A Langlands stack (under construction)

Updated Nov.12,2014: A new version of Arinkin-Gaitsgory on reformulation of the GLC

Leisure readings
Vergne  All what I wanted to know about Langlands program and was afraid to ask
Gelbart  An elementary introduction to the Langlands program
Frenkel  Recent Advances in the Langlands Program

I.Papers & Books

0.Tools
Gaitsgory  Geometric representation theory
Kac   Infinite dimensional Lie algebras
Frenkel, Ben-Zvi  Vertex algebras and algebraic curves
Beilinson, Drinfel’d   Chiral algebras
Arthur  An introduction to the trace formula

1.Langlands Program for number fields
1.1 Local Langlands for \Bbb C and \Bbb R

1.2 Local Langlands for p-adic fields
Harris, Taylor  The geometry and cohomology of some simple Shimura varieties
Henniart  Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

1.3 Fundamental Lemma
Waldspurger  Endoscopie et changement de caractéristique
Ngô  Fibration de Hitchin et endoscopie
Laumon, Ngô  Le lemme fondamental pour les groupes unitaires
Waldspurger  L’endoscopie tordue n’est pas si tordue
Ngô  Le lemme fondamental pour les algèbres de Lie

2.Langlands Program for function fields
Drinfel’d  Elliptic modules I,II
Laumon  Cohomology of Drinfeld modular varieties I,II
Lafforgue  Chtoucas de Drinfeld et applications
Lafforgue  Chtoucas de Drinfeld, formule des traces d’Arthur-Selberg et correspondance de Langlands

3.Geometric Langlands Program
Arinkin, Gaitsgory  Singular support of coherent sheaves, and the geometric Langlands conjecture
Witten  Quantum field theory, Grassmannians and algebraic curves
Kapustin, Witten  Electric-Magnetic Duality And The Geometric Langlands Program
Frenkel, Gaitsgory, Vilonen  On the geometric Langlands conjecture
Frenkel  Lectures on the Langlands Program and Conformal Field Theory
Donagi, Pantev  Lectures on the geometric Langlands conjecture and non-abelian Hodge theory

II.On-line Resources
The Work of Robert Langlands: the official on-line archive for Robert Langlands’ work
Northwestern Geometric Langlands Program maintained by Vilonen, Frenkel, Gaitsgory and Goresky
Chicago Geometric Langlands Seminar
Note by Dennis Gaitsgory
Homepage of Edward Frenkel
GRASP run by David Ben-Zvi

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The existence problem of Kähler-Einstein metrics: a grocery

Click here for a discussion on the problem itself.

I.Books on the classical theory

Aubin  Some nonlinear problems in Riemannian geometry

Futaki  Kähler-Einstein Metrics and integral invariants

Siu  Lectures on Hermitian-Einstein Metric for Stable Bundles and Kähler-Einstein Metrics

Tian  Canonical metrics in Kähler geometry

II.Papers

Calabi conjectures for c_1<0 and c_1=0 

Yau  Calabi’s conjecture and some new results in algebraic geometry

Yau  On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I

Lie-theoretical obstructions for c_1>0

Matsushima  Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété káhlérienne

Futaki  An obstruction to the existence of Einstein-Kähler metric

K-energy and Uniqueness for c_1>0

Bando, Mabuchi  Uniqueness of Einstein-Kähler metrics modulo connected group actions

Concrete constructions

Siu  The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and a suitable finite symmetry group

Nadel  Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature

c_1>0 for complex surfaces

Tian, Yau  Kähler-Einstein metrics on complex surfaces with c_1>0

Tian  On Calabi’s conjecture for complex surfaces with positive first Chern class

Hitchin-Kobayashi correspondence

Uhlenbeck, Yau  On the existence of Hermitian-Yang-Mills connections in stable vector bundles

Donaldson  Infinite determinants, stable bundles and curvature

Differential-geometric K-stability

Ding, Tian  Kähler-Einstein metrics and the generalized Futaki invariant

Tian  Kähler-Einstein metrics with positive scalar curvature

Tian  Kähler-Einstein metrics on algebraic manifolds

Cheeger, Coding, Tian  On the singularities of spaces with bounded Ricci curvature

Donaldson  Kähler metrics with cone singularities along a divisor

Berman  A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics

Li, Sun  Conical Kähler-Einstein metric revisited

Algebro-geometric K-stability

Donaldson  Scalar curvature and stability of toric varieties

Donaldson  Lower bounds on the Calabi functional

Stoppa  K-stability of constant scalar curvature Kähler manifolds

Donaldson  Stability, birational transformations and the Kähler-Einstein problem

Sun  Note on K-stability of pairs

Li, Xu  Special test configurations and K-stability of Fano varieties

Donaldson, Sun  Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry

Announced proofs for the Yau-Tian-Donaldson conjecture

Tian  K-stability and Kähler-Einstein metrics

Chen, Donaldson, Sun  Kähler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities

Chen, Donaldson, Sun  Kähler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 \pi

Chen, Donaldson, Sun  Kähler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2 \pi and completion of the main proof

My list of unsolved problems

数学上有很多未解决的问题。在Hilbert看来,这是整个学科兴旺发达的证据。

(1)Hilbert的23个问题中包含了精确的“问题”和模糊的“纲领”。第5问题(Lie群)、第9问题第12问题(代数数论)以及第16问题(代数几何)等至今仍是重要的研究课题。

(2)当下最重要的未知之谜或许当属千禧七问题。除Poincaré猜想外,Riemann猜想P vs NPHodge猜想Birch和Swinnerton-Dyer猜想Navier-Stokes存在性与光滑性Yang-Mills存在性与质量间隙等6个问题均未获解决。

(3)不少数学家试图模仿Hilbert,例如在世纪之交Smale提出了18个有待解决的问题

Smale  Mathematical problems for the next century

Erdős和Arnold等数学家以大量原创性的问题而知名。有兴趣者不妨参考以下著作

Erdős  Old and New Problems and Results in Combinatorial Number Theory

(以及Wiki索引页Erdős conjecture)

Arnold  Arnold’s Problems

(4)专门领域的清单是很常见的。除了Erdős清单(组合数论)和Arnold清单(主要涉及经典力学,微分拓扑和动力系统),著名的例子还包括丘成桐的微分几何问题集。我所知的最新版本是

Yau  Review of Geometry and Analysis

某些分支(例如数论)中有大量未解决的问题。这类似于星系:围绕一二中心(Riemann猜想、Langlands纲领)分布着无数同等重要(或者同等不重要)的小问题,参见

Guy  Unsolved Problems in Number Theory

(5)有些开问题(open problem)非常容易叙述和理解:是否每个大于2的偶数都能表示成2个素数之和(Goldbach猜想)?是否有无穷多对孪生素数?是否每个有限群都能实现为\Bbb Q的某个Galois扩张的Galois群(Galois问题)?S^6上是否有复结构?答案是:没有。)S^4上有多少个微分结构?是否每个截面曲率处处为正的偶数维紧Riemann流形都有正的Euler示性数(Hopf猜想)?

一个可能鼓舞不少“民科”的事实是:有时候甚至连这些问题的解决都是“初等”的,例如\zeta(3)的无理性(Apéry定理)和Bieberbach猜想(de Branges定理)。

(6)Wiki上有一份偏向数论和组合方向的清单

(7)我试图列出自己的一份清单,标准如下:

  • 在我看来这个问题必须是“有趣”的,如果“重要”,则更好;
  • 不应该过分冷僻,也不应该人尽皆知。这排除了(1)(2)(3)(4)中的某些问题。当然,我保留讨论例如Hodge猜想或者Langlands纲领的权利——它们实在太重要了;
  • 或涉及到较摩登的概念,或有更深广的背景:总之,必须复杂到“仅仅为了说清楚这个问题就值得专门写一篇甚至一系列post”,这排除了(5)中的某些问题,尤其是某些“孤立”的数论问题;

(8)这份清单仍有待补充。相关的post也会慢慢写好。

拓扑:

代数:

分析:

Riemann几何:

Kähler几何,辛几何:

数论,代数几何,算术几何:

Lie群,代数群和表示论: