Cohomology of Arithmetic Groups and Galois Representations

首席调查员(PI)与不同的合作者一起研究关于GL(n，Z)的有限指标子群的有理数和上同调的Galois群的表示的数论领域。它们之间的联系是由Hecke算子对上同调的作用和Frobenius元在Galois表示中的像给出的。PI探索了“ADPS”猜想，其中上同调和伽罗瓦表示都是mod-p值的。他提出了一种在某些三维情况下证明这一猜想的方法，在某些四维情况下测试它，并独立地验证它对丢番图问题的一些结果。PI提出了一个新的猜想，它将mod-p对象与自同构表示联系起来，并提出了一种在三维情况下证明该猜想的方法。最后，Pi和他的一位同事继续研究p-adic族的自同构上同调。一方面，对于某些类，例如GL(3)上的那些不是从GL(2)上提升的类，他们考虑p-进刚性问题。另一方面，对于可变形类，他们研究二变量p进L函数。求解代数方程组的能力一直是现代科学和工程的中心。它也一直是数学中的主要话题之一，受这些应用和它内在的审美吸引力的驱动。当方程具有整数系数时，它们的研究就成为数论的一部分。素数理论(除1以外没有更小因子的数，如2，3，5和7)是这里的核心。在过去的50年里，这些理论的应用已经成为密码学和通信理论的关键。首席研究员研究了关于多项式方程组解的精细结构、它们所展示的对称性以及它们与各种有趣的几何和拓扑对象的(通常令人惊讶的)关系的微妙问题。这些关系是由方程关于不同质数的行为来调节的。对这些关系有许多猜测的解释，PI对它们进行研究，在某些“简单”的情况下证明它们，在更复杂的情况下用计算机验证它们。他还研究了将这些结构的整族与固定素数进行比较的现象。

The Principal Investigator (PI), with various collaborators, studies anarea of number theory concerned with representations of Galois groups overthe rationals and cohomology of finite-index subgroups of GL(n,Z). Thelink between these is given by the action of Hecke operators on thecohomology and the image of Frobenius elements in the Galoisrepresentation. The PI explores the "ADPS" conjecture, where thecohomology and Galois representation are both mod-p valued. He outlinesan approach for proving this conjecture in certain 3-dimensional cases,testing it in certain 4-dimensional cases, and independently verifyingsome of its consequences for Diophantine problems. The PI states a newconjecture that links the mod-p objects with automorphic representationsand suggests an approach for proving this conjecture in the 3-dimensionalcase. Finally, the PI and a colleague continue their study of p-adicfamilies of automorphic cohomology. On the one hand, for certain classes,e.g those on GL(3) not lifted from GL(2), they consider questions ofp-adic rigidity. On the other hand, for deformable classes, theyinvestigate 2-variable p-adic L-functions.The ability to solve systems of algebraic equations has been central tomodern science and engineering. It has also always been one of the maintopics in mathematics, driven both by these applications and by itsintrinsic aesthetic appeal. When the equations have whole-numbercoefficients, their study becomes part of number theory. The theory ofprime numbers (numbers without smaller divisors except 1, such as 2,3,5and 7) is central here. In the last 50 years, applications of thesetheories have become crucial to cryptography and communications theory.The Principal Investigator studies delicate questions concerning the finestructure of sets of solutions to systems of polynomial equations, thesymmetries they exhibit, and their (often surprising) relationship tovarious interesting geometric and topological objects. Theserelationships are mediated by how the equations behave with respect to thevarious prime numbers. There are a number of conjectural explanations ofthese relationships, and the PI studies them, proving them in certain"easy" cases and verifying them by computer in more complicated cases. Healso investigates phenomena which compare whole families of thesestructures with respect to a fixed prime number.