Title: Mathematical Methods for Analyzing Genomic Data

Speaker: J. B. Nation, University of Hawaii

We describe how some simple modifications of basic methods

give a useful analysis of genetic data, and apply this to

ovarian cancer. A refined singular value decomposition

allows one to look for common biological signals in

gene expression, microRNA and methylation site data.

A clustering technique based on lattice theory identifies

microRNA signals that predict survival. A neural network

using a wavelet transform can be used to choose candidates

for different treatments. Finally, we can identify some

of the biology behind these results, indicating treatment

possibilities.

This is joint work with Gordon Okimoto, Ashkan Zeinalzadeh,

Jenna Maligro, Tammy Yoshioka, and and Tom Wenska of the

University of Hawaii Cancer Center.

Speaker: Anthony Walter, University of Hawaii

Title: Introduction to the Ising model.

Abstract: We will define the Ising model, one of the simplest statistical mechanics

models which exhibits a phase transition. We are looking at a configuration

space on a discrete finite lattice, which is extended to the infinite lattice limit. Depending on some boundary conditions and a parameter $\beta$, in two or more dimensions it can be shown there is a phase transition from a disordered to an ordered phase. In particular we will look at the ferromagnetic case with positive boundary conditions.

We will prove a lower bound of the critical phase transition point.

**Title:** * Bounds on the Number of Covers for Lattices and Related Posets *

**Abstract:** How many covers can there be in a lattice of order n? This question has gone unanswered for decades. In the pursuit of a couple of conjectures, one of which is nearly 40 years old, we have obtained novel results primarily related to asymptotic bounds for lattices and related posets. This talk will involve lattice theory, order theory, combinatorics, graph theory, and analysis.

Reconfiguration in Graph Coloring

In mathematics, as in life, there are often multiple solutions to a question.

Reconfiguration studies whether it is possible to move from one solution to

another following a given set of rules. Is it possible? How long will it take?

In this talk, we will consider reconfiguration of graph coloring.

A proper coloring of a graph is an assignment of a color to each vertex of the

graph so that neighboring vertices have different colors. Suppose we change the

color of just one vertex in a graph coloring. Can we get from one coloring to

another by a sequence of vertex changes so that each step along the way is a

proper coloring? The answer is yes, if we are allowed an unlimited number of

colors. But, what is the fewest colors we can have for this to work? How many

steps might it take? We will look at this, related questions and generalizations.