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**Survey Talk 1**

**Speaker: **Dong Uk Lee (IBS)

**Title: **Introduction to Langlands-Rapoport conjecture.

**Abstract:
**The Langlands-Rapoport conjecture, which was recently confirmed by M. Kisin, provides a group-theoretic description of the set of the points over finite fields of a (good) reduction of Shimura variety. It is an important big step in the Langlands program of expressing the Hasse-Weil L-function of a Shimura variety in terms of automorphic L-functions. In this survey, we introduce the conjecture and illustrate it in the modular curve case.

**Survey Talk 2
**

**Speaker:**Wansu Kim (Cambridge)

**Title:**Shimura varieties and their local analogues -- informal overview and examples

**Abstract:**

The (extremely ambitious) aim of this survey talk is to motivate "Rapoport-Zink spaces" (in the most general sense) and illustrate what we expect to hold in an ideal world we imagine. Most of the constructions, theorems, and conjectures were motivated by the studies of Shimura varieties, and often both theory have strong analogies, which we try to explain in this survey talk.

**Survey Talk 3**

**Speaker:**Sug Woo Shin (MIT/KIAS)

**Title:**Adic spaces and perfectoid spaces

**Abstract:**

Recently the theory of adic spaces and perfectoid spaces has become a powerful tool in arithmetic geometry and number theory. This expository talk aims to familiarize the audience with the basic formalism about those spaces, often in analogy with the theory of schemes, with a view towards understanding important arithmetic examples such as Shimura varieties and Rapoport-Zink spaces at infinite level.

**1. Speaker:**Przemyslaw Chojecki (Jussieu)

**Title:**Towards p-adic non-abelian Lubin-Tate theory

**Abstract:**

The goal of this mini-course is to show that the p-adic local Langlands correspondence for GL_2(Q_p) appears in the etale cohomology of the Lubin-Tate tower at infinity. I deliver three lectures:

1. Perfectoid modular curves (after Peter Scholze)

2. Overview of the p-adic Langlands correspondence for GL_2(Q_p)

3.P-adic non-abelian Lubin-Tate theory for GL_2(Q_p)

**2. Speaker:**Wansu Kim (Cambridge)

**Title:**Rapoport-Zink spaces of Hodge type

**Abstact:**

The aim of the lectures is to outline the idea behind my construction of a p-adic local analogue of Hodge-type Shimura varieties (or Hodge-type Rapoport-Zink spaces) under the "good reduction hypothesis".

Here is the rough plan for the 3 lectures:

1. Preliminaries:

2. Review of classical Rapoport-Zink spaces (of EL and PEL type)

3. Rapoport-Zink spaces of Hodge type (in the good reduction case)

**3. Speaker:**Keerthi Madapusi Pera (Harvard)

**Title:**Integral models for Shimura varieties of Hodge type and their mod-p fibers

**Abstract:**Our aim is to give an overview of the construction by Kisin of integral models for Shimura varieties of Hodge type, as well as his proof of

the Langlands-Rapoport conjecture for ther mod-p fibers.

**4. Speaker:** Yoichi Mieda (Kyoto)**
Title:** Perfectoid spaces and Rapoport-Zink space at infinite level

**Abstract:**Motivatied mainly by the global and local Langlands correspondence, we are interested in the I-adic etale cohomology of Shimura varieties (e.g. the modular curve) and Rapoport-Zink spaces (e.g. the Lubin-Tate space). They are towers (i.e. projective systems indexed bt levels) of algebraic varieties and rigid analytic spaces, respectively. Sometimes it is crucial to consider their "limits", namely, the spaces at infinite level. For example, the Hecke operators, being algebraic correspondences at finite level, become a group action at infinite level. More serious example is the Faltings-Fargues isomorphism between the Lubin-Tate space and the Drinfeld space at infinite level. This isomorphism does not descend to a finite level, so passing to infinite level is essential.