The Arbeitsgemeinschaft 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.
Spectral sequences and systems are tools from algebraic topology widely known and used in the study of homology and fibrations. These usually arise form an exact couple system, having a filtration on a space associated.
Parallel to this, connection matrices were developed in the realm of applied topology for qualitative study of dynamical systems. We have a Morse decomposition of a system, and the relative homologies being the Conley index provide us information about it. These two wildly different ideas turn out to be two faces of the same coin. We see how these connection matrices are at its core just a spectral system, and how given an exact couple system one can build a connection matrix and thus a space that generates it with the relative homologies.
Persistent homology is a popular method to compute topological features of metric data. Standard approaches based on the \v{C}ech or Vietoris-Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. Several alternative filtrations have been suggested to address this issue. However, these are only provably robust under relatively tame noise models. In this paper, we take a different perspective and consider the following question: Given metric data $Y = X \cup W$ consisting of uncorrupted data $X$ and a fixed fraction $\alpha \in (0, 1)$ of arbitrary outliers~$W$, which persistent features of~$Y$ can be guaranteed to reflect persistent features of~$X$? We formalize this question by introducing the notion of $\alpha$-robustness, and study the question of deciding whether a given bar in a barcode of $Y$ is $\alpha$-robust.
Join work (in progress) with Pepijn Roos Hoefgeest
The study of embedding spaces Emb(M,N) between smooth manifolds and its homotopy type is a central topic in modern differential topology and low-dimensional topology. In this talk I want to focus on one of the main homotopical tools for dealing with such problems, the Goodwillie-Weiss embedding tower: one associates to Emb(M,N) a sequence of spaces T_k Emb(M,N) whose limit should be a good approximation to the original embedding space. The upshot is that these approximations can be well understood via the formalism of operads, and therefore are more prone to homotopical methods, as I hope to explain in this talk.
[Joint work with Magnus Botnan] Let T(n,k+1) be the Turan graph with n vertices and k+1 partition classes. We study extremal Betti numbers and persistence in edgewise filtrations of flag complexes. For a graph G on n vertices, the kth Betti number of its flag complex is maximized when G = T(n,k+1). Extending this, we construct an edgewise filtration in which each graph attains the maximal kth Betti number among all graphs with the same number of edges. Moreover, the persistence barcode achieves the maximal number of intervals and total persistence among all edgewise filtrations with |E(T(n,k+1))| edges.
For k=1, we analyze edgewise filtrations of the complete graph. The maximal number of barcode intervals occurs precisely when T(n,2) appears in the filtration. Among these, we characterize those achieving maximal total persistence. We also prove that no filtration optimizes the first Betti number for all graphs in the filtration and conjecture that our constructions maximize total persistence over all edgewise filtrations of the complete graph.
Let k be a field and let C be a small category. A k-linear representation of C, or a kC-module, is a functor from C to the category of finite dimensional vector spaces over k. A motivating example for this work is the concept of a tame generalised persistence module, which can be reduced to the case where C is a finite poset. Unsurprisingly, it turns out that when the category C is more general than a linear order, then its representation type is generally infinite and in most cases wild. Hence the task of understanding such representations in terms of their indecomposable factors becomes difficult at best, and impossible in general. In a joint project with Jacek Brodzki and Henri Rihiimaki we proposed a new set of ideas designed to enable studying modules locally. Specifically, inspired by work in discrete calculus on graphs, we set the foundations for a calculus type analysis of kC-modules, under some restrictions on the category C. In this talk I will review the basics of the theory and describe some more recent advances.
Knotoids are a natural generalization of knots consisting of knot diagrams modelled on the interval rather than the circle. Knotoids have found applications in protein topology, but have proven difficult to classify even for low crossing numbers. In this talk I will define biframed knotoids, the proper knotoidal analogue for framed knots, and give two constructions of quantum invariants for biframed knotoids. I will show how these invariants can be used to improve slightly on the classification of planar knotoids, and that they give rise to Vassiliev invariants in analogy with the case of knots. Finally I will describe a combinatorial Kontsevich invariant for biframed spherical knotoids that constitutes a universal Vassiliev invariant.
Blowing up the diagonal of $M^2$ for a manifold $M$ yields a configuration space that remembers the collision axis of collided configurations. Fulton and MacPherson famously generalized this construction to configurations of more than two points. In joint work with A. del Pino, we build configuration spaces for jets of maps $M\to N$ in the same spirit, but adapted to the main structure on jet space: The Cartan distribution and the Lie filtration it generates. Higher jet orders necessitate the use of \textit{weighted} blow-ups, which we tackle within the recent differential-geometric framework of \textit{weightings} due to Loizides and Meinrenken. In this talk, I first intuitively illustrate this motivation and our construction. I will then give an introduction to weightings and discuss how we use them as far as time allows.
The celebrated Poincare-Birkhoff theorem on area-preserving maps of the annulus is of fundamental importance in the fields of Hamiltonian dynamics and symplectic topology. In this talk I will formulate a twist condition, inspired by the Poincare-Birkhoff theorem, which applies to the asymptotically linear Hamiltonian diffeomorphisms of Amann, Conley and Zehnder. When this twist condition is satisfied, together with some technical assumptions, the existence of infinitely many periodic points is obtained.
In 1974 Thurston, in one of his many proofs on foliations, used a technique called jiggling. This shows that, given a distribution, any embedding is homotopic to a piecewise transverse embedding. I will discuss the key ideas in the proof of jiggling and how we can generalize jiggling to certain first order differential relations on bundles. This allows us to make statements about the topology of the space of (piecewise) solutions of such a relation. I will give examples as to how these statements compare to other so-called h-principles.
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 Rham’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.