Arithmetic, Geometry and Representation Theory of Reductive Groups

在他们最近的两篇论文中，Gopal Prasad和Sai-Kee Yeung构造了所有算术“伪”射影空间，并发现了它们的一些几何性质。他们的工作导致了许多有趣的问题，这些空间，也对一些奇异表面的小几何不变量。普拉萨德建议就这些问题开展工作。他还与Andrei Rapinchuk合作，寻找有限体积的局部对称空间由其闭测地线或其谱的长度集决定的程度。他们的工作使他们定义了一个新的关系之间的“大”（Zagliki密集或算术）的小组，他们称之为“弱可分性”。他们证明了一个简单李群的两个算术子群的弱可交换性，其Dynkin图不具有对称性，暗示它们是可交换的。他们现在正在研究当Dynkin图确实具有对称性时的情况。Prasad和Rapinchuk利用超越数论中的定理，以及Schanual提出的一个被广泛相信的猜想，证明了如果一个绝对单真实的李群的对称空间的两个算术子群的乘积具有相同的闭测地线的长度集，或者具有相同的谱，那么这两个算术子群是弱可积的。因此，他们关于弱可分解算术子群的结果是可以利用的。普拉萨德建议获得类似的结果，局部对称空间所产生的复半单李群。在一个完全不同的方向，普拉萨德已经将一个自然列维子群与一个约化p进群的一个给定的不可约容许表示联系起来。他将调查什么样的作用，这一小组发挥代表性理论。普拉萨德与Rapinchuk正在进行合作，以简化，统一和完成同余子群问题的结果。他们计划在不久的将来写一本关于这个主题的书。最近的工作普拉萨德与赛记杨对某些有趣的几何对象被称为算术假射影空间已导致一个明确的建设所有这些和帮助确定许多他们的几何性质。他们的工作也引出了一些有关几何物体的重要问题。普拉萨德最近的工作与Rapinchuk有深刻的影响，一类特别重要的几何结构被称为局部对称空间。 这些空间起源于对称空间，顾名思义，对称空间有很多对称性。这项工作的普拉萨德和Rapinchuk已介绍了一个新的概念“弱可交换性”的大型子群的群的对称性和研究其后果在几何和群论。仍然有一些严重的未解决的问题，他们将工作。他们还计划写一本书，对著名的同余子群问题，以描述一个新的统一的方法来解决它。

In their two recent papers, Gopal Prasad and Sai-Kee Yeung have constructed all arithmetic "fake" projective spaces and have found some of their geometric properties. Their work has led to many interesting questions related to these spaces and also about some of the singular surfaces with small geometric invarients. Prasad proposes to work on these questions. He has also been working with Andrei Rapinchuk to find the extent a locally symmetric space of finite volume is determined by the set of lengths of its closed geodesics, or its spectrum. Their work led them to define a new relationship between "large" (Zariski-dense or arithmetic) subgroups which they call "weak commensurability". They have shown that weak commesurability of two arithmetic subgroups of a simple Lie group, whose Dynkin diagram does not have symmetries, implies that they are commensurable. They are now investigating the situation when the Dynkin diagram does have a symmetry. Prasad and Rapinchuk have used theorems in transcendental number theory, and a widely believed conjecture due to Schanual, to show that if the quotients of the symmetric space of an absolutely simple real Lie group by two arithmetic subgroups have same set of lengths of closed geodesics, or have the same spectrum, then the two arithmetic subgroups are weakly commensurable. So their results on weakly commensurable arithmetic subgroup can be used. Prasad proposes to obtain analogous results for locally symmetric spaces arising from comples semi-simple Lie groups. In a comepletely different direction, Prasad has associated a natural Levi-subgroup to a given irreducible admissible representation of a reductive p-adic group. He will investigate what role this subgroup plays in the representation theory. Prasad has an ongoing collaboration with Rapinchuk to simplify, unify and complete the results on the congruence subgroup problem. They plan to write a book on this topic in near future.Recent work of Prasad with Sai-Kee Yeung on certain interesting geometric objects known as arithmetic fake projective spaces has led to an explicit construction of all of them and helped to determine many of their geometric properties. Their work has also led to some important questions about related geometric objects. Prasad's recent work with Rapinchuk has deep implications for a particularly important class of geometric structures known as locally symmetric spaces. These spaces arise from symmetric spaces, which as the name suggests, have a lot of symmetries. This work of Prasad and Rapinchuk has introduced a new notion of "weak commensurability" of large subgroups of the group of symmetries and studies its consequences in geometry and group theory.There are still some serious unresolved questions on which they will work.They also plan to write a book on the famous congruence subgroup problem to describe a new unified approach to settle it.