Linear logic and monoidal categories

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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.