Chomology of Arithmetic Groups and Galois Representations

主要研究人员与多位合作者一起，将算术群的上同调作为Hecke算子代数上的模来研究。他考虑了这样的猜想：(1)mod-p上同调类同余群是所有Hecke算子的本征类；(2)Galois(对称)群的mod-p表示。这些猜想相当于说，应该存在连接这两类物体的互易律(通常是非阿贝尔定律)。对这些猜想进行了计算检验，从而进一步完善了这些猜想。首席调查员概述了在一些特殊案件中证明它们的方法。另一组计算机计算探索不是从较小的秩群提升的上同调型的自同构表示的存在。最后，作者和一位同事发展了同余群的p-adic族理论，其上同调类不一定是p-普通上同调类。他们利用这一点来研究非提升的自同构表示是否“p-adad刚性”的问题。代数论研究系数为整数的多项式函数和方程的性质。自从笛卡尔在17世纪初发明解析几何以来，S就一直是数学、科学和工程的核心。在过去的50年里，数论在密码学领域开辟了一系列新的应用。首席调查者研究涉及某些代数方程组解的精细结构和性质的微妙问题。虽然人们经常可以使用计算机获得近似解，但关于精确解的存在和形式存在许多非常微妙和困难的问题。桌子上有许多猜测来解释正在发生的事情，首席调查员贡献了其中的一些。他和他的同事们通过在某些情况下通过大量的计算机计算来验证这些猜想，在特殊情况下通过证明它们来研究这些猜想。这项研究的核心是与几何对象(“上同调类”)的惊人联系，这些几何对象可以在n维晶格空间中构造。

The principal investigator, together with various collaborators, studiesthe cohomology of arithmetic groups as modules over an algebra of Heckeoperators. He considers conjectures connecting (1) mod-p cohomologyclasses of congruence groups which are eigenclasses for all the Heckeoperators with (2) mod-p representations of the Galois (symmetry) groupof the algebraic numbers. These conjectures amount to saying there shouldexist reciprocity laws (usually non-abelian) connecting the two types ofobjects. The conjectures are tested computationally, which leads tofurther refinements of them. The principal investigator outlines anapproach for proving them in some special cases. Another set of computercalculations probe for the existence of automorphic representations ofcohomological type which are not lifted from smaller rank groups. Theseare conjecturally connected with reciprocity laws of Langlands type.Finally, the investigator and a colleague develop a theory of p-adicfamilies of not necessarily p-ordinary cohomology classes for congruencegroups. They use this to study the question whether non-liftedautomorphic representations are "p-adically rigid". Algebraic number theory studies the properties of polynomialfunctions and equations whose coefficients are whole numbers. Ever sinceDescartes invented analytic geometry in the early 1600's, the ability tosolve algebraic equations has been central to mathematics, science andengineering. In the last 50 years a new set of applications of numbertheory has opened up in the field of cryptography. The principalinvestigator studies delicate questions involving the fine structure andproperties of solutions to certain systems of algebraic equations.Although one can often obtain approximate solutions using computers, thereare many very subtle and difficult questions concerning the existence andform of the exact solutions. There are a number of conjectures on thetable to explain what is going on, and the principal investigator hascontributed some of them. He and his colleagues study these conjecturesby verifying them in some cases by extensive computer calculations, and byproving them in special cases. At the core of this research is thesurprising connection with geometric objects ("cohomology classes") thatcan be constructed in spaces of n-dimensional crystal lattices.