The 38th Annual

(Online) Workshop in
Geometric Topology

June 15 - 17, 2021

   

Abstracts of Talks

Henry Adams, Colorado State University
Title: Bridging applied and geometric topology
Abstract: I will advertise open questions in applied topology for which tools from geometric topology are relevant. If a point cloud is sampled from a manifold, then as more samples are drawn, the persistent homology of the Vietoris-Rips complex of the point cloud converges to the persistent homology of the Vietoris-Rips complex of the manifold. But what are the homotopy types of Vietoris-Rips complexes of manifolds? Essentially nothing is known, except that as the scale parameter increases, the Vietoris-Rips complex of the circle obtains the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., until finally it is contractible. I will survey emerging connections between Vietoris-Rips complexes of manifolds and the filling radius (which Gromov used to prove the systolic inequality in 1983), thick-thin decompositions, and sweepouts. This is joint work with Baris Coskunuzer.

Ulysses Alvarez, Binghamton University
Title: The Up Topology on the Grassmann Poset
Abstract: For a discrete poset \(\mathcal{X}\), McCord proved that there exists a weak homotopy equivalence from the order complex \(|\mathcal{X}|\) to where \(\mathcal{X}\) has the Up topology. Much later, Živaljević defined the notion of the order complex of a topological poset. For a large class of such topological posets we prove the analog of McCord's theorem, namely that there exists a weak homotopy equivalence from the order complex to the topological poset with the Up topology. An interesting example of a topological poset of said class is the Grassmann poset, that is, the collection of nonzero proper linear subspaces of \(\mathbb{R}^{n+1}\), whose order complex is understood to be homotopy equivalent to the \(m\)-sphere for some \(m\). In particular, there is a weak homotopy equivalence from the \(m\)-sphere to the Grassmann poset with the Up topology. (This is joint work with Ross Geoghegan)

Arka Banerjee, University of Wisconsin-Milwaukee
Title: Obstruction to coarse embedding
Abstract: Van Kampen developed an obstruction theory for embedding a finite n-complex K into Euclidean 2n-space. A modern approach to his theory uses (co)homology of the configuration space of K. We develop a coarse analog of Van Kampen's theory. More precisely, we produce an obstruction for coarsely embedding a metric space X into Euclidean n-space by using a notion of coarse configuration space of X.

Rima Chatterjee, Louisiana State University
Title: Knots and links in overtwisted manifolds
Abstract: Knot theory associated to overtwisted manifolds are less explored. There are two types of knots/links in an overtwisted manifold namely loose and non-loose. In this talk, I'll give a brief overview about these knots and talk about my work on coarse classification of loose null-homologos links. I'll also define an invariant called support genus for links and show it vanishes for loose links. If time permits, I will also talk about my recent work with Cahn- Chernov about strengthening this classification result.

Lizhi Chen, Lanzhou University
Title: Topological complexity of manifolds via systolic geometry
Abstract: We discuss homology and homotopy complexity of manifolds in terms of Gromov’s systolic inequality. The optimal constant in systolic inequality is usually called systolic volume. A central theorem in systolic geometry relates systolic volume to simplicial volume. Since for hyperbolic manifolds there exist proportionality principle, this theorem builds a bridge between systolic geometry and hyperbolic geometry. In the talk, we will present some applications of this theorem to the problem of homology and homotopy complexity of manifolds.

Arun Debray, The University of Texas at Austin
Title: Stable diffeomorphism classification of some unorientable 4-manifolds
Abstract: Kreck's modified surgery theory provides a bordism-theoretic classification of closed, connected 4-manifolds up to stable diffeomorphism, i.e. up to diffeomorphism after connect-sum with some number of copies of \(S^2 \times S^2\). For some classes of unorientable 4-manifolds with fundamental group \(\pi_1\) finite of order 2 mod 4, the classification question simplifies considerably, reducing to the case where \(\pi_1 = \mathbb{Z}/2\). In this talk, I'll explain the generalities of Kreck's theorem and the ingredients that go into it, then specialize and give the classification in the case where \(\pi_1\) is finite of order 2 mod 4.

Jialong Deng, University of Goettingen
Title: Enlargeable Length-structures and Scalar Curvatures
Abstract: We define enlargeable length-structures on closed topological manifolds and then show that the connected sum of a closed \(n\)-manifold with an enlargeable Riemannian length-structure with an arbitrary closed smooth manifold carries no Riemannian metrics with positive scalar curvature. We show that closed smooth manifolds with a locally CAT(0)-metric which is strongly equivalent to a Riemannian metric are examples of closed manifolds with an enlargeable Riemannian length-structure. Moreover, the result is correct in arbitrary dimensions based on the main result of a recent paper by Schoen and Yau.

We define the positive \(MV\)-scalar curvature on closed orientable topological manifolds and show the compactly enlargeable length-structures are the obstructions of its existence.

Ethan Farber, Boston College
Title: Constructing pseudo-Anosovs from expanding interval maps
Abstract: Pseudo-Anosov homeomorphisms of a surface are encoded by their action on a graph embedded in the surface. If this graph is isomorphic to an interval, then the graph map in question is conjugate to a piecewise-linear interval map. In this talk we partially classify which interval maps arise in this way, and that there is a natural way to assign a rational number to each such map. This rational number is the rotation number at infinity of the pseudo-Anosov, and grows monotonically in the dilatation of the pseudo-Anosov.

Ximena Fernández, Swansea University
Title: Intrinsic persistent homology via density-based metric learning
Abstract: Typically, persistence diagrams computed from a sample depend strongly on the distance associated to the data. When the point cloud is a sample of a Riemannian manifold embedded in a Euclidean space, an estimator of the intrinsic distance is relevant to obtain persistence diagrams from data that capture its intrinsic geometry.

In this talk, we consider a computable estimator of a Riemannian metric known as Fermat distance, that accounts for both the geometry of the manifold and the density that produces the sample. We prove that the metric space defined by the sample endowed with this estimator (known as sample Fermat distance) converges a.s. in the sense of Gromov-Hausdorff to the manifold itself endowed with the (population) Fermat distance. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this approach outperforms more standard methods based on Euclidean norm, with theoretical results and computational experiments [1].

[1] E. Borghini, X. Fernández, P. Groisman, G. Mindlin. 'Intrinsic persistent homology via density-based metric learning'. arXiv:2012.07621 (2020)

Jonah Gaster, University of Wisconsinsin-Milwaukee
Title: Vertical arcs and the Markov Uniqueness Conjecture
Abstract: The Markov Uniqueness Conjecture concerns a correspondence on the modular torus that ties together geometry, topology, and number theory. I will describe some new geometric reformulations of the conjecture, and present some intriguing experimental data.

Ryan Grady, Montana State University
Title: Persistence over the Circle
Abstract: In this talk we will construct algebraic topological invariants of persistence modules on the circle. In particular, we will discuss the K-theory of such modules. Examples from periodic data/functions will also be introduced. This talk is based on joint work with Anna Schenfisch.

Pawel Grzegrzolka, Stanford University
Title: Asymptotic dimension of fuzzy metric spaces
Abstract: In this talk, we will discuss asymptotic dimension of fuzzy metric spaces. After a short introduction to fuzzy metric spaces and large-scale geometry, we will show how fuzzy metric spaces can be studied from a large-scale point of view. In particular, we will define asymptotic dimension of fuzzy metric spaces and show a few of its interesting properties, including its relationship to asymptotic dimension of metric spaces.

Burns Healy, University of Wisconsin-Milwaukee
Title: Group boundaries under semidirect products with the integers
Abstract: Given a group G that admits a Z-structure, we demonstrate a way to explicitly build a Z-structure for any group of the form G semidirect product with the integers. This procedure preserves many desired structures of the space and gives an exact description of the Z-set. As applications, we show one natural way to define group boundaries for all finitely generated nilpotent groups and all 3-manifold groups. Under mild hypotheses, these results extend to EZ-structures.

Curtis Kent, Brigham Young University
Title: Lacunary CAT(0) groups
Abstract: We will discuss asymptotic cones of cocompact CAT(0) spaces. Some features of metric spaces/groups can be detected in a single asymptotic cone while others require one to consider all asymptotic cones. We will discuss several features of CAT(0) spaces/groups which can be detected by a single asymptotic cone. In particular, we will show that a proper cocompact CAT(0) space has isolated flats if and only if one asymptotic cone is tree-graded by flats. This relates to work of Hruska/Kleiner and Coulon/Hull/Kent. We will discuss why this does not follow from their results.

Vitaliy Kurlin, University of Liverpool
Title: Introduction to Periodic Geometry and Topology
Abstract: Motivated by applications in crystallography and materials science, the new area of Periodic Geometry studies continuous metrics on a space of periodic structures. The key object is a periodic point set obtained by lattice translations from a finite motif of points in a unit cell. The most natural equivalence of periodic point sets is rigid motion or isometry, which preserves all interpoint distances. Real crystal structures can be reliably distinguished only by isometry invariants that are also continuous under perturbations of points. Such recent invariants are density functions (Proceedings SoCG 2021, arXiv:2104.11046), average minimum distances (AMD, arXiv:2009.02488) and complete invariant isosets (Proceedings DGMM 2021, arXiv:2103.02749). The ultra-fast AMD enabled invariant-based visualizations of huge crystal datasets within a few hours on a modest desktop. The talk is based on joint papers with numerous colleagues at the Materials Innovation Factory, Liverpool, UK.

Jean-Pierre Magnot, University of Angers
Title: On diffeological gluing and the geometry of CW complexes
Abstract: In this short communication, we will review first the natural diffeologies of the n-simplex and of the space of triangulations of a smooth manifold. We will then generalize this construction to finite CW complexes. Two recent applications of this construction will be sketched at the end of the talk.

Sergey Melikhov, Steklov Math Institute (Moscow)
Title: Not all links are isotopic to PL links
Abstract: Two links in the 3-sphere are called (non-ambiently) isotopic if they are homotopic through embeddings. D. Rolfsen (1974) asked the following question: ``All PL knots are isotopic to one another. Is this true of wild knots? A knot in the 3-space which is not isotopic to a tame knot would have to be so wild as to fail to pierce a disk at each of its points. The `Bing sling' [R. H. Bing, 1956] is such a candidate.'' This problem has been mostly approached with geometric methods. M. Brin (1983) constructed a knot that at each point is locally equivalent to the Bing sling, but is isotopic to a PL knot. Giffen's shift-spinning construction (as presented in papers by Y. Matsumoto, 1979 and F. Ancel and C. Guilbault, 1996) shows that every knot is I-equivalent to the unknot.

We construct a 2-component link that is not isotopic to any PL link. In fact, we show that there exist uncountably many I-equivalence classes of 2-component links in the 3-sphere. The details appear in arXiv:2011.01409.

Atish Mitra, Montana Tech
Title: The space of persistence diagrams on n points coarsely embeds into Hilbert Space
Abstract: We prove that the space of persistence diagrams on n points (with either the Bottleneck distance or a Wasserstein distance) coarsely embeds into Hilbert space. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also discuss various non-embeddability results when the number of points is not bounded.

Allison Moore, Virginia Commonwealth University
Title: Essential Conway spheres and Floer homology via immersed curves
Abstract: We consider the problem of whether Dehn surgery along a knot in the three-sphere produces an L-space, which is a Floer-theoretic generalization of a lens space. The geometric characterization of these manifolds remains a difficult outstanding problem, and it is natural to ask whether the existence of certain essential surfaces in the complement of a knot can obstruct non-trivial surgeries yielding L-spaces. We will prove any knot in the three-sphere with a nontrivial L-space surgery admits no essential Conway spheres. As a corollary, we recover a classic result of Wu that states that if a knot K has an essential Conway sphere, then the fundamental group of rational Dehn surgery along K is never finite. Our proof uses the technology of peculiar modules, a Floer theoretic invariant for tangles due to Zibrowius, and the geometric realization of these modules as certain decorated immersed curves on the four-punctured sphere. This is joint work with Lidman and Zibrowius.

Tom Needham, Florida State University
Title: Decorated Merge Trees for Persistent Topology
Abstract: I will introduce the concept of a decorated merge tree (DMT), an invariant which tracks interactions between homological features in multiple degrees for a filtered space, in the context of topological data analysis. Intuitively, a DMT is a merge tree (a metric tree summarizing hierarchical connectivity of a dataset) overlaid with higher dimensional barcodes (summaries of the homology of the dataset at different resolutions). Formally, a DMT can be understood abstractly in terms of category theory or concretely as a barcode-attributed combinatorial graph. There is a natural extension of interleaving distance to the setting of DMTs; I will discuss stability properties of this metric as well as methods for computing it via Gromov-Wasserstein distance, a tool from optimal transport. This is joint work with Justin Curry, Haibin Hang, Washington Mio and Osman Okutan.

David Rosenthal, St. John's University
Title: Finitely F-amenable actions and decomposition complexity of groups
Abstract: In their groundbreaking work on the Farrell-Jones Conjecture for Gromov hyperbolic groups, Bartels, Lück and Reich introduced certain geometric conditions on a group that imply this conjecture. These conditions have since been reformulated by Bartels in terms of the existence of a “finitely F-amenable group action” (where F is a family of subgroups) on a suitable space in his work on the Farrell-Jones Conjecture for relatively hyperbolic groups. In this talk we will discuss some coarse geometric applications of finitely F-amenable group actions. One application states that if G is a countable group that is relatively hyperbolic with respect to peripheral subgroups that are contained in a collection of metric families that satisfies some basic permanence properties, then G is also contained in that collection. This is joint work with Andrew Nicas.

Lorenzo Ruffoni, Florida State University
Title: Graphical splittings of Artin kernels
Abstract: A main feature of the theory of right-angled Artin groups (RAAGs) consists in the fact that the algebraic properties of the group can be described in terms of the combinatorial properties of the underlying graph. We investigate how this can be exploited in the study of Artin kernels, i.e. subgroups of RAAGs obtained as kernels of integral characters. In the case of chordal graphs we obtain a sharp dichotomy for Artin kernels. For block graphs we obtain an explicit rank formula, and discuss some applications to the study of fibrations. (Joint work with M. Barquinero and K. Ye).

Ignat Soroko, Louisiana State University
Title: Groups of type FP: their quasi-isometry classes and homological Dehn functions
Abstract: There are only countably many isomorphism classes of finitely presented groups, i.e. groups of type \(F_2\). Considering a homological analog of finite presentability, we get the class of groups \(FP_2\). Ian Leary proved that there are uncountably many isomorphism classes of groups of type \(FP_2\) (and even of finer class FP). R.Kropholler, Leary and I proved that there are uncountably many classes of groups of type FP even up to quasi-isometries. Since `almost all' of these groups are infinitely presented, the usual Dehn function makes no sense for them, but the homological Dehn function is well-defined. In a joint paper with N.Brady, R.Kropholler and myself, we show that for any even integer \(k\ge4\) there exist uncountably many quasi-isometry classes of groups of type FP with a homological Dehn function \(n^k\). In particular there exists an FP group with the quartic homological Dehn function and the unsolvable word problem. In this talk I will give the relevant definitions and describe the construction of these groups. Time permitting, I will describe the connection of these groups to the Relation Gap Problem.

Eric Swenson, Brigham Young University
Title: Cuts and blobs
Abstract: We provide sharp conditions under which a collection of separators \(A\) of a connected topological space \(Z\) leads to a canonical \(\mathbb{R}\)-tree \(T\). Any group acting on \(Z\) by homeomorphisms will act by homeomorphisms on \(T\).

Courtney Thatcher, University of Puget Sound
Title: Classifying \((\mathbb{Z}_p)^2\) actions on products of spheres
Abstract: We consider free actions of \((\mathbb{Z}_p)^2\) on \(S^{2n-1}\times S^{2n-1}\) given by linear actions of \((\mathbb{Z}_p)^2\) on \(\mathbb{R}^{4n}\). Simple examples include a lens space cross a lens space, but \(k\)-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the \(k\)-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the \(k\)-invariants and the Pontrjagin classes from the rotation numbers.

Vera Tonic, University of Rijeka, Croatia
Title: Alternative proofs for the n-dimensional resolution theorems
Abstract: We present new, unified proofs for the cell-like-, \(\mathbb{Z}/p\)-, and \(\mathbb{Q}\)-resolution theorems in extension theory. Our arguments employ extensions that are much simpler than those used by our predecessors. We unify the proofs by providing a coordinated method for constructing the maps needed to yield the resolution theorems. This is joint work with Leonard Rubin of the University of Oklahoma.

Anderson Vera, Pohang University of Science and Technology (POSTECH - BK21 FOUR Mathematical Sciences Division)
Title: A double Johnson filtration for the mapping class group and the Goeritz group of the sphere
Abstract: I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere, the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of \(S^3\). In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group. A similar study could be done for the group of automorphisms of a free group. (joint work K. Habiro)

Thomas Weighill, University of North Carolina at Greensboro
Title: Coarse homotopy groups of warped cones
Abstract: Various versions of coarse homotopy theory have been around since the beginning of coarse geometry, and served important roles in early proofs of the coarse Baum-Connes Conjecture for certain spaces. More recently, coarse fundamental groups were formally defined by Mitchener, Norouzizadeh and Schick. In this talk I will apply a coarse geometric version of the theory of covering spaces to the coarse fundamental groups of warped cones, spaces which have seen a renewed interest as a source of counterexamples in coarse geometry.