Chomology of Arithmetic Groups and Galois Representations

主要研究者，连同各种合作者，studiesthe上同调的算术群作为模代数的赫克算子。 他认为连接（1）mod-p cohomologyclasses的同余组，这是本征类的所有Heckeoperators与（2）mod-p表示的伽罗瓦（对称）群的代数数。 这些理论相当于说，应该存在着连接这两种对象的互惠律（通常是非阿贝尔的）。 对这些结构进行了计算测试，从而进一步改进了它们。 主要研究者概述了在某些特殊情况下证明它们的方法。 另一组计算机计算探测存在的自守表示的上同调型，这不是解除较小的秩组。 最后，研究者和一个同事发展了一个理论的p-adicfamilies不一定p-普通上同调类的同余组。 他们利用这一点来研究非提升自守表示是否是“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世纪初发明解析几何以来，解代数方程的能力一直是数学、科学和工程学的核心。 在过去的50年里，数论在密码学领域的一系列新的应用已经展开。 主要研究涉及精细结构和某些代数方程组解的性质的微妙问题。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性和形式，还有许多非常微妙和困难的问题。桌子上有许多图表来解释正在发生的事情，主要研究者贡献了其中的一些。 他和他的同事们研究这些公式，在某些情况下通过广泛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可以在n维晶格空间中构造的几何对象（“上同调类”）的惊人联系。

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.