Counting faces on simplicial complexes: V-E+F and beyond

Start: 04/01/2015 - 4:15pm
End  : 04/01/2015 - 5:15pm


Steven Klee, Seattle University


A graph is a combinatorial object that is built out of vertices and edges.  More generally, a simplicial complex is a combinatorial object that is built out of vertices, edges, triangles, tetrahedra, and their higher-dimensional cousins.  The most natural combinatorial statistics to collect on a simplicial complex are its face numbers, which count the number of vertices, edges, and higher-dimensional faces in the complex.  

This talk will give a survey on face numbers of simplicial complexes, beginning with planar graphs and moving on to graphs on other surfaces, such as tori or projective planes.   From there, we will study spheres and manifolds of higher dimensions.  We will undertake two main questions in this talk: First, what is the relationship between the face numbers of a simplicial complex and its underlying geometric structure? Second, how can we infer extra combinatorial information from properties of the underlying graph of a simplicial complex, such as graph connectivity or graph colorability?

Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

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