2006 Talks

Computational Oncology: An Introduction to Cancer, with Simulations and Cool Results

Paul Macklin

November 8th, 2006 - 4:00 - 4:50pm - PSCB 120

Abstract

After giving an introduction to cancer, we shall present a nonlinear continuum model of cancer at the tissue scale. This extensible model includes nutrient transport, proliferation-induced biomechanical pressure/stress, degradation of the extracellular matrix (ECM), and angiogenesis. Reaction-diffusion equations are solved with advanced, highly-accurate nonlinear solvers including the ghost fluid method, and the tumor morphology is evolved with the level set method, which is capable of tracking arbitrarily complex shapes. We demonstrate the computer model in action (with multimedia animations), present tumor-microenvironment interaction studies, and discuss implications for cancer therapy.

About the Speaker

Paul Macklin is a fourth-year graduate student here at UCI. He earned his B.A. in mathematics and German at the U. of Nebraska-Lincoln in 1999 and his M.S. in industrial and applied mathematics at the U. of Minnesota in 2003. His past research interests include contaminant transport in groundwater flow and fiber optics.

Advisor and Collaborators

Paul's advisor is John Lowengrub. Collaborators include Natalia Komarova (UCI), Mark Chaplain (University of Dundee), "Sandy" Anderson (University of Dundee), and Steven McDougall (Heriot Watt University in Edinburgh).

Goodstein's Theorem, Euclid's Fifth Postulate, and Independence Proofs

Kyriakos Kypriotakis

October 25th, 2006 - 4:00 - 4:50pm - PSCB 120

Abstract

In mathematical logic, a sentence is called independent of a given set of axioms T if T neither proves nor refutes that sentence. Euclid's fifth postulate and Goodstein's theorem are two such sentences. Goodstein's theorem is a statement about the natural numbers that is undecidable in Peano arithmetic (PA), but can be proven to be true using the axiom system of set theory (ZFC). Euclid's fifth postulate says that two parallel lines do not intersect. We will prove the independence of the second (there is a surprisingly easy proof) and demonstrate the power of the Goodstein's theorem (i.e., explain how it gives us the relative consistency of PA; a set of axioms T is called consistent if T does not lead to any contradictions.)

About the Speaker

Kyriakos Kypriotakis is a fifth-year graduate student here at UCI. He earned his B.S. in mathematics at the U. of Salonica and his M.S. in mathematical logic at the U. of Athens in 2002.

Advisor and Collaborators

Kyriakos' advisor is Martin Zeman.

Exponentially Interseting Sums

Timothy Choi

October 11th, 2006 - 4:00 - 4:50pm - PSCB 120

Abstract

Like gamma and beta functions are ubiquitous in analysis, Gauss sums and Jacobi sums are everywhere in number theory. Gauss introduced the Gauss sums in his Disquisitiones Arithmeticae in 1801, and he wrote in his diary that he devoted some time every week for more than four years before he was able to prove his conjecture on the sign of (quadratic) Gauss sums. We will study motivation of exponential sums in number theory and look at Kloosterman sums, yet another exponentially interesting sums.

About the Speaker

Timothy Choi is a fifth-year graduate student here are UCI. He earned his B.A. in pure mathematics at UC San Diego in 2002. His past research has included study of exponential sums over finite fields. His current work continues his earlier work on exponential sums. When Timothy isn't researching, he enjoys studying differential geometry.

Advisor and Collaborators

Timothy's advisor is Daqing Wan.

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