New Unifying Structures in Lie Theory and the Cubic Dirac Operator

不同数学领域的一系列问题似乎都收敛于与半单李群的固定Weyl室中的一组凹室有关的问题。这些领域包括Kac-Moody理论、环群的同调、量子同调、Verlinde代数、MacDonald恒等式、schubert微积分、Borel子代数中的理想、对称空间理论和紧李群的Cartan-Weyl表示理论。实际上，我们的建议是整理出正在发生的事情，以统一我们认为在上面列出的主题中是统一的。群论，尤其是李群论，是数学的核心，也是数学在现实世界中应用的核心。后者包括经典和量子力学、控制理论、弦理论、化学和晶体学的应用。李群论是极其复杂的，它的适用程度在很大程度上取决于对其复杂性的了解。该项目有望在该学科中发现令人兴奋的新结构，并将许多现有结构统一起来。这将大大提高我们的理解和使用这些强大结构的能力。该奖项由几何分析与代数、数论和组合学项目共同资助。

A whole array of questions in different mathematical areas seemto be converge to questions having to do with the set of alcovesin a fixed Weyl chamber of a semisimple Lie group. The areasinclude Kac-Moody theory, homology of loop groups, quantumcohomology, the Verlinde algebra, MacDonald identities, Schubertcalculus, ideals in the Borel subalgebra, symmetric space theoryand the Cartan-Weyl representation theory of compact Liegroups. In effect our proposal is to sort out what is going on tounify what seems to us to be unifiable in the subjects listedabove.Group theory, and especially Lie group theory, lies at the heartof mathematics and the application of mathematics to problems inthe real world. Included in the latter are applications to bothclassical and quantum mechanics, control theory, string theory,chemistry and crystallography. Lie group theory is extremelyintricate and the extent to which it is applicable depends highlyon a knowledge of its intricacies. The proposed project expectsto discover highly exciting new structures in the subject andwould unify many existing structures. The effect that would be togreatly increase our understanding and our ability to use thesepowerful structures. This award is jointly funded by the programsin Geometric Analysis and Algebra, Number Theory, & Combinatorics.