# Colloquium: Fall 2013

(Tentative Schedule)

Time & location: All talks are on Thursday in Gibson 310 at 3:30pm unless otherwise noted. Refreshments in Gibson 426 after the talk.

Comments indicating vacations, special lectures, or change in location or time are made in green.

Organizer:  Mahir Can

August 29

## Topic

SpeakerInstitution

Abstract: TBA

September 5

## Generalized Symmetric Spaces and their Applications

Aloysius HelminckNC State

Abstract: Symmetric spaces occur in many areas of mathematics and physics, probably best known are their applications in Lie theory, differential geometry and harmonic analysis.  In this talk we will discuss a generalization of these symmetric spaces, which has become of importance in many areas of mathematics.

September 12

## Topic

SpeakerInstitution

Abstract: TBA

September 19

## Ramanujan's Lost Notebook

Bruce BerndtUniversity of illinois at urbana champaign

Abstract:

Srinivasa Ramanujan, generally regarded as the greatest mathematician in Indian history, was born in 1887 and died in 1920 at the age of 32.  Most of his work was recorded without proofs in notebooks.  In the spring of 1976, while searching through papers of the late G. N. Watson at Trinity College, Cambridge, George Andrews found a box containing 138 pages of Ramanujan's work.  In view of the fame of Ramanujan's earlier notebooks, Andrews naturally called this collection of papers Ramanujan's "lost notebook." This work, comprising about 650 results with no proofs, arises from the last year of Ramanujan's life and represents some of his deepest work. We begin with a history and origin of the lost notebook. Secondly, we describe the many topics that one can find in the lost notebook.  The remaining third portion of the lecture will be devoted to a survey of some of the most interesting entries in the lost notebook.  These include claims in q-series, theta functions, continued fractions, integrals, partitions, other infinite series, and Diophantine approximation.

September 26

## Mixed-type Problems for Transonic Flows and Isometric Embeddings

Dehua Wanguniversity of pittsburgh

Abstract:

Some mixed type problems of transonic flows in gas dynamics and isometric embeddings in geometry will be discussed. Connections between the two problems, and global existence of weak solutions will be presented. The talk is based on the joint works with Gui-Qiang Chen and Marshall Slemrod.

October 3

## Topic

SpeakerInstitution

Abstract: TBA

October 10

October 17

## The Lost Honor of 1_2-based Regularization

Uri Ascheruniversity of British Columbia

Abstract:

In the past two decades, regularization methods based on the $\ell_1$ norm, including sparse wavelet representations and total variation, have become immensely popular. So much so, that we were led to consider the question whether $\ell_1$-based techniques ought to altogether replace the simpler, faster and better known $\ell_2$-based alternatives as the default approach to regularization techniques.

The occasionally tremendous advances of $\ell_1$-based techniques are not in doubt. However, such techniques also have their limitations. Taking into account the considerable added hardship in calculating solutions of the resulting computational problems, $\ell_1$-based techniques must offer substantial advantages to be worthwhile. Such is not always the case in our experience.

In this talk we explore advantages and disadvantages of $\ell_1$- compared to $\ell_2$-based techniques using several practical case studies.  Our results suggest that, while $\ell_1$-based techniques often shine, in many applications $\ell_2$-based recovery may still be preferred.

October 24

## How Do Flows in Conduit and Porous Media Interact?

Xiaoming WangFlorida state university

Abstract:

Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), cloud and fog (water vapor, water and air).  Multiphase flows also play an important role in many engineering and environmental science applications.

In some applications such as flows in unconfined karst aquifers, karst oil reservoir, proton membrane exchange fuel cell, multiphase flows in conduits and in porous media must be considered together. How free flows in conduit/channel interact with flow in porous media is a challenge.

In this talk we present a family of phase field models that couples two phase flow in conduit with two phase flow in porous media. These models together with the associated interface boundary conditions are derived utilizing Onsager's extremum principle. The models derived enjoy physically important energy law. Numerical scheme that preserves the energy law will be presented as well.

October 31

## The Concavity and Subsolutions in Estimates for Fully Nonlinear Elliptic Equations

Bo GuanOhio State university

Abstract:

We report recent progresses in our effort of seeking methods to derive a priori second order estimates for fully nonlinear elliptic equations on real or complex manifolds under general structure conditions. We are concerned with both equations on closed manifolds, and the Dirichlet problem on manifolds with boundary without imposing geometric restrictions to the boundary except being smooth and compact. Even for domains in Euclidean space, our results are new and essentially optimal. In this talk the main topic will be the role of concavity and subsolutions.

November 7

## Wave Equations, Gas Dynamics and Multidimensional Conservation Laws

Barbara Keyfitzohio state university

Abstract:

Even to mathematicians who work in Partial Differential Equations, it is rather puzzling that the problem of proving existence of solutions to systems of quasilinear hyperbolic PDE has proved so difficult.  Well-posedness of the initial-value problem, and of many initial-boundary problems, for linear hyperbolic systems has been established, and passage from linear to quasilinear problems in the theory of elliptic equations is understood.  But for Conservation Laws (as quasilinear hyperbolic PDE are known), there is a satisfactory well-posedness theory only in a single space dimension, and extension to multidimensional problems appears intractable at present.

Despite the lack of progress, many fascinating phenomena have turned up during the two decades that my colleagues, students, and I, and other groups, have been trying to understand multidimensional conservation laws.  This talk will focus on some of our conclusions about differences between linear and non-linear behavior in hyperbolic PDE.  This is a technically demanding field, but the talk will avoid the details of technical results and will try to help a non-expert audience gain an understanding of what lies behind some of the paradoxes in conservation laws - to the extent that we the experts understand that ourselves.

November 8 - 11

November 14

## Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier-Stokes equations

Remi AbgrailINRIA Bordeaux-Sud-ouest and university of bordeaux

Abstract:

A robust and higher order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier-Stokes equation is presented. These methods can be interpreted as fully non linear, maximum principle satisfying, finite element methods.

The proposed method is very flexible; it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions.The approximation of the solution is obtained using standard Lagrangian finite elements, and we follow the traditional technique for designing RD scheme: evaluate, for any element, a total residual, split it into sub-residuals sent to the element degrees of freedom, solve the non linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries:this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations.Naive solution methods lead to very poor accuracy.

To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution. We design a method that can do this while reaching the optimal accuracy.  Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three dimensions.

November 21

## A Branching Process Model for Prion Dynamics

Peter OlofssonTrinity university

Abstract:

Prions are infectious agents composed of misfolded proteins, responsible for illnesses such as mad cow disease in cattle and Creutzfeldt-Jakob disease in humans. We create a branching process model for yeast cells to describe how prions grow inside the cell and how they are transmitted from mother to daughter cell. We compare our model predictions to simulated data and use it to estimate parameters.

December 5

## Identifying Genetic Variants for Alcohol and Nicotine Co-Addiction in Minorities

Huaizhen Qintulane university, department of biostatistics and bioinformatics, school of public health and topical medicine

Abstract:

Human twin studies suggest a shared genetic predisposition to develop alcohol and nicotine dependence. Specific pleiotropic genes, however, remain unknown and their function modes need to be elucidated. Existing statistical methods are underpowered to identify pleiotropic variants from DNA sequence data of minorities. Such methods neglect or oversimplify the causal mechanism of genetic determinants and their ancestries. It has become a standard practice, for example, to linearly adjust for local ancestries as traditional covariates. In this talk, I will introduce our (1) novel single-trait and bivariate trait causality models, (2) respective score tests and, (3) application to the DNA sequence data on co-addiction of African Americans. Our causality models allow traditional covariates to ensure that our score tests control false positive rate. Our score tests exploit ancestry-gene mechanism and prove more powerful than existing methods. Being applied to SAGE data of African Americans, our methods identified several susceptible pleiotropic genes for alcohol-nicotine co-abuse.

Keywords: Causality models; ancestry mechanism; score tests; pleiotropic variants.

This colloquium is being presented by the D. W. Mitchell Lecture Series and the Provost's Faculty Seminars in Interdisciplinary Research.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu