FET: Small: A triangle of quantum mathematics, computational complexity, and geometry

该项目的研究目标被组织成一个三角形，其角是三个不同的研究领域：量子现实基础的数学;计算问题的快速算法的存在或不存在;以及节点，曲率和高维空间及其对称性的现代几何。 考虑到研究三角的每一条腿，该项目将探索量子算法的存在或不存在，这意味着只能在量子计算机上运行的算法。 该项目将探索几何和拓扑问题的快速算法的存在或不存在;例如，当两个不同的外观结实际上是同一个结？该项目将探索量子物理学中作为相互作用图出现的网络的几何特性。 作为纯数学和理论计算机科学之外更广泛影响的一部分，该项目将揭示量子计算机和传统经典计算机的能力和局限性，特别是几何计算，但也涉及如何保护数据免受未来量子计算机的影响。 其他更广泛的影响将包括培训研究生和开发公共演示材料。该项目将探索代数问题的量子算法，如隐藏子群问题，该问题将Shor的整数分解算法推广到更一般的周期查找。 该项目将探索流形分类问题的计算复杂性，包括3-流形和高维单连通流形的同胚问题。该项目还将探索李群张量不变量的平面张量网络基的几何结构，推广了研究人员先前的工作，该工作建立了非正弯曲张量网络产生秩2李群张量不变量的基。该奖项反映了NSF的法定使命，并被认为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来支持。

The research aims of this project are organized as a triangle whose corners are three different research areas: the mathematics that underlies quantum reality; the existence or non-existence of fast algorithms for computational problems; and the modern geometry of knots, curvature, and higher-dimensional spaces and their symmetries. Considering each leg of the research triangle, the project will explore the existence or non-existence of quantum algorithms, meaning algorithms that can only be run on a quantum computer. The project will explore the existence or non-existence of fast algorithms for geometric and topological questions; for instance, when are two different-looking knots actually the same knot? And the project will explore the geometric properties of networks that arise as interaction diagrams in quantum physics. As part of the broader impacts outside of pure mathematics and theoretical computer science, the project will shed light on the capabilities and limitations of quantum computers and traditional classical computers, particularly for geometric calculations, but also concerning how to protect data from future quantum computers. Other broader impacts will include training graduate students and developing public expository materials.The project will explore quantum algorithms for algebraic problems such as the hidden subgroup problem, which generalizes Shor's algorithm for factoring integers and more generally for period-finding. The project will explore the computational complexity of classification problems for manifolds, including the homeomorphism problem for 3-manifolds and for higher-dimensional simply-connected manifolds. The project will also explore the geometry of planar tensor networks bases for tensor invariants of Lie groups, generalizing prior work by the investigator that established that non-positively-curved tensor networks yield bases of tensor invariants for rank 2 Lie groups.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.