2010 Talks
Stability Predictions of Tumor Growth and Evaluations Against Experimental Observations
Kara Pham
December 1st, 2010 - 4:00 - 4:50pm - RH 114
Abstract
We consider three constitutive relations to describe tumor growth: Darcy's law, Stokes law, and the combined Darcy-Stokes law. Darcy's law is used to describe fluid flow in a porous medium. Stokes law describes the flow of a viscous fluid. In this talk, we will discuss using linear theories to study tumor shape stability (the ability of the tumor to return to being spherical or exhibit protrusions) described by the three physical relations and to evaluate the consistency between theoretical model predictions and experimental data. The motivation behind this work is that shape instabilities (growing protrusions) are associated with local invasiveness, which is often a precursor to tumor metastasis (infiltration of the distant organs). We will discuss the results and further show that it is feasible to extract parameter values from a limited set of data and create a self-consistent modeling framework that can be extended to the multiscale study of cancer. Numerical methods are used to simulate the nonlinear effects of stress on solid tumor growth and invasiveness.
About the Speaker
Kara is in her sixth year and expects to graduate in June 2011. Her advisor is John Lowengrub, and her current projects involve studying tumor growth on the macroscale.
Advisor and Collaborators
Kara's advisor is John Lowengrub.
Perturbation Analysis of Slow Waves for Periodic Differential-Algebraic Equations of Definite Type
Aaron Welters
November 10th, 2010 - 4:00 - 4:50pm - RH 114
Abstract
In this talk we consider linear periodic differential-algebraic equations (DAEs) that depend analytically on a spectral parameter. In particular, we extend the results of M. G. Krein and G. Ja. Ljubarskii [Amer. Math. Soc. Transl. (2) Vol. 89 (1970), pp. 1--28] to linear periodic DAEs of definite type and study the analytic properties of Bloch waves and their Floquet multipliers as functions of the spectral parameter. Our main result is the connection between a non-diagonalizable Jordan normal form of the monodromy matrix for the reduced differential system associated with the DAEs and the occurrence of slow Bloch waves for the periodic DAEs, i.e., Bloch solutions of the periodic DAEs which propagate with near zero group velocity. We show that our results can be applied to the study of slow light in photonic crystals [A. Figotin and I. Vitebskiy, Slow Light in Photonic Crystals, Waves Random Complex Media, 16 (2006), pp. 293--382].
About the Speaker
Aaron Welters is currently pursuing his Ph.D. in Mathematics from the University of California at Irvine (UCI). Under his advisor, Professor Alexander Figotin, he has been studying mathematical problems related to Spectral and Scattering Theory, Slow Light, and Photonic Crystals. He expects to defend his thesis and receive his Ph.D. in June 2011.
Advisor and Collaborators
Aaron's advisor is Alexander Figotin.
Quantifying Metal Insulator Transitions - Lyapunov Exponent and Spectral Theory for Extended Harper's Model
Chris Marx
October 27th, 2010 - 4:00 - 4:50pm - RH 114
Abstract
Extended Harper's model arises in a quantum description of a 2d- crystal layer subjected to an external magnetic field. As a first step towards the spectral analysis we shall introduce the Lyapunov exponent and present a method of computation valid for any analytic cocycle with possible singularities. This enables us to give a description of the metal-insulator properties for extended Harper's model, which so far did not even exist on a heuristic level in physics literature. We finish the talk with some results on the spectral analysis of the model.
About the Speaker
Chris Marx is a third year grad student working in Mathematical Physics. He did his undergrad in Vienna, Austria, in Theoretical Chemistry and Mathematics. His research is focused on Spectral Theory of Random and Ergodic Operators.
Advisor and Collaborators
Chris' advisor is Svetlana Jitomirskaya.
Elliptic Curves, Cryptography, and Counting Points
Nick Alexander
October 6th, 2010 - 4:00 - 4:50pm - RH 440R
Abstract
We will introduce elliptic curves and their applications to cryptography and computer security. We will describe the computationally important "point counting problem" and survey some results and algorithms to solve this problem. We will finish with some recent results that "count points" on certain elliptic curves and generalizations to hyperelliptic curves.
About the Speaker
Nick Alexander is a sixth year student supervised by Alice Silverberg. He studied in Canada before coming to California.
Advisor and Collaborators
Nick's advisor is Alice Silverberg.
