Bruhat-Tits theory in the representations of p-adic groups, and post-quantum cryptography

我的研究项目集中在如何应用代数中的新思想来解决纯数学和应用数学中的问题。主要主题：p进群的表征理论。由于著名的朗兰兹计划（20世纪70年代-），这一数学领域已经成为研究的焦点。这是一系列巨大的猜想，它们将表征理论的力量引入到一些关于数字本质的最基本的问题上。在这里，p进数，对于任何质数p，作为无数不同的透镜进入画面，使这个理论成为焦点。我的工作集中在分支规则上：将复杂的表示分解成无限的小块集合，然后识别那些保留原始数论问题答案的集合。让我的研究令人兴奋的是我发现和开发了新的工具，使用一种叫做Bruhat-Tits building of G的组合对象，为这个无限的集合提供了一个几何结构，从而潜在地解决了数论问题本身。这是一个充满迷人的开放问题的领域，可以在多个层面上进行访问，并且是训练我的学生抽象，批判性思维和进行研究的理想选择。次要主题：后量子数学密码学。我们的互联网经济完全依赖于公钥基础设施（PKI），这是一种数学加密技术，它允许我们在不安全的通道上启动安全通信。随着量子计算机成为现实，现有的绝大多数PKI将被淘汰，因为它们基于易受量子算法攻击的问题。后量子密码学（PQC）指的是一类加密系统，旨在防止所有已知的攻击，使用经典和量子计算机。这些数学上复杂的算法的发展和彻底审查可以说是未来十年最紧迫和最引人注目的数学挑战。我的工作重点就是迎接这一挑战。在过去，我训练学生使用代数方法来分析、评估或攻击新的PQC。展望未来，我将通过量子信息和量子计算的新前沿跨学科项目加速量子算法的发展。我研究的一个特别的新方向与所谓的隐藏子群问题有关，它与我的第一个主题（表示理论）密切相关，令人兴奋的是，它是量子攻击和新PQC的来源。尽管PQC的研究本质上是高风险的，但它为学生提供了非常好的服务，既是进入更多理论数学研究的入口，也是培养编码和网络安全方面高度市场化技能的绝佳机会。

My research program centers on an overarching vision on how to apply new ideas in algebra to solve problems in both pure and applied mathematics. Primary theme: representation theory of p-adic groups. This area of mathematics has become a focal point of research largely due to the celebrated Langlands program (1970s-). This is a vast series of conjectures that bring the power of representation theory to bear on some of the most fundamental questions about the nature of numbers. Here, the p-adic numbers, for any prime p, enter the picture as the myriad of different lenses that bring this theory into focus. My work centers on branching rules: breaking up complex representations into an infinite collection of smaller pieces, and then identifying those that retain answers to the original number-theoretic questions. What makes my research exciting is my discovery and development of new tools, using a combinatorial object called the Bruhat-Tits building of G, to give a geometric structure to this infinite collection and thus potentially, to the number-theoretic questions themselves. This is a field rich in fascinating open questions that can be made accessible at multiple levels, and is ideal for training my students in abstraction, critical thinking, and conducting research. Secondary theme: post-quantum mathematical cryptography. Our internet economy relies utterly on public-key infrastructure (PKI), which is the mathematical cryptography that allows us to initiate secure communications over an insecure channel. As quantum computers become a reality, the vast majority of existing PKI will be rendered obsolete because they are based on problems vulnerable to attack by quantum algorithms. Post-quantum cryptography (PQC) refers to a class of cryptosystems intended to be secure against all known attacks, using either classical and quantum computers. The development and thorough scrutiny of these mathematically sophisticated algorithms is arguably the most urgent and compelling mathematical challenge of the coming decade. My work is focussed on meeting this challenge. In the past, I have trained students in the use of algebraic methods to analyse, evaluate or attack new PQC. Going forward, I will additionally embark of the development of quantum algorithms, accelerated by an interdisciplinary project through New Frontiers in quantum information and quantum computing. One particular new direction of my research relates to the so-called Hidden Subgroup Problem, which is strongly related to my first theme (representation theory) and is, excitingly, a source for both quantum attacks and new PQC. Although high-risk to its core, research in PQC serves students extremely well, both as an entry to more theoretical mathematical research, and as an excellent opportunity to develop highly marketable skills in coding and cybersecurity.