Topics in category theory

Context
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Pure mathematics is to a large extent about abstraction. Commonalities are spotted between seemingly different things and the essence of these commonalities are encapsulated in a mathematical entity. So it can be seen that multiplying numbers behaves similarly to composing transformations of the plane and this similarity can be captured in notion of a monoid (similar to a group of symmetries). Category theory gives you a language and a calculus for comparing and abstracting such mathematical abstractions. Category theory itself has commonalities with other mathematical ideas and these can be abstracted into enriched category theory. One can also go from plain category theory to ideas such as two-categories and monoidal two-categories or various kinds of double categories. Various combinations of these structures crop up in disparate parts of mathematics such as topological quantum field theory, representation theory and optimal transport. This project can be viewed as being about how these structures fit together.


Objective
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An initial objective, which may shift as the landscape becomes clearer, is to identify the correct notion of monoidal enriched fibrant double category with duals or internal homs that corresponds to examples seen in nature, in particular, examples consisting of things like bordisms, bimodules, enriched profunctors, Fourier-Mukai transforms, costing plans and homological matrix factorizations.


Approach
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The student will become an expert in the methods of low dimensional category theory and enriched category theory. He will also study many of the examples mentioned above from across mathematics, identify their commonalities and use these to model the appropriate abstraction. This should lead to further connections between these examples and a deeper understanding of the structures.


Alignment
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This research is aligned to algebra, logic and combinatorics, and to geometry and topology.