## 北京国际数学研究中心近期活动: (1) Towards the Locally Analytic Socle for GLn (2) Long-Term Special Program on Arithmetics and Geometry

Year: 2013

CAM-Net Digest, Vol. 10 (2013), Iss. 7 : p. 7

#### Abstract

(1) Towards the Locally Analytic Socle for GLn

Organizer: Ruochuan Liu

Venue: Room 82J04, Jiayibing Building, BICMR

Speaker: Christophe Breuil

Time: 4.29 - 5.11

Week 1: 11am-12am on Wednesday, Thursday and Friday.

Week 2: 11am-12am on Tuesday, Wednesday and Thursday.

General Abstract:

Let n be a positive integer. To any Hodge filtration with distinct Hodge-Tate weights on a diagonalizable generic n-diml representation of the Weil group of Qp, I will associate a semi-simple finite length locally analytic representation of GLn(Qp).

I will show several properties of this representation. For instance, if it has a lattice invariant under GLn(Qp), then the Hodge filtration is weakly admissible. The restriction to diagonalizable representations of the Weil group and to the base field Qp are only for simplicity.

Detailed schedule:

Lecture 1

1.1 Introduction and motivation

1.2 Quick review of locally analytic representations

We give a quick review of locally analytic representations of p-adic analytic groups and related material (mainly due to Schneider and Teitelbaum).

Lecture 2

2.1 Quick review of Verma modules

We review the BGG category O^p_alg and some of its most interesting objects:generalized Verma modules.

2.2 The representations F_P^G(M,pi_P) (after Orlik and Strauch)

We explain important results due to Orlik and Strauch which allow to understand the topological constituents of some locally analytic parabolic inductions.

Lecture 3

3.1 More on the representations F_P^G(M,pi_P)

We state, and sometimes prove, some useful statements about the representations F_P^G(M,pi_P).

3.2 Examples for GL2(Qp) and GL3(Qp)

We give the explicit socle fi ltration of some locally analytic principal series of GL2(Qp) and GL3(Qp).

Lecture 4

4.1 Necessary conditions for integrality

We give necessary conditions for the representations F_P^G(M,pi_P) to admit an invariant lattice (in the case of locally analytic parabolic inductions, these conditions are due to Emerton).

4.2 De finition of the representations pi(D,h)

We defi ne some locally algebraic representations pi(D,h) of GLn(Qp) and recall a (special case of a) conjecture of Schneider and myself predicting when they should admit an invariant lattice.

Lecture 5

5.1 Some preliminaries

We give several preliminaries which will be used afterwards to defi ne some semi-simple locally analytic representations pi(D,h,Fil ) containing pi(D,h).

5.2 De finition of the representations pi(D,h,Fil )

To any Hodge filtration Fil of Hodge-Tate weights h on a diagonalizable Deligne-Fontaine module D, we associate a fi nite length semi-simple locally analytic representation pi(D,h,Fil ) of GLn(Qp).

Lecture 6

6.1 The link with weak admissibility

We prove that if pi(D,h,Fil ) has an invariant lattice, then the Hodge filtration Fil on D is weakly admissible.

6.2 Examples for GL3(Qp) and open questions

We finish with the description of a few representations pi(D,h,Fil ) (with Fil weakly admissible) in the case of GL3(Qp), and with a few questions for possible future developments.

(2) Long-Term Special Program on Arithmetics and Geometry

Speaker: 陈华一, 田青春

Date: From 2013-05-02 To 2013-07-26

Venue: 北京国际数学研究中心 82J12 教室 | 北京国际数学研究中心全 9 教室

Geometric Measure Theory in Banach and Metric Spaces

Time: from 2013-5-2 to 2013-5-24, Thursday and Friday 14:00-16:30

Place: 北京国际数学研究中心全 9 教室

Speaker: Thierry De Pauw (Institut de Mathématiques de Jussieu)

Abstract: The Plateau problem asks for an m-dimensional possibly singular variety minimizing its area among all those bounded by a given boundary cycle in some ambient space X. In case m=2 and X is the 3 dimensional Euclidean space, this problem was solved in the 1930's by J. Douglas and T. Rado independently, using methods of complex analysis that do not generalize for m>2. In the 1960's H. Federer and W.H. Fleming developed their theory of normal and integral currents and solved the problem for arbitrary m when X is Euclidean space or a compact Riemannian manifold. They identified a subclass of those currents in the sense of G. de Rham that is closed under some appropriate weak* convergence and contains oriented manifolds or polyhedra. This is the class of integral currents, suitable for the Calculus of Variations. The theory has been generalized in recent years to the case when the ambient space X is a (possibly infinite dimensional) Banach space or a metric space, and to the case of so-called flat chains with coefficients in an arbitrary complete normed Abelian group G. Including such groups allows for applications to immiscible fluids and irrigation problems, for instance. In case X is an Lipschitz neighborhood retract of some Banach spaces, the homology of flat chains with coefficients in G coincides with the singular homology of X. In this graduate level course I will present the modern theory of flat chains with coefficients in G, with emphasis on the ambient space being an arbitrary Banach space. The theory was initiated by H. Whitney; W.H. Fleming; B. White; L. Ambrosio and B. Kirchheim; and more recently by R. Hardt and the present lecturer. I will cover basic material emphasizing the approximation procedure of flat chains by polyhedral chains, and of Lipschitz maps by PL maps, thereby illustrating the analogy with simplicial homology. Completion with respect to the flat norm adds powerful tools from analysis. As an example, I will treat the theory of slicing, originating in differential topology. The main results are the deformation theorem, the compactness theorem, the slice-nullity theorem, and the rectfiiability theorem.

Transcendence theory and slope method

Time: from 2013-7-4 to 2013-7-26, Thursday and Friday 9:00-11:00

Place: 北京国际数学研究中心 82J12 教室

Speaker: Mathilde Herblot (Université Paris VII)

Abstract: The slope method, invented by J.-B. Bost is a way of rewriting classical proofs of transcendence theory and of understanding them more geometrically. This course will first present some classical results in Diophantine approximation and transcendence theory and their classical proofs. Then we will introduce the basic notions of Arakelov geometry which are needed to present the slope method. We will see how this slope method can be applied to some `evaluation maps' to obtain applications to Diophantine approximation and transcendence theory.

An introduction to Berkovich geometry

Time: from 2013-7-4 to 2013-7-26, Thursday and Friday 13:00-15:00

Place: 北京国际数学研究中心 82J12 教室

Speaker: Amaury Thuillier (Université Lyon I)

Abstract: There are many (complementary) ways to define and study analytic spaces over a non-archimedean field. A specific aspect of Berkovich's approach is to produce topological spaces of classical type: locally compact and locally arcwise connected, with the homotopy type of a polyhedron (at least if one considers the analytification of an algebraic variety). This connection with usual topology has a lot of useful consequences, and it is behind most of the many applications of Berkovich geometry.

Among them: 1) Arakelov geometry, 2) Non-archimedean dynamical systems, 3) Asymptotic properties of linear systems, 4) Weight zero phenomena in algebraic geometry, etc. The lectures should give a flavor of the general theory and a more detailed description of applications 1 and 4.

Introduction to rigid analytic geometry

Time: from 2013-7-4 to 2013-7-26, Thursday and Friday 15:30-17:30

Place: 北京国际数学研究中心 82J12 教室

Speaker: 李彬，张原草 (北京大学)

Abstract: In these lectures we will give an introduction to classical rigid analytic geometry along Tate's approach. We will begin from Tate algebras, introduce affinoid algebras and affinoid spaces, study their subdomains and coverings, proof Tate's acyclicity theorem, and construct general rigid spaces using Grothendieck topology, study the coherent modules on these spaces. We will also discuss the relationship between formal geometry and rigid geometry, i.e. the Raynaud theory, briefly. In final, we will compare the classical rigid spaces and Berkovich spaces.

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###### Journal Article Details

Publisher Name: Global Science Press

Language: Multiple languages

DOI: https://doi.org/2013-CAM-15620

CAM-Net Digest, Vol. 10 (2013), Iss. 7 : p. 7

Published online: 2013-01

AMS Subject Headings: Global Science Press

Copyright: COPYRIGHT: © Global Science Press

Pages: 1