Atomistic-to-Continuum Coupling Methods

Speaker: 

Mitchell Luskin

Institution: 

University of Minnesota

Time: 

Thursday, November 15, 2012 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and the sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.

Tits geometry and positive curvature

Speaker: 

Fuquan Fang

Institution: 

Capital Normal Univ. Beijing and Notre Dame

Time: 

Thursday, January 17, 2013 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but one that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a topological building in the sense of Burns and Spatzier. Using this structure we classify all polar actions on (simply connected) positively curved manifolds of cohomegeneity at least two.
(Joint work with K.Grove and G. Thorbergsson)

CR geometry in dimension three

Speaker: 

Paul Yang

Institution: 

Princeton University

Time: 

Thursday, December 6, 2012 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

There is a good deal of resemblence of CR geometry in dimension three with conformal geometry
in dimension four. Exploiting this resemblence is quite fruitful. For instance, the presence of
several conformally covariant operators in both geometries allows us to formulate correct
conditions for the embedding problem as well as the CR Yamabe problem. There is also
large difference in the presence of pluriharmonic functions. I will also describe a new
operator which gives control of the pluriharmonics and allows a formulation of a
sphere theorem in this geometry.

The mathematics of Multiwave Imaging

Speaker: 

Professor Plamen Stefanov

Institution: 

Purdue University

Time: 

Thursday, March 1, 2012 - 4:00pm

Location: 

RH 306

 
Thermoacoustic (TAT) and Photoacoustic Tomography (PAT) are examples of multiwave imaging methods allowing to combine the high imaging contrast of one wave (an electromagnetic or a photoacoustic one) with the high resolution of ultrasound. We present recent results obtained in collaboration with Gunther Uhlmann, Jianliang Qian and Hongkai Zhao on the mathematical theory behind TAT, PAT and other multiwave methods. We allows the acoustic speed to be variable, and consider the partial data case as well. We will also discuss the case of a discontinuous speed modeling brain imaging.  Numerical reconstructions will be shown as well.
Most of the progress is due to the use of microlocal methods. One of the goals of the talk is to show the usefulness of microlocal methods to solving real life problems.

Getting stuck and breaking through: homogenization of free boundary problems

Speaker: 

Kaushik Bhattacharya

Institution: 

Caltech

Time: 

Thursday, April 19, 2012 - 4:00pm

Location: 

RH 306

 
This talk will discuss the problem of finding effective laws that the govern the overall evolution of free boundaries propagating in a heterogeneous media.  This is motivated by a number of phenomena in mechanics and materials physics including phase boundaries, peeling of adhesive tape, dislocations, fracture and wetting fronts.   While there is a rigorous mathematical theory of homogenization in the context of properties that are characterized by a variational principle, much remains unknown about equations that describe evolutionary processes.   The talk will discuss the mathematical issues, difficulties and results, and illustrate the implication on materials through selected examples.  The talk will conclude with current work on free discontinuity problems.

The Cauchy-Riemann operator in complex manifolds and L_2 serre duality

Speaker: 

Professor Mei-chi Shaw

Institution: 

University of Notre Dame

Time: 

Thursday, May 24, 2012 - 4:00pm

Location: 

RH 306

 
The Cauchy-Riemann operator for domains in a complex manifold is well understood for domains in complex spaces. However, much less is known for the solvability and regularity for the Cauchy-Riemann operator in a complex manifold which is not complex spaces  or Stein. Recently, some progress has been made for the L2 theory of the Cauchy-Riemann equations on product domains in complex manifolds. An analogous formula of the classical Kunneth formula for the harmonic forms are also obtained. We have also discuss an L2 version of the Serre duality for domains on complex manifolds. Furthermore, duality between the harmonic spaces and the Bergman space in complex manifolds will also be presented (Joint work with Debraj Chakrabarti).

Sums and products

Speaker: 

Professor Carl Pomerance

Institution: 

Dartmouth College

Time: 

Thursday, February 9, 2012 - 4:00pm

Location: 

RH 306

What could be simpler than to study sums and products of integers? Well maybe it is not so simple since there is a major unsolved problem: for arbitrarily large numbers N, can there be sets of N positive integers where both the number of pairwise sums and pairwise product is less than N^{3/2}?

No one knows. This talk is directed at another problem concerning sums and products, namely how dense can a set of positive integers be if it contains none of its pairwise sums and products? For example, take the numbers that are 2 or 3 more than a multiple of 5, a set with density 2/5. Can you do better?

Systemic Risk

Speaker: 

George Papanicolaou

Institution: 

Stanford University

Time: 

Monday, April 23, 2012 - 4:00pm

Location: 

RH 306

I will review briefly some recent developments in financial mathematics research, put them in a historical context, and then discuss the modeling and analysis of systemic risk phenomena.

Opportunities for Mathematical and Statistical Sciences

Speaker: 

Professor Sastry Pantula

Institution: 

NSF

Time: 

Friday, December 2, 2011 - 4:00pm

Location: 

RH 306

In this talk, I will be discussing various funding and other opportunities for mathematical and statistical sciences at NSF. New programs in computational and data-enabled sciences will be discussed. Also,this is an opportunity to hear feedback from the community regarding future needs.

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