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

这笔赠款的主要目的是为2014年5月12-16日(研讨会)和5月19-23日(会议)举行的研讨会和会议提供资金，主题为“表象理论、可积系统和量子场：西北大学的重点年”。按照拟议活动的总体模式，西北大学有很长一段成功的“重点年”历史。今年的重点将放在将新物理学与数学联系起来。由Nekrasov-Shatashvili、Alday-Gaiotto-Tchikawa和Kapustin-Witten开创的物理学最近的进步在全球范围内催生了数百篇论文和兴趣。Maulik-Okounkov提出了数学观点，Goncharov-Kenyon也有相关工作。这些结果不仅应该被理解为它们本身的优点，而且还应该被理解为可积系统、表象理论和量子场之间更广泛的新兴关系的一部分。研讨会和会议将做到这一点。拟议的大部分资金将直接支持研究生和博士后，通过资助他们的参与，或间接通过资助研究生研讨会上的演讲者。这些学生将主要是来自美国机构的美国公民，从而进一步实现了支持美国专业数学家发展的征集目标。该项目的总主题是探索对称群与其产生的物理系统之间的相互作用。物理现象有类似的数学描述，反之亦然：几何朗兰兹计划通过Kapustin和Witten的工作，在拓扑扭曲的超对称规范理论的降维方面有类似的描述。Alday-Gaiotto-Tchikawa猜想将二维共形场理论与四维超对称理论联系起来。这一猜想的一个数学表现是Virasoro代数和W-代数在由Maulik和Okounkov构造的复曲面上丛的模空间的上同调上的作用。与奇异Calabi-Yau三重弦紧化有关的可积系统在统计物理中被描述为二聚体问题，而这又与簇可积系统(Nekrasov-Shatashvili，Kenyon-Goncharov)有关。所有这些现象都将在研讨会和会议上进行探讨。会议网页的网址如下：http://www.math.northwestern.edu/emphasisyear/.

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/.