Polyhedral Combinatorics by Cesar Meza

Cesar Meza

SFSU | Jack Lund Endowment Fund

In a three-dimensional polyhedron, a “flag” is a vertex inside an edge inside a face; in higher dimensions, “flags” are defined similarly. Cesar Meza is studying the unsolved problem of counting the flags in interesting high-dimensional polyhedra.

ABSTRACT

Given two numbers, does there exist a polygon with exactly that many vertices and edges? If the number of vertices is at least 3 and the number of edges equals the number of vertices, then the answer is yes, such a polygon exists. Otherwise, no such polygon exists. This fact, in essence, characterizes all polygons. In three dimensions, every polyhedron satisfies Euler’s characteristic formula, # of vertices – # of edges + # of faces = 2. Remarkably, three-dimensional polyhedra are characterized by Euler’s characteristic formula and two inequalities. Characterizing higher-dimensional polyhedra by the number of their vertices, edges, faces, and higher-dimensional components remains an open problem. Counting the flags of polyhedra, which are a vertex inside an edge inside a face inside a higher-dimensional component, can be more informative. Cesar Meza’s research, at the interface of combinatorics and high-dimensional geometry, investigates the unsolved problem of counting the flags in interesting high-dimensional polyhedra.

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