Blyth 1999

The seventh annual
R.A. Blyth Lectures in Mathematics

 

Professor Victor Guillemin
Massachusetts Institute of Technology
 
will give three lectures on  

Equivariant deRham Theory and Graphs

 

Abstract:

Let G be a compact abelian Lie group. The objects we will discuss in these lectures are a class of compact G-manifolds called GKM manifolds (after Goresky, Kottwitz and MacPherson, who were the first to observe their unusual properties). The salient feature of a GKM manifold M, is its "one-skeleton", which is a necklace of embedded two-spheres on each of which the group, G, acts by rotation about an axis of symmetry. The intersection properties of the beads of this necklace are described by a graph , whose edges are the beads and whose vertices are their points of intersection. ( is a graph because each bead can intersect other beads only at the poles of its axis of symmetry; hence each edge has just two vertices.)

In our first lecture we will discuss some geometric structures on this graph which are inherited from the geometry of the manifold, M. The most interesting of these is a connection on the "tangent bundle" of the graph. We will describe this connection for some familiar graphs like the complete graph on N vertices and the permutahedron, and discuss some combinatorial invariants of graphs which can be defined by means of it.

Another important object which acquires from M is an axial function which describes how the beads associated with the edges of are rotated by the G-action. The main result of this lecture will be a Morse theoretical recipe for computing the Betti number of M from this axial function. (The statement and proof of this result, which is a theorem of Ethan Bolker on the combinatorics of "ordering edges of a graph", will not require any knowledge of standard Morse theory or of the theory of connections on manifolds.)

Our second lecture will be an introduction to the topic of GKM manifolds. In this lecture we will describe in more detail the one skeleton that we mentioned above and show how features of the topology of M are captured by it. The main result of this lecture will be a graph theoretical description of the equivariant cohomology ring of M, a result which is due to Goresky-Kottwitz-MacPherson.

Our third lecture will be concerned with the Smith conjecture: Suppose M is a compact G-manifold whose fixed point set consists of just two points, p and q. Let p and q be the isotropy representations of G on the tangent spaces of p and q. The Smith Conjecture conjectures that p and q are isomorphic. Using localization theorems in equivariant deRham theory, Atiyah, Bott and Milnor were able to prove this conjecture in the mid-1960's. However, one can ask: Suppose there are three (or more) fixed points. Are there interesting relations among their isotropy representations? In particular, what sorts of relations are predicted by the localization theorems? By converting these relations into identities on graphs one can say quite a bit about this question where M is a GKM manifold.

 

Morse Theory on Graphs

Monday, March 22, 1999 at 4:10 p.m.
Sidney Smith Hall
100 St. George Street
Room 2102

GKM Manifolds

Tuesday, March 23, 1999 at 4:10 p.m.
Sidney Smith Hall
100 St. George Street
Room 5017A

Graphical Versions of the Localization Theorems in Equivariant deRham Theory

Wednesday, March 24, 1999 at 4:10 p.m.
Sidney Smith Hall
100 St. George Street
Room 5017A