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

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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.