The Langlands Programme - p-adic and geometric methods.

数论的一个中心课题是研究丢番图方程：具有整数系数和变量的多项式方程。就像费马大定理一样，这些问题可能是基本的，但众所周知很难解决，成功往往依赖于与看似无关的数学领域的联系。朗兰兹纲领提供了这样一种联系，它设想了数论和表示论之间的深层结构关系。在这种关系的表示论方面，分析对象具有显著的对称性质，称为自守形式，经典模形式是一个普遍存在的例子。尽管最近有重大进展，朗兰兹纲领仍然在很大程度上是理论性的。许多进展背后的一个关键思想是p-adic变异的概念，将研究对象置于全等家庭中。这一概念在怀尔斯证明费马大定理的过程中发挥了至关重要的作用，并继续导致朗兰兹纲领所预测的关系的建立、它们的数论后果和新的研究方向的突破。几何思想和直觉也对最近的进展产生了重大影响，例如Laumon和Ngo对基本引理的证明。拟议的研究使用p-adic和几何方法来开发朗兰兹纲领及其应用的关键方面的新方法，包括它在不同数学领域之间建立的强大联系。

A central topic in number theory is the study of Diophantine equations: polynomial equations with integer coefficients and variables. Like Fermat's Last Theorem, the problems can be elementary to state, but notoriously difficult to solve, with success often relying on connections with seemingly unrelated areas of mathematics. One such connection is provided by the Langlands Programme, which envisions a deep structural relationship between number theory and representation theory. On the representation-theoretic side of this relationship are analytic objects with remarkable symmetry properties, called automorphic forms, classical modular forms being a ubiquitous example.Despite recent major advances, the Langlands Programme remains largely conjectural. A key idea underlying much of the progress has been the notion of p-adic variation, placing the objects of study in congruent families. This notion played a crucial role in Wiles' proof of Fermat's Last Theorem, and continues to lead to breakthroughs in establishing the relations predicted by the Langlands Programme, their number-theoretic consequences, and new research directions. Geometric ideas and intuition have also been a major influence in recent advances, such as the proof of the Fundamental Lemma by Laumon and Ngo.The proposed research uses p-adic and geometric methods to develop new approaches to key aspects of the Langlands Programme and its applications, including the powerful links it establishes among different areas of mathematics.