New Unifying Structures in Lie Theory and the Cubic Dirac Operator

不同数学领域的一整组问题似乎汇聚成与半单李群的固定Weyl腔中的凹穴集有关的问题。这些领域包括Kac-Moody理论，圈群同调，量子上同调，Verlinde代数，MacDonald恒等式，Schubert演算，Borel子代数中的理想，对称空间理论和紧Liegps的Cartan-Weyl表示理论。实际上，我们的建议是为了理清正在发生的事情，以统一在我们看来在上面列出的主题中可以统一的东西。群论，特别是李群理论，位于数学和数学在现实世界中的应用的核心。后者包括经典和量子力学、控制论、弦理论、化学和结晶学的应用。李群理论极其复杂，它的适用程度在很大程度上取决于对其复杂性的了解。拟议的项目预计将在该学科中发现非常令人兴奋的新结构，并将统一许多现有的结构。这将极大地提高我们的理解和使用相同结构的能力。该奖项由《几何分析与代数》、《数论与组合数学》等项目共同资助。

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.