Algebra & Geometry Seminar

Speaker: Tathagata Basak
Title: Building lattices from graphs
Abstract: Given a graph D such that the second largest eigenvalue of its adjacency matrix is a positive integer, one can define an free abelian group L together with a integer valued dot product of Lorentzian signature such that the symmetries of D act on L preserving the dot product. This kind of construction goes back to at least [Neumaier and Seidel, 1983] and is analogous to building a root lattice from a Dynkin diagram. Our aim is to illustrate some applications of this sort of construction.
We’ll discuss a “complexification” of our graph-to-lattice construction, that produces interesting “complex reflection groups”. One of these examples is related to the space of cubic surface. We’ll try to describe the new information we obtain about the space of cubic surfaces from this point of view.
We’ll try to keep the prerequisites to a minimum. In particular, no background knowledge of graph theory or hyperbolic geometry or cubic surfaces will be assumed.
Some of this is joint work with Daniel Allcock and Eduard Looijenga and some of this is joint work with Jon McCammond.