Speakers

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.

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.”

Weighted pullbacks in V-graded categories

As a replacement we have the talk on morphisms between bicategories

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.

Title: Systems of homotopy colimits

Abstract: Limits and colimit constructions are ubiquitous in category theory, and are one of the main tools used to understand how objects in a category relate to one another.  These are very concrete and easily stated in terms of universal properties: given a diagram in a category, the colimit is the initial cocone making the resulting diagram commute.  Homotopy colimits, on the other hand, have always been more difficult to define.  These so not satisfy a universal property in any category, and tend to be described in terms of a construction and properties.  In recent work with Brooks-Hess-Johnson-Rasmusen-Schreiner (BBHJRS for short) we attempted to enumerate the properties required for something to be a system of homotopy colimits.  Following a referee’s comments, we transformed the list into a more familiar categorical construction.  In this talk, I will offer an alternative definition of homotopy colimits using actegories.

Title: What can you buy with a bicategory?

Abstract: In this talk we’ll continue to get to know bicategories by learning about what you can do inside a bicategory. We’ll learn about pasting diagrams as well as adjoints and equivalences inside bicategories before presenting some examples of what these equivalences can look like in certain examples (and in doing so, we will recapture the notion of Morita equivalence of rings). Also Geoff had fun giving this talk.

Title: Saying Hello to Bicategories

Abstract: Bicategories are an important aspect of modern category theory and provide the first instance of “category theory up to coherent isomorphism” we see when hiking up Mount Higher Category Theory. In this talk I will introduce the notion of bicategories, what we can do with them, and explain some of their basic properties. There will also be many, many examples presented in order to both show people who are first becoming introduced to bicategories what flavours in which they can arise and also to keep the talk (somewhat) grounded.

This will be a 2 part talk on Friday Sept 13th and Friday Sept 20th.

Title: The scheaf of distributions

Abstract: Originally distributions were introduced in PDE-theory to generalize the notion of functions. However, this analytical concept also has some purely algebraic properties, which are worth considered.
In particular, they are an important example of sheaves in differential geometry. In this talk we are going to discuss two views on distributions and their connections. On one hand, distributions can be considered as generalized densities (a broader version of the determinant) and on the other hand as generalized functions. The latter, as we are going to show, form a sheaf, which gives us an insight on the sheaf-structure of distributions.

Title: Strict Monoidal Categories as Internal Monoids.

Abstract: Internalization is a paradigm for the generalization of classical mathematical definitions, e.g. monoid generalizes to internal monoid in a monoidal category, or group generalizes to internal group in a category with finite products. In this talk, we will demonstrate that strict monoidal categories may profitably be studied as internal monoids in the 1-category of 1-categories. We present detailed calculations pertaining to limits and colimits of internal monoid objects, while relying on little more than the material of an introductory course in category theory.