NSF-BSF Research in Representation Theory with Applications to Number Theory and Physics

表示论是研究线性对称性的。由于自然界中有许多对称性，并且线性方法是强大而有效的，因此这个主题在数学，物理，化学和计算机科学中有许多应用。经典上，研究了有限群或紧群的表示。目前的项目涉及无限，非紧，代数群的表示，PI和他的合作者将研究更近期的数学结构，如李群，量子群和双仿射Hecke代数。这些主题中的每一个都是现代数学研究多年来的活跃领域，但许多重要的问题仍然是开放的。该项目中所回答的问题的解决将导致数论，组合学和物理学的重要新应用。该项目的主要部分致力于研究自守表示，并将由PI和以色列合作者D.古雷维奇这是数论朗兰兹纲领中的一个重要课题，最近也是弦理论中的一个重要课题。PI和Gourevitch将采用Whittaker泛函的方法，这是研究大型自守表示的传统工具。然而，PI和合作者最近引入了一类更精细的泛函，它们非常适合研究小表示。这将应用于量子引力，其中小自守表示作为对爱因斯坦广义相对论的某些量子修正而出现。该项目的第二部分将研究Dirichlet级数的高维类似物，它推广了著名的Riemann zeta函数。PI将使用双仿射Hecke代数来研究这些级数，双仿射Hecke代数是理解麦克唐纳多项式对称性的传统工具，而麦克唐纳多项式本身也是代数组合学研究的中心对象。PI和合作者最近发现了这些代数的一类新的表示，称为元代数表示，这对研究多重狄利克雷级数有相当大的希望。本课程的第三部分将发展卡佩利微分算子的理论。这些算子起源于经典的不变量理论，它构成了世纪数学的很大一部分。他们还发挥了关键作用，在Atiyah-Bott-Patodi方法的指标定理，这是一个突出的世纪数学。PI和合作者将Capelli算子扩展到李超代数和量子群的现代领域。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。

Representation theory is the study of linear symmetries. Since there are many symmetries in nature, and since linear methods are powerful and efficient, this subject has found numerous applications in mathematics, physics, chemistry, and computer science. Classically, representations of finite or compact groups were studied. The current project concerns representations of infinite, non-compact, algebraic groups, and the PI and his collaborator will study more recent mathematical structures such as Lie supergroups, quantum groups, and double affine Hecke algebras. Each of these topics has been an active area of modern mathematical research for many years, yet many important questions remain open. The resolution of the questions answered in this project will lead to significant new applications in number theory, combinatorics, and physics.The main part of the project is dedicated to the study of automorphic representations and will be pursued by the PI and an Israeli collaborator D. Gourevitch. This is a subject of central importance in the Langlands program in number theory, and recently of much interest in string theory. The PI and Gourevitch will employ the method of Whittaker functionals, which are a traditional tool for studying large automorphic representations. However the PI and collaborators have recently introduced a more refined class of functionals, which are well-adapted to studying small representations. This will have applications to quantum gravity, where small automorphic representations arise as certain quantum corrections to Einstein's theory of general relativity. The second part of the project will study higher dimensional analogs of Dirichlet series, which generalize the celebrated Riemann zeta function. The PI will study these series using double affine Hecke algebra, which are a traditional tool for understanding the symmetries of Macdonald polynomials -- themselves a central object of study in algebraic combinatorics. The PI and collaborators have recently discovered a new class of representations of these algebras, called metaplectic representations, that hold considerable promise for studying multiple Dirichlet series. The third part of the project will develop the theory of Capelli differential operators. These operators have their origin in classical invariant theory, which comprised a large part of 19th century mathematics. They also played a key role in Atiyah-Bott-Patodi approach to the index theorem, which is a highlight of 20th century mathematics. The PI and collaborators will extend Capelli operators to the modern realms of Lie superalgebras and quantum groups.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.