P-adic Representation Theory and Geometry of the Lubin-Tate Tower

鲁宾-泰特塔的P进表示理论和几何

基本信息

项目摘要

The principal investigator studies two outstanding problems in p-adic representation theory. The first problem is analyzing a conjectural geometric construction of supercuspidal representations of p-adic groups proposed by Lusztig in 1979. Lusztig's construction can be viewed as a special case of automorphic induction. While automorphic induction has been constructed in many special cases, the existing approaches are quite complicated and often rely on global methods. Lusztig's construction is much more elegant, but the questions of formalizing it and comparing it to the more classical constructions of p-adic representation theory remain completely open. It is expected that the results obtained by the principal investigator will shed light on the geometry that underlies the known cases of automorphic induction. The second problem was formulated by M. Harris in 2002. It asks for a construction of Bushnell-Kutzko types for the general linear group of a local field K in the cohomology of suitable analytic subspaces of the Lubin-Tate tower of K. The principal investigator studies a family of open affinoids in the Lubin-Tate tower whose cohomology is expected to realize various special cases of the local Langlands and Jacquet-Langlands correspondences (which will partially answer Harris's question). The relevant cohomology computations have much in common with the examples of p-adic Lusztig induction that have so far been understood.The Langlands Program has dominated much of research in algebra during the last 40 years. It has connections to some of the most prominent results in number theory and other areas of mathematics, such as Fermat's Last Theorem. The principal investigator works in a branch of this field known as the local Langlands program. It is concerned with the representation theory of the so-called p-adic groups, and the main driving force is the search for a general proof of the local Langlands correspondence. Various special cases of this correspondence have been obtained by Henniart, Harris, Taylor and many other mathematicians. However, most of the existing proofs are not explicit and do not provide sufficient information for the desirable applications of the local Langlands correspondence. The principal investigator uses methods of geometric representation theory for unipotent groups, developed in his previous works, to give new explicit constructions of the local Langlands correspondence, and to simplify and clarify the existing ones. One of the main tools is a conjectural geometric construction of representations of p-adic groups formulated by Lusztig in 1979. Until recently this construction has remained relatively unknown because it was not clear how to compare it to the more classical constructions. The principal investigator developed general techniques for analyzing this construction, and is currently using it to shed light on the geometry behind the local Langlands correspondence.
主要研究者研究了p进表示理论中的两个突出问题。第一个问题是分析Lusztig在1979年提出的p进群的超尖表示的猜想几何构造。Lusztig的构造可以看作是自同构归纳法的一个特例。虽然自同构归纳法已经在许多特殊情况下得到了应用,但现有的方法非常复杂,而且往往依赖于全局方法。Lusztig的构造要优雅得多,但将其形式化并将其与更经典的p进表示理论结构进行比较的问题仍然是完全开放的。预计由首席研究员获得的结果将揭示已知自同构归纳案例的几何基础。第二个问题是哈里斯先生在2002年提出的。它要求在K的Lubin-Tate塔的合适解析子空间的上同调中,局部域K的一般线性群的Bushnell-Kutzko类型的构造。主要研究者研究了Lubin-Tate塔中的一个开放仿射族,其上同调有望实现局部Langlands和Jacquet-Langlands对应的各种特殊情况(这将部分回答Harris的问题)。相关的上同调计算与目前已知的p进Lusztig归纳的例子有很多共同之处。在过去的40年里,朗兰兹纲领在代数研究中占据主导地位。它与数论和其他数学领域的一些最突出的结果有关,比如费马大定理。首席研究员在这个领域的一个分支工作,被称为当地朗兰兹项目。它关注的是所谓p进群的表征理论,其主要推动力是寻找局部朗兰兹对应的一般证明。亨尼亚特、哈里斯、泰勒和许多其他数学家已经得到了这种对应关系的各种特殊情况。然而,现有的大多数证明是不明确的,不能为局部朗兰兹对应的理想应用提供足够的信息。主要研究者使用在他以前的工作中发展起来的单能群的几何表示理论方法,给出了局部朗兰兹对应的新的明确结构,并简化和澄清了现有的结构。其中一个主要的工具是由Lusztig在1979年提出的p进群表示的猜想几何构造。直到最近,这种结构仍然相对不为人所知,因为不清楚如何将其与更经典的结构进行比较。首席研究员开发了分析这种结构的一般技术,目前正在使用它来阐明局部朗兰兹对应背后的几何结构。

项目成果

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Sarah Kitchen其他文献

Sphere-Graph: A Compact 3D Topological Map for Robotic Navigation and Segmentation of Complex Environments
Sphere-Graph:用于机器人导航和复杂环境分割的紧凑 3D 拓扑图
Optimizing Heterogeneous Platform Allocation Using Reinforcement Learning
使用强化学习优化异构平台分配
  • DOI:
    10.1109/aero55745.2023.10115631
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Xavier Brumwell;Sarah Kitchen;P. Zulch
  • 通讯作者:
    P. Zulch

Sarah Kitchen的其他文献

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{{ truncateString('Sarah Kitchen', 18)}}的其他基金

P-adic Representation Theory and Geometry of the Lubin-Tate Tower
鲁宾-泰特塔的P进表示理论和几何
  • 批准号:
    1748706
  • 财政年份:
    2017
  • 资助金额:
    $ 13.8万
  • 项目类别:
    Standard Grant
Conference on Advances in Geometric Representation Theory
几何表示理论进展会议
  • 批准号:
    1600220
  • 财政年份:
    2016
  • 资助金额:
    $ 13.8万
  • 项目类别:
    Standard Grant

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通过 Bruhat-Tits 建筑物的 p-adic 群的表示和结构理论,以及在密码学中的应用
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