Topological Quantum Field Theory in dimensions two and three

提出的研究是在二维和三维的拓扑量子场论（TQFT）。tqft产生几何对象（如结和三维流形）的拓扑不变量。我们将研究tqft在经典拓扑问题中的应用，例如发现给定的表面（或更多，通常是在二维轨道上）上的Serre振动是否具有截面。这需要仔细研究作者在1999年引入的同伦qft。在另一个方向上，我们将给出三维状态和TQFT的一般代数框架，并将后者与三维手术TQFT联系起来如下：由有限半单单类C导出的状态和TQFT与由C的Drinfeld双元导出的手术TQFT同构。证明将使用由于bruguieres - vireliier - habiro的底缠结技术。量子物理学一直是新的数学见解和技术的来源。大约20年前，菲尔兹奖得主E. Witten和M. Atiyah提出了一种这样的见解，称为拓扑量子场论。它允许我们从一个新的物理角度来接近现实世界的几何物体，如固体的表面或空间中的打结的圆圈。我们将使用这个观点来解决有关曲面的某些几何问题。特别地，我们将回答几个经典的问题，早在引入量子场论之前提出的。我们还将研究量子场论和量子引力产生的数学技术之间的联系。这项研究将涉及另一位菲尔兹奖得主V.德林菲尔德引入的深度代数技术。这个项目将展示物理、几何和代数之间基本联系的新方面。

The proposed research is on Topological Quantum Field Theory (TQFT) in dimensions two and three. TQFTs produce topological invariants of geometric objects such as knots and 3-dimensional manifolds. We shall study applications of TQFTs to classical topological problems such as finding whether a given Serre fibration over a surface (or more, generally, over a 2-dimensional orbifold) has a section. This will require a careful study of Homotopy QFTs introduced by the author in 1999. In another direction, we will give a general algebraic framework for 3-dimensional state-sum TQFTs and connect the latter to 3-dimensional surgery TQFTs as follows: the state sum TQFT derived from a finite semisimple monoidal category C is isomorphic to the surgery TQFT derived from the Drinfeld double of C. The proof will use the technique of bottom tangles due to Bruguieres-Virelizier-Habiro.Quantum physics has invariably served as a source of new mathematical insights and techniques. One such insight introduced about 20 years ago by the Fields medalists E. Witten and M. Atiyah is called Topological Quantum Field Theory. It allows us to approach real-world geometric objects such as surfaces of solid bodies or knotted circles in the space from a new, physical perspective. We will use this viewpoint to address certain geometric problems concerning the surfaces. In particular, we will answer several classical questions raised well before the introduction of Quantum Field Theory. We will also study a connection between mathematical techniques arising from Quantum Field Theory and from Quantum Gravity. This study will involve deep algebraic techniques introduced by another Fields medalist V. Drinfeld. This project will exhibit new facets of the fundamental connection between physics, geometry, and algebra.