# Spring 2008

### On Floer homology and knots admitting lens space surgeries

#### Matthew Hedden, Professor

#### Department of Mathematics, Massachusetts Institute of Technology

#### Thursday, 31 January at 2:30 in AB 206

**Abstract:** There is a simple procedure called (Dehn) surgery which alters three dimensional manifolds using knots. The simplest three-manifolds are called lens spaces, and there is a conjecture regarding the knots on which one can perform surgery to obtain these manifolds. In this talk, I'll review these notions and discuss this conjecture, known as the Berge conjecture. I'll then discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.

### Modeling and Approximating the Distributions of Estimators of Financial Risk Under Asymmetric Laplace Laws

#### Alex Trindade, Professor

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

#### Thursday, 7 February at 2:30 in AB 206

**Abstract:** Explicit expressions are derived for parametric and nonparametric estimators of two measures of financial risk, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), under random sampling from the Asymmetric Laplace (AL) distribution. Asymptotic distributions are established under very general conditions. Finite sample distributions are investigated by means of saddlepoint approximations. The latter are highly computationally intensive, requiring novel approaches to approximate moments and special functions that arise in the evaluation of the moment generating functions. Plots of the resulting density functions shed new light on the quality of the estimators. Calculations for CVaR reveal that the nonparametric estimator enjoys greater asymptotic efficiency relative to the parametric estimator than is the case for VaR. An application of the methodology in modeling currency exchange rates suggests that the AL distribution is successful in capturing the peakedness, leptokurticity, and skewness, inherent in such data. We conclude with some extensions of the methodology when the sampling is under the framework of a stationary time series covariance structure. Preliminary results suggest that, in certain types of financial data, an ARMA model driven by IID AL noise provides a competitive fit to a classical ARMA driven by GARCH noise.

### Nonlocal Far-Field Artificial Boundary Conditions for 3D Computational Aerodynamics

#### Semyon Tsynkov, Professor

#### Department of Mathematics, North Carolina State University

#### Wednesday, 20 February at 4:00, AB 634

**Abstract:** Numerical solution of the problem originally formulated on an unbounded domain typically requires that the domain be truncated. Truncation, in turn, necessitates that the artificial boundary conditions (ABCs) be set at the far-field computational boundary. The issue of ABCs is critically important in many areas of scientific computing, for example, in computational fluid dynamics (CFD). In particular, the problems of external configuration analysis (fluid flows around aerodynamic shapes) represent a wide class of key practical applications in CFD, for which the proper treatment of artificial boundaries has a profound impact on the overall quality and performance of numerical algorithms, as well as on interpretation of their results. In the talk, we will describe the nonlocal ABCs that we have introduced for the computation of external compressible viscous flows. We will also provide an overview of other conceptually similar ABCs for various wave propagation problems. Our methodology combines the advantages relevant to both local and global methods proposed previously by other authors. It employs finite-difference counterparts to Calderon's pseudodifferential boundary projection operators and exploits the difference potentials method by Ryaben'kii. The resulting ABCs are accurate and robust, and at the same time inexpensive, geometrically universal, and easy to use. We will review the implementation of these ABCs along with the NASA-developed production multigrid flow solvers. We will show the results of configuration analysis for both two and three space dimensions (subsonic and transonic flows, turbulent flows with separation and reattachment, flows with jet exhaust), and demonstrate a consistent superiority of the proposed approach over the standard existing methods.

### Transonic problems in 2-dimensional conservation laws

#### Eun Heui Kim, Professor

#### Department of Mathematics, California State University Long Beach

#### Friday, 22 February at 1:00, AB 201

**Abstract:** Many practical problems in science and engineering, for example in aerodynamics, multi-phase flow and hemodynamics, involve conserved quantities, and lead to partial differential equations in the form of conservation laws. Understanding the mathematical structure of these equations and their solutions is essential to obtain physical insight into such practical problems. There are special difficulties associated (e.g. shock formation) with these equations that are not seen elsewhere and must be dealt with carefully. Moreover, in multidimensional conservation laws, there is little theory at present. One approach, the study of self-similar solutions, leads to the study of equations that change their type, namely, equations that are hyperbolic far from the origin and mixed near the origin. Some results have been obtained recently in this area, but there are still many open problems. In this talk, we discuss transonic problems in multidimensional conservation laws, present current results and ongoing research in this area.

### Real Zeros, Unimodality, Log-concavity, Normality, and All That Analysis Helping Combinatorics and Probability

#### Miklós Bóna, Professor

#### Department of Mathematics, University of Florida

#### Monday, 25 February at 4:00, AB 202

**Abstract:** If a combinatorially defined polynomial has only real roots, then that analytical property has a plethora of interesting combinatorial and probabilistic consequences. We will give a survey of some of these facts located on the borderline of various lines of research, such as enumerative combinatorics, extremal combinatorics, and probability. The talk is meant to be accessible for a general mathematics audience.

### Motion by Mean Curvature in Heterogeneous Medium

#### Aaron Yip, Professor

#### Department of Mathematics, Purdue University

#### Tuesday, 26 February at 2:30, WRB 2007

**Anstract:** The talk will discuss some mathematical questions motivated by the motion of materials phase boundaries in heterogeneous medium. The ultimate goal is to derive effective, homogenized equation and study the property of the solution in large space-time regime. Motion by mean curvature is used as a simple but illustrative example. It already involves interesting mathematical analysis due to the nonlinear interaction between the curvature and the background heterogeneity. In this talk, we will concentrate on periodic background environment. For some linearized version of motion by mean curvature flow, we derive the scaling behavior for the averaged velocity in some pinning and de-pinning transition regime. For the fully nonlinear version, we prove the existence, uniqueness and stability of pulsating waves (above the pinning threshold) for any normal direction. Furthermore, the effective speed of propagation is a Lipschitz continuous function of the normal. Connection with homogenization will be discussed. (This talk is based on joint works with Nicolas Dirr and Georgia Karali.)

### High-Fidelity Computer Methods for Large-Scale Multi-Physics Engineering Problems

#### Pavel Solin, Professor

#### Department of Mathematics

#### University of Texas at El Paso &

#### Academy of Sciences of the Czech Republic

#### Thursday, 28 February at 2:30, MSS 216

**Abstract:** Nowadays, the computer simulation of natural and industrial processes described by (systems of) partial differential and/or integral equations plays an essential role in engineering, science, biology, medicine, and many other fields. In contrast to ten years ago, results computed today need to be accompanied by information about their accuracy and the role of aleatory/epistemic uncertainty in input data. Such information cannot be provided without efficient self-adaptive computational methods equipped with reliable error control and uncertainty estimation mechanisms. From this point of view, the most difficult cases are multi-physics problems where phenomena spanning vastly different spatial and/or temporal scales interact in nonlinear ways. As examples of such processes let us mention microwave heating, induction heating, electromagnetic stirring of reactive liquids, magnetorheological fluids, magnetohydrodynamics, turbulent flows, fluid-structure interaction problems, thermoelasticity, or moisture and heat transfer in drying concrete. For many such problems, reliable computational methods equipped with error control are missing.

### Asymptotics of the connectivity function for percolation and spin systems

#### Gastão de Almeida Braga, Professor

#### Departamento de Matematica, Universidade Federal de Minas Gerais

#### Belo Horizonte - MG, Brazil

#### Thursday, 1 May at 2:30 in WRB 2007

**Abstract:** The outline of the presentation is as follows: We show several large-scale problems rooted in electrical, civil, mechanical and chemical engineering, and explain standard difficulties that practitioners face in their solution. We will explain what was done in this field so far and what are the limitations of standard methods. The advantages of modern higher-order methods will be discussed and main difficulties encountered in multi-physics problems will be described. In the second part of the talk, we will present our leading project aimed at universal self-adaptive higher-order computational methods applicable to a wide range of single- as well as multi-physics problems. Several practitioners have already adopted the new method and we hope that more will follow.

### Forecasting the occurrences of wildfires and earthquakes using point processes with directional covariates

#### Frederic Schoenberg, Professor

#### Dept. of Statistics, UCLA

#### Thursday, 8 May at 2:30, Location TBA

**Abstract:** Tremendous progress has been made in recent years in the development of methods for the description of directional data. However, one area that has remained largely unexplored is the inclusion of directional variables into point process models. This is an important subject, since such covariates appear to be very promising predictors in a variety of applications. For instance, forecasts of wildfire hazard could be improved by taking wind direction into account, and forecasts of earthquake hazard could benefit by using estimates of focal mechanisms of previous earthquakes as predictors. This talk explores methods for incorporating such information into the current best-fitting point process models for wildfires and earthquakes, in order to provide improved forecasts of wildfire and earthquake hazard for Southern California. Particular attention will be placed on extending separable models for wildfire hazard and branching point process models for earthquakes, such as the Epidemic-Type Aftershock Sequence (ETAS) models that are now commonly used in studies of seismic hazard, to include these important types of predictors.

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