Session I.4 - Computational Geometry and Topology
Talks
Monday, June 12, 14:00 ~ 14:30
Topological Parallax: a Geometric Specification for Deep Perception Models
Paul Bendich
Geometric Data Analytics, and Duke University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
For safety and robustness of AI systems, we introduce topological parallax as a theoretical and computational tool that compares a trained model to a reference dataset to determine whether they have similar multi-scale geometric structure.
Our proofs and examples show that this geometric similarity between dataset and model is essential to trustworthy interpolation and perturbation, and we conjecture that this new concept will add value to the current debate regarding the unclear relationship between ``overfitting'' and ``generalization'' in applications of deep-learning.
In typical DNN applications, an explicit geometric description of the model is impossible, but parallax can estimate topological features (components, cycles, voids, etc.) in the model by examining the effect on the Rips complex of geodesic distortions using the reference dataset. Thus, parallax indicates whether the model shares similar multiscale geometric features with the dataset.
Parallax presents theoretically via topological data analysis [TDA] as a bi-filtered persistence module, and the key properties of this module are stable under perturbation of the reference dataset.
Joint work with Abraham D. Smith, Gabrielle Angeloro, Michael Catanzaro and Nirav Patel.
Monday, June 12, 14:30 ~ 15:00
Inference in topological data analysis
Wolfgang Polonik
University of California, Davis, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
This talk presents some novel contributions to statistical inference in TDA. These are bootstrap based statistical inference methods for (persistent) Betti numbers and Euler characteristic curves along with their theoretical analyses. Beyond the formal presentation of these results, the presentation also aims at conveying the statistical thinking / interpretation that goes along with these developments. Indeed, while topological data analysis (TDA) has seen a huge increase in popularity, the more statistical aspects of TDA unfortunately tend to be less developed. This observation motivates the postulate underlying this presentation that the communication and the interaction between the statistics and the TDA communities deserves to be enhanced. The presented results are based on recent joint work with Benjamin Roycraft and Johannes Krebs.
Monday, June 12, 15:00 ~ 15:30
Brittany Fasy - TBA
Montana State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Monday, June 12, 16:30 ~ 17:30
Curse of dimensionality in persistence diagrams — How to characterize topology in single-cell resolution —
Yasu Hiraoka
Kyoto, Japan - This email address is being protected from spambots. You need JavaScript enabled to view it.
It is well known that persistence diagrams stably behave under small perturbations to the input data. This is the consequence of stability theorems, firstly proved by Cohen-Steiner, Edelsbrunner, and Harer (2007), and then extended by several researchers. On the other hand, if the input data is realized in a high-dimensional space with a small noise, the curse of dimensionality (CoD) causes serious adverse effects on data analysis, especially leading to inconsistency of distances.
In this talk, I will show several examples of CoD appearing in persistence diagrams and mappers (e.g., from single-cell RNA sequencing data in biology). Those examples demonstrate that the classical stability theorems are not sufficient to guarantee stable behaviors of persistence diagrams for high-dimensional data. Then I will show several mathematical results about the existence and the (partial) resolution of CoD in persistence diagrams. This is a joint work with Enhao Liu, Yusuke Imoto and Shu Kanazawa.
Tuesday, June 13, 14:00 ~ 14:30
Bridging Morse theory and persistent homology of geometric complexes
Ulrich Bauer
Technical University of Munich, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will discuss some recent results on the interplay between geometry and topology, and between Morse theory and persistent homology, in the setting of geometric complexes. This concerns constructions like Rips, Čech, Delaunay, and Wrap complexes, which are fundamental construction in topological data analysis. The tandem of Morse theory and homology shows the topological equivalence of several of these constructions, helps in speeding up their computation by a huge factor (in the software Ripser), reveals thresholds at which homology necessarily vanishes (with links to a classical result by Rips and Gromov), and relates optimal representative cycles for persistent homology to the industry-tested Wrap reconstruction algorithm.
Joint work with Fabian Roll (Technical University of Munich).
Tuesday, June 13, 14:30 ~ 15:00
Topological Optimization with Big Steps
Dmitriy Morozov
Lawrence Berkeley National Laboratory, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude.
Joint work with Arnur Nigmetov (Lawrence Berkeley National Laboratory, USA).
Tuesday, June 13, 16:30 ~ 17:00
Topology of spatiotemporal trajectories
Heather Harrington
University of Oxford, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
Many processes in the life sciences are inherently multi-scale and dynamic. Spatial structures and patterns vary across levels of organisation, from molecular to multi-cellular to multi-organism. With more sophisticated mechanistic models and data available, quantitative tools are needed to study their evolution in space and time. Topological data analysis (TDA) provides a multi-scale summary of data. Recent work by Kim and Memoli introduced an interlevel Rips filtration for the case of dynamic metric spaces, requiring three parameter persistence. In-progress work by Lesnick, Bender and Gäfvert combines F4 and F5 Groebner bases algorithms to compute minimal presentations of multiparameter persistence. Here we build on this work and present an algorithm, GBlandscapes, which computes 3-parameter persistent homology landscapes. We highlight its utility with concrete case studies of spatio-temporal trajectories arising in biological systems.
Joint work with Oliver Gäfvert (University of Oxford), Hamid Rahkooy (University of Oxford), Katherine Benjamin (University of Oxford) and Darrick Lee (University of Oxford).
Tuesday, June 13, 17:00 ~ 17:30
Chad Giusti - TBA
University of Delaware, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Tuesday, June 13, 17:30 ~ 18:00
Short decompositions and joint crossings of embedded graphs on surfaces
Arnaud de Mesmay
CNRS, Université Gustave Eiffel, France - This email address is being protected from spambots. You need JavaScript enabled to view it.
The joint crossing number of two graphs $G_1$ and $G_2$ embedded on a surface is the minimum number of crossings among all homeomorphic reembeddings of one of the graphs. Intuitively, it quantifies how much it would cost to decompose $G_1$, seen as a discrete metric for the surface, along a cutting graph whose shape is specified by $G_2$. An old conjecture of Negami states that this crossing number is always $O(|E(G_1) ||E(G_2) |)$, and it is still wide open, even in the case of one-vertex graphs, i.e., systems of loops.
In this talk,we will discuss this conjecture, its numerous connections with other problems, and survey recent progress, emphasizing recent work with Fuladi and Hubard in the non-orientable case.
Joint work with Niloufar Fuladi (Université Gustave Eiffel, France) and Alfredo Hubard (Université Gustave Eiffel, France).
Wednesday, June 14, 14:00 ~ 14:30
On the effectiveness of persistent homology
Nina Otter
Queen Mary University of London, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
Persistent homology (PH) is, arguably, the most widely used method in Topological Data Analysis. In the last decades it has been successfully applied to a variety of applications, from predicting biomolecular properties, to discriminating breast-cancer subtypes, classifying fingerprints, or studying the morphology of leaves. At the same time, the reasons behind these successes are not yet well understood. We believe that for PH to remain relevant, there is a need to better understand why it is so successful. In this talk I will discuss recent work that tries to take a first step in this direction. The talk is based on joint work with Renata Turkeš and Guido Montúfar.
Wednesday, June 14, 14:30 ~ 15:00
Geometry and physics of periodic tangling
Myfanwy Evans
University of Potsdam, Germany - This email address is being protected from spambots. You need JavaScript enabled to view it.
Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements, which are interesting in the context of physics, biomaterials and chemical frameworks. I will present a systematic way of enumerating and characterising new tangled periodic structures, using low-dimensional topology and combinatorics. In addition, I will show some geometric simulation strategies that are working towards understanding the form and function of these structures as materials, using the philosophy that geometry is at the heart of many physical processes.
Joint work with Stephen Hyde (University of Sydney, Australia) and Rhoslyn Coles (TU Berlin and TU Chemnitz, Germany).
Wednesday, June 14, 15:00 ~ 15:30
Magnitude, Alpha Magnitude and Applications
Sara Kalisnik
ETH, Switzerland - This email address is being protected from spambots. You need JavaScript enabled to view it.
Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known invariants of metric spaces. In recent work, Govc and Hepworth introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. Inspired by Govc and Hepworth’s definition, we introduced alpha magnitude. Alpha magnitude presents computational advantages over both magnitude as well as Rips magnitude, and is thus an easily computable new measure for the estimation of fractal dimensions of real-world data sets. I will also briefly talk about work in progress, a clustering algorithm based on the alpha magnitude. This is joint work with Miguel O'Malley and Nina Otter?
Joint work with Miguel O'Malley and Nina Otter.
Wednesday, June 14, 16:30 ~ 17:00
Distribution of links and their volume in new random link model based on meanders
Anastasiia Tsvietkova
Rutgers University, Newark, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
A link is an embedding of a union of circles in a 3-space, considered up to an ambient isotopy. It is among the main objects of study in low-dimensional geometry, topology, and knot theory. Random structures can be useful for proving statements about properties of a typical topological object. In this paper, we suggest a new random model for links based on meanders. We then prove that trivial links appear with vanishing probability in this model, no link L is obtained with probability 1, and there is a lower bound for the number of non-isotopic knots obtained on every step.
A random meander is obtained through matching pairs of parentheses, a well-studied problem in combinatorics. Hence tools from combinatorics can be used to investigate properties of random links in this model, and, moreover, of the respective 3-manifolds that are link complements in 3-sphere. One of the strongest invariants for such manifolds is hyperbolic or simplicial volume. We give expected twist number of a link diagram and use it to bound expected hyperbolic and simplicial volume of random links. The tools from combinatorics that we use include Catalan and Narayana numbers, and Zeilberger's algorithm.
Joint work with Nicholas Owad (Hood College).
Wednesday, June 14, 17:00 ~ 17:30
Gordian Unlinks
Jose Ayala Hoffmann
Universidad de Tarapacá, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
A Gordian unlink is a finite number of not topologically linked unknots with prescribed length and thickness that cannot be disentangled to the trivial link by an isotopy preserving length and thickness throughout.
In this talk, we present the first examples of Gordian unlinks. As a consequence, we detect the existence of isotopy classes of thick unknots different from those in classical knot theory. More generally, we present a one-parameter family of Gordian unlinks with thickness ranging in [1, 2) to conclude that thinner normal tubes lead to different rope geometry from the ones so far considered.
Wednesday, June 14, 17:30 ~ 18:30
The Gromov-Hausdorff distance between spheres
Facundo Mémoli
Ohio State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
The Gromov-Hausdorff distance is a fundamental tool in Riemannian geometry (through the topology it generates) and is also utilized in Applied Geometry and Topological Data Analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in Machine Learning Applications.
Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its precise value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology features and to Gromov's filling radius. However, these turn out to be non-sharp.
In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) ``correspondences" between spheres.
Posters
Extending Morse-Forman theory to vector functions
Guillaume Brouillette
Université de Sherbrooke, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
In the last decades, discrete Morse theory, or Morse-Forman theory, has proven itself to be useful for a wide range of applications, notably in topological data analysis. In particular, when analyzing a simplicial complex filtered by a real-valued function, the theory may be used to reduce its complexity and to optimize the computation of its associated persistent homology. Recent work has shown that it can be adapted in order to compute more efficiently the multipersistent homology of a complex induced by a vector function. Nevertheless, the theorical implications and geometrical insights of this adaptation have not yet been extensively investigated.
To gain some perspective on the matter, in this presentation, we extend Morse-Forman theory to vector-valued functions, thus defining the concept of multidimensional discrete Morse (MDM) functions. To do so, we use notions of combinatorial dynamics as defined in recent years to adapt the main definitions and theorems of Forman to the vectorial setting. This leads to a more general result than that of Forman concerning the sublevel sets of a MDM function and to an alternative approach on the matter. Moreover, we propose a way to derive a Morse decomposition in critical components from the critical points of a MDM function. This is specific to the multidimensional setting since critical points of real discrete Morse functions are isolated, and thus do not form components. Finally, from this Morse decomposition, we deduce new Morse equation and inequalities.
Joint work with Madjid Allili (Bishop's University, Canada) and Tomasz Kaczynski (Université de Sherbrooke, Canada).
Intrinsic persistent homology via density-based metric learning
Ximena Fernandez
Durham University, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this poster, I will explain a density-based method to address the problem of estimating topological features from data in high dimensional Euclidean spaces under the manifold assumption. The key of our approach is to consider a sample metric known as Fermat distance to robustly infer the homology the space of data points. I will show results that prove that such sample metric space GH-converges almost surely to the manifold itself endowed with an intrinsic (Riemannian) metric that accounts for both the geometry of the manifold and the density that produces the sample. This fact, joint with the stability properties of persistent homology, implies the convergence of the associated persistence diagrams, which overcome many weaknesses of the standard methods for homology inference. I will show that these intrinsic density-based diagrams are robust to the presence of (geometric) outliers in the input data and less sensitive to the particular embedding of the underlying manifold in the ambient space. Finally, I will exhibit a concrete application of these ideas to time series analysis, with examples in real data.
This poster is based on the article: X. Fernandez, E. Borghini, G. Mindlin and P. Groisman. Intrinsic persistent homology via density-based metric learning. Journal of Machine Learning Research (to appear, 2023) arXiv:2012.07621.
Joint work with Eugenio Borghini (Universidad de Buenos Aires, Argentina), Gabriel Mindlin (Universidad de Buenos Aires, Argentina) and Pablo Groisman (Universidad de Buenos Aires, Argentina).
Harder-Narasimhan and Persistence modules: single central charge
Marc Fersztand
Oxford University, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
The Harder-Narasimhan type of a quiver representation is a discrete invariant parameterised by a real-valued function (called a central charge) defined on the vertices of the quiver. We investigate the strength and limitations of the Harder-Narasimhan type for several families of quiver representations which arise in Topological Data Analysis. In order to evaluate the discriminative power of this invariant, we consider families of persistence modules whose irreducible decomposition is known: (1) for zigzag (and hence, ordinary) persistence modules, we completely characterise the set of central charges for which the Harder-Narasimhan type is a complete invariant (2) we extend the preceding characterisation to rectangle-decomposable multiparameter persistence modules of arbitrary dimension; and finally, (3) in the framework of persistence of circle-valued maps, we show that the barcode can be partially recovered using the Harder-Narasimhan type with a suitable choice of central charge.
This work is detailed in the following preprints https://arxiv.org/abs/2303.16075 (Sections 4 and 5.2) and https://arxiv.org/abs/2211.07553.
Joint work with Emile Jacquard (Oxford University), Vidit Nanda (Oxford University) and Ulrike Tillmann (Oxford University).
A Shape Characterization of Quadrics
Georgi Georgiev
Konstantin Preslavski University of Shumen, Bulgaria - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $D \subseteq \mathbb{R}^2$ be a domain and let $S: D\rightarrow \mathbb{R}^3$ be a surface of class $ C^3 $ with a parametrization $\{ S(u,v), (u,v)\in D\}$ and a unit normal vector field $\{N(u,v), (u,v)\in D \}$.There are two important types of surfaces which are associated to $S$. The surfaces of the first type form a one-parameter set of parallel surfaces $\{ S_{d} ,d \in \mathbf{R} \}$ with parameterizations $ S_{d}(u,v)=S(u,v)+ d N(u,v), (u,v)\in D $. The surfaces of the second type are two focal surfaces $S_{f1}$ and $S_{f2}$ of $S$ parameterized by \[ S_{f\textstyle{i}}(u,v)=S(u,v)+\frac{1}{f_{i}(u,v)} N(u,v), \quad (u,v)\in D , \, i=1,2, \] where $f_{1}(u,v)$ and $f_{2}(u,v)$ are the principal curvatures at the point S(u,v) of $S$. Any Euclidean motion of $\mathbb{R}^3$ preserves the pairs $\big(S, S_d\big)$, $\big(S, S_{f1}\big)$ and $\big(S, S_{f2}\big)$. Therefore, there is a large number of papers dealing with construction and investigation of $S_d$, $ S_{f1}$ and $S_{f2}$ for different classes of surfaces in $\mathbb{R}^3$.
Hans Hagen and Stefanie Hahmann introduced and studied particular cases of another associated surface $S_{g}$ to $S$ which is called a generalized focal surface of $S$. The surface $S_{g}$ is defined by the vector parametric equation \[ S_{\textstyle{g}}(u,v)=S(u,v)+g(u,v) N(u,v), \quad (u,v)\in D, \] where $g(u,v)$ is a function of $f_{1}(u,v)$ and $f_{2}(u,v)$.
The smallest extension of the group of Euclidean motions of $\mathbb{R}^3$ is the group of the direct similarities of $\mathbb{R}^3$ denoted by $Sim^{+}(\mathbb{R}^3)$. Any element of the last group is an affine transformation that preserves the orientation and the angles. The direct similarities preserve the shape of geometric objects.
In previous author's paper it was proved that the functions $g_1(u,v)$ and $g_2(u,v)$ on a surface $S$ with a non-vanishing Gaussian curvature $K$ and a non-vanishing mean curvature $H$ given by $g_1(u,v)=H/K=(f_{1}(u,v)+ f_{2}(u,v))/(2f_{1}(u,v) f_{2}(u,v))$ and $g_2(u,v)=1/H=2/(f_{1}(u,v)+ f_{2}(u,v))$ are similarity invariant functions of $S$. Then, for a surface $S$ with a non-vanishing Gaussian curvature $K$ and a non-vanishing mean curvature $H$, there are determined two generalized focal surfaces \[ S_{\textstyle{g1}}(u,v)=S(u,v)+g_1(u,v) N(u,v), \quad (u,v)\in D, \] and \[ S_{\textstyle{g2}}(u,v)=S(u,v)+g_2(u,v) N(u,v), \quad (u,v)\in D. \] Any element $\sigma$ of $Sim^{+}(\mathbb{R}^3)$ preserves the pairs $\big(S, S_{g1}\big)$, and $\big(S, S_{g2}\big)$. This means that $\sigma(S_{g1})$ is the generalized focal surface of $\sigma(S)$ of the same kind as $\big(S, S_{g1}\big)$.
In this poster presentation it is computed similarity invariant functions $g_1(u,v)$ and $g_2(u,v)$ for an ellipsoid, a hyperboloid of one sheet,a hyperboloid of two sheets, an elliptic paraboloid and a hyperbolic paraboloid. For the same quadrics, the parametric equations of the generalized focal surfaces $ S_{g1}$ and $ S_{g2}$ are calculated. Both similarity invariant functions and the considered generalized focal surfaces are related to the shape of the mentioned quadrics.
Harder-Narasimhan and persistence module: multiple central charges
Emile Jacquard
University of Oxford, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
The work presented here is best illustrated by Sections 3, 5.1 and 6 of the preprint \url{https://arxiv.org/abs/2303.16075}. This was written by Ulrike Tillmann, Vidit Nanda, Marc Fersztand and Emile Jacquard, all affiliated with the University of Oxford.
The Harder-Narasimhan type of a quiver representation is a discrete invariant parameterised by a real-valued function (called a central charge) defined on the vertices of the quiver. We introduce the skyscraper invariant, which amalgamates the HN types along central charges supported at single vertices, and generalise the rank invariant from multiparameter persistence modules to arbitrary quiver representations. Our main results are as follows: (1) we show that the skyscraper invariant is strictly finer than the rank invariant in full generality, (2) we show that although no single central charge is complete for nestfree ladder persistence modules, a finite set of central charges is complete.
A central charge on a quiver $Q$ is a function $\alpha \colon Q_0 \mapsto \mathbb{R}$. The $\alpha$-slope of a representation $V$ of $Q$ is the ratio $$ \mu_\alpha(V) = \frac{\sum_{x \in Q_0 }{\alpha(x) \cdot \dim V_x}}{\sum_{x \in Q_0} \dim V_x}. $$ $V$ is $\alpha$-semistable if all its subrepresentations have smaller $\alpha$-slope. Every nonzero representation $V$ admits a unique {\em Harder-Narasimhan} filtration of finite length \[ 0 = HN^0_\alpha(V) \subsetneq HN^1_\alpha(V) \subsetneq \cdots \subsetneq HN^{n-1}_\alpha(V) \subsetneq HN^n_\alpha(V) = V \] whose successive quotients $S^i := HN^i_\alpha(V)/HN^{i-1}_\alpha(V)$ are $\alpha$-semistable and satisfy $\mu_\alpha(S^i) \gt \mu_\alpha(S^{i-1})$ for all $i$. The main discrete invariant of interest is the {\bf HN type} of $V$ along $\alpha$ \[ \mathbf{T}[\alpha][V]= \left(\underline\dim_{S^1},\underline\dim_{S^2} \ldots, \underline\dim_{S^n}\right). \] Given a collection of central charges, the collection of associated HN types is also a discrete invariant. Consider, for each vertex $x$, the central charge $\delta_x:Q_0 \to \mathbb{R}$ which maps $x$ to $1$ and all other vertices to $0$. The skyscraper invariant $\delta_V$ is the collection of HN types $\mathbf{T}[\delta_x][V]$ indexed by the vertices of $Q$. Our main result is
$\textbf{Theorem 1}$: The skyscraper invariant is finer than the rank invariant on $Rep(Q)$ for any finite quiver $Q$.
We prove this using the spanning subrepresentation of $V$ at a vertex $x$ --- this is defined up to isomorphism as the smallest subrepresentation $\langle V_x \rangle \subset V$ containing $V_x$. The function $\rho_V:Q_0 \times Q_0 \to \mathbb{N}$ that sends each $(x,y)$ to the dimension of $ \langle V_x \rangle_y$ vastly generalises the rank invariant of Carlsson and Zomorodian.
Next, we consider nest-free ladders (commuting representations of the ladder quiver with no nested bars in the top and bottom rows), and show that no single central charge yields a complete invariant, but that we may find a collection of central charges whose associated HN-types yield a complete invariant.
$\textbf{Theorem 2}$: There is no complete central charge on the category of nestfree ladder persistence modules of length $\ell \geq 4$; however, for all $\ell$ there exists a finite set $A = A(\ell)$ of central charges which is complete on this category.
Joint work with Ulrike Tillmann, Vidit Nanda and Marc Fersztand.
Discrete group actions on 3-manifolds and embeddable Cayley complexes
George Kontogeorgiou
University of Warwick, United Kingdom - This email address is being protected from spambots. You need JavaScript enabled to view it.
A theorem of Maschke [2, p. 287] states that a finite group acts discretely and topologically on $\mathbf{S}^2$ if and only if it has an alternative Cayley graph that embeds equivariantly in $\mathbf{S}^2$. Recently, Georgakopoulos [1] generalised this theorem to finitely generated groups. We extend the above results to three dimensions. Namely, we prove that a finitely generated group $\Gamma$ admits a discrete topological action on a simply connected 3-manifold if and only if $\Gamma$ has a generalised Cayley complex that embeds equivariantly in one of the following four 3-manifolds: (i) $\mathbf{S}^3$ , (ii) $\mathbf{R}^3$ , (iii) $\mathbf{S}^2 \times \mathbf{R}$, and (iv) the complement of a tame Cantor set in $\mathbf{S}^3$. In the process, we derive a combinatorial characterization of the finitely generated groups that act discretely and topologically on simply connected 3-manifolds.
[1] Georgakopoulos, A. On planar Cayley graphs and Kleinian groups. Trans. Amer. Math. Soc. Vol. 373, pp. 4649-4684, 2020.
[2] Gross, J.L. and Tucker, T.W. (1987). Topological Graph Theory. John Wiley & Sons.
Joint work with Agelos Georgakopoulos (University of Warwick).
A topological and 3D-geometrical study about an initial skeleton of the 2mass pleiades nucleus with DR2 distance
Francisco J. Marco Castillo
Universitat Jaume I, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
There are many works about the Pleiades, despite which there are still unknowns about them, including their exact number. The Pleiades are subjected to external and internal forces, which, if identified, can help determine their origin and evolution. Our goal does not pretend to be that ambitious. We will study the issue from a different perspective requiring several phases, being the first the one we will study in this poster. It consists of identifying some subset of their core by imposing geometric restrictions on the "Pleiadest" set contained in Vizier. We will consider only those belonging to 2MASS but using the stellar distances estimated by the DR2. The type of restrictions has a lot to do with the distances between stars (related to gravitational properties). Since we are concerned about the nucleus, the discrimination of stars will also consider high densities in their spatial distribution. After the adoption of such a subset (which is not unique, although it is preferable to take it small and then increase it with more 2MASS and DR2 stars that are not in 2MASS), it will be treated as a three-dimensional point cloud and applied Analysis techniques in Topological Data, in order to probe the topological structure of the nucleus of such an open cluster. This subject will be dealt with from a classical point of view and also from a dual point of view, using a net of points to which nill densities are assigned through calculation. These points are also likely to be studied using TDA techniques. The results of both approaches should necessarily coincide for the conclusions to be acceptable. Finally, we will try to relate the astrophysical properties to different topological-geometric elements obtained in the first part. Summing up, the main goal of the poster is the deduction of a primary skeleton for the nucleus of the Pleiades that, on the one hand, adjusts well to astrophysical parameters and, on the other, can be expanded in subsequent studies.
Joint work with María José Martínez Usó (Universitat Politècnica de València, Spain) and José Antonio López Ortí (Universitat Jaume I, Spain).
Real Polarization via Bohr Sommerfeld Geometric Quantization for Quadrics
Som Sadashiv Phene
University of Michigan Ann Arbor, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Following Miranda-Presas-Solha's sheaf cohomology approach to compute Geometric Quantization for almost toric manifolds, in particular K3 surfaces, we establish whether counting Bohr Sommerfeld leaves in the interior of polytope in the case of Quadrics gives the same result as Kahler Quantization. This forms an important model for Gelfand-Cetlin type flags studied by Guillemin-Sternberg.