Representation Theory of Finite Groups

主要研究方向是有限群的表示理论。他主要研究与Alperin、Dade和Broue的猜想有关的问题。这些猜想表明，有限群和固定素数的某些不变量可以由与该素数相关的子群的相同不变量确定。第一个猜想关注的是整数不变量，而第三个猜想预测了范畴的等价性。前两个猜想和第三个猜想的结果已经通过大量令人信服的例子得到了验证。这三种猜想很可能是由一个尚不为人知的事实、对象、理论或结构所导致的结果。也许比解决这些特殊的猜想更重要的是发现这个隐藏的特征。首席研究员研究如何处理这些猜想的一般思路。研究生们通过编写计算机程序来参与这项工作，这些程序将抛弃或提供更多证据来证明所建议的方法之一的适用性。首席研究员研究群体的表征。群可以看作是对称概念的数学抽象。它们是许多数学领域中最基本的概念之一，被广泛应用于密码学、物理学、化学和计算机科学等领域。群的表示是对其他数学对象的一种特殊类型的对称的表现，例如，平面上的正方形允许的四次旋转和四次反射，在二维向量空间上形成了一个有8个元素的群的表示。更具体地说，主要研究者研究了有限群体表征理论中的神秘巧合，这些巧合大约在15年前就被发现了，但迄今为止还无法解释。这项研究的目的不在于数学以外的直接应用。然而，在数学与其他科学，特别是物理学相互作用的历史中，有一种反复出现的模式，即在数学体系中被认为重要的理论和结果，恰恰成为其他科学描述我们真实世界所需要的东西。例如，一个世纪以前研究过的小组陈述，在很久以后才成为描述核物理中粒子的正确工具。

The principal investigator is working in the field ofrepresentation theory of finite groups. Mainly he studiesquestions related to the conjectures of Alperin, Dade, and Broue.These conjectures state that certain invariants of a finite groupand a fixed prime number can be determined by the same invariantsof subgroups which are again related to that prime. The firstconjectures are concerned with invariants that are integers,whereas the third conjecture predicts equivalences of categories.The first two conjectures and consequences of the third have beenverified for a convincing number of examples. It is likely thatthese three conjectures are consequences of a single fact, object,theory, or construction which is still hidden. Probably moreimportant than solving these particular conjectures would be thediscovery of this hidden feature. The principal investigator workson general ideas of how to approach these conjectures. Graduatestudents are involved in this effort by writing computer programswhich will either discard or give more evidence to theapplicability of one of the suggested approaches.The principal investigator studies representations of groups.Groups can be viewed as mathematical abstractions of the notion ofsymmetry. They are among the most basic notions in many fields ofmathematics and are applied widely, for example in cryptology,physics, chemistry, and computer science. Representations ofgroups are manifestations of a particular type of symmetry onother mathematical objects, as for example the four rotations andfour reflections, that a square in the plane allows, form arepresentation of a group with 8 elements on a two-dimensionalvector space. More specifically, the principal investigatorstudies mysterious coincidences in the representation theory offinite groups which have been discovered about fifteen years agobut could not be explained so far. This research is not aimed atimmediate applications outside mathematics. However, in thehistory of the interplay between mathematics and other sciences,in particular with physics, it is a repeated pattern that theoriesand results which were considered as important within the edificeof mathematics became precisely what was needed in the othersciences in order to describe our real world. For example, grouprepresentations which have been studied a century ago became muchlater the right tool to describe particles in nuclear physics.