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构造了所有的算术“伪”射影空间，并发现了它们的一些几何性质。他们的工作引出了许多与这些空间有关的有趣问题，也导致了一些具有小几何不变量的奇异曲面的问题。普拉萨德建议解决这些问题。他还一直在与Andrei Rapinchuk合作，寻找有限体积的局部对称空间由其闭测地线的长度集或谱决定的程度。他们的工作导致他们定义了“大”(Zariski稠密或算术)子群之间的一种新的关系，他们称之为“弱可公度性”。他们证明了单李群的两个算术子群的弱可公约性，其动态图不具有对称性，意味着它们是可公度的。他们现在正在调查动态金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和Schanual提出的一个被广泛相信的猜想，证明了如果一个绝对单实李群由两个算术子群构成的对称空间的商具有相同的闭测地线长度集，或者具有相同的谱，则这两个算术子群是弱可公度的。因此，他们关于弱可公度算术子群的结果是可以使用的。Prasad建议对复半单李群产生的局部对称空间得到类似的结果。在一个完全不同的方向上，Prasad把一个自然的Levi-子群与一个可约p-ady群的给定的不可约可容许表示联系起来。他将研究这个子群在表象理论中扮演的角色。Prasad与Rapinchuk正在进行合作，以简化、统一和完善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Prasad和Sai-Kee Yeung在某些有趣的几何对象(称为算术伪射影空间)上的工作导致了对所有这些对象的显式构造，并帮助确定了它们的许多几何性质。他们的工作还引出了一些关于相关几何对象的重要问题。Prasad最近与Rapinchuk的工作对一类特别重要的几何结构--局部对称空间--有着深刻的影响。这些空间源于对称空间，顾名思义，对称空间有很多对称性。Prasad和Rapinchuk的这项工作引入了对称群的大子群的“弱可公度性”的新概念，并研究了它在几何和群论中的结果。他们还计划写一本关于著名的同余子群问题的书来描述解决这个问题的新的统一方法。

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.