## 2023 Thurston Lectures

The Thurston Lectures are named after Fields Medalist and former UC Davis mathematician William Thurston. The next Thurston Lecture series will be given by Richard Kenyon (Yale University) on May 10-12, 2023.

**Series title:** *Dimers, webs and SL(n) local systems I,II,III*

**Detailed schedule:**

Lecture I. Dimers and tensor networks — Wed May 10, 3:10-4 in MSB 1147

Lecture II. 3-webs and the boundary measurement matrix — Thu May 11, 3:10-4 in MSB 1147

Lecture III. From higher rank dimers to scalar dimers — Fri May 11, 3:10-4 in MSB 1147

**Abstract:**

*One of Bill Thurston’s many side projects was a result on domino tilings (tilings with 2x1 and 1x2 rectangles): he gave an algorithm for deciding when a planar polygon could be tiled with dominos. The domino tiling model (and more generally the planar dimer model) turns out to have remarkable connections with other parts of mathematics, from conformal field theory to integrable systems to representation theory.*

*Webs are representation-theoretic objects, defined by Greg Kuperberg to study invariants in tensor products of SL(3) representations. In these lectures I will discuss recent results on large-scale structures (webs) in random multiple-domino tilings, and their conformal invariance properties. *

*The lectures are based on joint work with David Wilson, Daniel Douglas, Haolin Shi.*

*Abstract for Lecture I. We will motivate the study of tensor networks in a number of areas of mathematics including statistical physics, representation theory, and graph theory. We define the dimer model, webs and multiwebs, their traces, and show how Kasteleyn theory is adapted to compute sums of traces of multiwebs. (Based on joint work with Daniel Douglas, Haolin Shi, and David Wilson.)*

*Abstract for Lecture II. We study 3-multiwebs (triple dimer covers) in planar graphs with boundary. Greg Kuperberg famously defined graphical presentations of SL3-invariants, called "spiders", in 1996, and a 3-multiweb is an example of such an object. We discuss how spiders arise in triple-dimer covers of graphs, and how the "boundary measurement matrix" can be used to compute connection probabilities for these objects. We give certain "SL3" versions of the well-known Lindstrom-Gessel-Viennot theorem. In the scaling limit these spider probabilities have appropriate conformally invariants limits. (Based on joint work with Haolin Shi.)*

*Abstract for Lecture III. Various "vertex models" such as the 6-vertex model, 20-vertex model and many others have special choices of interactions for which the model is "free-fermionic", that is, solvable by determinants. We discuss a general set-up for such vertex models, and how to solve them. In particular we show how higher-rank dimer models (dimer models on graphs with a GL_n -connection in particular) can be modeled by scalar dimers on decorated versions of the graph. (Based on joint work with Nicholas Ovenhouse.)*

## About the speaker

Richard Kenyon received his Ph.D. from Princeton University in 1990 under the direction of William Thurston. After a postdoc at IHES, he held positions at CNRS in Grenoble, Lyon, and Orsay and then became professor at UBC, Brown University and then Yale where he is currently Erastus L. Deforest Professor of Mathematics. He was awarded the CNRS bronze medal, the Rollo Davidson prize, the Loève prize, is a member of the American Academy of Arts and Sciences, and is a Simons Investigator.

### About William Paul Thurston

William Paul Thurston (October 30, 1946 – August 21, 2012) was among the most original and influential mathematicians of the twentieth century. He transformed the mathematics of foliations, low-dimensional topology, hyperbolic manifolds, the theory of rational maps, and geometric group theory. His work led to a fundamental rethinking of the structure of 3-dimensional spaces.

Thurston received a bachelor's degree from New College in 1967 and a Ph.D. in Mathematics from UC Berkeley in 1972. He spent a year at the Institute for Advanced Study and a year at MIT before being appointed Professor of Mathematics at Princeton University in 1974. In 1991 he moved to UC Berkeley and in 1993 he was appointed Director of the Mathematical Sciences Research Institute. In 1996 he moved again, this time to UC Davis, where he was a Professor of Mathematics until 2003, when he moved to Cornell. Bill was in the process of returning to UC Davis in 2012 when he tragically passed away.

Thurston’s work revealed the unexpectedly central role played by hyperbolic geometry in the study of low-dimensional manifolds. His Geometrization Conjecture, which he solved in many cases, changed the fundamental viewpoint from which mathematicians approached the study of manifolds.

Thurston was awarded the Veblen Prize in Geometry in 1976, the Fields Medal at the 1983 International Congress of Mathematicians in Warsaw and the Leroy P. Steele Prize for seminal contribution to research in 2012. Thurston had numerous Ph.D. students, many of whom became leading mathematicians themselves.

*"Mathematics is a process of staring hard enough with enough perseverence at the fog of muddle and confusion to eventually break through to improved clarity."*

–Bill Thurston, "About me" on MathOverflow