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.

Analysis and control of coupled slow and fast dynamics

Speaker: 

Professor Zvi Artstein

Institution: 

Weizmann Institute

Time: 

Thursday, February 16, 2012 - 4:00pm

Location: 

RH 306

In this talk, which will be will be aimed at a general math audience, we shall overview an approach to analyze coupled slow-fast dynamics via Young measures, namely, via probability measure valued maps. The framework is ordinary differential equations, possibly controlled.

We shall address the modeling issue, motivating examples, the mathematical analysis and, in brief, numerical prospects.

Mathematical Methods for Cardiovascular Treatment

Speaker: 

Professor Suncica Canic

Institution: 

University of Houston

Time: 

Thursday, January 19, 2012 - 4:00pm

Location: 

RH 306

Mathematical modeling, analysis and numerical simulation, combined with imaging and experimental validation, provide a powerful tool for studying various aspects of cardiovascular treatment and diagnosis. This talk will address two examples where such a synergy led to novel results. The first example concerns a mathematical study of fluid-structure interaction (FSI) in blood flow with clinical application to 2D and 3D Doppler assessment of mitral regurgitation (MR). Our computational studies, performed in collaboration with several experts in echocardiography, addressed current imaging challenges in Doppler assessment of MR, which led to refinement and reinforcement of the emerging 3D echocardiographic applications. The second example concerns a novel dimension reduction/multi-scale approach to modeling of endovascular stents as 3D meshes of 1D curved rods forming a 3D network of 1D hyperbolic conservation laws. Our computational studies, motivated by the questions posed to us by cardiologists at the Texas Heart Institute, provided novel insight into the mechanical properties of 4 currently available coronary stents on the US market, and suggested optimal stent design for a novel application of stents in transcatheter aortic valve replacement.
The applications discussed above gave rise to new mathematical problems whose solutions required a development of sophisticated mathematical ideas. They include a design of a novel unconditionally stable, loosely coupled partitioned scheme for numerical simulation of solutions to FSI in blood flow, and the development of the theory and numerics for nonlinear hyperbolic nets and networks arising in dimension reduction of the stent problem. An overview of the basic mathematical ideas associated with this research, and application to the two related problems in cardiovascular diagnosis and treatment, will be presented. This talk will be accessible to a wide scientific audience.

Active Scalar Equations and a Geodynamo Model

Speaker: 

Professor Susan Friedlander

Institution: 

USC

Time: 

Thursday, December 1, 2011 - 4:00pm

Location: 

RH 306

We discuss an advection-diffusion equation that has been proposed by Keith Moffatt as a model for the Geodynamo. Even though the drift velocity can be strongly singular, we prove that the critically diffusive PDE is globally well-posed. We examine the nonlinear instability of a particular steady state and use continued fractions to construct a lower bound on the growth rate of a solution. This lower bound grows as the inverse of the diffusivity coefficient. In the Earth's fluid core this coefficient is expected to be very small. Thus the model does indeed produce very strong Geodynamo action.

This work is joint with Vlad Vicol.

Regularity of rotational travelling water waves

Speaker: 

Professor Joachim Escher

Institution: 

Leibniz University, Germany

Time: 

Thursday, November 10, 2011 - 4:00pm

Location: 

RH 306

Several recent results on the regularity of streamlines beneath a rotational travelling wave, along with the wave profile itself, will be discussed.

The talk includes the classical water wave problem in both finite and infinite depth, capillary waves, and solitary waves as well. A common assumption in all models to be discussed is the absence of stagnation points.

Betting on Monte Carlo: Stochastic Computational Methods for Finance

Speaker: 

Professor of Mathematics and Director of the Institute for Pure and Applied Mathematics (IPAM) Russel Caflisch

Institution: 

UCLA

Time: 

Thursday, April 28, 2011 - 4:00pm

Location: 

NS2 1201

Monte Carlo is a computational workhorse for valuation of financial securities and risk. It is directly applicable to almost all types of financial securities and is robust in that it is insensitive to the complexities of a security. On the other hand, Monte Carlo can be terribly slow and inaccurate. This talk will review the basics of Monte Carlo quadrature in the context of finance and methods for its acceleration, including variance reduction, quasi-Monte Carlo and ad hoc methods. American options, for which the exercise time is chosen by the option holder, are a class of securities to which Monte Carlo is not directly applicable. The talk will also describe the recently developed Least Square Monte Carlo (LSM) method for American options, some generalizations of LSM, and methods for estimating the accuracy of Monte Carlo for American options.

Critical Scaling Limits and Measure Ensembles

Speaker: 

Professor Charles Newman

Institution: 

Courant Institute of Mathematical Sciences, NYU

Time: 

Thursday, March 31, 2011 - 4:00pm

Location: 

NS2 Room 1201

In statistical physics, systems like percolation and Ising models are of particular interest at their critical points. Critical systems have long-range correlations that typically decay like inverse powers. Their continuum scaling limits, in which the lattice spacing shrinks to zero, are believed to have universal dimension-dependent properties. In recent years critical two-dimensional scaling limits have been studied by Schramm, Lawler, Werner, Smirnov and others with a focus on the boundaries of large clusters. In the scaling limit these can be described by Schramm-Loewner Evolution (SLE) curves.

In this talk, I'll discuss a different but related approach, which focuses on cluster area measures. In the case of the two-dimensional Ising model, this leads to a representation of the continuum Ising magnetization field in terms of sums of certain measure ensembles with random signs. This is based on joint work with F. Camia (PNAS 106 (2009) 5457-5463) and on work in progress with F. Camia and C. Garban.

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