BPS states from number theory to knot homology

在一般的科学领域里，数学和物理是相邻的学科。纵观历史，这两门学科交流思想，取得成果，相互受益。传统上，信息可以在两个方向上流动--物理理论的要求可以利用现有的数学结构，或者指向新的结构，而数学结果可以证实物理直觉，或者建立物理理论的一些迄今为止未知的属性。 最年轻的、可以说是最有前途的数学物理学分支之一，它体现了这种混合，这是弦理论：高能物理学家试图理解量子引力，统一基本相互作用的努力，需要使用复杂的数学机器，反过来，影响或直接贡献了纯数学的新结果。也许有些令人惊讶的是，这种相互作用是与数学领域的，在世纪的大部分时间里，数学领域与理论物理的发展脱节，最著名的是代数几何和低维拓扑。新的联系不仅丰富了这两个领域，而且还提供了一个可持续的信心来源，即弦理论正走在正确的道路上，成为人类理解宇宙、粒子物理学和宇宙学的下一个里程碑。 这个研究计划的重点是两个具体的愿景，在数学弦理论领域的未来进展：（1）拓扑量子理论的统一作为一种简化版本的无所不包的理论本身。解开结和链的奥秘是这种统一最有可能首先走到一起的领域;（2）与数论的联系。 最近的进展领域，从三个流形的分类，朗兰兹对偶镜像对称是证据，许多相似之处发现多年来将进一步延伸到心脏的两个主题。 一个庞大而迅速增长的世界范围内的数学家和物理学家社区正在研究这两个学科的弦理论接口。一个强有力的证明，越来越多的公共利益是建立了几个跨学科的研究中心，奖项，并在世界各地的会议系列。近年来，加拿大大学已经做出了在这一领域扩大的战略决策，因为我们社会的声誉将从这两个最基础科学的知识进步中获得。

In the general field of science, mathematics and physics are neighbouring disciplines. Throughout history, the two subjects have exchanged ideas and results, and benefitted from each other. Traditionally, information can flow in both directions -- requirements of physical theories can draw on an existing mathematical structure, or point to new ones, and mathematical results can confirm physical intuition or establish some hitherto unknown property of the physical theory. The youngest and arguably one of the most promising branches of mathematical physics which manifests this mixture is the one stemming from string theory: Efforts of high-energy physicists trying to understand quantum gravity, and to unify the fundamental interactions, have required the use of sophisticated mathematical machinery and, conversely, influenced or directly contributed new results in pure mathematics. Perhaps somewhat surprisingly, the interaction has been with areas of mathematics that, for large parts of the 20th century, have been disconnected from developments in theoretical physics, most notably algebraic geometry and low-dimensional topology. The new connections not only enrich both fields, but also provide a sustainable source of confidence that string theory is on the right track to becoming the next milestone in humanity's understanding of the Universe, particle physics, and cosmology. This research program is focused on two concrete visions for future progress in the field of mathematical string theory: (1) The unification of topological quantum theory as a kind of simplified version of the all-encompassing theory itself. Unravelling the mysteries of knots and links is the area in which this unification is most likely to first come together; and (2) The connections with number theory. The recent progress in areas ranging from three-manifolds to categorification to Langlands duality to mirror symmetry is evidence that the many parallels discovered over the years will extend much further to the heart of the two subjects. A large and rapidly growing worldwide community of mathematicians and physicists is working at the string-theoretic interface of the two disciplines. A strong testimony to the growing public interest is the establishment of several interdisciplinary research centres, prizes, and conference series around the world. Canadian Universities have made strategic decisions to expand in this area in recent years, as the reputation of our society stands to gain from the advancement of knowledge in those two most fundamental sciences.