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1/27: Organizational meeting
2/1: Burns Healy (UWM)
Title: Topological and geometric data in auxetic meta-materials
Abstract: When materials such as plastic, concrete, polymer, etc. are used to build structures, it is possible to arrange the material in such a way that the macro system has very different mechanical properties than the material itself; this is called meta-material design. Achieving nontrivial meta-material design is typically done by leaving empty space in a regular way throughout the construction. One particular application of interest of this procedure is the production of useful, and counter-intuitive, behavior of a system when it undergoes strain. We look at some ways this desirable behavior has been achieved, and discuss the investigation into how the study of the geometry and topology of these systems can predict their deformation properties.
2/3: Yulan Qing (Fudan University)
Title: The large scale geometry of big mapping class groups
Abstract: In this talk, we introduce the framework of the coarse geometry of non-locally compact groups in the setting of big mapping class groups, as studied by Rosendal. We will discuss the characterization results of Mann-Rafi and Horbez-Qing-Rafi that illustrate big mapping groups' rich geometric and algebraic structures. We will outline the proofs in these results and their implications. If time permits, we will discuss some open problems in this area.
2/8: Nathan Fisher (Tufts)
Title: Sub-Finsler metrics, random walks, and the Heisenberg group at infinity
Abstract: In this talk, we will discuss the real Heisenberg group and a class of sub-Finsler metrics which arise as the asymptotic cones of word metrics on the integer Heisenberg group. We will describe new results on the boundaries of these polygonal sub-Finsler metrics and explore applications to the study of random walks. Parts of this work are joint with Sebastiano Nicolussi Golo.
2/10: Abdalrazzaq Zalloum (Queen's University)
Title: Hyperbolicities in CAT(0) cube complexes
Abstract: For a CAT(0) cube complex X, there are many competing notions of what is meant by a "hyperbolic-like" geodesic ray in X. Work of Charney-Cordes-Sultan, Cashen-Mackay and Qing-Rafi-Tiozzo shows that such hyperbolic-like (quasi)-geodesic rays could be grouped together to form "hyperbolic-like" boundaries for CAT(0) cube complexes. On the other hand, there are a few natural hyperbolic spaces that can be built out of a CAT(0) cube complex such as the contact graph of Hagen and the well-separation space of Genevois. The main theorem I will discuss states that for every known proper cocompact CAT(0) cube complex X, there exists a hyperbolic graph Γ, depending only on X, such that:1) Every "hyperbolic-like" geodesic ray in X projects to an infinite unparametrized quasi-geodesic in Γ.2) All the "hyperbolic-like" boundaries of X continuously inject in the Gromov boundary of Γ.
This work is joint with Medici.
This work is joint with Medici.
2/15, 2/17, 2/22: Chris Hruska (UWM)
Title: Canonical splittings of relatively hyperbolic groups
Abstract: (Joint with Matt Haulmark)A JSJ decomposition is a graph of groups decomposition that allows one to classify all splittings of a group over certain subgroups. I will discuss a JSJ decomposition for relatively hyperbolic groups splitting over elementary subgroups that depends only on the topology of its boundary. This decomposition could potentially be of use for understanding groups that have homeomorphic boundaries, but are not necessarily quasi-isometric.
2/24: Sami Douba (McGill)
Title: Virtually unipotent elements of 3-manifold groups
Abstract: An element of an abstract group is virtually unipotent if every finite-dimensional linear representation of the group maps some power of that element to a unipotent matrix. Since unitary matrices are diagonalizable, and since a unipotent matrix is torsion if its entries lie in a field of positive characteristic, a group containing an infinite-order virtually unipotent element does not admit a faithful finite-dimensional unitary representation, nor is such a group linear over a field of positive characteristic. We discuss the existence (or absence) of infinite-order virtually unipotent elements in various finitely generated groups, with an emphasis on 3-manifold groups.
3/1, 3/8: Prayagdeep Parija (UWM)
Title: Random quotients of hyperbolic groups and Property (T)
Abstract: What does a typical quotient of a group look like? Gromov had looked at quotients of free groups. He used a 'density' parameter d measuring the rate of exponential growth of the number of relators compared to the length of relators to model a typical quotient of a free group .Using this model he had proved that "most" groups are hyperbolic. Expanding on this idea Żuk proved that for density d>1/3 "most" groups have Property(T).We show that Żuk's result can be extended to hyperbolic groups, answering a question of Gromov(and Ollivier).
This is work in progress.
This is work in progress.
3/3: Binbin Xu (Université de Luxembourg)
Title: Equivalent curves on surfaces
Abstract: We consider a closed surface of genus at least 2. To describe curves on it, one natural idea is to choose once for all a collection of curves as a reference system and to hope that the intersection numbers of any other curve with curves in the reference system will determine that curve. For simple curves, using the work of Dehn and Thurston, it is possible to find such a reference system consisting of finitely many simple curves. The situation becomes more complicated when curves have self-intersections. In particular, for any non negative integer k, it is possible to find a pair of curves that intersect every curve with k self-intersections the same number of times, which are called k-equivalent curves. In this talk, I will discuss the general picture of a pair of k-equivalent curves and the relation between k-equivalence relations for different k's. This is a joint-work with Hugo Parlier.
3/10: Siddhi Krishna (Georgia Tech)
Title: Taut foliations and Dehn surgery along positive braid knots
Abstract: The L-space conjecture has been in the news a lot lately. It predicts a surprising relationship between the algebraic, geometric, and Floer-homological properties of a 3--manifold Y. In particular, it predicts exactly which 3-manifolds admit a ``taut foliation". In this talk, I'll discuss some of my past and forthcoming work investigating these connections. In particular, I'll discuss a strategy for building taut foliations manifolds obtained by Dehn surgery along knots realized as closures of ``positive braids". As an application, I will show how taut foliations can be used to obstruct positivity for cable knots, and produce a new unknot detector. All are welcome; no background in foliation or Floer homology theories will be assumed.
3/15: Yvon Verberne (Georgia Tech)
Title: The asymptotic dimension of big mapping class groups
Abstract: In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.
3/17: Eduard Einstein (UIC)
Title: Relatively geometric actions on CAT(0) cube complexes
Abstract: The study of hyperbolic and relatively hyperbolic groups acting on CAT(0) cube complexes has produced exciting recent results in geometric group theory. I will talk about a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex called a relatively geometric action.In joint work with Daniel Groves, we develop analogues of tools used to construct and study geometric actions of hyperbolic and relatively hyperbolic groups on CAT(0) cube complexes, including a relatively geometric version of Agol's Theorem.I will also discuss some of the structural theorems we hope to prove and a potential application to the Relative Cannon Conjecture.
3/29: Elia Fioravanti (Max Planck)
Title: Coarse-median preserving automorphisms
Abstract: We study fixed subgroups of automorphisms of right-angled Artin and Coxeter groups. If \phi is an untwisted automorphism of a RAAG, or an arbitrary automorphism of a RACG, we prove that Fix(\phi) is finitely generated and undistorted. Up to replacing \phi with a power, we show that Fix(\phi) is even quasi-convex with respect to the standard word metric. This implies that Fix(\phi) is separable and a special group in the Haglund-Wise sense. Some of our techniques may be applicable in the more general context of Bowditch's coarse median groups. Based on arXiv:2101.04415.
3/31: Ross Geoghegan (Binghamton University)
Title: Geometric topology meets tropical phased matroids
Abstract: This talk is about an interesting class of topological partially ordered sets (posets). With each member of the class is associated (1) a CW complex called the “order complex”, and (2) a rather strange topology on the topological poset itself, called the “Up topology”. To make this understandable I’ll first discuss familiar Grassmannian examples.
There is a “Comparison Map” from (1) to (2). I will discuss a theorem, joint with my student Ulysses Alvarez, that (under mild hypotheses) the Comparison Map is a weak homotopy equivalence. The proof involves topics from geometric topology, eg cell-like maps and shape theory, but the interest comes from topological combinatorics. For that, I’ll describe some of Alvarez’s thesis work. It involves topics that are not always familiar to topologists: eg the tropical phase hyperfield, and tropical phased matroids over that hyperfield. I’ll explain these to some extent. Each of these matroids gives rise to an interesting topological poset in the class being discussed. Alvarez computes the homeomorphism type of the corresponding order complex. He finds it is always an odd-dimensional sphere, which proves a conjecture of Laura Anderson. Using our theorem, he thus gets the homology of the relevant poset when it is equipped with the Up topology.
There is a “Comparison Map” from (1) to (2). I will discuss a theorem, joint with my student Ulysses Alvarez, that (under mild hypotheses) the Comparison Map is a weak homotopy equivalence. The proof involves topics from geometric topology, eg cell-like maps and shape theory, but the interest comes from topological combinatorics. For that, I’ll describe some of Alvarez’s thesis work. It involves topics that are not always familiar to topologists: eg the tropical phase hyperfield, and tropical phased matroids over that hyperfield. I’ll explain these to some extent. Each of these matroids gives rise to an interesting topological poset in the class being discussed. Alvarez computes the homeomorphism type of the corresponding order complex. He finds it is always an odd-dimensional sphere, which proves a conjecture of Laura Anderson. Using our theorem, he thus gets the homology of the relevant poset when it is equipped with the Up topology.
4/5, 4/7: Ric Ancel (UWM)
Title: The isometrization of groups of homeomorphisms
Abstract: Let G be a group of homeomorphisms of a topological space X. G is (properly) isometrizable if there is a G-invariant (proper) gauge structure on X. Simple topological properties of the action of G on X – equiregularity and near properness – are defined. These notions lead to the following theorems.
The Isometrization Theorem: Suppose X is a Hausdorff space and G\X is a paracompact regular space. Then G is isometrizable if and only if G is equiregular.
The Proper Isometrization Theorem: Suppose X is a locally compact σ-compact Hausdorff space and G\X is a regular space. Then G is properly isometrizable if and only if G is equiregular and nearly proper. The Proper Isometrization Theorem has the following corollary:
Theorem (Abel-Manoussos-Noskov, 2011): If X is a locally compact σ-compact Hausdorff space and G acts properly on X, then G is properly isometrizable.
The Isometrization Theorem: Suppose X is a Hausdorff space and G\X is a paracompact regular space. Then G is isometrizable if and only if G is equiregular.
The Proper Isometrization Theorem: Suppose X is a locally compact σ-compact Hausdorff space and G\X is a regular space. Then G is properly isometrizable if and only if G is equiregular and nearly proper. The Proper Isometrization Theorem has the following corollary:
Theorem (Abel-Manoussos-Noskov, 2011): If X is a locally compact σ-compact Hausdorff space and G acts properly on X, then G is properly isometrizable.
4/12: Nima Hoda (École Normale Supérieure)
Title: Hierarchical hyperbolicity, coarse Helly spaces and the strong shortcut property
Abstract: This talk concerns three classes of metric spaces defined bynonpositive curvature-like properties: hierarchically hyperbolicspaces, coarse Helly spaces and strongly shortcut spaces. Roughlyspeaking, hierarchically hyperbolic spaces are metric spaces whosegeometry can be understood through a structured family of projectionsonto Gromov-hyperbolic spaces; coarse Helly spaces are metric spaceswhose balls satisfy a coarse version of the classical 1-dimensionalHelly property; and strongly shortcut spaces are metric spaces whosesubspaces cannot approximate long circles with arbitrary precision.
I will discuss recent joint work with Thomas Haettel and Harry Petytin which we prove that hierarchically hyperbolic spaces are coarselyHelly and that coarse Helly spaces of uniformly bounded geometry arestrongly shortcut. This work has important consequences forhierarchically hyperbolic groups: it shows that they aresemihyperbolic, have solvable conjugacy problem, are of typeFP-infinity, have finitely many conjugacy classes of finite subgroupsand that their finitely generated abelian subgroups are undistorted.
I will discuss recent joint work with Thomas Haettel and Harry Petytin which we prove that hierarchically hyperbolic spaces are coarselyHelly and that coarse Helly spaces of uniformly bounded geometry arestrongly shortcut. This work has important consequences forhierarchically hyperbolic groups: it shows that they aresemihyperbolic, have solvable conjugacy problem, are of typeFP-infinity, have finitely many conjugacy classes of finite subgroupsand that their finitely generated abelian subgroups are undistorted.
4/14: Daniel Gulbrandsen (UWM)
Title: Group Actions and the Cubical Boundary
Abstract: In this talk we will discuss a certain boundary of CAT(0) cube complexes. This boundary, which we call the cubical boundary, can be thought of as a hybrid of the visual and Roller boundaries. It resembles the visual boundary in that it can be used to Z-compactify the CAT(0) cube complex, and it resembles the Roller boundary in that it contains the Roller boundary as a canonical subspace. Our current interest in the cubical boundary will be to show that a group action by cubical isomorphisms on the cube complex extends equivariantly to its cubical boundary. Examples and possible avenues of further exploration will be discussed.
4/19, 4/21: William Braubach (UWM)
Title: Introduction to Coarse Algebraic Topology
Abstract: In recent years there has been a lot of developments in the field of Coarse Geometry. One such area of interest is the creation of tools for Coarse Algebraic Topology. In this talk I plan to introduce some general concepts of Coarse Geometry such as coarse structures and coarse maps between spaces. Then I plan to define two very important tools of Coarse Algebraic Topology, Coarse Homology and Coarse Homotopy Groups. Lastly, I will define the homomorphism between higher Coarse Homotopy Groups and Coarse Homology Groups and its properties.
4/26: Liam Axon (Oberlin College)
Title: End-sum uniqueness for surfaces
Abstract: The end-sum operation is a way of combining two manifolds of the same dimension. It is tailored for non-compact manifolds, and has been employed, among other uses, in creating non-compact manifolds with specific properties. It is analogous to the more well-known operations of connected sum and boundary connected sum. Unlike these operations, however, the end-sum is known to be very non-unique in all dimensions n > 2. In a recent honors thesis, I proved that the end-sum of two surfaces with compact boundary is unique up to a choice of "ends" used in the end-sum construction. In this talk, I will go over ends and the end-sum, as well as my recent uniqueness result.
4/28: Matthew Durham (UC-Riverside)
Title: Stable cubulations of hulls in mapping class groups
Abstract: A defining property of Gromov hyperbolic spaces is that the hulls of finite sets of points are quasiisometric to trees. Recently, Behrstock-Hagen-Sisto proved that the hulls of finite sets of points in mapping class groups (and hierarchically hyperbolic spaces, more generally) are quasiisometric to CAT(0) cube complexes.
I will discuss their construction and describe joint work with Yair Minsky and Alex Sisto in which we stabilize this cubulation process. Novel applications of this stabilization include semihyperbolicity of the group and the existence of coarse barycenters.
I will discuss their construction and describe joint work with Yair Minsky and Alex Sisto in which we stabilize this cubulation process. Novel applications of this stabilization include semihyperbolicity of the group and the existence of coarse barycenters.
5/5: Jonah Gaster (UW-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 different points of view on this conjecture, a new geometric reformulation, and present some intriguing experimental data.
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