Algebraic structures in mathematical physics and category theory

自然界中的对称性自从文明开始以来就吸引着人类，因为它们帮助我们理解其最基本的规律。一个简单的例子是旋转对称性，如果我们观察它与视角无关，则现象不变性。一个对称性越不常见，从物理学的角度看它就越有趣，描述它们的数学结构也就越奇特和新颖。在过去的几十年里，人们一直在努力理解那些与共形对称有关的对称性，这种对称性特别罕见，它保留了角度（但不是例如大小）。这些进展构成了一个活跃、丰富和前沿的世界性研究领域。这个博士项目将研究系统的方法来检测和分类在这种设置中产生的某些代数结构，并描述某些物理实体，特别是从顶点算子代数的表示中获得的模张量类别中的代数对象。这些对象有一个美丽的物理描述，并连接到其他数学形式的物理理论，如r-自旋拓扑场论。

Symmetries in Nature have fascinated the mankind ever since the beginning of civilization, since they help us understanding its most fundamental laws. An easy instance is rotational symmetry, invariance of a phenomenon if we observe it independent of the viewing angle. The more uncommon a symmetry is, the more interesting it is from a physics point of view and more exotic and original mathematical structures describe them. In the last decades there has been a substantial effort to understand those related to conformal symmetry, a symmetry particularly rare which preserves angles (but not e.g. sizes). These advances have constituted an active, rich and cutting-edge field of world-wide research. This PhD project will study systematic ways of detecting and classifying certain algebraic structures arising in this setting and describing certain physical entities, in particular algebra objects in modular tensor categories obtained from representations of vertex operator algebras. These objects have a beautiful physical description and are connected to other mathematical formalizations of physical theories like e.g. r-spin topological field theories.