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。拓扑学是研究表面和更高维度的形状，以及它们可以被拉伸，变形和映射到彼此的方式。 一种常见的技术是通过用环、球和其他几何规则空间来探测空间来研究空间。 这通常导致代数不变量。最近，拓扑学的思想在其他领域，如代数、代数几何和物理学中得到应用。 有许多问题和技术，使在所有这些设置的意义，我的工作在拓扑学的主题是研究这样的全球性问题。学科之间的这种相互作用已被证明是非常富有成效的，我将继续朝着这个方向努力。物理学中最重要的未解决的问题之一是爱因斯坦非常成功的引力理论广义相对论与描述所有其他已知力的量子力学的统一。 我的工作集中在解决这个问题的一个有希望的方法上，一个叫做圈量子引力的理论。目前，圈量子引力理论中两个最大的开放问题是研究它的经典极限和如何将物质引入理论。 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。将物质引入到理论中是很重要的，以便了解该理论对现有粒子物理学理论的修正，并看看这些预测中是否有任何一个是可以实验检验的。我将通过分析和计算机模拟来研究这些问题。这样的工作是对一个理论的重要检验，可以导致改进和其他发展。

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.