The goal of our research is to identify the conditions under which ratios of totally-positive matrix minors are bounded. My presentation will seek primarily to explain the concepts of boundedness and total-positivity. In addition, I hope to discuss how these concepts come together in our research and, as time permits, offer a brief sketch of the techniques we use to study them.

Imagine you had a piece of string and you wound it into a knot and then you fused the two ends together. This is what mathematicians call a knot. Given two such knots when can you turn one into the other without cutting the string? How can you describe different knots? It turns out that these questions are very interesting ones in the subject of topology. In this talk, I will discuss knots and how I studied them at the Claremont Colleges REU in 2006, including the elementary definition of a Lissajous Knot, and what we found out about them.

If one takes the unit interval, then deletes the open middle third, one is left with two line segments, [0, ^1 /_3 ] and [^2 /_3 , 1]. If one deletes the open middle third of the remaining intervals, one gets 4 segments: [0, ^1 /_9 ] , [^2 /_9 , ^1 /_3 ] , [^2 /_3 , ^7 /_9 ] , and [^8 /_9 , 1]. Continuing this process infinitely many times, one obtains a collection of points known as the Cantor Set. In this talk I will describe my work this past summer Cornell's REU where I worked with four other students to study R. Thompson's group V, which can be viewed as the group of maps from the Cantor Set to itself. I will focus on the visualization and intuition we used in working with the maps, and briefly present our results which include a complete classification of the centralizer any group element.

A forward problem is a problem where we know the entire question, and part of the solution, and determine the rest of the solution. For example, solving for y in ay''+by'+cy=0 given a, b, c, y(0), and y'(0) is a common example of a forward problem. For any forward problem, there is a corresponding inverse problem, which is deriving the whole question from the solution and part of the question, for example, solving for a in the above equation given b, c, and y. In this talk, I will discuss one specific example of a forward problem and an inverse problem in graph theory, along with some basic techniques for solving the inverse problem.

I will talk about different types of topological spaces and how they can be folded up and glued together to create new and interesting spaces. Specifically, I will give examples of groups acting on spaces and the orbit spaces that result from these group actions. I will also talk about dimension and how techniques from topology can be used to identify the dimension of these orbit spaces. I will present simple examples involving familiar groups and spaces and introduce the question of the smallest possible dimension for the orbit space for an action of a given group.

An important concept in Abstract Algebra is groups, which is a set with a specific structure. The talk will cover the definition of a group and many examples of groups, including groups with a finite number of elements and infinite groups. Examples will include cyclic groups, dihedral groups, symmetric groups, and integers viewed as a group.

Within the last century, topology as a branch of mathematical study has been a hotbed for research and discovery. The beginnings of topology can be traced to the 1700s in the research of Leonard Euler, and renowned mathematicians the likes of Georg Cantor, Henri Poincare, and Karl Weierstrass have made significant contributions to the sub ject. Rather than being an offshoot of analysis, topology is a type of geometry, one that lays the foundation for all other geometries. The results of research in topology have had profound, intriguing, and exciting effects on both classical and more modern branches of mathematics. In the hope of alluding to the elegance and geometric nature of the field of topology, we will construct a basic foundation of topological notions and definitions with the goal of discussing the important concept of compactness. From compactness we will state the Heine-Borel Theorem and its converse in order to give a more intuitive and visual meaning to compactness. Finally, we will conclude by stating discussing the Weierstrass Maximum Theorem, thereby touching on the work of one of the pioneers of topology and connecting compactness with the notion of continuous functions.

I am going to give an overview of several projects that Mathematical and Computational Biology Group is working on and talk about how the REU students we had during the last few summers contributed to these projects. In particular, I will talk about myxobacteria movement and aggregation; microtubule dynamics; and blood clot formation.