Coding Theory and Toric Geometry

Josh Whitney

December 3rd, 2008 - 12:00 - 12:50pm - RH 440R

Abstract

We will begin by giving a very brief introduction to coding theory and toric varieties in general. In particular we will discuss toric codes which are obtained from evaluating certain polynomials at points of a torus. We will also examine the interesting combinatorial and algebro-geometric questions these codes provide and possible methods for estimating their associated parameters. The talk will be accessible to everyone: no prior algebraic geometry is required.

About the Speaker

Josh Whitney is a fifth year math grad at UCI. Josh got his BS from Arizona State. He once performed yoyo tricks on the JumboTron at an Arizona Diamondbacks game.

Advisor and Collaborators

Josh's advisor is Vladimir Baranovsky.

Spatial Patterns of Cancer Mutant Spread

Erin Brown

November 13th, 2008 - 12:00 - 12:50pm - RH 440R

Abstract

We discuss the role of spatial dynamics in cancer growth. We will begin with the non-spatial model. We find the probability of invasion, the time to invasion and other characteristics of this process. We examine the obstacles that arise when including space in our model as well as preliminary results about how space influences the probability of a cancerous invasion.

About the Speaker

Erin is a fifth year math grad at UCI. She got her BS at UCLA in 2003. She loves her two German Short Hair Pointers, Mandy (a puppy) and Noel.

Advisor and Collaborators

Erin's advisor is Natalia Komarova.

Modeling Thin Film Morphology: A Geometric Evolution Law from Anisotropic Surface Energy

Chris Ograin

May 28th, 2008 - 4:00 - 4:50pm - RH 190

Abstract

The morphology of the solid-vapor interface of a nano-scale thin crystalline film is influenced by many factors. We consider an interface whose evolution is driven by anisotropic surface diffusion. It is known that in cases of strong anisotropy, the equilibrium shape of the energy-minimizing interface will have sharp corners. The equations used in this case are ill-posed leading to difficulty in numerical simulation. As a result, researchers have proposed to regularize the problem by adding a high-order term to the energy that is proportional to the curvature squared (Willmore energy). We derive the corresponding evolution law and perform a linear stability analysis.

About the Speaker

Chris earned a B.S. in mathematics in 1992, and an M.Ed. (education) degree in 1993, both at UCLA. He taught math at the high school level from 1992-2003. Chris is a fifth-year grad student working in applied math, who plays quarterback (flag football), center (basketball) and first base (softball).

Advisor and Collaborators

Chris' advisor is John Lowengrub.

Effects of Surface and Line Tension on the Shape of a Vesicle

Geoff Cox

May 14th, 2008 - 4:00 - 4:50pm - RH 190

Abstract

Why is a red blood cell squished in the center? Many believe that the key to understanding the bioconcave shape of a red blood cell lies in considering vesicles. Vesicles are simpler models of red blood cells that have been studied for several decades.

The shape of such a biological entity is driven by nature's desire to be efficient. In other words, the equilibrium shape of a vesicle will assume a shape that which minimizes the total energy associated with the physical model.

Geoff will formulate the energy to be minimized and derive a system of ODE's which describe the vesicle shape. The talk will conclude with results and future projects.

About the Speaker

Geoff is a fifth year who went to UCLA and got a BS in Math. His idea of happiness is "enjoying a pint during an angel game followed by a candlelight dinner and a romantic walk on the beach". I can't make that stuff up.

Advisor and Collaborators

Geoff's advisor is John Lowengrub.

Probability and Random Polynomials

Jeff Matayoshi

April 30th, 2008 - 4:00 - 4:50pm - RH 190

Abstract

A random polynomial is a polynomial whose coefficients are given by a specific probability distribution. Random polynomials have found many applications in quantum chaos and random matrix theory.

The study of random polynomials is a broad area, with work begin done on all the usual topics of interests for polynomials, including distribution of roots and multiplicities. Our discussion will focus on the properties of the roots of certain families of random polynomials. Several historical results will be presented, followed by a brief description of our current research in the field. Along the way, several interesting results from probability theory will be discussed.

About the Speaker

Jeff Matayoshi is a fourth-year graduate student here at UCI. He completed his B.S. at UCLA and lives and dies Bruin's blue and gold. He has been solid at third base for the math softball team for three years.

Advisor and Collaborators

Jeff's advisor is Michael Cranston.