Cohomology of Arithmetic Groups and Galois Representations

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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.