Distinguished Lecture Series of the Courant Research Centre, Göttingen, November 4-6, 2008
Russell Lyons (Indiana University, Bloomington): Determinants: Probability, Combinatorics, Topology.
Everybody is invited to attend the first Distinguished Lecture Series of the Courant Research Centre in Göttingen. For any further information about the lecture series please contact Andreas Thom (firstname.lastname@example.org).
1st Talk (Tuesday, November 4, 16.15 - 17.15)
Asymptotic Enumeration of Spanning Trees via Fuglede-Kadison Determinants
ABSTRACT: Methods of enumeration of spanning trees in a finite graph and relations to various areas of mathematics and physics have been investigated for more than 150 years. We will review the history and applications. Then we will give new formulas for the asymptotics of the number of spanning trees of a graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call ``tree entropy", which we show is a logarithm of a Fuglede-Kadison determinant of the graph Laplacian for infinite graphs. Proofs involve new traces and the theory of random walks.
2nd Talk (Wednesday, November 5, 16.15 - 17.15)
Stationary Determinantal Processes (Fermionic Lattice Gases)
ABSTRACT: Given a measurable function f on the d-dimensional torus with values in the unit interval, there is a 2-state stationary random field on the d-dimensional integer lattice that is defined via minors of the d-dimensional Toeplitz matrix of the function f. The variety of such systems includes certain combinatorial models, certain finitely dependent models, and certain renewal processes in one dimension. Among the interesting properties of these processes, we focus mainly on whether they have a phase transition analogous to that which occurs in statistical mechanics. We describe necessary and sufficient conditions on f for the existence of such a phase transition and give several examples to illustrate the theorem. We also give some idea of the proofs, which are based on harmonic analysis, functional analysis, real analysis, and complex analysis. This is joint work with Jeff Steif.
3rd Talk (Thursday, November 6, 15.30 - 16.30)
Random Complexes via Topologically-Inspired Determinants
ABSTRACT: Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. We present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes. On finite complexes, they relate to (co)homology, while on infinite complexes, they relate to $\ell^2$-Betti numbers. One use is to get uniform isoperimetric inequalities.