Talk

Smooth representations of p-adic groups are of interest as they appear on one side of the local Langlands correspondence. Over the complex numbers, they decompose into explicit blocks, which are in turn known in many cases to be equivalent to modules over explicit Hecke algebras. Over an algebraically closed field of characteristic l not equal to p, neither the decomposition nor the algebras are known. For GLn, the same block decomposition holds, but the Hecke algebra is now too small to describe the whole block.

In topology, cohomology is an invariant we can assign to spaces. In algebra, there are also cohomology theories for various algebraic structures, such as group cohomology, Lie algebra cohomology, and André-Quillen cohomology of commutative rings. Quillen cohomology provides a cohomology theory for any algebraic structure. It has been studied notably for divided power algebras and restricted Lie algebras, both of which are instances of divided power algebras over an operad: the commutative and Lie operad respectively.

In this follow-up to yesterday's Topology Seminar, we will look more closely at some applications, some homotopical machinery such a weak adjunctions and weak (co)tensoring, and examples of weak monoidal Quillen pairs.

Most results on the Hasse principle for surfaces concern conic bundles, that is, families of conics parameterised by the projective line. Beyond this case, one needs to work with families of torsors for abelian varieties. Assuming finiteness of relevant Tate-Shafarevich groups, we use a method originally due to Swinnerton-Dyer to prove the Hasse principle for certain Kummer surfaces. As a somewhat unexpected application, we obtain the Hasse principle for sufficiently general quartic del Pezzo surfaces. This is joint work with Adam Morgan.

Based on the recent result of Peter Koymans and Adam Morgan, who
proved that there are infinitely many nonisomorphic hyperelliptic curves of
any genus whose Jacobian has rank 1 over any number field, we prove the
same result for rank 2; however, our curves do not have simple Jacobians,
instead they factor into two rank 1 Jacobians. We also discuss results
about higher rank Jacobians of small genus curves. This is joint work with
Sun Woo Park.

We introduce Kazhdan’s property (T) of groups and explore its main features. After the classical definition and basic examples, we examine its connections to finite generation and presentability. We then outline some proof strategies and finish with several applications, including constructions of expanders and generating random elements from finite groups.