DescriptionGraphs are simple combinatorial objects used in everyday life in countless contexts for illustrative purposes (examples are family trees or a map of the London underground). My passion is geometry and its interaction with other fields in mathematics. So I have a natural curiosity whether deep and fundamental concepts of geometry can be introduced and successfully exploited in the combinatorial setting of graphs. One way to get from curved spaces like surfaces to the combinatorial world of graphs is to tessellate the surface. But the world of graphs is much richer and many graphs are not surface tessellations. Very interesting fundamental concepts from topology/geometry are invariants like the Euler characteristic, suitable notions of curvature, eigenvalues of the Laplace operator (related to a famous classical question "Can one hear the shape of a drum?") and even the dimension of the space of meromorphic functions on a surface with prescribed zeroes and poles (Theorem of Riemann-Roch).In this project, we will investigate how to transfer some of these geometric concepts from the smooth world of surfaces to the combinatorial world of graphs. We will study a discrete Laplace operator (closely related to the adjacency matrix of the graph), curvature notions on graphs motivated by the Theorem of Gauss-Bonnet and the Euler-characteristic (combinatorial curvature) or based on geometric ideas stemming from the highly active Theory of Optimal Transport (Ollivier Ricci curvature) or on Bochner's formula, an important identity in Riemannian Geometry (Bakry-Emery curvature), and how a fundamental result from Algebraic Geometry, the Riemann-Roch Theorem, can be applied to decide about winning strategies of certain chip-firing games on the vertices of a graph. For students interested in this topic it would be useful but not obligatory to have taken the 3H module Differential Geometry and the current 4H module Riemannian Geometry. With regards to resources, there are not yet textbooks available covering these very timely topics, and we have to consult articles. We will start with chapters in the book of Normal Biggs (see below), which is a classic and allows us to become familiar with the area of algebraic/spectral graph theory. Then students will branch out in specific topics, sources for some of which are represented below. It is natural, at present, if readers understand only little of these sources, but as the background knowledge develops, they will become increasingly accessible. Some Resources
email: N Peyerimhoff |