The current organizers are Christopher Dean and Andre Kornell. Information about winter 2023 seminars can be found here.
Ski lifts, Mobius transformations, and quantum chemistry
One of the promising applications of quantum computing is solving problems in quantum chemistry. I will attempt to explain quantum chemistry in 15 minutes. Then I will talk about a method for constructing very efficient circuits for quantum chemistry using fun ideas from mathematics and winter sports. Specifically, we will be using ski lifts and Mobius transformations over finite fields. This is joint work with Andre Kornell.
The Grothendieck Construction for Dummies
The Grothendieck construction for a pseudo functor into the category of categories has lots of interesting properties. In this talk I will start by describing the category of elements and then giving and in some cases proving its properties. For those that I don’t prove I will give you good references to look up the proofs.
Quantum chromatic numbers categorically
A coloring of a simple graph is a function that assigns colors to vertices such that adjacent vertices are assigned distinct colors. In quantum information science, one also speaks about a simple graph having a quantum coloring. Using categories, I will explain what this phrase usually means and how it can be interpreted literally. This is joint work with Bert Lindenhovius.
The Groethendieck construction for tangent categories
Recently, given the importance of this subject in my field of studies, my supervisor Geoff Cruttwell explained to me the notion of fibrations. I have to confess I was quite reluctant at the beginning: fibrations appeared to me like an equivalent but more mysterious object than indexed categories, so why bother? Well, this talk is me reconsidering this position and actually realizing how interesting and natural (tangent) fibrations are!
In their paper on differential bundles, Robin Cockett and Geoffrey Cruttwell introduced the notion of tangent fibrations and proved that the fibres of a tangent fibration inherit a tangent structure. In the perspective of extending the Groethendieck construction in the context of tangent categories, one could hope to see this as part of an equivalence between tangent fibrations and indexed tangent categories. However, in the process of splitting the fibres into different tangent structures part of the information of the tangent structure over the total category is lost. The solution we provide is to introduce a new approach to the tangent world that could lead to a formal theory of tangent categories: tangent objects.
I would like to thank Geoff Vooys and Dorette Pronk for the great discussion we had about tangent fibrations, and Geoff Cruttwell for introducing me into the world of fibrations.
Pseudocones, Pseudolimits, and Equivariant Descent or How I Learned to Stop Worrying and Embrace the Abstraction
Yes, but it's fun! In this talk we'll discuss how working with categories of pseudocones generalizes and captures the formalism needed to define the equivariant derived category both of topological spaces and of varieties. After going through these motivating examples we'll discuss some properties of these categories, some basic ideas on how to study them, and some properties they inherit from their corresponding pseudofunctor.
Categorical composable cryptography
We formalize the simulation paradigm of cryptography in terms of category theory and show that protocols secure against abstract attacks form a symmetric monoidal category, thus giving an abstract model of composable security definitions in cryptography. Our model is able to incorporate computational security, set-up assumptions and various attack models such as colluding or independently acting subsets of adversaries in a modular, flexible fashion. We use the one time pad as our main example: in abstract terms, its composable security follows from the axioms of a Hopf algebra with an integral, which concretely speaking corresponds to a group structure on the message space and a uniformly random key.
Joint work with Anne Broadbent
The Categorical Spin-Statistics Theorem: A Topological Field Theory Perspective
The "spin-statistics theorem" is a fundamental principle in quantum field theory, stating that a particle has half-integer spin if and only if it is a fermion. This talk provides a category-theoretic approach to elucidate and rigorously prove this theorem in the context of topological field theories. Our method draws upon the perspective on dagger categories by anti-involutions and Hermitian forms, which I jointly developed with Jan Steinebrunner. This approach not only provides a clearer understanding of the spin-statistics theorem, but also offers valuable insights into the classification of unitary dual functors on symmetric monoidal dagger categories.
Representable PROs
The definition of monoidal category intimidates people. The good news is, there is an equivalent definition that is relatively self-explanatory. Not only is this definition beginner-friendly, but in my opinion it makes thinking about monoidal categories easier. There are also similar tricks for thinking about symmetric monoidal categories, bicategories, and doubly weak double categories.
If you don't know anything about monoidal categories, come along to learn what they are. If you know everything about monoidal categories, come along to unlearn what they are.
Categorical Quantum Groups, Braided Monoidal 2-Categories and the 4d Kitaev Model
It is well-known since the late 20th century that Hopf algebra quantum groups play a signification role in both physics and mathematics. In particular, the category of representations of quantum groups are braided, and hence captures invariants of knots. This talk is based on arXiv:2304.07398 & JHEP 2023 141, where we develop a categorification of the theory of quantum groups/bialgebras, including homotopy refinements, and prove that their 2-representations form a cohesive braided monoidal (tensor) 2-category. If time allows, I will discuss an application of this framework to describe the 4D toric code and its spin variant, uniting the 2-categorical and 2-group gauge theory description of topological orders.
Thoughts on (higher) dagger categories
In this impromptu talk, I will discuss some recent perspectives on dagger categories, including a definition of dagger category that respects the principle of categorical equivalence, and suggest a definition of dagger n-category.
This is based on joint work with B. Bartlett, G. Ferrar, B. Hungar, C. Krulewski, L. Müller, N. Nivedita, D. Penneys, C. Scheimbauer, L. Stehouwer, and C. Vuppulury.
Representation theorem for enriched categories
Universal properties play an important role in mathematics, as they allow us to make many constructions such as (co)limits, Kan extensions, adjunctions, etc. In particular, a universal property is often formulated by requiring that a certain presheaf is representable. The representation theorem gives a useful characterization of these representable presheaves in terms of terminal objects in their category of elements. Going one dimension up and considering 2-categories, with tslil clingman we showed that the straightforward generalization of the representation theorem does not hold in general, but instead one needs to pass to a double categorical setting. In this talk, after reviewing the case of 2-categories, I will explain how to generalize the representation theorem to the more general framework of V-enriched categories, where V is a cartesian closed category. This is joint work with Maru Sarazola, and Paula Verdugo.
Updated November 26, 2023.