# Math 676

**Instructor:** Dr. Clayton Shonkwiler

**Time:** Monday, Wednesday, Friday 2:00–2:50

**Location:** Engineering E204

**Office:** Weber 216

**Office Hours:** Wednesday 1:00–2:00, Thursday 1:00–2:00, and by appointment

**Email Address:** clay@shonkwiler.org

**Syllabus**

## Overview

A foundational model in the study of polymers is the random walk model, in which a polymer is modeled by a collection of unit-length steps in space whose directions are chosen independently and uniformly on the sphere. While this model is not particularly physically realistic (real polymers have thickness, stiffness, charge, solvent interactions, etc.), it is a surprisingly decent approximation for open polymer chains, and serves as the foundation of all more advanced models.

The mathematics of random walks with independent steps has been well-understood for at least a century. However, many biologically interesting polymers (including bacterial DNA) form closed loops rather than open chains, and the mathematics of random walks which are conditioned to return exactly to their starting points is considerably more challenging. Applied mathematicians and polymer physicists have been simulating such closed random walks (a.k.a. *random polygons*) and doing numerical experiments for almost 50 years, but without rigorous justification for their algorithms.

Very recently, fast and provably correct algorithms for simulating random polygons – as well as very precise theorems about their geometry and topology – have been developed using the theory of symplectic geometry. Specifically, the moduli space of *n*-edge random polygons arises as the symplectic reduction of the product of *n* copies of the 2-sphere by the diagonal action of *SO*(3), and turns out to be a toric symplectic (almost) manifold.

The first major goal of this course is to understand the previous paragraph, developing and building on the basic concepts of symplectic geometry, Hamiltonian group actions, moment maps, and symplectic reduction.

Symplectic reduction is the natural quotient operation in symplectic geometry. A fact which is either surprising or obvious, depending on your perspective, is that there is a natural correspondence between symplectic reduction and the natural quotient operation for projective varieties: the Geometric Invariant Theorey (GIT) quotient. Consequently, the moduli space of random polygons also has an interpretation as a projective variety, and turns out to be a particular choice of compactification of the moduli space of pointed curves of genus 0.

The second major goal of this course, in conjunction with Renzo Cavalieri’s section of MATH 676, is to understand the previous paragraph and its consequences for polymer models and to explore whether other moduli spaces from algebraic geometry have interesting applications.

Since the first connections between the random walks/polymers world and the algebraic/symplectic geometry world are just beginning to be established, this is an exciting time to get involved, and this course is a unique opportunity to do so.

Prior familiarity with topology and advanced linear algebra will be very helpful, but the course will be reasonably self-contained and should not require too much specialized knowledge.

Some useful resources:

*Introduction to Symplectic Topology*, by Dusa McDuff and Dietmar Salamon*Lectures on Symplectic Geometry*, by Ana Cannas da Silva*Cohomology of Quotients in Symplectic and Algebraic Geometry*, by Frances Kirwan- Deligne, P., & Mostow, G. D. (1986). Monodromy of hypergeometric functions and nonlattice integral monodromy.
*Publications Mathématiques de l’Institut des Hautes Études Scientifiques***63**, 5–89. http://www.numdam.org/item?id=PMIHES_1986__63__5_0 - Kapovich, M., & Millson, J. J. (1996). The symplectic geometry of polygons in Euclidean space.
*Journal of Differential Geometry***44**(3), 479–513. http://projecteuclid.org/euclid.jdg/1214459218 - Hausmann, J.-C., & Knutson, A. (1997). Polygon spaces and Grassmannians.
*L’Enseignement Mathématique***43**, 173–198. http://dx.doi.org/10.5169/seals-63276 or https://arxiv.org/abs/dg-ga/9602012 - Hu, Y. (1999). Moduli Spaces of Stable Polygons and Symplectic Structures on \(\overline {\mathcal{M}} _{0,n}\).
*Compositio Mathematica*,**118**(2), 159–187. http://doi.org/10.1023/A:1001055409867 or https://arxiv.org/abs/dg-ga/9701011 - Kamiyama, Y., & Yoshida, T. (2002). Symplectic toric space associated to triangle inequalities. Geometriae Dedicata,
**93**(1), 25–36. http://doi.org/10.1023/A:1020393910472 - Thomas, R. P. (2005). Notes on GIT and symplectic reduction for bundles and varieties.
*Surveys in Differential Geometry***X**, 221–273. http://dx.doi.org/10.4310/SDG.2005.v10.n1.a7 or https://arxiv.org/abs/math/0512411 - Cantarella, J., & Shonkwiler, C. (2016). The symplectic geometry of closed equilateral random walks in 3-space.
*The Annals of Applied Probability***26**(1), 549–596. http://doi.org/10.1214/15-AAP1100 or https://arxiv.org/abs/1310.5924