The Simons Symposium on Families of Automorphic Forms and the Trace Formula provided an opportunity for researchers to study families of automorphic representations of higher rank groups. During the symposium participants investigated the trace formula, plancherel equidistribution and other recent techniques from harmonic analysis and representation theory with the goal to set an agenda for the field.
This symposium will focus on setting a clear agenda for future developments in the related fields of tropical and nonarchimedean analytic geometry. One of the goals of the meeting will be to produce high quality expository material presenting the methods, results, and ambitions of these active research areas. Another will be to identify problems in other fields of mathematics that could be amenable to tropical and nonarchimedean analytic methods and establish new rigorous links with those neighboring fields.
The Kardar-Parisi-Zhang equation is a nonlinear stochastic partial differential equation widely used in the physics literature as a model of randomly growing interfaces, but until recently very poorly understood from the mathematical point of view.
Many of the striking advances in theoretical computer science over the past two decades concern approximation algorithms, which compute provably near-optimal solutions to NP-hard optimization problems. Yet the approximability of several fundamental problems such as TSP, Graph Coloring, Graph Partitioning etc. remains an open question. For other problems the so-called PCP Theorems and more recently, the Unique Games Conjecture (UGC), provide a complexity-theoretic explanation for the failure to design better algorithms.
This meeting explored exotic quantum states of matter, with a particular focus on the role of quantum entanglement. A variety of new tools arising from both the study of quantum field theoretic techniques in condensed matter physics, and from the “holographic” duality between field theories and gravity theories coming out of string theory, make this an excellent time to try and rise to the challenge of understanding exotic quantum states.