# Research

## Discrete Mathematics

### Thursday, September 6, 2001

Title: An Anti-Ramsey Theorem on Posets
Speaker: Greg McColm
Time: 2:00pm‐3:00pm
Place: LIF 267

#### Abstract

One of the generalizations of Ramsey Theory has been to posets. It is known that for any finite poset $$P$$, there is a finite poset $$Q$$ such that any 2-coloring of the nodes of $$Q$$ yields a monochromatic copy of $$P$$. We will explore some of the results surrounding this fact, including the fact that for any trees $$S$$, $$T$$, and any 2-coloring of the Cartesian product $$S\times T$$ into red nodes and blue nodes, there is either a red copy of $$S$$ or a blue copy of $$T$$. Then we will find a pair of finite posets $$P$$, $$Q$$, and a 2-coloring of $$P\times Q$$ admitting no red copy of $$P$$ nor a blue copy of $$Q$$.

### Thursday, September 13, 2001

Title: An Anti-Ramsey Theorem on Posets, Part II
Speaker: Greg McColm
Time: 2:00pm‐3:00pm
Place: LIF 267

### Thursday, September 20, 2001

Title: Solutions of Bethe Ansatz Equations in Some Physics Models
Time: 2:00pm‐3:00pm
Place: LIF 267

#### Abstract

The Bethe Asatz equations are nonlinear algebraic equations satisfied by the eigenvalues of a physical system. Stieltjes solved these equations for the Coulomb gas model. This work is also connected to earlier work of Heine who counted the number of polynomial solutions to second order differential equations with polynomial coefficients. $$Q$$-analogues of these results will be described and I will show the connection with Bethe Ansatz equations for the XXX and XXZ models. In doing so one needs to develop a new theory of singuarities of second order equations in the Askey-Wilson operators.

### Thursday, September 27, 2001

Title: Difference Equations, Orthogonal Polynomails, and Rogers-Ramanujan Identities
Time: 2:00pm‐3:00pm
Place: LIF 267

#### Abstract

We show how the Rogers-Ramanujan identities follow from studying difference equations motivated by orthogonal polynomials. In particular this explains and gives infinite families of generalizations of a list of Rogers-Ramanujan identities developed in the 1960's by L. J. Slater, who claimed it was a complete list.

### Thursday, October 4, 2001

Title: A Non-Monotonic Propositional Logic
Speaker: Richard Stark
Time: 2:00pm‐3:00pm
Place: LIF 267

### Thursday, October 11, 2001

Title: A Non-Monotonic Propositional Logic, Part II
Speaker: Richard Stark
Time: 2:00pm‐3:00pm
Place: LIF 267

### Thursday, October 18, 2001

Title: A Non-Monotonic Propositional Logic, Part III
Speaker: Richard Stark
Time: 2:00pm‐3:00pm
Place: LIF 267

### Thursday, October 25, 2001

Title: Graph Homomorphisms and Graph Automorphisms
Speaker: Brian Curtin
Time: 2:00pm‐3:00pm
Place: LIF 267

#### Abstract

Let $$G$$ and $$H$$ denote finite simple graphs. An automorphism of $$H$$ is a permutation of its vertices which maps adjacent vertices to adjacent vertices (and nonadjacent vertices to nonadjacent vertices). Let $$\operatorname{Aut}(H)$$ denote the full group of automorphisms of $$H$$. A homomorphism of $$G$$ into $$H$$ is a map from the vertex set of $$G$$ to that of $$H$$ which maps the endpoints of edges of $$G$$ to the endpoints of edges of $$H$$. We discuss the use of graph homomorphisms in determining the automorphisms of a graph.

More precisely, we do the following. Fix a natural number $$n$$, and let $$p$$ and $$q$$ denote $$n$$-tuple of vertices of $$H$$. We show that if $$p$$ and $$q$$ belong to distinct orbits under the action of $$\operatorname{Aut}(H)$$ then there is a graph $$G$$ and an $$n$$-tuple $$r$$ of vertices of $$G$$ such that the number of homomorphisms from $$G$$ into $$H$$ maping $$r$$ to $$p$$ element-wise differs from the number of homomorphisms from $$G$$ into $$H$$ mapping $$r$$ to $$q$$ element-wise. To prove this result we shall use some classical results on polynomial invariants of finite groups.

### Thursday, November 1, 2001

Title: Ph.D. Program Review Planning Session
Time: 2:00pm‐3:00pm
Place: LIF 267

#### Abstract

We will discuss what we will say to the Ph.D. committee next week.

### Thursday, November 14, 2001

Title: Graph Homomorphisms and Graph Automorphisms, Part II
Speaker: Brian Curtin
Time: 2:00pm‐3:00pm
Place: LIF 267