Topology Festival

Kirby diagrams give us a way to record a smooth four manifold up to diffeomorphism, and Kirby calculus tells us how to modify a Kirby diagram without changing the four manifold it represents. In this talk, I will show how to use Kirby calculus to build a 2-sphere representing the homology class (3,0), which is smoothly embedded in the 4-manifold CP^2 # CP^2.

Khovanov's homology is a strong knot invariant, but very difficult to compute in its initial formulation. In this expository talk, I will present Bar-Natan's ideas that lead to a local version of Khovanov homology (that is, a version for tangles). This local version is both conceptually satisfying and computationally efficient. This is, in particular, the case for 4-ended tangles, for which Kotelskiy, Watson and Zibrowius provided a beautiful geometric interpretation of Bar-Natan's invariant, which we will mention if time allows. This talk is intended to provide some groundwork and context for the plenary talk by Liam Watson.

In the 1970s, Per Martin-Lof introduced his dependent type theory (MLTT) as a constructive foundation for mathematics. Unexpectedly, this type theory allows for a foundational system of mathematics where the most basic objects are the homotopy types of spaces instead of sets. In this talk, I will informally introduce Martin-Lof's type theory with an eye toward these topological connections. Finally I will introduce the univalence axiom, which informally states that two types are equal if and only if they have equivalent structures. No background on mathematical logic is expected.

Homotopy, pseudo-isotopy, and smooth isotopy give three ways to compare self-diffeomorphisms of a smooth manifold. These notions may not coincide, but they can be related in a natural way. In this talk, I will show how to build a pseudo-isotopy from an orientation preserving self-diffeomorphism of the 4-sphere to the identity and discuss how pseudo-isotopy can be used to study the smooth isotopy class of a diffeomorphism.

Differential topologists working in and around dimension 4 enjoy using "diagrams" and some form of "diagrammatic calculus", the prototype being Kirby diagrams and Kirby calculus. Mathematicians working in other fields sometimes seem a little suspicious, or at least feel that it's "not quite fair" that we get to prove things by "just drawing pretty pictures". In this talk I will attempt to think carefully about what diagrams really mean, explain some standard and not-so-standard examples, and pose some questions about just how much mileage we can expect to get using purely diagrammatic methods. In particular, the usual diagram is some 2-dimensional combinatorial data that describes a smooth 4-dimensional object up to a natural smooth isomorphism - but what if we want to understand the isomorphisms themselves? Isomorphisms up to what? Can diagrams help us understand not just objects but also morphisms? Morphisms between morphisms? The main goal is to open a can of worms and leave the audience to clean up the mess!

I will describe some of my joint work with Claudius Zibrowius and Artem Kotelskiy, which recasts Bar-Natan’s invariants for Conway tangles in terms of immersed curves. This gives rise to some unexpected structural results, and I will attempt to place these in context by explaining what they tell us about Khovanov homology for knots admitting an essential Conway sphere.

I will describe the beginnings of a project to understand and visualize singular integral affine structures on the two-sphere via rational polyhedra, integral affine unfoldings of them, and integral affine coordinate charts. Integral affine structures on the two-sphere are induced by special Lagrangian fibrations of K3 surfaces, which are studied in the context of mirror symmetry in dimension four. (The polyhedral presentations are motivated the Gross-Siebert approach to mirror symmetry on K3 surfaces.) More generally, the integral affine geometries are induced by almost toric structures on a symplectic K3 surface (symplectic four-manifold diffeomorphic to a quartic surface in CP3). I will explain how the integral affine structures arise, what some of them look like, and what they reveal about the fibered four-manifolds that produced them.

Contact structures, like foliations, are intimately sensitive to the topology of their ambient manifolds, and have been used to prove deep results in 3-manifold topology. The contact condition is an example of a "topologically stable" condition for k-plane distributions on n-manifolds. Such conditions were classified by Cartan at the turn of the previous century. The one topologically stable condition that is still poorly understood is the Engel condition. Engel structures are 2-plane fields that only exist on 4-manifolds. It is still unknown what, if anything, they might tell us about 4-manifold topology. After discussing background, we will take a first step in this direction by exploring an Engel analogue of transverse knots in contact 3-manifolds.

The Equivalence Principle is an informal principle asserting that equivalent mathematical objects have the same properties. For example, group theory has been developed so that isomorphic groups have the same group-theoretic properties, and category theory has been developed so that equivalent categories have the same category-theoretic properties (though sometimes other, ‘evil’ properties are considered). Vladimir Voevodsky established Univalent Foundations as a foundation of mathematics (based on dependent type theory) in which the Equivalence Principle for types (the basic objects of type theory) is a theorem. Later, versions of the Equivalence Principle for set-based structures such as groups and categories were shown to be theorems in Univalent Foundations.

In joint work with Ahrens, Shulman, and Tsementzis, we formulate and prove versions of the Equivalence Principle for a large class of categorical and higher categorical structures in Univalent Foundations. Our work encompasses bicategories, dagger categories, opetopic categories, and more.

Early versions and our generalization of the Equivalence Principle in Univalent Foundations rely on the fact that the basic objects -- the types -- can be regarded as spaces. That is, Univalent Foundations can be viewed as an axiomatization of homotopy theory and as such is closely related to Quillen model category theory. Univalent Foundations can also be viewed as a foundation of mathematics based not on sets, but on spaces. It is only the homotopical content of this foundation of mathematics that allows us prove something like the Equivalence Principle, something which is not possible in set-based foundations of mathematics, such as ZFC.

The strict hyperbolization process developed by Charney and Davis inputs a simplicial complex, and outputs a piecewise hyperbolic, locally CAT(-1) space. This process has been used to produce aspherical manifolds with various interesting properties. I'll briefly review the strict hyperbolization process, and then explain how to construct an action of the fundamental group of the resulting space on a CAT(0) cube complex with controlled cell stabilizers. This is joint work with Lorenzo Ruffoni (Tufts).

A metric space is called injective if any family of pairwise intersecting balls has a nonempty global intersection. Such injective metric spaces enjoy many properties typical of nonpositive curvature, notably the existence of a convex geodesic bicombing. We will show that many groups of geometric nature have an interesting action by isometries on an injective metric space, including Gromov-hyperbolic groups, braid groups, lattices in Lie groups and mapping class groups of surfaces.

I will talk about a group which might be the trivial group, in which case either the results I will discuss are pointless or are a useful step in the direction of proving triviality! But the group might not be trivial, which is one of the many examples of how little we really understand about dimension 4. The group in question is the smooth mapping class group of the 4-sphere, namely the group of smooth isotopy classes of orientation-preserving self-diffeomorphisms of S^4. Using ideas going back to Cerf I will explain why every element of this group can be described by loops of 2-spheres in connected sums of S^2XS^2's, that under favorable conditions (which might always hold), this description can be reduced to a loop of circles in S^1XS^3 and that the subgroup of the smooth mapping class group that arises under these favorable conditions is at worst a cyclic group of order 2. The last part involves finding explicit generators and relations for the subgroup in question.

Minimum genus bounds for surfaces representing specific homology classes in some 4-manifolds can be used to show that certain knots in S^3 are not slice. For example, one genus bound due to Bryan in the 1990s can be used to show that the (2,1)-cable of the figure eight knot is not slice, recovering a result of Dai—Kang—Mallick—Park—Stoffregen from last summer based on Heegaard Floer (and related) homology. I’ll talk about this construction, underlying motivation, and some interesting open questions. This is joint work with Paolo Aceto, Nickolas A. Castro, JungHwan Park and András Stipsicz.