Integrated Calendar

Speakers: Omri Bar-Elli, Ayelet Uzan

Consider the correspondence f(x,y) &#8594 f(X,Y), where f is a function, x,y &#8712 &#8476 are coordinates, and X,Y are linear operators on an auxiliary space that satisfy the commutation relation XY-YX=i &#8712 &#8465, for some real number &#949. Making sense of f(X,Y) can require great care.
In physics one encounters this problem with the phase space picture of quantum mechanics commonly used in quantum optics, where X,Y are conjugate position and momentum operators. Such notions also occur in the strong field limit of the Landau problem and its string theory equivalent for D-branes where X,Y are the position operators for coordinates in a spatial plane. In mathematics one finds these problems addressed by harmonic analysis on the Heisenberg group along with general pseudo-differential operators and symbol calculus.
The ubiquity of the f(x,y) &#8594 f(X,Y) question has led many researchers in many different fields to perennially rediscover some formulas and miss others. This talk will discuss a review paper which aims to help the disparate communities speak to one another and cover new ground. This talk will focus on explicit representations and useful formulas. We will emphasize otherwise unexpected correspondences between continuous PDE and lattice difference equations, the likes of which are often found in models of nonlinear optical waveguide array lattices.