Workshop on Representation Theory, Combinatorics, and Geometry

该奖项将支持主要是早期职业研究人员在表示理论，组合数学和几何研讨会的参与，以及杰出的讲座系列“弗吉尼亚数学讲座”将于10月19日至21日和22日至24日分别在弗吉尼亚大学举行。该研讨会将由领先的专家就量子群，量子变种和正则基之间的联系进行会谈，以及早期职业参与者的简短会谈。参与者将接触到令人兴奋的新方向表示理论，几何和组合，并将获得饲料为未来的项目。数学研究人员和领域的多样性将促进新的合作和新的研究项目。更详细地说，半单李代数的幂零锥的Springer分辨率为几何表示理论提供了肥沃的土壤。Maulik和Okounkov在辛分解的框架中引入了稳定包络的概念（其中Springer分解和N-S簇是最重要的例子），并使用它提供了R-矩阵的第一个几何构造和在广泛背景下使用等变上同调的Yangians的不同构造。它们的构造与辛分解的量子上同调密切相关。这些学科在现代数学的几个领域中发挥了重要作用。一些最新的研究方向（如i-bismuth品种）是在去年左右才出现的，目前正在与Maulik-Okounkov的理论保持一致。在弗吉尼亚大学的活动将提供一个及时的平台，交流新兴的新思想，这将有助于他们的进一步发展。该研讨会的网站是http://math.virginia.edu/ims/workshop-fall-2018/.This奖反映了NSF的法定使命，并已被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

The award will support participation of primarily early-career researchers in the Workshop on Representation Theory, Combinatorics, and Geometry, and the Distinguished Lecture Series "Virginia Mathematics Lectures" which will take place on October 19-21 and 22-24, 2019, respectively, at the University of Virginia. The workshop will consist of talks by leading experts on connections between quantum groups, quiver varieties, and canonical bases, as well as short talks by early-career participants. Participants will be exposed to exciting new directions in representation theory, geometry, and combinatorics, and will gain fodder for future projects. The diversity of mathematical researchers and areas represented at the workshop will foster new collaborations and new research projects.In more detail, Springer resolutions of nilpotent cones of semisimple Lie algebras provide a fertile ground for geometric representation theory. Maulik and Okounkov introduced a notion of stable envelope in the framework of symplectic resolutions (with Springer resolutions and quiver varieties as most important examples), and used it to provide a first geometric construction of R-matrices and a different construction of Yangians using the equivariant cohomology in a wide context. Their construction is intimately related to quantum cohomology of the symplectic resolutions. These subjects have played an important role in several areas of modern mathematics. Some of the most recent research directions (such as i-quiver varieties) have emerged only in the last year or so, and are currently aligning with Maulik-Okounkov's theory. The events at University of Virginia will provide a timely platform for exchange of emerging new ideas that will facilitate their further development. The website of the workshop is http://math.virginia.edu/ims/workshop-fall-2018/.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.