# New York Psi

Iona College

2007-2008 Chapter Report

**Advisor and Officers**

Advisor:
Robert Lavelle

President:
Janine Tepedino

Vice President:
Cheyne Miller

Treasurer:
Matthew Krammer

Secretary:
Jessica Beyer

**Number of New Members Inducted:**

4

**Number of Continuing Members: **

4

**Describe the two most interesting or unusual chapter activities this year: **

Lecture: A Brief Introduction to Knot Theory and the Jones Polynomial Peter L. Gregory Abstract: Knot theory, combinatorial knot theory in particular, can be explored and understood without much technical or high powered mathematics This talk hopes to display this relative accessibility. The notion of knot and knot equivalence will be discussed so that we may address the fundamental questions of the theory: When are two knots the same? When are they different? How can we tell? For an answer, we will need to define some spatial isotopies, in particular Reidemeister Moves. From these definitions we will have a method for determining when knots are equivalent The opposite, determining when Knots are not the same, will not come as easily, and will require the development of a knot invariant. One such invariant is the Jones Polynomial. We will construct this polynomial from scratch via Kaufmann’s Bracket Polynomial and the Writhe, show that it is indeed a knot invariant, and perform a sample calculation. Lecture: The Fundamental Group: An Introduction to Homotopy Theory David Allen Abstract: In this talk, basic notions in the study of Topology will be discussed. From a “Space” X one constructs an algebraic invariant called the Fundamental Group of X which allows one to detect differences in geometry. The notion of continuous deformation as well as loops at a fixed point in X will be explored. Roughly speaking, the fundamental group of X, denoted !1(X) is the set of all maps from the circle into X satisfying some additional properties. The basic ideas behind the mathematics and computations of this important invariant will be sketched. Cheyne Miller "Algebraic Properties Associated to Certain Simple Convex Polytopes in Lower Dimensions" Abstract: Let P be an n-dimensional simple convex polytope and hp be its associated h-vector. It is natural to ask whether or not the standard vector operations on h-vectors yield any interesting results. For two polytopes, P1 and P2, one has operations such as the product, the connected-sum, and the box product. In the case of the connected-sum, the h-vector of the resulting polytope is defined and can be explicitly computed. This paper will give a detailed example of the connected-sum operation, as this operation is relatively new and explicit examples are not easily found. Moreover, it will be shown that the set of h-vectors associated to simplex convex n-polytopes is a semi-group under the connected-sum for n=2 or n=3.

**Do any of your chapter members plan to attend or present a paper at the PME national meeting at MathFest in Madison, Wisconsin? **

**Do you have any news about current students or recent graduates, for example, scholarships or other awards, acceptances to graduate school, paper presentations at conferences?**