I will give an overview of a three decade long project of novel symmetries in quantum field theory, with the emphasis on the most recent development concerning the realisation of stable envelopes (proposed by A. Okounkov and collaborators) and R-matrices via supersymmetric interfaces in 1-2-3 dimensional supersymmetric gauge theories. Based on the recent paper arXiv:2109.10941 with Mykola Dedushenko.

Two dimensional sigma models describe (harmonic) maps of Riemann surfaces to Riemannian manifolds. I will present the motivations to study the complexification of this problem. I will present the novel approach, developed in my work with Igor Krichever, allowing to construct essentially all twisted complexified harmonic maps of two-torus to spheres and projective spaces. -- Part of the London Integrability Journal Club. If you are a new participant please register at integrability-london.weebly.com. The link will be emailed.

I will review the intermediate long wave equation which interpolates between the celebrated Korteweg-de Vries and Benjamin-Ono equations, describing the one-dimensional waves in a shallow and deep water, respectively. The quantum version of this equation is of some interest to the theory of quantum wires and other quasi-one dimensional condensed matter systems. I will explain the unexpected connection of this equation to the dynamics of gauge theories in two, four, and six dimensions. This connection is one of the applications of the BPS/CFT correspondence. Based on the work in progress with Andrei Okounkov

First of two lectures. For details, please consult the webpage www.ma.ic.ac.uk (slash) (tilde) rpwt (slash) LMS.html (sorry, triangle server does not allow special characters)