The Graduate Student Seminar is a 1 unit for credit participating seminar (MATH 589). Particpants are required to attend and give one 45 minute presentation on their research or a mathematical topic of their interest. The public is invited to attend these lectures as well.

**May 01, 2008**

16:00-16:55, JR214

Option-I graduate student Yeranuhi Semerdjian

Title: Noetherian Rings and Affine Algebraic Sets

ABSTRACT: The basic idea behind algebraic geometry is to equate geometric questions with algebraic questions involving ideals in such rings as k[x1, ... , xn]. The polynomial ring k[x1,...,xn] is Noetherian, and the properties associated with this type of rings allows one to reduce many geometric questions into finitely many algebraic equations. I will prove that there is a bijective correspondence between Morphisms from V to W as affine algebraic sets & k-algebra homomorphisms from the coordinate ring of W to the coordinate ring of V.

**April 24, 2008**

16:00-16:55, JR214

Option-I graduate student Erin Colberg

Title: The Knot Group

ABSTRACT:A fundamental objective in the mathematical study of knots is to find invariants that can distinguish non-isotopic knots and links. In this talk, we will define the group of a knot or link, show how to compute it using the Wirtinger presentation, and demonstrate its application as an invariant.

**April 17, 2008**

16:00-16:55, JR214

Option-I graduate student Elizabeth Leyton

Title: The Mapping Class Group of the Torus.

ABSTRACT: In this expository talk, I will introduce the idea of the mapping class group of a topological space. The mapping class group is a topological invariant that is essentially the group of symmetries of the space. The main focus will be to display the mapping class group of the torus using various group theoretical and topological ideas.

**April 10, 2008**

16:00-16:55, JR214

Option-I graduate student Vahe Khachadoorian

Title: Application of Banach’s Theorem to Integral Equations.

ABSTRACT: In this expository talk, I will consider the Banach fixed theorem as a source of existence and uniqueness theorems for integral equations. Integral equations can be considered on various function spaces. However, I will consider Fredholm’s equation of the second kind on C[a,b], the space of all continuous functions. Next, I will discuss the Volterra integral equation without any restriction on its parameter and conclude that a Volterra equation can be regarded as a special Fredholm equation.

**April 3, 2008**

16:00-16:55, JR214

Option-I graduate student Aime Jones

Title: Riemannian Metrics.

ABSTRACT: Intuitively, a Riemannian metric is a way of measuring the length of vectors tangent to a manifold. This measurement should change differentiably from point to point. In this talk we present the rigourous definition of a Riemannian metric and show how to construct a Riemannian metric on a Hausdorff and Second Countable differentiable manifold.

**March 27, 2008**

16:00-16:55, JR214

Option-II graduate student Ali Ashar

Title: Ranking Schemes, Paired Comparisons, and the Perron-Frobenius Theorem.

ABSTRACT: Ranking schemes and paired comparisons have various applications ranging from ranking of football teams, tennis ladders, to the ordering of a search result in Google and the ratings of videos on YouTube. But these processes (based on rigorous Mathematics) have always been obscured and shrouded in mystery for an average sports fan and a Google user. My goal will to be introduce and explain four different ranking procedures used in the real world today and conclude that all four schemes actually depend (one way or another) on a powerful result in Linear Algebra, known as the famous Perron-Frobenius Theorem.

**March 6, 2008**

16:00-16:55, JR214

Option-I graduate student Cynthia Shepherd

Title: Backlund Transformations and Surfaces of Constant Negative Curvature.

ABSTRACT: A classical result connecting solition equations
and differential geometry, dating back to nineteenth century, is the Backlund Theorem. It asserts a correspondence between solutions of the sine-Gordon equation (SGE) and surfaces of constant negative curvature. Moreover, transformations that take one such surface to another, are, from the analytic point of view, maps taking a solution of the SGE to another solution of the same equation. These transformations have
been called (classical) Backlund transformations}. In this
talk we will prove Backlund Theorem and discuss some of its possible generalizations to hypersurfaces of R^{4}.

**February 21, 2008**

16:00-16:55, JR214

Option-I graduate student David Berkowitz

Title: A Proof of Gauss's Theorema Egregium.

ABSTRACT: The curvature of a 2 dimensional surface in 3 dimensional Euclidean space was defined by Gauss in a way that quantitatively demonstrates how much the surface deviates from being flat. However, this definition of "extrinsic" curvature assumes the existence of knowledge of how the surface sits in space, something unknown to anyone constrained to the surface itself. However, much like a map of the Earth distorts distances and must have a "key" to interpret the map, every surface has a metric associated with it, and any metric different from the usual Euclidean metric can be interpreted as "intrinsic" curvature. A remarkable Theorem of Gauss, a theorem even he regarded as remarkable, shows that one can determine the extrinsic curvature of a surface from a knowledge of the metric alone. In this talk, I will prove Gauss's theorem. A basic understanding of Calculus and Linear Algebra is all that is required.

**February 14, 2008**

16:00-16:55, JR214

Option-I graduate student Emily Mcleod Schnitger

Title: Pisot Numbers and Pisot-2 Pairs.

ABSTRACT: A Pisot number is a real algebraic integer greater than 1 whose remaining conjugates have modulus strictly less than 1. A great deal is known about Pisot numbers, including that the set is closed. We have defined a new set of generalized Pisot numbers called Pisot-2 pairs and proved various propositions about the properties of this set that are similar to the properties of the set of Pisot numbers. Our research culminated in the statement and proof of a theorem involving the limit points of Pisot-2 pairs.

**February 7, 2008**

16:00-17:00, JR214

Option-I graduate student Cynthia Flores

Title: Numerical Simulation of Potential Flow using Finite Element Methods.

ABSTRACT: The finite element method is a general and powerful technique used extensively in Applied Mathematics, especially in the fields of fluid dynamics and in wave propagation such as acoustic problems. Mathematically speaking, the finite element method deals with constructing approximate solutions to boundary-value problems by dividing the domain of the solution into a finite number of simple subdomains called finite elements. Then, using variational concepts, construct an approximation of the solution over the collection of finite elements.

We have shown that the finite element method gives a very close estimate of the solution to differential problems when we use the Galerkin approximations. We have used this method to simulate velocity and pressure vector fields and potential flow around obstacles with Dirichlet boundary conditions.

**January 30, 2008**

16:00-17:00, JR214

Gk-12 Fellow Will Yessen

Title: Uniquely Associating Algebraic Curves With Irreducible Polynomials Over Algebraically Closed Fields.