Santa Clara University

Mathematics and Computer Science department

Colloquium Series

Spring 2012

Unless otherwise noted, talks will be at 3:50 PM in O'Connor 204.  Also, there will be refreshments before each talk in O'Connor 31 at 3:40 PM.

Tuesday April 17, O'Connor 204

Speaker: Kay Giesecke, Stanford University

Title: Large Portfolio Asymptotics for Loss From Default

Abstract: We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. The density of the limiting measure is shown to solve a non-linear stochastic PDE, and certain moments of the limiting measure are shown to satisfy an infinite system of SDEs. The solution to this system leads to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustrate the accuracy of the approximation, and highlight its computational advantages over a direct Monte Carlo simulation of the original stochastic system. This is joint work with Kostas Spiliopoulos, Richard Sowers, and Justin Sirignano.

Tuesday, April 24, O'Connor 204

Speaker: Nicolette Meshkat, Santa Clara University

Title: The Differential Algebra Approach to Identifiability

Abstract: Parameter identifiability analysis for dynamic system ODE models concerns finding which unknown parameters can be quantified from given input-output data.  If all the parameters of a model have a finite number of solutions, then the model is said to be identifiable.  A model is called unidentifiable if the parameters can take on an infinite number of values and yet result in identical input-output data.  The differential algebra approach has been an effective method for analyzing identifiability properties, in particular, for proving global identifiability.  In this talk, we examine unidentifiable models and a method for finding globally identifiable parameter combinations using Groebner Bases.  We show that a set of algebraically independent identifiable parameter combinations can always be found from the Groebner Bases and can be used to reparameterize the model's input-output equations.  We also discuss computational difficulties in finding these identifiable parameter combinations and explore possible improvements to our algorithm.

Tuesday, May 1, O'Connor 204

Speaker: Nancy Rodriguez, Stanford University

Title: Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity

Abstract: Recently, there has been much interest in the use of mathematical tools to obtain insight into the phenomena of crime.  In this work we study a reaction-diffusion system of PDEs, which can be taken as a basic model for criminal activity, to explore the effect of a populations "natural tendencies'' towards crime. In this talk, I will first discuss the effect these natural tendencies have in the propagation of crime by studying the existence of traveling wave solutions.  Secondly, I will consider the problem of preventing the propagation of crime using a minimum number of resources.

Tuesday, May 15, O'Connor 204

Speaker: Kelli Talaska, UC Berkeley

Title: Determinants and path counting

Abstract: If we have a network with some sources and sinks (think of a
collection of starting points connected to some ending points by a
system of one-way roads), we can use a matrix to encode how many paths
we have from each source to each sink.  A classical result from
algebraic combinatorics tells us that subdeterminants of such a matrix
count certain families of paths in our network, assuming there are no directed cycles.  We will explore this result and a recent theorem which tells us what happens in networks with cycles.

Background needed: Very basic linear algebra (determinants).  A little
exposure to multivariable polynomials would be helpful, but is not

If you have a disability and require a reasonable accommodation,
please call/email Rick Scott 408-554-4460/rscott at scu dot edu (or
use 1-800-735-2929 TTY—California Relay).

Abstracts of previous talks are available here.
Printer-friendly format