## EVENT DETAILS AND ABSTRACT

**Colloquium**
**Title:** Fixed-energy harmonic functions and acyclic orientations

**Speaker:** Rick Kenyon

**Speaker Info:** Brown University

**Brief Description:**

**Special Note**:

**Abstract:**

The discrete Laplacian on graphs is one of the most
fundamental and useful operators in graph theory, discrete
probability, and network science. One naturally assigns positive
weights, or conductances, to edges. The Dirichlet problem in this
context is to find a harmonic function with specified boundary
values. Here we change the problem slightly: we solve the Dirichlet
problem for finite networks, fixing ``edge energies" rather than
fixing conductances. More precisely, we show that for any given
choice of edge energies there is a choice of conductances for which
the resulting harmonic function realizes those energies. In fact the
set of solutions is the number of compatible acyclic orientations of
the graph. For rational data, the Galois group of the totally real
algebraic numbers acts naturally on this set of solutions. We also
consider scaling limits on large graphs which lead to novel PDEs. As
a discrete geometry application we study fixed-area rectangulations
of planar domains. This is joint work with Aaron Abrams.

**Date:** Wednesday, May 06, 2015

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Antonio Auffinger

**Contact email:** auffing@math.northwestern.edu

**Contact Phone:** 847-491-5466

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