Affine Deligne--Lusztig varieties (ADLV) naturally arise in the study of Shimura varieties and Rapoport--Zink spaces. Their irreducible components provide an interesting class of cycles on the special fiber of Shimura varieties. We prove a conjecture of Miaofen Chen and Xinwen Zhu, which relates the number of irreducible components of ADLV's to a certain weight multiplicity for a representation of the Langlands dual group. Our approach is to count the number of F_q points as q goes to infinity; this boils down to computing a certain twisted orbital integral. After applying techniques from local harmonic analysis, we reduce to computing a particular coefficient of the matrix for the inverse Satake transform. Using an interpretation of this coefficient in terms of a q-analogue of Kostant's partition function, we are able to reduce the problem to the previously known special cases of the conjecture proved by Hamacher--Viehmann and Nie. This is joint work with Yihang Zhu.

An etale p-adic local system on a rigid analytic variety can be regarded as a family of p-adic Galois representations parametrized by the variety, and p-adic Hodge theory has brought many results and applications on such objects, including a p-adic Riemann-Hilbert correspondence by Diao, Lan, Liu and Zhu. I will discuss constancy of a key invariant (generalized Hodge-Tate weights) of general p-adic local systems

We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q when p is at least 5. In the first talk, I will present a generalization of Emerton's local-global compatibility result. In the second talk, I will use this compatibility result to make a patching argument for completed homology in this setting.

A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field. This has applications to the equicharacteristic case of some conjectures on l-independence. I will also discuss the relationship with compatible wild ramification. This is joint work with Qing Lu.

In this series of two talks, we will introduce the recent progress on Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives of arbitrary rank. We will discuss an important technique used in the proof, namely, the arithmetic level raising for unitary groups of even rank. We will also mention other interesting results we obtained during the course of proof. This is based on a joint work with Y. Tian, L. Xiao, W. Zhang, and X. Zhu

We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary strength supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest as a model for the coherent vortex filament structures observed in 3d fluids, these results are the first to give well-posedness (in a certain sense) in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions which are locally approximately self-similar. Joint work with Pierre Germain and Benjamin Harrop-Griffiths.