Viscosity Solutions for Forward SPDEs and PPDEs

Speaker: 

Jin Ma

Institution: 

USC

Time: 

Tuesday, May 20, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

In this talk we introduce a notion of stochastic viscosity solution

for a class of fully nonlinear SPDEs and the corresponding Path-dependent

PDEs (PPDEs). The definition is based on our new accompanying work

on the pathwise stochastic Taylor expansion, using a variation of the path-

derivatives initiated by Dupire. As a consequence this new definition of the

viscosity solution is directly in the pathwise sense, without having to invoke

the stochastic characteristics for the localization. The issues of consistency,

stability, comparison principles, and ultimately the well-posedness of the

stochastic viscosity solutions will be discussed under this new framework.

This is a joint work with Rainer Buckdahn and Jianfeng Zhang.

A spatial evolutionary model of field cancerization

Speaker: 

Jasmine Foo

Institution: 

University of Minnesota

Time: 

Monday, May 5, 2014 - 10:00am to 11:00am

Location: 

340M

Cancer often arises through a sequence of genetic alterations.  Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion.  To model this process we consider a generalization of the biased voter process on a lattice which incorporates successive mutations modulating individual fitness.  We will study the rate of mutant spread and accumulation of oncogenic mutations in this process.  We then investigate the geometry and extent of premalignant fields surrounding primary tumors, and evaluate how the risk of secondary tumors arising from these fields may depend on the cancer progression pathway and tissue type.  (joint work w/K. Leder, R. Durrett, and M. Ryser).

Some Large-Scale Fractals

Speaker: 

: Davar Khoshnevisan

Institution: 

University of Utah

Time: 

Tuesday, May 6, 2014 - 11:00am to 12:00pm

Location: 

RH 306

We present a case study of the large-scale “fractal” behavior of concrete families of random processes that arise in complex systems. Among other things we will exhibit two random functions both of which are “multifractal” on large scales, but only one of which shows “intermittency.” This contradicts  the commonly-held view that “multifractality” and “intermittency” can be used interchangeably. This is based on joint work with K. Kim and Y. Xiao.

The Gaussian Free Field, Conformal Field Theory, and Schramm-Loewner Evolution

Speaker: 

Tom Alberts

Institution: 

Cal Tech

Time: 

Tuesday, April 15, 2014 - 11:00am to 12:00pm

Location: 

RH 306

 I will review the recent mathematical approach to Conformal Field Theory proposed by my colleagues Nam-Gyu Kang and Nikolai Makarov. Their construction defines a certain class of algebraic operations on correlation functions of the Gaussian Free Field, and these operations can be used to give meaning to "vertex observables" and other well known objects in CFT. Using conformal transformation rules for the GFF these objects can be defined on any simply connected domain, and using Lie derivatives they can be analyzed when the domain evolves according to an infinitesimal flow. Using the flow of Loewner's differential equation produces a connection with the random curves of the Schramm-Loewner evolution, which I will describe along with some recent work in the case of multiple SLE curves.

Self-adjoint extensions, point potentials, and pinned polymers

Speaker: 

Mike Cranston

Institution: 

UCI

Time: 

Tuesday, March 4, 2014 - 11:00am to 12:00pm

Location: 

RH 306

 
In this talk we discuss closed self adjoint extensions of the Laplacian and fractional Laplacian on L2 of Euclidean space minus the origin. In some cases there is a one parameter family of these operators that behave like the original operator plus a potential at the origin. Using these operators, we can construct polymer measures which exhibit interesting phase transitions from an extended state to a bound state where the pinning at the origin due to the potential takes over. The talk is based on joint works with Koralov, Molchanov, Squartini and Vainberg.
 

On Self-Similarity and some Interacting Diffusions

Speaker: 

Leif Doring

Institution: 

ETH

Time: 

Tuesday, February 11, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH306

We recall some classical results on self-similar Markov processes and in particular explain how those relate to a very particular class of stochastic differential equations. Those SDEs have an infinite dimensional counter part which arise naturally in interacting diffusions and can be interpreted as generalized voter process.

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