CAREER: Factorization homology and quantum topology

这个职业奖资助了一个研究项目，该项目将开发因子分解同源性，这是一种用于组织量子物理学中长距离/长时间观测的数学设备。 尺度无关性和局域性是量子观测量的两个有利的和成立的性质。 尺度独立性有助于物质的稳定相，可以通过称为拓扑学的数学领域进行编码。 如果在一个大的时空区域上的每一个观测都是由那些在较小的区域上的观测所决定的，例如由点或线的观测所决定的，那么观测就具有定域性。 点、线、面、......可观测量被组织成一个数学实体，称为更高类别，这是一个更高维的图。 因式分解同调是一种将这种局部化的观测量组装成时空上的全局观测量的方法，它具有几个结构特征，一旦这样做，就会给理论带来优势。 一般来说，它提供了概念和组合的访问长距离/长时间量子理论，特别是那些不是由他们的点可观测。 为了整合学生和研究人员之间的研究和教育，并增加蒙大拿州及其周边州的EPSCoR的数学和物理活动，PI将组织两次会议。 这些会议将汇集著名的数学物理研究人员，并将吸引区域参与。 主要研究者将汇编和传播此类事件的偏倚敏感选择过程的文献。 PI将制作并公开发布两个视频系列。 一个系列将看到通过与该奖项相关的活动产生的研究水平的数学。 另一个系列将是实验性的课堂教学，为高年级数学班。 该项目继续发展和明确的识别理论的因式分解同源性较高的类别，部分合作与约翰弗朗西斯。 因式分解同调可以被认为是研究分层的模空间上的层，很像贝林森-德林费尔德在Ran空间上用层构造共形块。 一个目标是实现这个理论的一个重要方面：正交群和更高范畴群之间的确切关系。 这种关系的一个后果是证明了协边假说，贝兹多兰和卢里，阿蒂亚后。 另一个目标是恢复已知的，并建立新的，流形和纠缠不变量表示理论的数据。 这样的预期例子包括琼斯多项式和Reshetikhin-Turaev不变量。 现在，因式分解同调连续依赖于同构/合痕;它拥有微妙的函子性和局部到整体原则;它被期望携带自然过滤以及一般形式的庞加莱/科苏尔对偶。 这样一个目标，从而有利于计算访问流形和纠缠不变量产生通过表示理论。 这样的目标也有助于状态和型拓扑量子场论和西格玛模型型场论之间令人惊讶的二元性。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This CAREER award funds a research project that will develop factorization homology, a mathematical device for organizing long-distance/long-time observables in quantum physics. Scale-independence and locality are two favorable and tenable features of quantum observables. Scale-independence lends to stable phases of matter that can be codified via a field of mathematics called topology. Observables possess locality if each observation on a large region of space-time is determined by those on smaller domains, such as by point- or line-observables. Point-, line-, plane-, … -observables organize as a mathematical entity known as a higher category, which is a higher-dimensional graph. Factorization homology is a device that assembles such localized observables to global observables on space-time and it possesses several structural features that give a theory advantage once cast as so. Generally, it offers conceptual and combinatorial access to long-distance/long-time quantum theory, notably even to ones that are not determined by their point-observables. To integrate research and education between students and researchers, and to increase mathematics and physics activity in the EPSCoR State of Montana and its surrounding states, the PI will organize two conferences. These conferences will bring together renowned researchers in mathematical physics, and will draw regional participation. The PI will compile and disseminate literature for bias-sensitive selection processes for such events. The PI will produce and publicly post two video series. One series will behold research-level mathematics generated through activities related to this award. The other series will be for experimental in-class instruction for upper-division mathematics classes. The project continues the development and explicit identification of the theory of factorization homology for higher categories, partly in collaboration with John Francis. Factorization homology can be conceived as the study of sheaves on moduli spaces of stratifications, much like Beilinson-Drinfeld’s construction of conformal blocks in terms of sheaves on Ran spaces. One goal is to fulfill an essential aspect of this theory: the exact relationship between orthogonal groups and higher categorical adjunctions. A consequence of such a relationship is a proof of the cobordism hypothesis, of Baez-Dolan and Lurie, after Atiyah. Another goal is to recover known, and establish new, manifold and tangle invariants from representation theoretic data. Expected examples of such include the Jones polynomial, and Reshetikhin-Turaev's invariant. Now, factorization homology depends continuously in diffeomorphisms/isotopies; it possesses subtle functoriality and local-to-global principles; it is expected to carry natural filtrations as well as a general form of Poincare/Koszul duality. Such a goal thusly lends advantage for computational access to manifold and tangle invariants arising through representation theory. Such a goal also lends to surprising dualities between state-sum type topological quantum field theories and sigma-model type field theories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.