Mathematical Sciences: Presidential Young Investigator Award

本研究涉及组合学、表示论和同调代数等领域。一个基本的方面是超级计算机的使用。算法是混合使用代数和组合方法设计的。这些计算的结果对目前正在进行的研究问题的解决方案提供了有价值的见解。第一个问题是计算有限维复李代数L与某多项式商环张量积的李代数上同调。如果L是半简单的，这个问题的解决方案将隐含麦克唐纳根系统猜想。部分结果和强有力的计算机证据表明，当L是半单质或幂零时，存在完全解。第二个问题是确定Richard Brauer定义的辛扶正代数和正交扶正代数的代数结构。这些代数是辛群和正交群表示理论的基础，具有丰富的组合结构。同样，强有力的计算机证据和部分结果表明，该结构问题的完整解决方案将具有优雅的组合语句。本研究探讨代数问题的组合结构。这些问题中的许多可以被描述为确定某些多项式的某些系数，这些多项式一方面以复杂的代数方式定义，另一方面有简单的组合解释。汉隆在他短暂的职业生涯中成为这一重要课题的世界主要权威之一。他利用超级计算机来协助这项工作是很有创意的。他的研究肯定会产生最深远的影响。

This research overlaps the areas of combinatorics, representation theory and homological algebra. A fundamental aspect is the use of supercomputers. Algorithms are designed that use a mix of algebraic and combinatorial methods. The results of these computations give valuable insight into the solutions of the ongoing research problems described now. The first problem is to compute the Lie algebra cohomology of the tensor product of a finite dimensional complex Lie algebra L with a certain polynomial quotient ring. If L is semisimple, a solution to this problem would imply the Macdonald root-system conjectures. Partial results and strong computer evidence points to a complete solution when L is either semisimple or nilpotent. The second problem is to determine the algebra structure of the symplectic and orthogonal centralizer algebras defined by Richard Brauer. These algebras are fundamental to the representation theory of the symplectic and orthogonal groups and are rich in combinatorial structure. Again, strong computer evidence and partial results suggest that a complete solution to this structure problem will have and elegant combinatorial statement. This research investigates the combinatorial structure of algebraic problems. Many of these questions can be described as determining certain coefficients of certain polynomials which on the one hand are defined in a complicated algebraic way but on the other hand have a simple combinatorial interpretation. Hanlon has in his short career become one of the world's leading authorities on this important subject. His use of the supercomputers to aid this work is very innovative. His research will certainly have the most profound affect.