Chomology of Arithmetic Groups and Galois Representations

主要研究者与其他合作者一起，研究了算术群作为Heckeoperators代数上的模的上同调。他考虑了连接(1)所有heckeoperator的同余群的模p上同调类与(2)代数数的伽罗瓦（对称）群的模p表示的猜想。这些猜想相当于说，应该存在连接这两类物体的互惠定律（通常是非阿贝尔定律）。这些猜想经过了计算检验，从而导致了它们的进一步完善。首席研究员概述了在一些特殊情况下证明它们的方法。另一组计算机计算探讨了上同调型的自同构表示的存在性，这些自同构表示不是从较小的秩群中提取的。这些与朗兰兹型的互易律推测有关。最后，研究者和他的同事发展了一个关于同余群的不一定是p-普通上同类的p-adicfamilies的理论。他们利用这一点来研究非提升自同构表示是否为“p-基刚性”的问题。代数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世纪早期发明解析几何以来，求解代数方程的能力一直是数学、科学和工程的核心。在过去的50年里，数论在密码学领域中出现了一系列新的应用。首席研究员研究涉及某些代数方程组的精细结构和解的性质的微妙问题。虽然人们经常可以用计算机得到近似解，但关于精确解的存在和形式，有许多非常微妙和困难的问题。有很多猜测可以解释发生了什么，首席研究员提出了其中的一些。他和他的同事们研究这些猜想，在某些情况下通过大量的计算机计算来验证它们，并在特殊情况下证明它们。这项研究的核心是与几何物体（“上同调类”）的惊人联系，这些物体可以在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.