Representation Theory, Integrable Systems and Quantum Fields: Emphasis Year at Northwestern University, May 19-23, 2014

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The main purpose of this grant is to fund the workshop and conference, "Representation Theory, Integrable Systems, and Quantum Fields: Emphasis Year at Northwestern University," running May 12-16 (workshop) and May 19-23 (conference) in 2014. Northwestern has a long history of successful 'emphasis years' following the broad pattern of the proposed activities. This year's emphasis will be placed on connecting new physics to mathematics. Recent advanced in physics pioneered by Nekrasov-Shatashvili, Alday-Gaiotto-Tachikawa, and Kapustin-Witten have spawned hundreds of papers and interest around the globe. A mathematical perspective has been offered by Maulik-Okounkov and there is related work of Goncharov-Kenyon. These results should be understood not only on their own merit, but also as a part of a broader emerging relationship between integrable systems, representation theory and quantum fields. The workshop and conference will do just that. The bulk of the proposed funding will support graduate students and postdocs either directly, by funding their participation, or indirectly by funding speakers at the graduate workshop. These students will largely be US nationals from US institutions, thus furthering the goal of the solicitation to support the development of professional mathematicians in the United States.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). All of these phenomena will be explored in the workshop and conference. The conference webpage is located at the following address: http://www.math.northwestern.edu/emphasisyear/.