Homotopy theory and quantum gravity

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。一种常见的技术是通过用回路、球面和其他几何规则空间来探测空间来研究空间。这通常会产生代数不变量。最近，拓扑学的思想在其他领域得到了应用，如代数、代数几何和物理。在所有这些背景下，都有许多有意义的问题和技术，而我在拓扑学方面工作的主要主题就是研究这样的全局问题。这种学科之间的相互作用已被证明是非常富有成效的，我将继续朝着这个方向工作。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非常成功的引力理论，量子力学描述了所有其他已知力。我的工作集中在解决这个问题的一种有前途的方法上，这是一种被称为圈量子引力的理论。目前，圈量子引力中最大的两个开放问题是对其经典极限的研究，以及将物质引入理论的方法。经典极限问题询问理论的大尺度行为是什么，并将其与已知的宇宙学数据进行比较。将物质引入理论是很重要的，这样才能了解该理论对现有粒子物理理论的修正，并看看这些预测中是否有任何是可通过实验验证的。我将通过分析和使用计算机模拟来研究这些问题。这样的工作是对理论的重要检验，可以带来改进和其他发展。

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.  A common technique is to study spaces by probing them with loops, spheres and other geometrically regular spaces.  This generally results in algebraic invariants. More recently, the ideas of topology are finding applications in 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 global problems. This interplay between disciplines has proven to be extremely fruitful, and I will continue work in this direction. One of the most important unsolved problems in physics is the unification of general relativity, Einstein's highly successful theory of gravity, with quantum mechanics, which describes all the other known forces.  My work focuses on a promising approach to this problem, a theory called loop quantum gravity. Currently two of the biggest open problems in loop quantum gravity are the study of its classical limit and the ways in which matter can be introduced into the theory.  The classical limit problem asks what the large scale behaviour of the theory is, and compares it to known cosmological data. Introducing matter into the theory is important in order to learn what corrections the theory makes to the existing theory of particle physics, and to see if any of these predictions are experimentally testable. I will study these problems both analytically and through the use of computer simulations. Such work is an important test of a theory and can lead to improvements and other developments.