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). 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.
Abstract:
In this talk I will try to demystify the notion of twisted theories
using the language of parametrized spectra, all while seeing how an
equivariance group plays into the picture.
I will talk about orientations, K-theory, local coefficient systems, and
Thom spectra, and how each concept admits a natural enhancement or
description in terms of twists.
Finally, I will talk about the natural home for equivariant twisted
(co)homology theories, the category of parametrized G-spectra.
We consider two non-diffeomorphic three-dimensional lens spaces that are of a certain "generic" type and calculate the cohomology of the diffeomorphism group of their connected sum. Our approach builds on techniques of Hatcher, and Boyd, Bregman and Steinebrunner, who recently proved that classifying spaces of diffeomorphism groups of closed oriented 3-manifolds are always of finite type.
Abstract: In this thesis, we outline the methodology for determining the CW-decomposition of Sobolev spaces through the Morse theory of the energy functional. We examine the resulting information about the harmonic maps, including stability and indices as determined by the Jacobi operator. Additionally, we address obstacles to CW decomposition and review recent research aimed at overcoming challenges with these spaces. As the main challenge is that the CW-decomposition cannot be determined through Morse theory, further analysis of the Jacobi operator may be pursued by adopting a Floer-theoretic perspective on harmonic maps from the torus to Kähler manifolds. This approach follows the theory of Oliver Fabert and Ronen Brilleslijper on polysymplectic geometry and the CRPS action functional, which generalizes symplectic geometry. With this, we aim to formulate a relation between the Morse index and the Conley-Zehnder index for minimal surfaces, as was done for geodesics through the symplectic action functional.
I present my master thesis about Configuration Homology, a homology theory I constructed that allows us to geometrically represent homology groups in terms of the homotopy groups of formal sum spaces. In this seminar, I will explain what these formal sum spaces are and how they relate to homology. If time permits, I will also discuss applications of configuration homology: notably, configuration homology gives intuitive models for K(G,n) spaces.
For more information see here
For more information see here
Abstract: One of the notions of h-principle Gromov introduced in his book Partial Differential Relations is that of an h-principle near a submanifold. In this talk I will introduce an alternative perspective on such h-principles by introducing the triangular jet bundle, a geometric object that we can use to reduce to a problem over the submanifold rather than the ambient manifold
One of the classes I remember the most in high school was a biology lesson. We dissected an octopus and studied its innards. In todays talk I will dissect eta, the hopf map using is a classical method, due to Postnikov and Moore. I will discuss the general method and show that it can be used to compute cohomotopy sets. On the Wednesday talk in the spectral sequence seminar, I will discuss the actual spectral sequence computations that are necessary.
Abstract: The clearing method is a well-known algorithmic optimization of computing persistent homology in TDA. In this talk, I will introduce the application of clearing optimization into the computation of persistent Laplacian. This is an ongoing joint work with Magnus Bakke Botnan.
Abstract: Gromov-Witten invariants are important invariants in symplectic and algebraic geometry, with applications in string theory. They roughly count the number of (complex) curves in a space that intersect with some cohomology classes. To spare the audience a long tangent on stacks and virtual fundamental classes, we will avoid the rigorous construction, and instead focus on their role in quantum cohomology. This is a deformation of the cup product whose 'classical limit' is ordinary cohomology. For dessert, we discuss the physical motivation behind Gromov-Witten invariants, and their connection to Floer theory. We heuristically introduce the A-twisted superstring, focusing on the relevance of Gromov-Witten invariants and quantum cohomology. This also makes the classical limit rigorous. Finally, we sketch a proof that Floer cohomology with the pair-of-pants product is isomorphic to quantum cohomology.
Abstract: Laplacian operators are classical objects that are fundamental in both pure and applied mathematics and are becoming increasingly prominent in computational and data science fields and application areas such as machine learning and network science. I will present a unifying operator-theoretic framework of generalized Laplacians that we introduced in our recent paper and that encompasses all existing constructions, from discrete combinatorial settings to de Rham complexes of smooth manifolds. Within this framework, I will introduce and study a generalized notion of persistent Laplacians. While the persistent Laplacians fails to satisfy the desirable properties of monotonicity and stability, I will demonstrate that their component maps, the up- and down-persistent Laplacians, satisfy these properties individually. Moreover, I will provide a condition for full monotonicity and show that the non-zero spectra of these separate components determine the non-zero spectra of the full Laplacians.
Abstract: Morse spacetimes are Lorentz spacetimes with singularities where time has a Morse critical point. The intention of the definition is that the spacetimes are "nice" but still allow the topology of "space" to change with time. Borde and Sorkin conjectured that such Morse spacetimes are causally continuous if and only if neither the index nor the coindex of any critical point is 1. This has been recently confirmed by Garcia-Heveling for the case of small anisotropy and Euclidean background metric. Here, we provide a complementary counterexample: a four dimensional Morse spacetime whose critical point has index 2 and large enough anisotropy is causally discontinuous. Thus, the Borde-Sorkin conjecture does not hold. The proof features a low regularity causal structure and causal bubbling.
The Morse complex is a chain complex that can be assigned to a gradient-like Morse-Smale vector field on a smooth manifold. It consists of vector spaces generated by the fixed points of the vector field and the differential is defined by counting the flow lines between points of adjacent indices. The homology of this chain complex is isomorphic to the singular homology of the manifold, while the acyclic parts describe the topology of the vector field. In this talk we explore how we can go beyond the gradient-like case. After recalling the relevant definitions, we present a method by Franks to replace a closed orbit by a pair of fixed points and explain why there are different, non-equivalent ways to follow this procedure. Then, we present a pipeline to construct a chain complex also in the presence of closed orbits. To this end, we consider a filtration of the manifold by unstable manifolds first described by Smale and the resulting spectral sequence in Cech homology. The first page of this spectral sequence can be endowed with canonical bases, where every fixed point corresponds to one basis element and every closed orbit corresponds to two basis elements. We show how the algebraic information of this spectral sequence can be rearranged into a chain complex whose homology is isomorphic to the singular homology of the underlying manifold. If time permits, we go back to the gradient-like case and describe a method to decompose the Morse complex in order to obtain an invariant which is related to the persistence barcode. This is joint work with Claudia Landi.
Holonomic approximation is the main tool used in proving that the h-principle holds on open manifolds for a large class of partial differential relations. In the first part of this talk I discuss a generalization of holonomic approximation which also works on closed manifolds, using mild singularities known as wrinkles. In the second part, we use this result to carry out the first step of a strategy proposed by Laudenbach and Meigniez to study h-principles via a certain type of singular foliations known as Haefliger structures. We end with an application to the classifying space of principal groupoid bundles which encode Haefliger structures with a transverse geometry.
Analysing the topology or geometry of data is a common task in data
analysis, using methods such as clustering, networks, and persistent
homology.
Often, however, we wish to embellish data with extra local information,
which mathematically can take the form of a quiver representation or
cellular sheaf.
One example of this enriched data comes from analysis of single-cell
epigenetics, and this situation can be further abstracted to a question
about local versus global feature selection.
Graph Laplacians have had great success in the analysis of networks. We
introduce the quiver Laplacian as an analogue of the graph Laplacian for
quiver representations, and formulate a pipeline for its application to
local versus global feature selection.
Applying it to our case study in single-cell epigenetics, we show that
the eigenvalues of the quiver Laplacian can aid in feature selection to
recover biologically meaningful results. To show stability of this
method, we provide explicit bounds on how the spectrum of a quiver
Laplacian changes when the representation and the underlying quiver are
modified in certain natural ways.
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.