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

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，从表征理论到通信理论。 在我的研究计划项目的多样性是语言和代数工具的普遍性的反映。我的研究计划的第一个主要部分涉及p进群的表示理论。 表示论试图通过将群线性化来理解群，也就是说，描述它们在（很好理解的）复数上的线性向量空间上的作用。在这里，群是p进数上的矩阵群，取决于素数p。p进群的表示是当今数论中一些最大的开放问题的基础，包括所谓的朗兰兹纲领。 我的研究旨在通过确定这些表示在被限制为较小的子组时如何分解来加深我们的理解。 例如，如果子群是一个极大紧开子群，则有无穷多个有限维分量，除了极少数情况外，这些分量是未知的。 在这个领域中，一个令人兴奋的开放问题是发现当两个明显不相关的群表示在限制下共享共同成分时，表明它们之间有更深层次的联系。我的研究的第二个方面涉及到一个p-adic群在其李代数上的作用，李代数是一个p-adic数上的向量空间。 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最近证明了这些轨道的理论分类，根据与p-adic群密切相关的组合和几何对象，称为Bruhat-Tits建筑。我正在研究的一个重要的开放性问题是具体地实现这种分类，通过产生每个轨道的代表（对于一个固定的群），并从DeBacker参数中推导出各种幂零轨道的维度和属性，包括它们之间的接近度。这有无数的应用;例如，Harish-Chandra-Howe特征公式通过一组幂零轨道上的加权和来参数化表示（的特征）。最后，我的研究的第三个方面是数学密码学，这部分是由于我对群及其相关格的专业知识。 密码学是一门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-如NTRU，椭圆曲线密码和同态加密系统。 我们的目标是更深入地了解密码算法，从而了解其潜在的意外漏洞，通过学习它如何作为其所基于的代数对象的函数而变化。

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.**