Modeling Cancer and Drug Resistance

Allen Katouli

November 7th, 2007 - 5:00 - 5:50pm - MSTB 124

Abstract

Drug resistant cancer cells are a big problem in treatment. In this introductory talk, we will formulate and analyze a stochastic model describing multi-drug resistance. We will introduce CML and the new miracle drugs known as small molecule inhibitors. Our main question is, how many small molecule inhibitors must be simultaneously prescribed to treat a tumor of fixed size?

About the Speaker

TBA

Advisor and Collaborators

Allen's advisor is Natalia Komarova.

Modern Number Theory: Ranks of Elliptic Curves

Sunil Chetty

October 24th, 2007 - 5:00 - 5:50pm - MSTB 124

Abstract

The rank of an elliptic curve measures how many rational solutions a cubic equation has. The mystery of rank has motivated many deep questions in the theory of elliptic curves over the past century. In particular, the Birch-Swinnerton-Dyer (BSD) Conjecture has been a guide in developing a bridge between the analytic and algebraic aspects of the theory and has led to many subsequent (and possibly more attainable) conjectures. We will explore the setting of the BSD conjecture, a simple consequence, and discuss an approach to one subsequent conjecture, the Parity Conjecture.

About the Speaker

Sunil is a fifth year graduate student at UCI. He received his B.S. in mathematics from Cal Berkley in 2002. Sunil is the fastest man in the department and welcomes challengers. Sunil's research is interested in ranks of elliptic curves.

Advisor and Collaborators

Sunil's advisor is Karl Rubin.

Applications of the Gradient Estimate on Noncompact Manifolds

Ovidiu Munteanu

October 10th, 2007 - 5:00 - 5:50pm - MSTB 124

Abstract

We will discuss an estimate that is essential in the study of harmonic functions on manifolds. The gradient estimate was proved by S. T. Yau (Fields Medal recipient and distinguished guest of the UCI math department last year) more than 30 years ago using a maximum principle argument. An important application is that manifolds with nonnegative Ricci curvature satisfy the Liouville property. The maximum principle method has since been used in many other situations, for example to prove eigenvalue estimates for the Laplace operator or to prove parabolic versions of the gradient estimate. Recently, P. Li and J. Wang improved the gradient estimate of Yau and obtained a sharp version. We will give a short account of these results and their applications.

About the Speaker

Ovidiu is a fourth year graduate student at UCI. He received his B.S. in mathematics at Transylvania University in Romania in 2002. Ovidiu's research interest is in Differential Geometry and Partial Differential Equations.

Advisor and Collaborators

Ovidiu's advisor is Peter Li.

An Example of Classification: Discrete Spectrum Transformations

Andrew-David Bjork

May 23rd, 2007 - 4:00 - 4:50pm - MSTB 114

Abstract

Transformations on probability spaces arise in many branches of mathematics. Classifying such transformations is a fundamental question asked by researchers, not yet completely solved. Finding a way to classify transformations is a growing field of study. We would like to be able say exactly when two measure preserving systems are isomorphic in ways that do not involve explicitly checking the definition. Therefore, we look for invariants that classify systems. Most often, we seek a numerical invariant. Sometimes a more complex classification is required. In this talk I will present a successful classification of some measure preserving transformations by an algebraic structure. This result was a triumph of early Ergodic Theory. We will also touch on more recent results that have a much less optimistic feel to them.

About the Speaker

Andrew-David went to High School at Lycee Victor Hugo, in Caen, France. He graduated in June of 1995. He then went to a small liberal arts college in Santa Barbara called Westmont where he met his wife, Naomi. He graduate Magna cum Laude with a BS in Math in 1998, and enrolled in a Masters program at UCSB, which he completed in 2000. After working in accounting for 2 years, he completed a Teaching Credential from CSUN in 2003. He is currently in his 4th year of graduate work at UCI.

Advisor and Collaborators

Andrew-David's advisor is Matthew Foreman.

Morphological Stability Analysis of the Epitaxial Growth of a Circular Island

Zhengzheng Hu

May 9th, 2007 - 4:00 - 4:50pm - MSTB 114

Abstract

After introducing the epitaxial growth of thin films, we shall present the Burton-Cabrera-Frank (BCF) model of the adatom concentration at the nano scale. This widely used semi-discrete model, supplemented with later modifications and extensions, captures many key features (e.g., diffusion, attachment-detachment kinetics, surface diffusion and so on) in epitaxial growth of the thin films. We shall then move on to the stability analysis, which is a powerful tool to understand the behavior of a solution to a nonlinear system. We close by demonstrating how stability results may be exploited to control the shape of a thin film island.

About the Speaker

Zhengzheng Hu is a fourth-year graduate student here at UCI. She holds a BS degree in Computational Mathematics from Sichuan University in China, and a MS degree in Mathematics from UCI. Zhengzheng's current research includes the shape control of the epitaxially growing circular island during the thin film growth.

Advisor and Collaborators

Zhengzheng's advisor is John Lowengrub.

Newton Polygons of Zeta Functions

Phong Le

April 18th, 2007 - 4:00 - 4:50pm - MSTB 114

Abstract

Given a set of polynomial equations over a finite field, one may construct an associated zeta function and its L function. Both of these functions are known to be rational, i.e., they are the quotient of polynomials. The L function carries deep number theoretical information about the solutions of the set of equations. This information is geometrically presented in the Newton polygon and an associated polygon, the Hodge polygon. We present the basic objects and tools used to determine when the Newton polygon coincides with the Hodge polygon and what this means number theoretically. We close with some current applications to mirror symmetry.

About the Speaker

Phong is a fourth year graduate student at UCI. He received his B.A. in mathematics from Goucher College (Baltimore, Maryland) in 2003. His current research involves mirror symmetry and zeta functions. In his free time he likes to cook and play Tetris.

Advisor and Collaborators

Phong's advisor is Daqing Wan.

Relative Consistency in Set Theory

Sean Cox

March 7th, 2007 - 4:00 - 5:00pm - MSTB 114

Abstract

The Zermelo-Fraenkel axioms with Choice (ZFC) are the standard axioms of set theory. Assuming ZFC is consistent, Gödel proved that there are statements which are independent of ZFC (the statement S is independent of ZFC if both (ZFC + S) and (ZFC + "not S") are consistent; equivalently, ZFC cannot prove either S or its negation). The Continuum Hypothesis (CH) turned out to be such a statement. Gödel showed CH is consistent with ZFC, and Cohen showed its negation is consistent with ZFC. I will give very rough sketches of these proofs. These sketches will introduce the main tools used to establish relative consistency between collections of axioms which extend ZFC. Finally, I will discuss large cardinals and my own research.

About the Speaker

Sean Cox is a fifth-year Ph.D. student at UCI. He received his B.A. in economics from North Carolina State University (Raleigh) in 1999. His research involves locating the consistency strength (on the large cardinal scale) of various set-theoretic statements. He specializes in inner model theory, which is often used to find lower bounds for the consistency of such statements.

Advisor and Collaborators

Sean's advisor is Martin Zeman.

The WEB-Spline Finite Element Method

Yanping Cao

February 21st, 2007 - 4:00 - 4:50pm - MSTB 114

Abstract

In this talk, we introduce WE(weighted-extended)B-splines, which we use to implement the finite element method for numerically solving PDEs. After discussing the advantages and disadvantages of such methods, we explore the technical implementation details and derive the theoretical convergence rate. We close by applying the method to several sample 2-D problems, including elliptic equations on the unit circle and irregular domains. In doing so, we shall see the beauty of the "mesh-free" idea we present.

About the Speaker

Yanping Cao is a fourth-year graduate student here at UCI. She received her B.S. in applied mathematics at Shanghai Jiatong University in China in 2003, and she completed her M.S. in mathematics at UCI in June 2005. Her research interests include partial differential equations, dynamics and turbulence models.

Advisor and Collaborators

Cao's advisor is Edriss Titi.

A Free-Boundary Problem with Applications in Pharmaceuticals

Micah Webster

January 31st, 2007 - 4:00 - 4:50pm - MSTB 114

Abstract

In controlled release pharmaceuticals, it is important to understand how a drug will diffuse through its storage device, a polymer. In many cases, the interaction between the polymer and chemical exhibits a phenomenon termed "Case II" diffusion. After a brief introduction to free-boundary problems and Case II diffusion, we give motivation for a new two-dimensional model. Next, using the Case II diffusion model as an example, we focus on the questions (and hopefully the answers) graduate students encounter when dealing with mathematical models. Furthermore, we relate graduate course work to the specific research questions posed in the presentation.

About the Speaker

Micah Webster is a fifth-year graduate student here at UCI. He earned his B.S. in mathematics at the University of North Carolina (Chapel Hill) in 2002 and his M.S. in mathematics at UCI in 2004. His research interests include free boundary problems with onset of a new phase, stability analysis of special solutions, and numerics.

Advisor and Collaborators

Micah's advisor is Patrick Guidotti.

Financial Mathematics: Is There Such Thing as a Free Lunch?

Michail Kortiotis

January 24th, 2007 - 4:00 - 4:50pm - MSTB 114

Abstract

We will start by introducing some basic terminology used in the field of mathematical finance. In particular, we will present ideas such as bonds, options, forward contracts, volatility, arbitrage, hedging, and replicating portfolios. Based on these, we will see how one can formulate financial problems (e.g., option pricing) into mathematical equations and discuss what mathematical tools can be used in order to get solutions.

Disclaimer: The speaker can not guarantee that you will have any financial profit from this talk, other than the refreshments served.

About the Speaker

Michail Korniotis is a fourth-year graduate student here at UCI. He completed his B.S. at the University of Cyprus in 2003 and his M.S. at UCI in 2005. His research interests include stochastic modeling of financial markets, interest rate derivatives, credit risk, and derivative pricing.

Advisor and Collaborators

Michail's advisor is Knut Solna.