Topology Seminar Monday 16 January, 14:15-15:15, VA2 Varmeteknisk

Kim Anders Frøyshov (Universitetet i Oslo): 4-manifolds and instanton homology

Abstract: The starting point for this talk is the by now classical diagonalization theorem of Donaldson, which says that if the intersection form of a closed oriented smooth 4-manifold X is definite, then the intersection form is diagonal over the integers. It is natural to ask which unimodular definite forms may occur as intersection form if X is allowed to have a fixed homology sphere Y as boundary. Attempts at adapting proofs of Donaldson's theorem to this situation (using either instantons, Seiberg-Witten monopoles, or Heegaard Floer theory) naturally lead to numerical invariants of Y defined in terms of various kinds of Floer homology of Y. These invariants constrain the "exoticness" of any definite form bounded by Y. I will review both classical and more recent results in this area and also discuss ongoing work of my own using mod 2 instanton homology.

Topology Seminar Monday 06 February, 14:15-15:15, VA2 Varmeteknisk

Sira Gratz (Aarhus): Thick Subcategories and Lattices

Abstract: The computation of lattices of thick subcategories has emerged as a popular topic and serves as a more achievable analogue of classifying objects. Often one understands such lattices by describing them in terms of some associated topological space. However, in many representation theoretic examples this is not possible. I’ll explain what the obstruction is and introduce possible methods of addressing this issue. This talk is based on joint work with Greg Stevenson.

Topology Seminar Tuesday 14 February, 11:15-12:15, EL2 Gamle elektro

Lisbeth Fajstrup (Aalborg): Directed Algebraic Topology – collapsing results and issues.

Abstract: The objects of Directed algebraic topology are topological spaces with a direction, a selected subset of the paths. Here, the space is a finite Euclidean cubical complex – a union of unit cubes (of varying dimension) in a Euclidean space and the directed paths are those which are non-decreasing in all coordinates. The space of directed paths from a given initial vertex is determined by the past links at all vertices in the sense that if all past links a contractible/connected, then all these path spaces are contractible/connected. And these obstructions, i.e. non-trivial past links, can be realized in a certain sense. This leads to conditions for collapsing of cubes. Unlike the non-directed case, where collapsing corresponds to a homotopy, here directed homotopy is much too rigid to use as a condition. We will give an introduction to directed topology and then dive into the combinatorial examples and see why these collapsibility results cannot be as satisfying as in the non-directed case.

Topology Seminar Monday 6 March, 14:15-15:15, Simastuen 656

Tashi Walde (TU München): Lax additivity in categorified homological algebra

Abstract: Recent developments in representation theory, algebraic geometry and symplectic geometry have motivated the study not just of individual stable infinity-categories but also of chain complexes thereof, hinting at a deeper theory of “categorified homological algebra”. Just like classical homological algebra is most naturally formulated in the context of abelian categories, one expects the natural setting of such a theory to be “2-abelian" (infinity,2)-categories—a concept that is yet to be defined.

In this talk I will focus on an easier notion that lies partway towards “abelian categories” and is fundamental in its own right, namely “additivity”. We introduce the corresponding 2-categorical notion of _lax additivity_ and show that the (infinity,2)-category of stable infinity-categories is lax additive. Using this basic example as a guide, we show how the setting of lax additivity suffices to categorify many basic constructions/notions from homological algebra, such as mapping complexes, chain homotopies and mapping cones.

This talk is based on joint work with Merlin Christ and Tobias Dyckerhoff (Hamburg).

Topology Seminar Wednesday 8 March, 14:15-15:15, Simastuen 656

Luca Pol (Regensburg): Global homotopy theory via partially lax limits

Abstract: Global homotopy theory is the study of equivariant objects which exist uniformly and compatibly for all compact Lie groups in a certain family, and which exhibit extra functoriality. In this talk I will present new infty-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limit to formalize the idea that a global object is a collection of G-objects, one for each compact Lie group G, which are compatible with the restriction-inflation functors. This is joint work with Sil Linskens and Denis Nardin.

Topology Seminar Monday 20 March, 14:15-15:15, Simastuen 656

Ambrus Pal (Imperial College London): An arithmetic Yau-Zaslow formula

Abstract: Enriched enumerative geometry is a new area in which we take results in enumerative geometry over the complex numbers and refine them to give results over any base field. The "refinements" in question recover the classical results over algebraically closed fields but may also include arithmetic information about the base field. In this talk, I'll give an overview of a proof of an enriched refinement of the Yau-Zaslow formula for counting rational curves on K3 surfaces. Joint work with Jesse Pajwani.