Representation and structure theory of p-adic groups via Bruhat-Tits buildings, and applications to cryptography

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

• 批准号: RGPIN-2015-06294
• 负责人: Nevins, Monica
• 金额: $ 1.02万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=660820
• 关键词: Representation; structure; theory; adic; groups

Algebra is a fundamental domain of the mathematical sciences. Its richness manifests itself in its diversity of applications, from representation theory to communications theory. The diversity of the projects in my research program is a reflection of the universality of the language and tools of algebra.***A first, major part of my research program concerns the representation theory of p-adic groups. Representation theory seeks to understand groups by linearizing them, that is, characterizing their actions on linear vectors spaces over the (well-understood) complex numbers. Here, the groups are matrix groups over the p-adic numbers, depending on a prime p.  Representations of p-adic groups are fundamental to some of the biggest open questions in number theory today, including the so-called Langlands program. My research seeks to deepen our understanding by determining how these representations decompose when they are restricted to a smaller subgroup. For example, if the subgroup is a maximal compact open subgroup, then there are infinitely many finite-dimensional components, which are unknown except in very few cases. One of the exciting open problems in this area is to discover when two apparently unrelated representations of the group share common constituents upon restriction, suggesting a deeper connection between them.***A second aspect of my research relates to the action of a p-adic group on its Lie algebra, which is a vector space over the p-adic numbers. This action decomposes the Lie algebra into orbits, of which the finitely-many nilpotent orbits are the most interesting. DeBacker has recently proven a theoretical classification of these orbits in terms of a combinatorial and geometric object closely related to the p-adic group, called the Bruhat-Tits building. The important open problem which I am working on is to realize this classification concretely, by producing representatives of each orbit (for a fixed group), and deriving the dimension and attributes of the various nilpotent orbits, including their proximity to each other, from the DeBacker parameters. This has myriad applications; for example, the Harish-Chandra-Howe character formula parametrizes (characters of) representations by what are essentially weighted sums over a set of nilpotent orbits.***Finally, a third aspect of my research, arising in part from my expertise with groups and their associated lattices, is mathematical cryptography. Cryptography is the art and science of obfuscating messages.  Our research in this area centers on analyzing and extending cryptographic protocols based on algebraic systems---such as NTRU, elliptic curve cryptography and homomorphic encryption systems. The goal is to gain a deeper understanding of a cryptographic algorithm, and thus of its potential unintended loopholes, by learning how it changes as a function of the algebraic object on which it is based.**

代数是数学科学的一个基本领域。它的丰富性体现在其应用的多样性上，从表征理论到传播理论。我研究计划中项目的多样性反映了代数语言和工具的普遍性。*我研究计划的第一个，也是主要部分，涉及p-进群的表示理论。表示理论试图通过线性化群来理解群，即刻画群在(众所周知的)复数上的线性向量空间上的作用。在这里，群是p-进制数上的矩阵群，取决于素数p。p-进制群的表示是当今数论中一些最大的公开问题的基础，包括所谓的朗兰兹程序。我的研究试图通过确定这些表征被限制在一个较小的子群中时如何分解来加深我们的理解。例如，如果子群是极大紧开子群，则存在无限多个有限维分支，这些分支除了极少数情况外是未知的。这一领域最令人兴奋的公开问题之一是发现群的两个明显不相关的表示在限制时共享公共成分，这暗示了它们之间更深层次的联系。*我研究的第二个方面涉及p-进群在其李代数上的作用，李代数是p-进数上的一个向量空间。这个作用将李代数分解成轨道，其中有限多个幂零轨道是最有趣的。DeBacker最近证明了一种关于这些轨道的理论分类，其依据是一个与p-ady群密切相关的组合和几何对象，称为Bruhat-Tits建筑。我正在研究的重要公开问题是具体地实现这种分类，方法是产生每个轨道的表示(对于固定的群)，并从DeBacker参数推导出各种幂零轨道的维度和属性，包括它们彼此之间的接近程度。这有无数的应用；例如，Harish-Chandra-Howe特征标公式通过本质上是一组幂零轨道上的加权和的表示来参数化(特征)表示。*最后，我的研究的第三个方面，部分源于我对群及其相关晶格的专业知识，是数学密码学。密码学是混淆信息的艺术和科学。我们在这一领域的研究集中在分析和扩展基于代数系统的密码协议-例如NTRU、椭圆曲线密码和同态加密系统。其目标是通过了解密码算法如何随其所基于的代数对象的函数而变化，从而更深入地了解密码算法以及其潜在的意外漏洞。**