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Local tomography (or tomographic locality) is the principle that the state of a composite systems is determined by the probabilities it assigns to outcomes of experiments performed separately on the component systems. It's well known that complex quantum theory enjoys, and real quantum theory lacks, this feature. This means that a composite of two real quantum systems has additional "global" degrees of freedom. What if we could simply factor these out? In this talk, I'll describe how this can be done, not only for real quantum theory, but for essentially any probabilistic theory. The result is a locally tomographic theory we call the "locally tomographic shadow" of the original. I will also discuss what this shadow theory looks like in the case of real quantum theory. (This is joint work with Howard Barnum and Matthew Graydon).

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This talk explains the theory of "factorisation" or "vertex algebra" analogues of the theory of quantum groups, (2312.07274).

We will first recap the theory of quantum groups and their connections to wider mathematics, due to Drinfeld, Etingof, Kazhdan, Jimbo, Reshetikhin, Turaev, and many others. We will then explain how to give a natural definition of "factorisation" version of these objects, give the basic structure theory about their categories of representations, and derive an explicit model for them. This relates to previous work of Etingof-Kazhdan and Frenkel-Reshetikhin.

We will sketch some physics heuristics for the above theory; briefly, our structures should appear on the category of line operators in 4d theories due to Costello, Witten and Yamazaki, e.g. on the representations Rep U_q(g^) of affine quantum groups.

We will then give some examples. The last part discusses the conjectures that naturally suggest themselves, for instance by analogy to the connections referred to above.

Prerequisites on chiral and vertex algebras will be given at a talk earlier in the week. (1-2pm Tuesday, Bob Room)

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Historically, the synergy between physics and geometry, from the times of Archimedes and Newton to the era of Einstein, has repeatedly been the catalyst for pivotal breakthroughs in physics and mathematics. In this talk, I will introduce a new narrative demonstrating how physics and geometry intertwine, leading to unexpected and significant results in critical phenomena in physics. Specifically, I will elucidate how non-commutative geometry—a mathematical framework born from the insights of physicists—offers fresh perspectives on conformal field theory, a subject with profound applications across various physics domains, from condensed matter to quantum gravity, and string theory.

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