**Abstract:**

I shall describe some conjectures and theorems about classical spin systems with continuous symmetry at low temperature. Examples of such spin systems include the XY and Heisenberg models as well as the SUSY hyperbolic sigma model which arises from random matrix theory. Symmetry plays a key role in the description of their long distance behavior.

**Biography:**

I received my PhD in 1972 under the direction of James Glimm at NYU. I was a professor at NYU from 1980-86 and have been at the Institute for Advanced Study since then. My primary interests have been in constructive quantum field theory, spectral theory for random and quasi-periodic operators, phase transitions and critical phenomena for interacting spin systems.

*Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai*