Model Theory, Quantum Complexity, and Embedding Problems in Operator Algebras

这个项目位于三个看似不相关的领域的交叉点：冯·诺伊曼代数、量子复杂性理论和模型理论。冯·诺伊曼代数是由约翰·冯·诺伊曼在他的量子力学的数学描述中引入的，它由在各种自然运算下闭合的无限大的矩阵组成。量子复杂性理论考虑了使用量子计算模型解决或验证决策问题解决方案的难度，也就是说，根据量子物理定律运行的计算机，而不是经典物理。最近，量子复杂性理论的一个里程碑式的结果表明，经典的不可解决策问题可以通过量子计算机可靠地验证。这个量子复杂性的结果为算子代数中一个著名的问题--所谓的Connes嵌入问题建立了一个负解，这个问题是在1976年提出的，它询问每个von Neumann代数是否都可以被一个简单的von Neumann代数近似，这个简单的von Neumann代数被称为超有限II_1因子。利用模型理论的技术，PI和一个合作者极大地简化和阐明了量子复杂性结果和Connes嵌入问题的解决方案之间的联系。模型理论是数理逻辑的一个分支，通过使用一阶逻辑来研究结构类。该项目计划通过分离量子复杂性结果背后的确切模型理论内容并进一步推导冯·诺伊曼代数结果来加深这三个领域之间的联系。更具体地说，PI计划扩展对量子复杂性结果的模型理论分析，以了解超有限II_1因子的全一阶理论的复杂性；PI与Hart的工作为单量词理论建立了这种联系。此外，PI计划通过使用存在闭合模型和Robinson强迫的模型论概念来寻求避免使用量子复杂性结果的Connes嵌入问题的失败的证明；由于证明量子复杂性结果的难度，沿着这些路线的新的证明将大大简化Connes嵌入问题的解决。该项目还将研究模型理论在von Neumann代数理论中的其他应用，包括进一步研究Popa的嵌入问题，该问题提出了II_1因子在超幂中的某些遍历嵌入的存在性。PI还计划在Connes嵌入问题的C*代数版本上取得进展，该C*代数版本被称为Kirchberg嵌入问题，该问题询问是否每个C*代数都由Cuntz代数近似，Cuntz代数是核C*代数分类程序中非常重要的代数。最后，虽然von Neumann代数的大部分模型论研究都集中在所谓的有限代数上，但PI计划通过W*-概率空间的透镜来研究任意von Neumann代数的模型论性质。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project lies at the intersection of three seemingly unrelated areas: von Neumann algebras, quantum complexity theory, and model theory. Von Neumann algebras were introduced by John von Neumann in his mathematical account of quantum mechanics and consist of infinite-sized matrices closed under various natural operations. Quantum complexity theory considers the difficulty of solving or verifying solutions to decision problems using the quantum model of computation, that is to say, computers that run according to the laws of quantum physics as opposed to classical physics. Recently, a landmark result in quantum complexity theory showed that classically unsolvable decision problems could be reliably verified by a quantum computer. This quantum complexity result established a negative solution to a famous problem in operator algebras, the so-called Connes Embedding Problem, posed in 1976, which asks whether or not every von Neumann algebra can be approximated by a simple von Neumann algebra known as the hyperfinite II_1 factor. Using techniques from model theory, a branch of mathematical logic that studies classes of structures by examining what is expressible about them using first-order logic, the PI and a collaborator greatly simplified and elucidated the connection between the quantum complexity result and the solution to the Connes Embedding Problem. This project plans to deepen the connection between these three areas by isolating the exact model-theoretic content behind the quantum complexity result and deducing further von Neumann algebraic consequences. More specifically, the PI plans on extending the model-theoretic analysis of the quantum complexity result to understand the complexity of the full first-order theory of the hyperfinite II_1 factor; the PI's work with Hart established this connection for the one-quantifier theory. In addition, the PI plans to pursue proofs of the failure of the Connes Embedding Problem which avoid the use of the quantum complexity result by using the model-theoretic notions of existentially closed models and Robinson forcing; due to the difficulty in proving the quantum complexity result, a new proof along these lines would serve as a great simplification of the resolution of the Connes Embedding Problem. The project will also study other uses of model theory in von Neumann algebra theory, including furthering progress on Popa's embedding problem, which asks about the existence of certain kinds of ergodic embeddings of II_1 factors into ultrapowers. The PI also plans on making progress on the C*-algebra version of the Connes Embedding Problem known as the Kirchberg Embedding Problem, which asks if every C*-algebra is approximated by the Cuntz algebra, an algebra of extreme importance in the classification program for nuclear C*-algebras. Finally, while the majority of the model-theoretic study of von Neumann algebras has focused on so-called finite algebras, the PI plans on studying the model-theoretic properties of arbitrary von Neumann algebras through the lens of W*-probability spaces.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.