# Yang-Baxter and reflection equations: unifying structures behind quantum and classical integrable systems

## GAP Seminar

## Meeting Details

For more information about this meeting, contact Eyal Subag, Ping Xu, Mathieu Stiénon, Nigel Higson.

**Speaker:** Vincent Caudrelier, City University of London

**Abstract:** The Yang-Baxter equation (YBE) is central in the theory of quantum integrable systems. For decades, together with its companion for problems with boundaries (the quantum reflection equation), it has been studied and used in the quantum realm, leading to the area of quantum groups. But it was suggested by Drinfeld in 1990 that the general study of the so-called Â«set-theoretical YBE Â» is also important. This can be understood as the problem of finding nonlinear representations of the braid group on arbitrary sets. It turns out that classical integrable PDEs provide a means to construct certain types of such representations, called Yang-Baxter maps, by looking at soliton collisions. I will use the vector nonlinear SchrÃ¶dinger (NLS) equation as the main example to illustrate the idea. It has its origin in the physics of wave phenomena in fluid dynamics, nonlinear optics, plasma physics or quantum cold gases. After reviewing this, I will show how the new concept of set-theoretical reflection equation naturally emerges by studying solitons in integrable PDEs with a boundary. As before, the problem of finding solutions to this equation can be understood as the question of finding nonlinear representations of the finite Coxeter group of type BCn. I will show how to construct such representations using solitons.

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 09/01/2015

**Time:** 2:30pm - 3:30pm