Homotopy theory, representation theory and mathematical physics

我的研究涉及拓扑学、物理学、计算，以及介于两者之间的各种主题。 拓扑学是研究曲面和高维形状，以及它们可以被拉伸、变形和相互映射的方式。我们通过用环、球面和其他简单的几何形状来探测它们来研究这种空间。这通常导致提供关于空间的定性和计算信息的代数不变量。拓扑学在许多其他领域都有应用，如代数、代数几何和物理。在所有这些设置中，有许多问题和技术是有意义的，我在拓扑学中工作的主要主题是研究这种跨学科问题。学科之间的这种相互作用已被证明是非常富有成效的，通常提供了洞察力，导致解决开放问题的方案。我将继续解决这些问题，重点是涉及代数和几何的问题。 物理学中最重要的悬而未决的问题之一是广义相对论和量子力学的统一，广义相对论是爱因斯坦的引力理论，量子力学描述了所有其他已知力。我的工作重点是解决这个问题的一种很有前途的方法，一种名为环圈量子引力的理论。我打算研究这一领域的一个主要悬而未决的问题：如何将物质纳入理论。我将通过分析和使用计算机模拟来研究这个问题，目的是找到一个可行的物质理论，并看看它是否做出了可检验的预测。 我还从事格点规范理论的工作，这是一种粒子物理的计算方法，使用离散的时空近似。来自表象理论和量子引力的方法可以用来以规范不变的方式理解这些计算，我正在开发新的算法，我希望能让它们更快。

My research involves topology, physics, computation, and various topics in between. Topology is the study of surfaces and higher-dimensional shapes, and the ways in which they can be stretched, deformed and mapped into each other. We study such spaces by probing them with loops, spheres and other simple geometric shapes. This generally results in algebraic invariants which provide qualitative and computational information about a space. Topology has applications in many other fields, such as algebra, algebraic geometry and physics. There are many questions and techniques that make sense in all of these settings, and the main theme of my work in topology is the study of such interdisciplinary problems. This interplay between disciplines has proven to be extremely fruitful, often providing insight that leads to solutions to open problems. I will continue to tackle such problems, with a focus on problems involving algebra and geometry. One of the most important unsolved problems in physics is the unification of general relativity, Einstein's theory of gravity, with quantum mechanics, which describes all of the other known forces. My work focuses on a promising approach to this problem, a theory called loop quantum gravity. I intend to study one of the main open problems in this field: how to incorporate matter into the theory. I will study this problem both analytically and through the use of computer simulations, with the goal of finding a viable theory of matter and seeing whether it makes testable predictions. I also work on lattice gauge theory, which is a computational approach to particle physics that uses a discrete approximation to space-time. Methods from representation theory and quantum gravity can be used to understand these computations in a gauge invariant way, and I am developing new algorithms that I hope will make them faster.