computational methods, this approach relies only on the eigenspectrum of the network adjacency matrix and does not require free parameters to be tuned or trained. I will describe the method, explain how the number of communities can be estimated from the spectral information, prove the validity of the approach in an ideal setting, and demonstrate its performance through numerical simulations on weighted graphs with random edge perturbations. The other part of the talk focuses on dynamics in multilayer networks of Kuramoto oscillators. The Kuramoto model describes synchronization processes in many natural systems, including interacting neural populations. I will show how the dynamics of a large multilayer Kuramoto system can be related to two smaller systems: one describing intra-layer interactions and the other describing inter-layer interactions. This decomposition makes it possible to construct solutions for the full multilayer system and study their linear stability. Together, these projects show how spectral methods can provide analytical insight into both the organization of networks and the collective dynamics they support.

Abstract: Inspired by the work of Long, Margalit, Pham, Verberne, and Yao, we prove that the fine separating curve graph of a surface of genus at least 3 has automorphism group isomorphic to the group of self-homeomorphisms of the surface. To obtain this result, we prove that the natural map from the group of homeomorphisms to the group of automorphisms of the fine inessential curve graph is an isomorphism. The first result is analogous to that of Brendle and Margalit on the separating curve graph and its relationship to the mapping class group. The main results of this project serve as evidence in partial support of an Ivanov-type metaconjecture in the context of the homeomorphism group.

In this presentation, we will introduce surfaces and curves, the extended mapping class group and its action on curve graphs, Ivanov's metaconjecture, analogous results on fine curve graphs, and the Ivanov-type metaconjecture for the group of homeomorphisms. We will then encode points in a surface using convergent sequences and configurations of separating curves and show that these are preserved by automorphisms of the fine separating curve graph. In other words, we will show that, despite being restricted to separating curves, the graph-theoretic data of the fine separating curve graph is sufficient to encode the topological properties of the surface and is preserved by automorphisms of the graph, allowing us to prove the desired group isomorphism.