The PDEs of mathematical finance

Speaker: 

Jerry Goldstein

Institution: 

University of Memphis

Time: 

Thursday, April 28, 2016 - 4:00pm

Location: 

RH 306

We will discuss three one space dimensional time dependent linear parabolic equations: the heat equation, the Black-Scholes equation (describing stock options) and the Cox-Ingersoll-Ross equation (describing bond markets).  New results will involve representation of the solution semigroups, chaotic properties of the semigroups, and a new kind of Feynman-Kac type representation of the solution for the CIR equation.

Random matrix type fluctuations: how to see them in the Ising model?

Speaker: 

S. Shlosman

Institution: 

CNRS, Marseille

Time: 

Thursday, March 10, 2016 - 4:00pm

Location: 

RH 306

I will talk about the Ising model -- the drosophila of the rigorous statistical physics. It turns out that some of the new phenomena which appear in modern mathematical physics can still be observed in the Ising model as well. 
One example which I will focus on is the size of typical fluctuations of the extended systems. If the size of the system is N, then the usual (Gaussian) fluctuations are of the order of N^{1/2}. Bit in the random matrix theory one sees the fluctuations of the order N^{1/3}. I will explain that one can see them already in the Ising model -- one just needs to know where to look.

Journey to the Center of the Earth

Speaker: 

Gunther Uhlmann

Institution: 

University of Washington

Time: 

Thursday, January 28, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

     We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.
     The problem can be recast as a geometric problem: Can one determine a Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.
     We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed. 

Birkhoff Conjecture and ''spectral rigidity'' of planar convex domains

Speaker: 

Vadim Kaloshin

Institution: 

Maryland University

Time: 

Thursday, January 21, 2016 - 4:00pm to 5:00pm

Host: 

Location: 

Rowland Hall 306

The classical Birkhoff conjecture states that the only integrable convex planar domains are circles and ellipses. In a joint work with A. Avila and J. De Simoi we show that this conjecture is true for perturbations of ellipses of small eccentricity. It turns out that the method of proof gives an insight into deformational spectral rigidity of planar axis symmetric domains and a partial answer to a question of P. Sarnak. The latter is a joint work with J. De Simoi and Q. Wei.

Gluing constructions in differential geometry

Speaker: 

Nicolaos Kapouleas

Institution: 

Brown University

Time: 

Thursday, October 22, 2015 - 4:00pm to 5:00pm

Host: 

Location: 

Rowland Hall 306

Abstract:

I will discuss various geometric gluing constructions. First I will discuss constructions for Constant Mean Curvature hypersurfaces in Euclidean spaces including my earlier work for two-surfaces in three-space which settled the Hopf conjecture for surfaces of genus two and higher, and recent generalizations in collaboration with Christine Breiner in all dimensions. I will then briefly mention gluing constructions in collaboration with Mark Haskins for special Lagrangian cones in Cn. A large part of my talk will concentrate on doubling and desingularization constructions for minimal surfaces and on applications on closed minimal surfaces in the round spheres, free boundary minimal surfaces in the unit ball, and self-shrinkers for the Mean Curvature flow. Finally I will discuss my collaboration with Simon Brendle on constructions for Einstein metrics on four-manifolds and related geometric objects.

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