**abstract:**
Simplicial complexes associated with graphs form a rich and diverse area of research with numerous interesting results and connections to various fields of mathematics. These complexes provide a natural combinatorial framework that relates graph-theoretic properties to algebraic topology concepts. Following the introduction, we will focus on two classes of graph complexes: cut complexes and matching complexes. Cut complexes are a new family of graph complexes, that we defined motivated by a famous theorem of Ralf Fröberg, connecting commutative algebra and graph theory through topology. The k-cut complex ∆k(G) is the pure simplicial complex whose facets are complements of disconnected vertex sets of size k. We extend the results that motivated us and determine the topological and combinatorial properties of these complexes for various families of graphs. The second complex of our interest is the matching complex. For a simple graph G, the matching complex M(G) is the simplicial complex on the set of edges, with faces given by matchings in the graph, where a matching is a set of edges no two of which share a vertex. The topology of matching complexes has been the subject of much research over the years. We will present some recent work on matching complexes of polygonal line tilings and similar graphs. The talk is based on several papers with two groups of co-authors: one including M. Bayer, M. Denker, R. Rowlands, S. Sundaram, and L. Xue, and the other with M. Bayer and J. Vega.

Wed 2 Oct, 9:30 - 10:30, Aula Dini

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