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.**

代数是数学科学的基本领域。它的丰富性在其应用多样性中表现出来，从表示理论到通信理论。我的研究计划中项目的多样性反映了代数的语言和工具的宇宙。***我的研究计划的第一个主要部分涉及P-Adic群体的代表理论。表示理论试图通过线性化来理解它们，也就是说，在（良好理解）复数上表征了他们对线性向量空间的行为。在这里，这些组是P-ADIC数字上的矩阵组，具体取决于prime p。 P-ADIC群体的代表是当今数字理论中一些最大的开放问题的基础，包括所谓的Langlands计划。我的研究旨在通过确定这些表示形式如何局限于较小的子组时如何分解来加深我们的理解。例如，如果亚组是最大的紧凑型子组，则有无限的许多有限维成分，除非在很少的情况下，这是未知的。该领域令人兴奋的开放问题之一是发现何时该组的两个显然无关的表示构成限制，这表明它们之间的更深层次的联系。该动作将谎言代数分解为轨道，其中最终的尼尔植物轨道是最有趣的。最近，Bebacker通过与P-Adic组密切相关的组合和几何对象（称为Bruhat-titt building）对这些轨道进行了理论分类。我正在努力的重要开放问题是通过产生代表​​每个轨道（对于固定组）并得出各种nilpotent轨道的维度和属性来具体意识到这种分类，包括从脱发者参数中，包括它们的近距离。这有无数的申请；例如，Harish-Chandra-How-How-How targuare公式参数（表示表示）是通过一组nilpotent Orbits在基本加权的总和上进行的。密码学是混淆信息的艺术和科学。我们在该领域的研究集中在分析和扩展基于代数系统的加密协议上 - 例如NTRU，椭圆曲线加密和同型加密系统。目的是通过学习如何随着代数对象的函数的函数来了解其潜在的意外漏洞，从而更深入地了解其潜在的意外漏洞。**