2014 Talks

Modeling Cooperation and Competition

Karen Wood

December 3rd, 2014 - 12:00 - 12:50pm - RH 306

Abstract

In this talk I will discuss a mathematical model using ODEs to describe cooperation and competition in a population. The model was motivated by the theorized cooperation of cancer cells. I will discuss several models with growing complexity as well as present the requirements for cooperating subspecies to perform better than an indiviual in the different scenarios. I will also discuss a study on E coli and a model used to describe the observed sympatric speciation.

About the Speaker

Karen received her bachelor's degree in mathematics from Cal Poly Pomona and her masters degree in mathematics from the University of California, Irvine. In her free time, Karen enjoys baking and camping, but not baking while camping.

Advisor and Collaborators

Karen's advisor is Natalia Komarova.

Intro to Kahler Geometry / Asymptotic Expansion of the Bergman Kernel on Polarized Manifolds

Sho Seto

November 20th, 2014 - 4:00 - 4:50pm - RH 440R

Abstract

I will give an introductory talk on Kahler geometry aimed for beginning graduate students in hopes to motivate them to do research in geometry. We will begin with a brief introduction to Riemannian geometry so that first and second year graduate students will be able to understand the talk, at least heuristically. We will then cover the necessary materials to define the Bergman kernel and its asymptotic expansion on polarized manifolds. We will apply the Bergman kernel to prove Tian's theorem on the convergence of polarized metrics. Come for the pizza, stay for the geometry!

About the Speaker

Sho is a 6th year graduate student who studies complex geometry. He is a graduate of UC Berkeley. Sho enjoys music and coffee.

Advisor and Collaborators

Nikki's advisor is Zhiqin Lu.

Bounding the Regularity of a Graded Module

Roger Dellaca

November 4th, 2014 - 4:00 - 4:50pm - RH 440R

Abstract

Given a set X of d points in the complex plane, every complex-valued function on X is the restriction of a polynomial of degree at most d-1; some configurations of points allow the maximal degree of such polynomials to be strictly smaller than d-1, while others do not. This is an example of the notion of (Castelnuovo-Mumford) regularity of the algebraic set. Regularity measures the complexity of the geometric object, and its associated algebraic object as well. A general definition of regularity can be given in terms of a minimal free resolution of a graded module, which is used in several explicit techniques in Commutative Algebra, including the Hilbert function and Hilbert polynomial. A theorem of Gotzmann takes an odd-looking way of writing a Hilbert polynomial, and uses it to bound the regularity of all ideals with that Hilbert polynomial. This was used to give an explicit construction of the set of all such ideals as a geometric space, called the Hilbert scheme. I will show how this technique can be used to bound the regularity of certain classes of modules with a given Hilbert polynomial. This allows an explicit construction of a generalization of the Hilbert scheme.

About the Speaker

Roger is a 6th-year student. He has a M.S. in Math from CSULB and a M.S. in Management Science from CSUF. His research is in Commutative Algebra and Algebraic Geometry. He enjoys cycling.

Advisor and Collaborators

Ali's advisor is Vladimir Baranovsky.

Analysis of Stochastic Stem Cell Models with Control

Jay Yang

October 20th, 2014 - 3:00 - 3:50pm - RH 440R

Abstract

Understanding the dynamics of stem cell lineages is of central importance both for healthy and cancerous tissues. We study stochastic population dynamics of stem cells and differentiated cells, where cell decisions, such as proliferation vs differentiation decisions, or division and death decisions, are under regulation from surrounding cells. The goal is to understand how different types of control mechanisms affect the means and variances of cell numbers. We use the assumption of weak dependencies of the regulatory functions (the controls) on the cell populations near the equilibrium to formulate moment equations. We derive simple explicit expressions for the means and the variances of stem cell and differentiated cell numbers. We demonstrate that these findings are consistent with numerical results via some novel examples. This methodology is formulated without any specific assumptions on the functional form of the controls, and thus can be used for any biological system.

About the Speaker

Jay is a 4th-year math Ph.D. student, a graduate of UC Berkeley, with current research interest in math computational biology and applied/computational math.

Advisor and Collaborators

Taiji's advisor is Natalia Komarova.

Complex Modeling of Epidermal Stratification, An Intersection of Mathematics, Biology, and Computer Science

Mary Lee

January 22nd, 2014 - 4:00 - 4:50pm - RH 440R

Abstract

Modeling the effects of cell shape and asymmetric cell division on epidermal stratification requires modeling at the sub-cellular level. To meet these modeling requirements we turn to differential equations inspired by molecular dynamics to provide the mathematical framework, and cutting edge computer science to provide the tools to run large scale simulations.

About the Speaker

Alex has a bachelors degree in physics and computational math from Rensselaer Polytechnic Institute. His field of research is mathematical biology with an emphasis on computational modeling.

Advisor and Collaborators

Mary's advisor is Qing Nie.

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