Tue, Sep 8

What is...? Seminar

Gino Biondini

Nonlinear waves, solitons and integrable systems

4:00 PM, 250 Mathematics Building

The mathematical study of natural phenomena is an essential component of physical applied mathematics.  In a large number of cases, the systems under consideration lead to a certain class of nonlinear partial differential equations, which, remarkably, are also completely integrable infinite-dimensional Hamiltonian systems.  The development of the mathematical theory of nonlinear wave equations, solitons and integrable systems has been one of the great achievements of mathematical physics of the second half of the twentieth century.  The study of these equations is especially attractive because it offers a unique combination of beautiful mathematics and concrete physical and technological applications.  This talk is aimed at providing a first introduction to the subject.  Specifically, after introducing the concept of a Lax pair, I will try to provide a concise overview of the inverse scattering transform, which is an exact generalization of the Fourier transform for nonlinear systems, and which, when available, provides a way to solve the initial value problem for these systems.  We will then look at how solitons naturally arise in these systems and at some of their properties.  Time permitting, the last part of the talk will be devoted to a brief discussion of some of the current research directions in this field.

Fri, Sep 18

Topology and Geometry Seminar

Vasudha Bharathram (Princeton)

4:00 PM, 122 Mathematics Building

Tue, Sep 22

What is...? Seminar

Bill Menasco

TBA

4:00 PM, 250 Mathematics Building

TBA

Wed, Sep 23

Special Event

Myhill Lectures, Thanasis Fokas Cambridge University.

The Beauty and Usefulness of the Complex Plane

4:00 PM, Mathematics Building

An attempt will be made to demystify complex analysis. This naturally leads to the introduction of the Riemann-Hilbert and d-bar formalisms. For many years, the employment of the Wiener-Hopf technique to acoustics and other physical problems, was the only manifestation in applications of the Riemann-Hilbert formalism. However, in the last 50 years this formalism has appeared in many problems in mathematics and mathematical physics. In addition, in the early 1980s the d-bar formalism also began to emerge in applications. Several of these applications will be mentioned, including the solution of integrable nonlinear evolution PDEs in one, two, and three dimensions, as well as medical imaging

Thu, Sep 24

Special Event

Myhill Lectures, Thanasis Fokas Cambridge University.

The Unified Transform Method for Linear and Integrable Nonlinear PDEs

4:00 PM, Mathematics Building

The Unified Transform, also known as the  Fokas method,  provides a new and powerful method for analysing boundary value problems for linear and integrable nonlinear PDEs. For linear PDEs, it provides an unexpected extension of classical works of D’Alembert, Fourier, Laplace, Green and Kelvin. For nonlinear PDEs, perhaps the most striking application of the new method is the solution of  the x-periodic problem  for integrable PDEs, which was first considered in the early 1970s, and which attracted the involvement of many distinguished mathematicians. This lecture will review the above novel method with emphasis on linear evolution PDEs.