From Representation Theory to Mathematical Physics and Back

从表示论到数学物理并返回

• 批准号: 2149565
• 负责人: Joshua Sussan
• 金额: $ 3.07万
• 依托单位: CUNY Medgar Evers College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-05-01 至 2023-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2149565&HistoricalAwards=false
• 关键词: Representation; Theory; Mathematical; Physics; Back

The theme of the conference "From Representation to Mathematical Physics and Back" is the continued influence of mathematics and physics on each other. This symbiotic relationship has played a significant role in these disciplines for centuries. For instance, Newton's discovery of gravity necessitated his development of calculus. While the interactions between mathematics and physics have evolved over time, some of the deepest aspects of these fields are still intertwined. In particular, the discovery of quantum physics in the twentieth century led mathematicians to study difficult problems in analysis, topology, and algebra. In turn, solutions to these questions have then been reinterpreted by physicists, and have led to new physics questions. Understanding how gravity fits into our quantum mechanical universe remains one of the most important open questions in physics and mathematics. Recent advances in these areas will be presented at this conference. A primary goal of the meeting is to introduce a young generation of researchers, including those from typically underrepresented groups, to these fundamental mathematical and physical problems. For more information see the conference website: http://scgp.stonybrook.edu/archives/34420The conference will highlight progress in the areas of vertex operator algebras, conformal field theory, categorification, low dimensional topology and representation theory of affine Lie algebras, loop groups, and quantum groups. String theory gave rise to the mathematical theory of vertex operator algebras, which led to the construction of representations of affine Lie algebras and the Moonshine module of the Monster group. These mathematical constructions have in turn led to ideas about 3-dimensional quantum gravity. In another direction, the discovery of the Jones polynomial led to a physical construction of 3-dimensional TQFTs, which in turn led to many mathematical discoveries in quantum groups and low dimensional topology. Crane and Frenkel introduced the categorification program with the goal of upgrading 3-dimensional TQFTs coming from representation theory of quantum groups to 4-dimensional TQFTs. They suggested that one should try to construct categorical quantum group actions to achieve this aim. These ideas led to the discovery of Khovanov homology. Other link homologies have been constructed from representation-theoretic, algebraic-geometric, combinatorial, and physical structures. Consequently, categorification has become a truly interdisciplinary subject.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.

会议的主题“从表示到数学物理和背部”是数学和物理对彼此的持续影响。 几个世纪以来，这种共生关系在这些学科中发挥了重要作用。 例如，牛顿发现万有引力使他发展微积分成为必要。 虽然数学和物理学之间的相互作用随着时间的推移而发展，但这些领域的一些最深层次的方面仍然相互交织。 特别是，量子物理学的发现在二十世纪导致数学家研究困难的问题，在分析，拓扑学和代数。 反过来，这些问题的解决方案又被物理学家重新解释，并导致了新的物理问题。 理解引力如何适应我们的量子力学宇宙仍然是物理学和数学中最重要的开放问题之一。 这些领域的最新进展将在本次会议上介绍。 会议的主要目标是向年轻一代的研究人员，包括那些代表性不足的群体，介绍这些基本的数学和物理问题。欲了解更多信息，请参阅会议网站：http://scgp.stonybrook.edu/archives/34420The会议将突出在顶点算子代数，共形场论，分类，低维拓扑和仿射李代数，回路群和量子群的表示理论领域的进展。 弦论产生了顶点算子代数的数学理论，这导致了仿射李代数和Monster群的Moonshine模的表示的构造。 这些数学构造反过来又导致了关于三维量子引力的想法。 在另一个方向上，琼斯多项式的发现导致了三维TQFT的物理构造，这反过来又导致了量子群和低维拓扑的许多数学发现。 起重机和Frenkel引入了分类程序，目的是将来自量子群表示论的三维TQFT升级为四维TQFT。 他们建议，人们应该尝试构建分类量子群作用来实现这一目标。 这些想法导致了Khovanov同源性的发现。 其他链接的同源性已经从表示理论，代数几何，组合和物理结构。 该奖项反映了NSF的法定使命，并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，被认为值得支持。