Algebra and Geometry of q-Simplicial Complexes

A Grassmannian complex is a family of linear subspaces of a given linear space, closed under inclusion. In the talk we will explore the properties of Grassmannian complexes over a finite field and define boundary maps that give rise to notions of connectivity and high dimensional expansion. In contrast to the simplex, where all the homology groups are trivial, the complete Grassmannian (consisting of all subspaces of a given linear space) may have a non-trivial homology, and other exciting phenomena.

We will show analogues to the theorems of Linial, Meshulam and Wallach on the expansion of the complete Grassmannian, and to the phase transition of the connectivity of a random complex.

If time permits, we will discuss related extremal problems and topological overlap.

Based on joint work with Ran Tessler.