The Arbeitsgemeinschaft Differential Topology commenced in fall 2022. AG Meetings are on Tuesdays, unless explicitly mentioned, in the seminar room of the ninth floor of the NU building from 11:00 until max 12:30 (we will typically finish sooner). Fishy AG (joined with Utrecht) is held on Friday 14:00-17:00. Please send me an email if you are interested in attending or show just up at one of the meetings. There is a mailing list you can join.

FISHY AG: The friday sessions are joined with Utrecht. We will be reading on factorization homology.

The Link of a singularity is naturally a contact submanifold of S^{2n-1} with the standard contact structure. In this talk, I will try to sketch the symplectic and contact phenomena that appear in the study of Isolated Singularities, going over some key results that motivate current research. Time permitting, I will also mention joint work with F. Pasquotto and A. Zanardini.

Abstract: A net is a connected simple infinite graph. It is n-periodic if it is periodic in n-directions, i.e. its automorphism group contains a subgroup that is the group of n independent translations. A crystallographic net is a n-periodic net whose maximal symmetry can be realized in an embedding. Group theory, geometric topology (for their entanglements), tilings, and (minimal) surfaces are used for their study. The aim of this presentation is to introduce crystallographic nets, which are the objects of interest in topological crystal chemistry.

Abstract: Every abstract graph can be embedded on a closed oriented surface and the genus range for the surface is known. If, instead of an abstract graph, an embedding of a graph is considered, it is still easy to find a closed oriented surface on which the spatial graph is embedded on by placing the graph on the boundary of a tubular neighbourhood of the graph. But in general there is no good control over the complement of the graph in a surface, it could be a union of discs with any number of punctures. We are interested in finding surfaces for a spatial graph, such that the complement of the graph in the surface is a set of open discs, so called cellular embeddings. To this end, we introduce a new family of spatial graphs, called levelled embeddings. The defining feature of levelled embeddings is their decomposition into planar subgraphs, all of which are interconnected through a common cycle within the graph. This structure allows for a systematic exploration of their embedding possibilities. We prove that levelled embeddings of low complexity can always be cellular embedded. We extend this result in a sufficient condition for finding cellular embeddings of levelled graphs with arbitrarily high complexity.

Abstract: Two links are called link-homotopic if they are transformed to each other by a sequence of self-crossing changes and ambient isotopies. The notion of link-homotopy is generalized to spatial graphs and it is called component-homotopy. The link-homotopy classes were classified by Habegger and Lin through the classification of the link-homotopy classes of string links. In this talk, we classify colored string links up to colored link-homotopy by using the Habegger-Lin theory. Moreover, we classify colored links and spatial graphs up to colored link-homotopy and component-homotopy respectively. This research is joint work with Atsuhiko Mizusawa.

Abstract: Stability problems appear in various forms throughout geometry and algebra. For example, given a vector field $X$ on a manifold that vanishes in a point, when do all nearby vector fields also vanish somewhere? As an example in algebra, we can consider the following question: Given a Lie algebra $\mathfrak g$, and a Lie subalgebra $\mathfrak h$, when do all deformations of the Lie algebra structure on $\mathfrak g$ admit a Lie subalgebra close to $\mathfrak h$? I will show that both questions are instances of a general question about differential graded Lie algebras, and under a finite-dimensionality condition which is satisfied in the situations above, I will give a sufficient condition for a positive answer to the general question. I will then discuss the application to fixed points of Lie algebra actions.

Abstract: In this talk we will introduce various aspects of non-Hausdorff manifolds, constructed from first principles. Typically, the Hausdorff property is included in the definition of a manifold for technical convenience, and the alternative may seem somewhat daunting: without the Hausdorff property we do not have access to partitions of unity in their full generality, and thus various structures may or may not exist in the non-Hausdorff case. However, as we will see, certain topological representations allow us to circumvent this issue and recreate differential geometry without the need of arbitrarily-existent partitions of unity. To illustrate this idea, we will start from the topology of non-Hausdorff manifolds and then introduce more and more structure of various interest, finally finishing with a proof of de Rhamâ€™s Theorem.

Abstract:I will survey some homotopy theoretic techniques (based on two flavours of functor calculus) to study the space of embeddings of manifolds. One approach is very well understood, the other largely conjectural. Time permitting I will lay out a strategy for remedying some of the conjectural nature of the latter approach.

Orbifolds and sub-Riemannian geometry are interesting generalizations of the concept of manifold. Orbifolds generalize manifolds by incorporating singularities, while sub-Riemannian manifolds exclude specific geodesics and restrict movement to chosen subsets. But how to define a sub-Riemannian structure on an orbifold? I will talk about parking cars, falling cats, barber shops and teardrops in order to discuss these generalizations.

First I focus on the example of lens space, which are quotient spaces without singularities and where a unique "Cartan decomposition" can be defined. This decomposition yields intriguing properties for the sub-Riemannian dynamics. Defining sub-Riemannian orbifolds in general poses several challenges. In the talk I address these challenges and show cases where we can define a sub-Riemannian structure on an orbifold.

Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In recent work that will be presented in this talk, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations.

Abstract: Many partial differential equations are encoded by proper Fredholm maps between (infinite dimensional) Hilbert spaces. By the Pontryagin-Thom construction these maps correspond to finite dimensional framed submanifolds. This gives a connection between finite and infinite dimensional topology. In this talk, I will use this relation to classify proper Fredholm maps (up to proper homotopy) between Hilbert spaces in terms of the stable homotopy groups of spheres. This is based on work in progress with Thomas Rot.

Cut and paste or SK groups of manifolds are formed by quotienting the monoid of manifolds under disjoint union with the relation that two manifolds are equivalent if I can cut one up into pieces and glue them back together to get the other manifold. Cobordism cut and paste groups are formed by moreover quotienting by the equivalence relation of cobordism. We categorify these classical groups to spectra and lift two canonical homomorphisms to maps of spectra. This is joint work with Mona Merling, Laura Murray, Carmen Rovi and Julia Semikina.