Embedded Files
9/9: Organizational meeting
9/14, 9/16, 9/21: Craig Guilbault (UWM)
Title: Coarse Z-boundaries for groups
Abstract: (Joint with Molly Moran). Bestvina developed an axiomatic approach to group boundaries under which Gromov boundaries of hyperbolic groups and visual boundaries of CAT(0) groups become special cases. He called these "Z-boundaries". In this talk we will generalize Bestvina's definition to that of a "coarse Z-boundary". We will show that most of the established theorems about Z-boundaries carry over nicely to the more general setting, and that some wished-for properties of Z-boundaries become a reality. Most significantly, the property of admitting a coarse Z-boundary is a pure quasi-isometry invariant. In the process, we will streamline both new and existing theory by introducing the notion of a "model Z-geometry". For the sake of completeness, we also develop an equivariant version of the above---that of a "coarse EZ-boundary".
9/23, 9/28, 9/30: Geoffrey Hruska (UWM)
Title: Planar boundaries and parabolic subgroups
Abstract: (Joint with Genevieve Walsh). Relatively hyperbolic groups are a family of groups that generalize the geometrically finite Kleinian groups. Each relatively hyperbolic group has an associated “Bowditch boundary” on which it acts. In the Kleinian setting this Bowditch boundary is simply the limit set of the action on the 2–sphere at infinity of hyperbolic space H³.
This talk is motivated by the question: What properties of the Bowditch boundary are sufficient to imply that the group G is a Kleinian group?
We do not have a conclusive answer to this difficult question (which includes the Cannon Conjecture). Towards a better understanding of the above question, we prove that in certain key cases, every homeomorphism of the planar Bowditch boundary extends to a homeomorphism of the 2–sphere, and we show that the parabolic subgroups must be virtual surface groups.
This talk is motivated by the question: What properties of the Bowditch boundary are sufficient to imply that the group G is a Kleinian group?
We do not have a conclusive answer to this difficult question (which includes the Cannon Conjecture). Towards a better understanding of the above question, we prove that in certain key cases, every homeomorphism of the planar Bowditch boundary extends to a homeomorphism of the 2–sphere, and we show that the parabolic subgroups must be virtual surface groups.
10/5, 10/7: Jonah Gaster (UWM)
Title: Curves on a torus intersecting at most k times
Abstract: (Joint with Tarik Aougab). A well known counting problem in surface topology asks for the maximum number of simple closed curves that can fit on a closed surface of genus g so that every pair intersect at most k times -- such a collection of curves is called a "k-system". Answers to this problem, even just for order of growth in k or g, are still unknown. The case k=1 is a notoriously difficult problem, with notable recent progress from Przytycki and Greene.
Somewhat surprisingly, the complementary case g=1 remains mysterious. Ian Agol made the elegant observation that, on the torus, the size of a k-system is bounded by one more than the smallest prime larger than k, and via the Prime Number Theorem one can deduce that this quantity is asymptotic to k. We will discuss joint work with Tarik Aougab, in which we tighten the available upper bounds for g=1, showing that a k-system on the torus has size at most k+O(\sqrt{k}\log k). (Curiously, this matches the bound one would obtain via Agol's bound with the assumption of the Riemann hypothesis.) Our methods involve analysis of some combinatorial aspects of the associahedron and the hyperbolic geometry of the Farey complex.
Somewhat surprisingly, the complementary case g=1 remains mysterious. Ian Agol made the elegant observation that, on the torus, the size of a k-system is bounded by one more than the smallest prime larger than k, and via the Prime Number Theorem one can deduce that this quantity is asymptotic to k. We will discuss joint work with Tarik Aougab, in which we tighten the available upper bounds for g=1, showing that a k-system on the torus has size at most k+O(\sqrt{k}\log k). (Curiously, this matches the bound one would obtain via Agol's bound with the assumption of the Riemann hypothesis.) Our methods involve analysis of some combinatorial aspects of the associahedron and the hyperbolic geometry of the Farey complex.
10/12, 10/14: Prayagdeep Parija (UWM)
Title: How to avoid cancellation when concatenating words in Hyperbolic Groups
Abstract: For density d>1/3, Żuk showed (explained by Kotowski and Kotowski) that with overwhelming probability random quotients of free groups have Kazhdan's property (T). What about quotients of an arbitrary hyperbolic group G?An approach to attack this problem is to show that a random quotient of a hyperbolic group is the homomorphic image of a random triangular group, which are already shown by Żuk to have property (T) with overwhelming probability. To construct this homomorphism we need to ensure that the relators of the triangular groups (which are reduced words of length three) are mapped to elements of length L of G (which are the relators we quotient G by). This may be done by sending the generators of the triangular groups to geodesic words of length L/3 in the hyperbolic group. However, concatenating arbitrary geodesic words of length L/3 may not give you a geodesic word. We address this issue by showing that are a "large" number of geodesic words of length L/3,any three of which when concatenated with each other produce geodesic words of length L, which is enough for our purposes.
10/19, 10/26: Daniel Gulbrandsen (UWM)
Title: Showing that the Cubical Boundary of a CAT(0) Cube Complex is a Z-Set
Abstract: CAT(0) cube complexes possess a rich combinatorial structure one can access by considering collections of halfspaces. Starting with a CAT(0) cube complex, X, we will use this combinatorial structure to define a map of X into a hilbert cube. Taking the closure of the image under this map allows us to define an object called the cubical boundary of X. In this talk, we will define a contracting homotopy of the hilbert cube and, using this homotopy, we will show that the cubical boundary of X is a Z-set.
10/21: Michael Harrison (Carnegie Mellon)
Title: Totally nonparallel immersions
Abstract: An immersion from a smooth n-dimensional manifold M into R^q is called totally nonparallel if, for every distinct points x and y in M, the tangent spaces at f(x) and f(y) contain no parallel lines. The simplest example is the map R \to R^2 sending x to (x,x^2). Given a manifold M, what is the minimum dimension q = q(M) such that M admits a totally nonparallel immersion into R^q? I will discuss one method of studying this problem using the h-principle, a powerful tool in differential topology which is used to study spaces of functions with certain distinguished properties (immersions, submersions, embeddings, free maps, etc.). I will give a neat proof of a special case of the Smale-Hirsch Theorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. I will discuss how to adapt this argument to study totally nonparallel immersions.
11/2: Arunima Ray (Max Planck)
Title: Embedding surfaces in 4-manifolds
Abstract: I will present a surface embedding theorem for 4-manifolds with good fundamental group in the presence of potentially unframed, immersed dual spheres. The essential obstruction is the Kervaire-Milnor invariant and a goal of the talk is to describe how it may be computed. This is based on joint work with Daniel Kasprowski, Mark Powell, and Peter Teichner.
11/4: Arka Bannerjee (UWM)
Title: Coarse embedding obstruction
Abstract: If a group G acts properly discontinuously on a metric space X, it induces a coarse embedding of the group into X by sending G to the orbit of a point in X. In other words, the existence of a coarse embedding to X is a necessary condition for the group G to act on X with a properly discontinuous action. This motivates one to ask the following question:
Q. When does a given metric space Y coarsely embed inside a metric space X?
We investigate this question when X is uniformly contractible n-manifold (that is, for all r there exist R such that every ball of radius r is contractible inside a concentric ball of radius R). Van Kampen showed that certain cohomology classes of the configuration space of Y obstruct topological embedding into contractible n-manifolds. In this talk I will briefly review Van Kampen's theory and present work in progress to develop a coarsened version of this theory to produce an obstruction for coarse embedding into uniformly contractible n-manifolds. As an application I will show uniformly contractible n-manifolds cannot coarsely embed into uniformly contractible (n-1)-manifolds. This implies an earlier result by Yoon, which says that the fundamental group of an n-dimensional aspherical manifold cannot act cocompactly and properly discontinuously on a contractible (n-1)-manifold.
Q. When does a given metric space Y coarsely embed inside a metric space X?
We investigate this question when X is uniformly contractible n-manifold (that is, for all r there exist R such that every ball of radius r is contractible inside a concentric ball of radius R). Van Kampen showed that certain cohomology classes of the configuration space of Y obstruct topological embedding into contractible n-manifolds. In this talk I will briefly review Van Kampen's theory and present work in progress to develop a coarsened version of this theory to produce an obstruction for coarse embedding into uniformly contractible n-manifolds. As an application I will show uniformly contractible n-manifolds cannot coarsely embed into uniformly contractible (n-1)-manifolds. This implies an earlier result by Yoon, which says that the fundamental group of an n-dimensional aspherical manifold cannot act cocompactly and properly discontinuously on a contractible (n-1)-manifold.
11/9: Brice Loustau (Heidelberg University)
Title: The hyper-Kähler geometry of minimal hyperbolic germs
Abstract: In this joint work with F. Bonsante, A. Sanders, and A. Seppi, we study the space of minimal immersions of a closed orientable surface in a hyperbolic 3-manifold. We show that it admits a hyper-Kähler metric with particularly attractive properties, for instance: it is invariant under the mapping class group action and admits the area functional as a Kähler potential. Moreover, it admits complex symplectic maps to the character variety and the cotangent bundle of Teichmüller space. I will present these ideas without assuming any expertise from the audience on the subject.
11/11: Didac Martinez-Granado (UC-Davis)
Title: From curves to currents
Abstract: Geodesic currents are the closure of the space of closed curves on a surface, analogous to measured laminations for simple closed curves. In 1986, Bonahon introduced geodesic currents and proved that the notion of hyperbolic length for curves extends continuously to currents. Since then, many other functions defined on the space of curves have been proven to extend to currents, such as negatively curved lengths, lengths from singular flat structures or some word lengths. One of the key properties of these functions is that they decrease under surgery of an essential crossing of a curve, a phenomenon we refer to as the ``smoothing property''. In this talk, we explain how a function defined on the space of curves that satisfies the smoothing property can be extended to geodesic currents continuously. Our theorem subsumes previous extension results and, furthermore, it gives an extension of extremal length of curves. Finally, via a result of Rafi and Souto, it provides a new counting result for curves on surfaces relative to extremal length. This is joint work with Dylan Thurston.
11/18: Tarik Aougab (Haverford)
Title: Characterizing covers from simple closed curves
Abstract: It is well known that any closed curve on a surface S admits an embedded lift to some finite cover. In this talk, we ask to what extent a covering map is determined by the list of closed curves that admit such a lift to that particular cover. In previous work joint with Max Lahn, Marissa Loving, and Sunny Yang Xiao, we showed that a regular cover is completely determined (up to the obvious natural equivalence for covering maps) by this information. In new work with Lahn, Loving, and Nick Miller, we remove the regularity assumption. We’ll mention a “goal” application (we are working on it!) related to Sunada’s construction of isospectral hyperbolic surfaces, and discuss some of the ideas behind the arguments.
11/23: Georg Essl (UWM)
Title: Sound of circular drums 2: Wave front dynamics
Abstract: In this seminar we will specifically look at the topological and geometric content of the dynamics of wave fronts on a flat disk domain. Overall the seminar will serve as a survey of wave front dynamics with an emphasis on singularities. Our main aim will be to highlight the connection of the phenomenon of caustic formation and the presence of cusps on wave fronts to a crossing condition of neighboring characteristic rays. This hints at the possibility of gleaning interesting information even from ray bundles of countable subsets of angles of billiard reflections from the boundary (such rational periodic orbits).
[NB: This is the second talk in a series; the first talk will be 11/20 in the UWM Analysis Seminar.]
[NB: This is the second talk in a series; the first talk will be 11/20 in the UWM Analysis Seminar.]
11/30: Talia Fernós (UNC-Greensboro)
Title: The Roller, Simplicial, and Tits Boundaries of CAT(0) Cube Complexes
Abstract: The universe of CAT(0) cube complexes is rich and diverse thanks to the ease by which they can be constructed and the many natural metrics they admit. As a consequence, there are several associated boundaries, such as the Roller, simplicial, and Tits boundaries. In a joint work-in-progress with Futer and Hagen, we show that the Roller boundary can be simplicialized to be homotopy equivalent to both the simplicial and Tits boundaries and that these maps are (up to homotopy-) equivariant under the automorphism group of the cube complex. In this talk we will introduce the objects in question and discuss some of the techniques for proving this result.
12/2: Rebecca Winarski (College of the Holy Cross)
Title: Polynomials, branched covers, and trees
Abstract: Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.
12/7, 12/9: Burns Healy (UWM)
Title: Homogeneous spaces not admitting geometric (quasi)-actions
Title: Homogeneous spaces not admitting geometric (quasi)-actions
Abstract: Thurston's geometrization answers the question of what homogeneous Riemannian manifolds occur as the universal cover of a closed 3-manifold. The ``reverse'' question is also of interest; that is, when does a given homogeneous Riemannian manifold cover a finite volume quotient? When the upstairs space is strictly negatively curved, there is a complete classification of this behavior due to Heintze. In this talk, we will examine a generalization of these Heintze spaces (negatively curved homogeneous Riemannian manifolds), a particular class of solvable Lie groups which admit CAT(0) metrics. We give a characterization of when these metric spaces admit finite volume quotients. Furthermore, we produce a wide array of examples of these spaces which do not admit any geometric quasi-actions by discrete groups, meaning they are cocompact but not QI to any discrete group. This talk represents joint work with Mark Pengitore.
Page updated
Google Sites
Report abuse