Linear logic and monoidal categories

我的项目的目标是继续探索一元范畴的理论，特别是使用线性逻辑的技术。从计算机科学中的并行计算理论到物理学中的拓扑量子场论，在许多环境中都出现了单胞范畴的例子。一元体结构对于理解这些应用是至关重要的，而且来自线性逻辑的思想已经成功地阐明了这些一元体结构。在接下来的四年里，我的工作将是继续进行这种分析并进一步开发这些应用程序。我特别感兴趣的是继续从事离散量子因果动力学方面的工作，正如我与合著者普拉卡什·帕南加登和伊万·伊万诺夫共同工作中所定义的那样。我们构建的结构类似于通常的拓扑量子场论的Cobordism范畴，但使用了更精细的线性逻辑连接词。因此，我们能够捕捉到在通常的TQFT公式中所缺乏的纠缠概念。我希望，我们发现的新例子，使用吉拉德的概率相干空间等结构，也将为TQFT带来新的曙光。我最近的工作导致了一些与代数拓扑学的意想不到的联系，我也打算探索这一点。对某些线性逻辑模型的考察导致了形状理论的一元版本，这是博尔苏克提出的同调不变理论。新的例子具有这样的性质，即它们的一元体结构规范地提升到相关的形状类别。我打算探索一下这种现象有多普遍。

The goal of my project will be to continue to explore the theory of monoidal categories, especially using the techniques of linear logic. Examples of monoidal categories have arisen in a number of settings, from the theory of concurrent computation in computer science to topological quantum field theory in physics. The monoidal structure is crucial in understanding these applications, and furthermore ideas from linear logic have successfully shed light on these monoidal structures. My work over the next four years will be to continue with this analysis and to develop these applications further. I am especially interested in continuing work on discrete quantum causal dynamics, as defined in joint work with my coauthors Prakash Panangaden and Ivan Ivanov. The structure we constructed is similar to the usual cobordism category of topological quantum field theory, but uses the more refined connectives of linear logic. We are thus able to capture a notion of entanglement which is lacking in the usual formulation of TQFT. I am hopeful that the new examples we have discovered, using such structures as Girard's probabilistic coherence spaces, will shed new light on TQFT as well. My more recent work has led to some unexpected connections to algebraic topology, which I also intend to explore. Examining certain models of linear logic led to a monoidal version of shape theory, a homology-invariant theory due to Borsuk. The new examples have the property that their monoidal structure lifts canonically to the associated shape category. I intend to explore how common this phenomenon is.