Peeling High-Dimensional Oranges -
A `simplicial complex' is a space that one can obtain by gluing together triangles, tetrahedra, and higher dimensional analogues called `simplices'.  Simplicial complexes are important objects in topology, algebra, and combinatorics, as they can model a wide variety of spaces in a way that is accessible to calculation. One way to study a simplicial complex X is via a `shelling': a way to order the top dimensional faces of X so that each piece intersects the previous facets in a nice way, similar to the way one would (un)peel an orange. The existence of a shelling has important consequences for its topological and algebraic properties of X.  A well-known conjecture of Simon posits that certain complexes admit a greedy shelling, where one cannot get stuck in the process of lining up the facets. We will discuss ideas surrounding Simon's conjecture, including our proof of a special case which involves an application of `chordal graphs'.  Much of this is joint work with Suho Oh and past REU students Michaela Coleman and Nathan Geist.