Algebra Seminar

DATE:     Thursday, 2 September 2004
SPEAKER:  David Glickenstein (Arizona)
TITLE:    Combinatorial Yamabe flow: between geometric analysis and
          graph theory 

ABSTRACT: We shall study a piecewise linear geometry which lies somewhere 
between the geometry of graphs and the geometry of Riemannian manifolds. 
In our context, the geometry comes from a simplicial complex whose 
vertices are given weights which determine the lengths of edges (so the 
vertices and edges form a weighted graph), and hence the area and volume 
of higher dimensional simplices. Combinatorial Yamabe flow is a way to 
deform the geometry into something less complicated via an ordinary 
differential equation, an analogue of the Ricci or Yamabe flow in 
Riemannian geometry designed for a piecewise-linear object instead of a 
smooth manifold. Such equations may be helpful in applying the successful 
methods of geometric evolution equations to new realms of problems in 
physics, topology, algebraic geometry, numerical analysis, graph 
theory, and other fields. The methods will involve basic Euclidean 
geometry as well as the application of simple ideas from partial 
differential equations to functions on graphs. This talk will be 
self-contained and should be easily accessible to graduate students.