Representation Theory, Integrable Systems, and Quantum Fields: Emphasis Year at Northwestern University 2013-14

Emphasis Year in Representation Theory, Integrable Systems, and Quantum Fields: Project Summary The full Project Summary is located in internal documents. The Department of Mathematics at Northwestern University plans to hold an emphasis year in Representation Theory, Integrable Systems, and Quantum Fields in the 2013-14 academic year. This will be part of a long and successful tradition of emphasis years at Northwestern. This will entail having a program of senior visitors in related fields throughout the year, holding a one-week workshop with mini-courses on relevant topics aimed towards graduate students, and culminating with a one-week conference at the end of the year. In parallel, Northwestern will offer a selection of graduate courses to provide the necessary background and foundations to students. These courses will be advertised widely in the Chicago area and will be available for audit to any interested graduate students. There will also be a special lecture for undergraduates. The general theme of the program is to explore the interplay between symmetry groups and the physical systems in which they arise. Physical phenomena have analogous mathematical descriptions, and vice versa: the geometric Langlands program has an analogue in a dimensional reduction of a topologically twisted supersymmetric gauge theory, through the work of Kapustin and Witten. The Alday-Gaiotto-Tachikawa conjecture relates two-dimensional conformal field theory to four-dimensional supersymmetric theories. A mathematical manifestation of this conjecture is the action of Virasoro and W-algebras on the cohomology of moduli spaces of bundles on a complex surface constructed by Maulik and Okounkov. Integrable systems relating to string compactification on singular Calabi-Yau threefolds have descriptions as dimer problems in statistical physics, which in turn relate to cluster integrable systems (Nekrasov-Shatashvili, Kenyon-Goncharov). The central aim of the emphasis year is to introduce exciting areas of research to graduate students, postdocs and mathematicians at the early stages of their careers. The four-pronged approach should serve interested participants at all levels. (1) Local graduate students will be free to come to Northwestern to audit (or enroll in) graduate courses which provide the foundations for these topics, including topics aimed at a deeper exploration of the subjects. (2) The workshop will consist of mini-courses and selected talks aimed at new researchers. Speakers will be selected for their pedagogical skills. (more)