# Spring 2010

### Four-manifolds and mapping class groups

#### Prof. Thomas Mark

#### Department of Mathematics, University of Virginia and MSRI

#### Thursday, April 22, 2:30-3:45pm, AB 102

**Abstract:** The mapping class group of a manifold S is the group of self-diffeomorphisms of S$ considered up to isotopy (smooth deformation). If S is a closed surface, its mapping class group is generated by the collection of Dehn twists around simple closed curves in S. Through the notion of a Lefschetz fibration, a sequence of Dehn twists can be used to determine a four-dimensional manifold, and many interesting four-manifolds can be obtained this way. This group-theoretic description of four-manifolds has been of great interest in recent years, but many questions remain: for example, can one understand natural geometric operations on 4-manifolds in terms of the sequence of Dehn twists that encode them? We will describe some progress on this question based on the notion of monodromy substitution, which uses relations in the mapping class group to change one sequence of Dehn twists into another and thus one 4-manifold into another. In particular we describe some new families of relations in the mapping class group (generalizing the well-known lantern relation), where the corresponding substitution yields a particularly interesting operation on 4-manifolds known as rational blowdown.

### Testing Homogeneity in Clustered (Longitudinal) Count Data Regression Model with Over-dispersion

#### Sudhir Paul, Professor

#### Department of Mathematics and Statistics, University of Windsor

#### Thursday, April 8, 2:30-3:45pm, AB 102

**Abstract:** Clustered (longitudinal) count data arise in many bio-statistical practices in which a number of repeated count responses are observed on a number of individuals. The repeated observations may also represent counts over time from a number of individuals. One im- portant problem that arises in practice is to test homogeneity within clusters(individuals) and between clusters (individuals). As data within clusters are observations of repeated responses, the count data may be correlated and/or over-dispersed. Jacqmin-Gadda and Commenges (1995) derive a score test statistic HS by assuming a random intercept model within the framework of the generalized linear mixed model and a score test statistic HT using the generalized estimating equation (GEE) approach (Liang and Zeger, 1986; Zeger and Liang, 1986). They further show that the two tests are identical when the covariance matrix assumed in the GEE approach is that of the random-effects model. In each of these cases they deal with (a) the situation in which the dispersion parameter φ is assumed to be known and (b) the situation in which the dispersion parameter φ is assumed to be unknown. The second situation, however, is more realistic as φ will be unknown in practice. For over- dispersed count data with unknown over-dispersion parameter we derive two score tests by assuming a random intercept model within the framework of (i) the negative binomial mixed effects model, and (ii) the double extended quasi-likelihood mixed effects model (Lee and Nelder, 2001). These two statistics are much simpler than the statistic obtained from the statistic HS derived by Jacqmin-Gadda and Commenges (1995) under the framework of the over-dispersed generalized linear model. The first statistic takes the over-dispersion more directly into the model and therefore is expected to do well when the model assumptions are 1satisfied and the other statistic is expected to be robust. Simulations show superior level property of the statistics derived under the negative binomial and double extended quasi- likelihood model assumptions. Further, a score test is developed to test for over-dispersion in the generalized linear mixed model and some simulations are conducted. A data set is analyzed and a discussion is given.

### The Kakimizu complex of a knot

#### Jennifer Schultens, Professor

#### Department of Mathematics, UC Davis

#### Thursday, April 1, 2:30-3:45pm, AB 102

**Abstract:** We will discuss knots, Seifert surfaces, why every knot admits a Seifert surface and how to build a complex, called the Kakimizu complex, from the collection of all Seifert surfaces of a given knot. This complex was first defined and studied by Osamu Kakimizu who described much of its structure. Using results of Kakimizu

along with minimal surface theory allows us to obtain additional structural information concerning the Kakimizu complex. This work is joint with Piotr Przytycki.

#### Chris Kees

#### Coastal and Hydraulics Laboratory, U.S. Army Corps of Engineers

#### Thursday, March 25, 2:30-3:45pm, AB 102

**Abstract:** Standard level-set methods do not conserve volume in incompressible multi-phase flows or mass for more general flows. The conservation errors are the result of describing the interface using a level set formulation and are not associated with any particular discretization. Since conservation errors accumulate over time to produce qualitatively incorrect approximate solutions, several researchers have attempted to address this issue by using hybrid level-set/volume-of-fluid and hybrid level-set/particle-tracking approaches. We will present a method for correcting the level set in such a way that the conservation error in the fully discrete model can be controlled, and we compare this approach to other recently developed approaches. The correction is defined as the solution of a nonlinear reaction-diffusion equation and is easy to apply in the context of unstructured finite element methods. We will present numerical results for two- and three-dimensional air/water flows where each fluid is governed by the Navier-Stokes equations. Numerical results will be presented for several standard level-set test problems as well as surface water applications of interest to the U.S. Army Corps of Engineers.

### Generalized Gamma Frailty Model and Applications

#### N. Balakrishnan, Professor

#### Department of Mathematics and Statistics, McMaster University

#### Thursday, March 4, 2:30-3:45pm, AB 102

**Abstract:** Frailty models are usually used to model correlation in clustered survival data. Some of the most popular frailty models are based on gamma, Weibull, lognormal and stable distributions. In this talk, I will introduce a generalized gamma frailty model that includes gamma, Weibull and lognormal as special cases. I will then discuss inferential issues for this frailty model. This parsimonious model is then used for model discrimination between different special cases for a given data. Finally, I will take some real-life data sets to illustrate the practical usefulness of this model.

### What is a physical measure in stochastic dynamical systems?

#### Mickaël D. Chekroun, Professor

#### University of California, Los Angeles

#### Thursday, February 25, 2:30-3:45pm, AB 102

**Abstract:** The geometric and ergodic theory of dynamical systems represents a significant achievement of the last century. The global attractor of a deterministic dynamical system provides crucial geometric information about its asymptotic regime as t → ∞, while the Sinaï-Ruelle-Bowen (SRB) measure provides, when it exists, the statistics of the flow over this attractor and for most of the initial conditions. In the meantime, the foundations of the stochastic calculus also led to the birth of a rigorous theory of time- dependent random phenomena. Under the influences of Baxendale, Le Jan, Kunita, and Stroock among others, the notion of a stochastic flow — extending the related deterministic concept — has emerged in the 80’s opening the door for a geometrical description of the dynamics in stochastic dynamical systems.

During the past two decades, the mathematical theory of random dynamical sys- tems (RDS) and of nonautonomous dynamical systems (NDS) has laid out the foun- dation of a geometric theory of systems subject to random influences. In particular, the RDS and NDS theory has made substantial progress in describing the asymp- totic behavior of open systems, subject to time-dependent forcing, via the concept of pullback (random) attractor. The pertinent mathematical literature, however, is fairly technical and opaque for the non-specialist. Its concepts and methods have, therefore, not become widely applied to the physical sciences in general.

The main objective of this introductory talk is twofold: (i) to introduce the key concepts and tools of RDS theory to a wider audience; and (ii) to present how these concepts offer a natural way for extending the concepts of ergodic theory in random dynamical systems via the notion of random SRB measure. A particular attention will be paid on the connection with the more familiar and traditional approach based on the Fokker-Planck equation. Numerical results on a stochastic Lorenz system and on alow-dimensional,nonlinearstochasticElNin~o/SouthernOscillationmodel—from climate dynamics — will illustrate the discussion. This talk is based on joint works with Eric Simonnet and Michael Ghil.

### Elliptic Curves and Cryptography

#### Paul Duvall, Professor

#### Department of Mathematics and Statistics, University of North Carolina Greensboro

#### Thursday, February 18, 2:30-3:45pm, AB 102

**Abstract:** We will present definitions and some elementary properties of elliptic curves and then discuss some of the ways that elliptic curves arise in public key cryptography. This talk is intended to be accessible to upper level undergraduate mathematics students.

### Theta functions and knots (joint with Alejandro Uribe, U. of Michigan)

#### Razvan Gelca, Professor

#### Department of Mathematics and Statistics, Texas Tech University

#### Thursday, February 11, 2:30-3:45pm, AB 102

**Abstract: ** In the representation theoretic point of view introduced by A. Weil, the theory of classical theta functions consists of: the space of theta functions, viewed as the holomorphic sections of a line bundle over the Jacobian variety of a Riemann surface, an irreducible representation of the Heisenberg group on theta functions which arises by applying the Weyl quantization procedure to exponential functions, and a projective representation of the mapping class group on theta functions induced by the representation of the Heisenberg group via a Stone-von Neumann theorem.

We show that these objects can be modeled combinatorially using algebraic topology based on knots (the so called skein modules). We also show how one can arrive at the Reshetikhin-Turaev formula for invariants of 3-manifolds based on the linking number from strictly quantum mechanical considerations, without using Witten's quantum field theoretic insight. Moreover, we explain how A. Weil's point of view can be generalized to non-abelian theta functions. As an application, we explain why the cocycle of the metaplectic representation and the change in the signature of 4-manifolds under glueings are both expressed in terms of the Maslov index.

### The Significant-digit Phenomenon, or Benford's Law

#### Theodore P. Hill, Professor

#### School of Mathematics, Georgia Institute of Technology

#### Thursday, February 4,

**Abstract:** A century-old empirical observation now called Benford's Law says that the significant digits of many real datasets are logarithmically distributed,

rather than uniformly distributed, as might be expected. This talk will briefly survey some of the colorful history and empirical evidence of Benford's law, and recent discoveries that Benford sequences are typical in many deterministic sequences such as dynamical systems and differential equations, and in many stochastic processes such as mixtures of data. The talk will include concrete examples, applications to fraud detection and diagnostic tests for mathematical models, and open Benford-related problems in dynamical systems, probability, number theory, and differential equations. The talk is aimed for the non-specialist.

### Online Numerical Methods Laboratory and Its Use in Classroom Teaching

#### Pavel Solin, Professor

#### Department of Mathematics and Statistics, University of Nevada, Reno

#### Thursday, January 28,

**Abstract: ** Wouldn't it be nice to demonstrate all numerical methods taught in a Numerical Methods course on a computer? This is often easier said than done for various reasons. Should we teach the students to use expensive commercial software such as Maple, Matlab, Mathcad or Mathematica? What after they graduate and may have no free access to these packages? To provide free, anywhere, anytime access to numerical methods for both the students and the instructors, we developed an interactive web browser tool we call "Online Numerical Methods Laboratory". The application does not require anything to be purchased or installed, and it does not even require a computer. Any iPhone, PDA or netbook with Internet access will work (PCs and laptops work as well). Additionally, our tool will stay accessible to everyone long after they graduated and got their jobs. We will demonstrate the tool using elementary numerical methods such as the Taylor polynomial, root-finding, numerical integration, polynomial interpolation, least squares approximation, Fourier series, and elementary numerical linear algebra topics.

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