Faribault Woolen Mill:  the Effect of Different Speed Levels on Overall Yarn Performance

Design an experiment, collect and analyze data from Faribault Woolen Mill.

Seminar in Probability

Description can be found here.

Topics:

Polya Urn Models
Random Triangles on a Sphere
The Birthday Problem and Generalizations

Combinatorial Species and Graph Enumeration

The theory of graphs has diverse applications in pure and applied mathematics, computer science, and elsewhere. These applications often contribute back to the core theory notions of “interesting” properties a graph might have—for example, a graph might be connected, acyclic, or regular. To an enumerative combinatorialist like me, the natural question is then: how many graphs are there satisfying a given property and with a given number of vertices?

Historically, such questions have often been answered using the algebra of generating functions, which is powerful but often rather mysterious. In 1985, Canadian mathematician André Joyal introduced the theory of “combinatorial species”, which rigorizes and illuminates the theory of generating functions using the very modern language of category theory. This theory provides a uniform (and often very helpful!) framework for tackling questions related to graph enumeration.

In the first weeks of this project, I’ll give you a guided tour of category theory and its application to enumerative combinatorics, using an excellent text by Bergeron, Labelle, and Leroux. We’ll then move on to consider a few interesting graph properties (including point-determination, bipartitionability, and self-complementarity) and apply the theory you’ve learned to try to enumerate these graphs, possibly answering open problems (in which case we might even pursue publication!). Along the way, we’ll make extensive use of computer algebra to get computers to do the hard work for us in your choice of languages: Python, Haskell, and Maple are all useful in this field.

It is recommended that the students who undertake this project have some background in combinatorics and at least the basic theory of finite groups. It will also be very helpful if at least one student already knows one of the above programming languages or is interested in learning.

Galois Groups, Plane Trees, and Special Polynomials

One of the fundamental objects in number theory is the Galois group of the algebraic closure of the rational numbers, which we’ll call G for short. This group encodes information about every algebraic extension of the rationals, and in turn these extensions describe fundamental properties of prime numbers. Understanding the structure of G remains a crucial goal of many researchers. To better understand any group, it can be useful to have a collection of objects on which the group acts faithfully, that is, so that each group element can be uniquely identified by how it transforms the collection. A surprising action of G is on the collection of bicolored plane trees, seemingly innocuous objects that are simple graphs in the two-dimensional plane. This action comes about because each bicolored plane tree can be associated to a special kind of polynomial — one that maps all of its critical points to just two values. This subject combines ideas from number theory, group theory, and geometry, with some optional tangents into combinatorics and complex analysis. The specific focus will be determined in part by the interests of the group, though the main goal is to understand the faithful action of G on the set of plane trees.

The project will revolve around a close reading of the paper “Plane trees and algebraic numbers” by G. Shabat and A. Zvonkin, which details in an approachable style the surprising interrelations of these various ideas. You’ll work out some examples, and perhaps study new questions about how these polynomials behave under iteration.

Some familiarity with group theory and Galois theory is recommended but not required. The paper itself assumes very little background.

Directed Reading in Elliptic Functions and Modular Forms

Description can be found here.