# Math Colloquia: Fall 2012

#### Friday September 7

 Title Analysis and approximation of a constrained optimization problem representing a rod with hard self-contact Speaker Thomas Seidman UMBC http://www.math.umbc.edu/~seidman/

Abstract:
Minimizing the integrated pointwise potential of a stationary elastic rod dates back to Euler, but only recent work includes analysis of the self-contact problem of the rod with itself. In the case of an elastic rod with hard contact, the minimization is complicated by the requirement that the rod is treated as a solid tube without permitting self-penetration. This is a nonconvex inequality constraint and requires construction of the basic normal cone at a minimizer to compute contact forces. One may view the constraint as the imposition of an infinitude of nonlinear scalar constraints. We formulate the variational formulation of the elastic rod with hard contact in this framework and prove existence and regularity of minimizers. We further approximate both the functional and the minimizers of this problem by a sequence of elastic rods with soft contact, which are modeled of by unconstrained optimization problems. We prove convergence of both the energy and the minimizers to the appropriate hard contact energy and minimizers. This is ongoing work with K. Hoffman.

#### Friday September 14

 Title A Finite Element-Discontinuous Galerkin method for coupled flow-transport problems Speaker Stefan Kopecz University of Kassel, Germany http://www.mathematik.uni-kassel.de/~kopecz/indexe.html

Abstract:
The simulation of cavitation in the context of micro foams can be modeled as a combination of a flow model with a second model for transport. The flow equations generalize the incompressible Navier-Stokes equations in terms of variable viscosity and density, plus a pressure and void fraction dependent divergence constraint. The transport of the void fraction is given by an advection equation. A numerical method for the solution of these equations will be presented. It utilizes the Finite Element (FE) method for the solution of the flow equations and a Discontinuous Galerkin (DG) method for the transport. Main focus is on a nonlinear projection scheme for the flow equations, the satisfaction of a maxium principle within the DG method and a post processing strategy to overcome the incompatibilty of the FE and DG discretizations.

#### Friday September 21

 Title Physical transformations between quantum states Speaker Chi-Kwong Li College of William and Mary http://people.wm.edu/~cklixx/

Abstract:

#### Friday November 9

 Title Modeling of Microbial Biofilm Communities Speaker Isaac Klapper Temple University http://www.math.temple.edu/~klapper/

Abstract:
Single-celled, microbial organisms are estimated to make up a large fraction of extant biomass. Many of these microbial communities exist in the biofilm form. (A biofilm is a dense aggregation of microorganisms that are embedded in a hydrated polymer matrix of their own secretion.) The distinction between microorganisms in the biofilm state and those in free aqueous suspension (i.e., planktonic) is important. Microorganisms in biofilms function very differently because they are subject to physical, chemical, and biological phenomena that have less impact on conventional planktonic cultures. Multicellular phenomena such as diffusion gradient formation, intercellular communication, differentiation, and extracellular electron transfer operate in biofilms and make them scientifically rich topics of investigation and also inherently complex. Mathematical models are therefore valuable complementary approaches to analyzing and understanding these systems. Resulting models are inherently interdisciplinary; the rich interaction of microbiology, chemistry, and physics requires theory grounded in the mathematics. In this talk, I will discuss a class of biofilm models based on continuum mechanics principles that present a natural platform for combining the relevant biology, chemistry and physics, and will present a few important implications that these models predict.

No talk today

#### Monday November 19

 Title MCMC for improper target distributions Speaker Krishna B. Athreya Iowa State University http://iowastate.edu/

Abstract:
MCMC for estimating the means w.r.t a proper target distribution(ie, a probability distribution) are well known. In this talk we consider improper targets and present some MCMC methods for them. We apply this to some Bayesian contexts including Gibbs sampler, importance sampling. We also show how to reduce the improper case to a proper one.(This is joint work with my colleague Vivek Roy of ISU stat dept)

Thanksgiving

#### Friday November 30

Lecture Hall 1 (joint with Biology)

 Title The Mathematics of Life — Decisions, Decisions Speaker James P. Keener University of Utah http://www.math.utah.edu/~keener/

Abstract:
In order to survive, living organisms must constantly make decisions, about what to eat, when and where to move, when to reproduce, when to build, when to destroy, etc. In this talk I will give an overview of the mathematics of decision making, namely the mathematical principles that underlie biological processes of measurements, switches, and signals. The short answer to how decisions are made is that the rate of molecular diffusion contains information that can be transduced by biochemical reactions to give control over behavior. These processes can be given quantitative descriptions using diffusion-reaction equations, and the study of these equations gives valuable insights into how organisms work as well as an opportunity to learn and develop new mathematics. I will illustrate this dual role of quantitative reasoning by way of several specific examples from cell biology.

#### Friday December 6

 Title Capturing Community Behavior in Very Large Networks using MapReduce Speaker Tamara G. Kolda Sandia National Laboratories, Livermore, CA http://www.sandia.gov/~tgkolda

Abstract:
Graphs and networks are used to model interactions in a variety of contexts, and there is a growing need to accurately measure and model large-scale networks. We consider especially the role of triangles, which are useful for measuring social cohesion. This talk will focus on two topics: (1) Accurately estimating the number of triangles and clustering coefficients for very large-scale networks, and (2) Generating very large-scale artificial networks that capture the degree distribution and clustering coefficients of observed data. Triangles form a basic pattern of interest for understanding networks because they reflect friend-of-friend relationships. However, counting triangles can be extremely expensive in terms of both computational time and memory requirements. We explain how sampling can be used to accurately estimate the number of triangles and the number of triangles per degree (or degree range) in an undirected graph, as well as the number of each type of directed triangle in a directed graph. Using Hoeffing’s inequality, we can predict exactly how many samples we need for a desired error and confidence level, and this sample size is independent of the size of the graph. We describe a MapReduce implementation and provide examples demonstrating its utility. Once we know how to measure triangles, we can calculate the clustering coefficients and use this data as input to a generative model. Ideally, a good model should be able to reproduce the observations, yet few models are able to do so. We hypothesize that any graph with a heavy-tailed degree distribution and large clustering coefficients must contain a scale-free collection of dense ER subgraphs. From this, we propose the Block Two-Level Erdos-Renyi (BTER) model, and demonstrate that it accurately captures the observable properties of many real-world social networks. Moreover, the BTER model can scale to very large networks, and we describe our MapReduce implementation and recent experimental results. This is joint work with Ali Pinar, Todd Plantenga, C. Seshadhri (Sandia National Labs) and Christine Task (Purdue University).