**Title & Abstract**

**Miaofen Chen** (East China Normal University)

**Title:**Fargues-Rapoport conjecture in the non-basiccase

**Abstract:**Rapoport and Zink introduce the p-adic perioddomain (also called the admissible locus) inside the rigid analytic p-adic flagvarieties. Over the admissible locus, there exists a universal crystalline Qp-local system which interpolates a family of crystalline representations. The weakly admissible locus is an approximation of the admissible locus in the sensethat these two spaces have the same classical points. The Fargues-Rapoport conjecture for basic local Shimura datum gives a group theoretic characterization when the admissible locus andthe weakly admissible locus coincide. Inthis talk, we will give a similar characterization for non-basic local Shimuradatum. If time permits, we will also discuss the question about where lives theweakly admissible points outside the admissible locus in general.

**Laurent Fargues** (IMJ-PRG)

**Title:**Minimality of integral models of Rapoport-Zinkspaces

**Abstract:**I will explain the main result of my article"Groupes analytiques rigides p-divisibles II" that says that in somesense integral Rapoport-Zink spaces are minimal models of their generic fiber

**Toby Gee** (Imperial College London)

**Title:**Moduli stacks of (phi,Gamma)-modules

**Abstract:**I will discuss the basic properties of certain moduli stacks of (phi,Gamma)-modules, and some applications of them toquestions about Galois deformations. This is joint work with Matthew Emerton.

**Ulrich Görtz** (University of Duisburg-Essen)

**Title:**Stratifications of affine Deligne-Lusztigvarieties and extremal cases of Rapoport-Zink spaces

**Abstract:**In this talk, I will report on joint work withX. He and M. Rapoport on parahoric Rapoport-Zink spaces with ``minimal'' and ``maximal'' dimension,respectively. Interestingly, in addition to the well-known cases (Lubin-Tateand Drinfeld spaces), there are certain "exotic" cases with similarproperties.

The proofs use,among other ingredients, the properties and classification of the "fullyHodge-Newton decomposable cases" (joint work with X. He and S. Nie).

**Xuhua He** (University of Maryland)

**Title:**Affine Deligne-Lusztig varieties and the irirreducible components, I

**Abstract:**The notion of affine Deligne-Lusztig varietieswas introduced by Rapoport in 90's. It plays an important role in the study ofreduction of Shimura varieties. In 2014, I discovered a relation between the dimensions of affine Deligne-Lusztig varieties and the degrees of class polynomials of affine Hecke algebras. In this talk, I will explain an upgradedversion of the ``dimension=degree'' theorem using motivic counting and explainhow it will be used to study the irreducible components of affine Deligne-Lusztig varieties. The talk is based on a joint work in progress withRong Zhou and Yihang Zhu.

**Eugen Hellmann** (University of Muenster)

**Title:**On the derived category of the Iwahori-Hecke algebra

**Abstract:**Emerton and Helm have proposed that there exitsa family of smooth representations on a space of L-parameters that interpolatesthe local Langlands correspondence. We use this family in order to construct afunctor from the derived category of the Iwahori-Hecke algebra to the derivedcategory of coherent sheaves on a space of L-parameters. We conjecture thatthis functor is fully faithful and satisfies a certain compatibility withparabolic induction. Finally we prove the conjecture in the case of GL_2.

**Naoki Imai** (University of Tokyo)

**Title:**Deligne-Lusztig stack

**Abstract:**We will explain a construction ofDeligne-Lusztig stack which we expect to realize a local automorphic inductionfrom an unramified maximal torus in the formulation of the geometrization ofthe local Langlands correspondence.

Further, we willexplain some conjectures and discuss relations with related topics.

This is a joint work in progress with Laurent Fargues and Ildar Gaisin.

**Chao Li** (Columbia University)

**Title: **On the Kudla-Rapoport conjecture

**Abstract: **The Kudla-Rapoport conjecture predicts an identity between the arithmetic intersection numbers of special cycles on unitary Rapoport-Zink spaces and the derivatives of local representation densities of hermitian forms. We discuss a proof of this conjecture.

This is joint work with Wei Zhang.

**Yifeng Liu** (Yale University)

**Title:**More arithmetic fundamental lemma

**Abstract:**In this talk, we will introduce some newarithmetic fundamental lemma that arise from the study of heights of so-calledFourier-Jacobi cycles in the context of the arithmetic Gan-Gross-Prasad conjecture. We will also discuss its relation with the arithmetic fundamental lemma studied by Rapoport-Terstiege-Zhang and Li-Zhu.

**Eduard Looijenga** (Tsinghua University)

**Title:**On the stable cohomology of the minimal compactification of A_g

**Abstract:**we review what it is known about this stable cohomology, emphasizing themotivic aspects (and treating all characteristics).

**Xu Shen** (Morningside Center of Mathematics, CAS)

**Title:**EKOR strata for the Kisin-Pappas integral modelsof Shimura varieties

**Abstract:**EKOR (Ekedahl-Kottwitz-Oort-Rapoport) stratifications were introduced by Xuhua He and Michael Rapoport to study thegeometry of reduction modulo p of Shimura varieties with parahoric levelstructures, under certain axioms on the integral models. Such a stratification interpolates between the Kottwitz-Rapoport stratification in the Iwahori caseand the Ekedahl-Oort stratification in the hyperspecial case. In this talk, wewill explain some geometric constructions of EKOR strata for the Kisin-Pappas integral models. This is joint work with Chia-Fu Yu and Chao Zhang.

**Sug Woo Shin** (UC Berkeley)

**Title:**Discrete part of the Hecke orbit conjecture

**Abstract:**Oort defined central leaves in the special fiber of Shimura varieties as the locus on which the isomorphism class of the universal p-divisible group is constant (when Shimura varieties parametrize abelian varieties with additional structure). He formulated the Hecke orbit(HO) conjecture asserting that the prime-to-p Hecke orbit of a point is Zariskidense in the central leaf through the point.

By employing geometric methods, C.F. Yu proved irreducibility of leaves for Hilbert modular varieties, and Chai and Oort proved the full HO conjecture for Siegel modular varieties. An. Shankar proved the HO conjecture for Deligne's strange models.In this talk I review the HO conjecture and report on recent progress with A.Kret to prove the irreducibility of leaves (a.k.a. discrete part of the HOconjecture) for Hodge-type Shimura varieties via a different approach using automorphic forms.

**Brian Smithling** (Johns Hopkins University)

**Title:**On Shimura varieties for unitary groups

**Abstract:**Shimura varieties attached to unitary similitude groups are a well-studied class of PEL Shimura varieties. There are also natural
Shimura varieties attached to (honest) unitary groups; these lack a moduli interpretation, butthey have other advantages (e.g., they give rise to interesting cycles of thesort that appear in the arithmetic Gan-Gross-Prasad conjecture). I will describe some variant Shimura varieties which enjoy good properties from both of these classes. This is joint work with M. Rapoport and W.Zhang.

**Yichao Tian** (University of Bonn)

**Title:**An Ihara’s Lemma for even unitary groups andlevel raising of automorphic forms

**Abstract:**Let $E$ be an imaginary quadratic field, and $G$be the unitary group attached to a totally definite hermitian space over $E$ of even rank. Let $p$ be a prime inert in $E$ such that $G$ is non-quasi-split at $p$.

In this talk, Iwill explain an analogue of Ihara’s Lemma for $G$ at $p$, and its applicationsto level raising of automorphic forms on $G$. This is one of the key ingredientin my joint on-going work with Yifeng Liu, Liang Xiao, Wei Zhang and Xinwen Zhion the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives.

**Yihang Zhu** (Columbia University)

**Title:**Affine Deligne-Lusztig varieties and the irirreducible components, II

**Abstract:**The irreducible components of affineDeligne-Lusztig varieties provide interesting algebraic cycles on Shimura varieties. A natural symmetry group J acts on the set of irreducible components, and it is desirable to determine the number of orbits and toidentify the stabilizers. In joint work with Rong Zhou, we prove a formula forthe number of orbits, conjectured by Miaofen Chen and Xinwen Zhu. (This conjecture was also proved by Sian Nie by a different method.) Our method is to estimate the number offinite field points of the variety using the Base Change Fundamental Lemma and the q-analogue of Kostant partitions. We establish an identity that relates thevolumes of the stabilizers to certain weight multiplicities of the Langlands dual group, which implies the Chen-Zhu Conjecture. In joint work in progress with Xuhua He andRong Zhou, we combine this identity with motivic counting, to prove that allthe stabilizers are very special parahoric subgroups. In many cases this already characterizes the stabilizers.