# Spring 2015 Applied Mathematics Colloquium

## spring 2015 schedule

Colloquia are held in Phillips 332, Fridays at 4:00 PM unless otherwise noted. Tea is served at 3:30 PM in Phillips 330.

- Feb 06, Paul Vos, East Carolina University, hosted by Sorin Mitran
*Applications of Information Geometry to Statistical Inference*An information geometric manifold is a Riemannian manifold together with a third order tensor that defines a family of connections, or a family of geometries, on the manifold. There is a unique connection in which geodesics are curves of minimum length and in classical differential geometry it is this metric connection that is studied. In statistics and other fields a distance-like quantity, called a divergence, is of interest and the nonmetric connections can be more useful than the metric connection. For the Kullback-Leibler information, and more generally for any Bregman divergence, there is a pair of geometries that provide a Pythagorean theorem for the divergence. This Pythagorean relationship will be applied to several statistical ideas including relative information loss, sufficiency, and generalized linear models. The geometries describing the Pythagorean relationship are particularly simple (they are flat) and can be considered generalizations of Euclidean geometry so that many technical details necessary for general Riemannian manifolds will not be required for this talk. (Slides will be available here http://turl.ca/unc15)

- Feb 13, William Layton, University of Pittsburgh, hosted by Sorin Mitran

*Turbulence Not at Statistical Equilibrium*

Standard eddy viscosity models, while robust, cannot represent backscatter and have severe difficulties with complex turbulence not at statistical equilibrium. This talk will:

(1) prove that on (time) average the 1877 conjecture of Boussinesq (that turbulent fluctuations are dissipative on the mean) is true;

(2) give a new derivation of eddy viscosity models from an equation for the evolution of variance in a turbulent flow;

(3) show how to correct eddy viscosity models to successfully exhibit intermittent backscatter.The talk will be understandable provided one knows the formal derivation of the energy equality for the Navier-Stokes equations. (There will also be a short movie about kayaking the Grand Canyon to explain the proof.) - Feb 20, JianFeng Lu, Duke University, hosted by Sorin Mitran

Understanding the coupling of physical models at different scales is important and quite challenging. In this talk, we focus on the issue of kinetic-fluid coupling, in particular, the half-space problems for kinetic equations coming from the boundary layer. We will present some recent progress in algorithm development and analysis for the linear half-space kinetic equations, and its application in coupling of neutron transport equations with diffusion equations. (joint work with Qin Li and Weiran Sun)

Numerical methods for linear half-space kinetic equations

- Mar 06, Gary Huber, UCSD, hosted by Jingfang Huang
*Hybrid Particle-Continuum Method for Diffusing Charged Particles*Many biological systems at the cellular level exhibit a wide range of length and time scales, which presents a challenge for designing algorithms to simulate such systems. When simulating diffusion, one can either perform a direct simulation of the diffusing bodies using the Langevin Equation, or one can solve a continuum diffusion equation; which one to use depends on the length and time scales. We present a hybrid particle-continuum method that has recently been updated to use charged particles, which are challenging because of their long-ranged interactions.

- Mar 20, Daphne Klotsa, Cambridge University. Special joint Applied Mathematics and Applied Physical Sciences Colloquium. (Chapman 125)
*Packing Polyhedra: From Ancient Math to Advanced Materials*The densest way to pack objects in space, also known as the packing problem, has intrigued scientists and philoso- phers for millenia. Today, packing comes up in various systems over many length scales from batteries and catalysts to the self-assembly of nanoparticles, colloids and biomolecules. Despite the fact that so many systems’ properties depend on the packing of differently-shaped components, we still have no general understanding of how packing varies as a function of particle shape. Here, we carry out an exhaustive study of how packing depends on shape by investigating the packings of over 55,000 polyhedra.

By combining simulations and analytic calculations, we study families of polyhedra interpolating between Platonic and Archimedean solids such as the tetrahedron, the cube, and the octahedron. Our resulting density surface plots can be used to guide experiments that utilize shape and packing in the same way that phase diagrams are essential to do chemistry. The properties of particle shape indeed are revealing why we can assemble certain crystals, transi- tion between different ones, or get stuck in kinetic traps.

- Mar 27, Thilo Strauss, Clemson University
*Statistical Inversion In Electrical Impedance Tomography*Electrical Impedance Tomography (EIT) is a well known technique to estimate the conduc- tivity distribution σ of a body Ω with unknown electromagnetic properties. EIT is a severely ill-posed inverse problem. In this presentation, we briefly discuss the analytical setting of the for- ward problem which consists of solving an elliptic partial differential equation. Then we focus on the Bayesian framework for the inverse problem for estimating σ. We use the Markov Chain Monte Carlo (MCMC) method to reconstruct the conductivity distribution σ of a body Ω. We present reconstructions of the conductivity from experimental data. Particularly, the data has been taken from a circular concrete object with direct current injections. This is particularly interesting be- cause of its potential application for testing the degradation of concrete.

- Apr 03,
- Apr 10, Tim Chartier, Davidson University
*Who’s number one? from ranking to bracketology*

“Who’s number one?” is an inherent and often debated question in sports. Ranking algorithms supply mathematical answers to such questions. They can and are used to choose teams for the playoffs. They can also be used in predictive analysis. Who will win the next game? Who will win a tournament? This talk will present current and recent sports analytics research. A variety of questions will be explored. For example, how can one integrate late season momentum? Does it help to consider home field advantage? Such questions will be explored in the context of sports like soccer and basketball. In particular, we will see how such research created brackets for March Madness that beat over 90% of over 8 million brackets submitted to ESPN¹s online tournament. - Apr 17, Robert Lipton, Louisiana State University, hosted by Greg Forest
- Apr 24, Johanna Rosman, UNC, hosted by Laura Miller
*Flow and Mixing over Complex Biological Substrates: Understanding Linkages between Organism- and Reef-Scales*One of the biggest challenges in addressing questions about physical-biological interactions on reefs is that topography varies over a wide range of spatial scales. Currents predicted by circulation models usually represent spatially-averaged conditions over large areas. However, in these types of systems, topography varies at scales as small as cm, and flow occurs through layers containing solid obstacles as well as above them. Two major challenges are therefore: 1) how to properly represent small-scale processes in circulation models, and 2) how to interpret circulation model results if questions require knowledge of flow conditions at a particular point on a reef. This talk will draw examples from two systems: coral reefs and oyster reefs. We will present results from field measurements and computer simulations that provide insight into spatial and temporal patterns in flow at scales ranging from a single organism to clusters of organisms. We will discuss the ways in which organism-scale processes affect and are affected by flow at larger (reef) scales, and some feedbacks that exist between reef structure, hydrodynamics, and reef development.