Speakers

Title: Functor calculus for multi-persistent homology

Abstract: Functor calculus is a general framework for studying functors, and comes in many flavors. We construct a variant of functor calculus, called poset cocalculus, for studying functors out of posets. Our motivation is to better understand multi-persistence modules, as these modules can be viewed as functors from R^k into a category of chain complexes. We demonstrate some properties of poset cocalculus, and apply these in concrete examples with multi-persistence modules. We finish by giving examples of poset cocalculus in other contexts, such as filtrations of simplicial complexes.

Title: Moduli Spaces of Arrows

Abstract: Many moduli spaces parametrize isomorphism classes in some category, and we can wonder if the morphisms of said category can be similarly parametrized. The relevant moduli stack encodes families of morphisms, but its points are still objects and not morphisms. The idea of this talk is to consider a moduli stack whose points are the morphisms involved in the original moduli problem. We will develop this idea for the specific case of the moduli problem of vector bundles over a fixed base and observe that this has a generalization: if we can parametrize arrows of vector bundles, we can parametrize diagrams thereof of more general shapes. We will then discuss some implications of this line of thought for the moduli problem of Higgs bundles and non-Abelian Hodge theory. Time permitting, we will discuss abstract moduli theory and homotopy theory in this light. This talk reports on joint work with Steven Rayan.

Title:  Weighted pullbacks in V-graded categories

Abstract:  Introduced by Richard Wood in 1976, categories graded by a monoidal category V generalize both V-actegories and V-enriched categories.  In this talk, we review some basics of V-graded categories, and then we introduce a notion of weighted pullback in V-graded categories.  Weighted pullbacks are certain weighted limits that generalize the usual (conical) pullbacks, yet they also specialize to certain notions of universal quantification and certain dependent products.  Indeed, weighted pullbacks generalize simple products in the codomain fibration of a cartesian closed category with finite limits and, in particular, simple universal quantification in the subobject fibration of such a category.  Generalizing the latter example, we introduce notions of simple product and simple universal quantification in V-actegories as special cases of the notion of weighted pullback.  In particular, weighted pullbacks thus give rise to a notion of simple universal quantification in monoidal categories.

Title: Higher groupoids and higher generalized morphisms

Abstract: Higher groupoids play a crucial role in the active research area of interplay between higher categorical structures and other fields of mathematics. We give the notion of a Good Geometric category, where one can define and study these higher structures with applications to geometry, for example the category of smooth manifolds. We define the notion of higher groupoids in Good Geometric categories and organize them into an (∞, 1) categorical framework. The morphisms between the higher groupoids are given by bibundles which are Kan fibrations over the interval. Higher morphisms will be modelled by Kan fibrations over the higher simplices. This approach gives a more combinatorial and geometric way of approaching anafunctors and higher generalised morphisms between groupoids. This is of particular interest in higher gauge theory and string theory, where the higher connection on higher bundles will give the notion of parallel transport of strings and surfaces.

Title: All Concepts are Kan Extensions… and Kan Extensions are Concepts: A Categorical Theory of Vector Symbolic Architectures

Abstract: It is widely understood by category theorists that Kan extensions subsume all other notions in category theory (limits, ends, initial objects, adjoints, etc.). In this talk, I introduce vector symbolic architectures (VSAs)—a distributed model of data storage and manipulation that have gained popularity alongside the neuromorphic hardware upon which engineers hope to implement VSAs. Broadly, VSAs populate a vector space with a collection of “primitive” vectors, then use two operations, ‘binding’ and ‘bundling’, to create new vectors (ex. “red” and “car” bind as “red car”). Despite their many implementations, VSAs have little unifying theory. I discuss my recent work generalising and unifying VSAs. I focus on extending from vectors to co-presheaves and demonstrating that bind and bundle arise as right Kan extensions of the external tensor product and direct sum.

Title: Introduction to equivalences between bicategories and 2-categories
Abstract: This is part four of our series of introductory talks on bicategories. This time we will be considering different ways in which bicategories can be considered to be equivalent. While sets are considered “the same” when they are isomorphic, an equivalence of categories (the right way of “same-ness” for categories) can be seen as being isomorphic up to isomorphisms. Analogously we will see that two bicategories are biequivalent if they are isomorphic up to isomorphisms up to isomorphisms. In practice it is often easier to check biequivalence using the Whitehead theorem, which I will present and apply to examples.

Title: An Introduction to Embedding Calculus and the Role of Automorphisms of the (Framed) Little Disk Operad

Abstract: Embedding calculus is a powerful tool which is useful in making quantitative and qualitative conclusions about the topology of embedding spaces. In this talk, we will give a geometric description of the fiber of $\operatorname{Emb}(M,M) \rightarrow T_\infty \operatorname{Emb}(M,M)$ using smooth structures as well as an algebraic description of its delooping using automorphisms of the little disk operad. In the second part of the talk, we will describe some known properties of that automorphism space and will then explain some ideas that go into the proof of showing that the endomorphisms of the framed little disk operad agree with the automorphisms.

Weighted pullbacks in V-graded categories

As a replacement we have the talk on morphisms between bicategories

Title: The Three Flavours of Morphisms Between Bicategories

Abstract: In everybody’s first course in category theory, we learn that categories have two different flavours of morphism: 0-morphisms (which are categories themselves), 1-morphisms (which are functors), and 2-morphisms (which are natural transformations). In this talk I’ll continue with our theme of “Geoff teaches everyone the magic of bicategories” by showing that this pattern of having multiple flavours of arrows continues with bicategories by introducing us to the 0-morphisms (which are bicategories), the 1-morphisms (which are pseudofunctors), the 2-morphisms (which are pseudonatural transformations), and the 3-morphisms (which are called modifications). Depending on time, we’ll also see various examples of these gadgets and how they can arise “in nature.”

Title: The Intersection between Number Theory, Abstract Algebra, and Cryptography

Abstract: In this talk we will cover the mathematics behind a security system that is thought to be secure against quantum computer hacking; the Supersingular Ell Isogeny Cryptosystem. Specifically the Supersingular Ell Isogeny Graph. This abstract object is made of isomorphic elliptic curves and equivalent homomorphisms between them. The Supersingular Ell Isogeny graph is an extremely complicated graph that is used for cryptosystems (for the future), but we will use a simpler approach to find patterns within this convoluted graph.