Representation Theory of Finite Groups

主要研究人员从事有限群表示论领域的工作。他主要研究与Alperin、Dade和Broue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来确定。第一个猜想涉及的是整数不变量，而第三个猜想预测的是范畴的等价性。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。这三个猜想很可能是一个仍然隐藏的单一事实、对象、理论或结构的结果。也许比解决这些特殊猜想更重要的是发现这个隐藏的特征。主要研究人员提出了如何处理这些猜想的总体思路。研究生通过编写计算机程序来参与这项工作，这些程序将抛弃所建议的方法之一的适用性，或者提供更多的证据。主要研究人员研究群的表示。群可以被视为对称性概念的数学抽象。它们是许多数学领域中最基本的概念之一，并被广泛应用于密码学、物理学、化学和计算机科学。群的表示是其他数学对象上特定类型的对称性的表现，例如，平面上的正方形允许的四个旋转和四个反射，在二维向量空间上表示一个有8个元素的群。更具体地说，主要研究人员研究了大约15年前发现但至今无法解释的岩石群表示理论中的神秘巧合。这项研究的目的不是为了数学以外的中间应用。然而，在数学与其他科学，特别是与物理学相互作用的历史上，一种反复出现的模式是，在数学教育中被认为重要的理论和结果，恰恰变成了其他科学为了描述我们的现实世界而需要的东西。例如，一个世纪前已经研究过的群预演后来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.