Cannon Seminar: Fall 2017/Winter 2018

Seminars will be on Wednesdays at 2:00 - 2:50 pm in 292 TMCB, unless otherwise noted.

Wednesday, Sept 13, 2017

Eric Swenson (Brigham Young University)

Title: CAT(0) Spaces

Abstract: We will explain some of the basics necessary to understand the different boundaries associated with CAT(0) spaces.

Wednesday, Sept 20, 2017

Chris Cashen (University of Vienna)

Title: The contracting boundary of a group.

Abstract: We construct a bordification of a proper geodesic metric space by adding a ‘contracting boundary’ consisting of equivalence classes of rays satisfying a contraction property enjoyed by rays in a hyperbolic space. We think of these as the distinct ways of going to infinity through hyperbolic directions. The topology we introduce on this contracting boundary is invariant under quasi-isometries and is homeomorphic to the Gromov boundary when the space is hyperbolic. If the space admits a geometric group action then our topology on the boundary is metrizable

This is joint work with John Mackay.

Wednesday, Sept 27, 2017

Mark Hughes (Brigham Young University)

Title: Branched coverings in low-dimensional topology

Abstract: In this talk, I will discuss branched coverings, which are covering maps of manifolds away from a singular set.  These branched covering maps are useful in a number of contexts in low-dimensional topology, in part because they occur relatively frequently and can be used to construct objects like open book decompositions and Lefschetz fibrations on 3- and 4-manifolds.  I will construct a number of examples, and outline some classical results in this area.

Wednesday, Oct 4, 2017

Sudipta Kolay (Georgia Institute of Technology)

Title: Braided embeddings of manifolds

Abstract: The theory of braids has been very useful in the study of classical knot theory. One can hope that higher dimensional braids will play a similar role in higher dimensional knot theory. In this talk we will introduce the concept of braided embeddings of manifolds, and discuss existence, lifting and isotopy problems for braided embeddings. We will talk about generalizations of Alexander's theorem about closed braids in the piecewise linear category.

Wednesday, Oct 11, 2017

Michael Andersen (Brigham Young University)

Title: Inscrutability

Abstract: A standard tool in algebraic topology is to pass between a continuous map between topological spaces and the corresponding homomorphism of fundamental groups using the $\pi_1$ functor. It is a non-trivial question to ask when a specific homomorphism is induced by a continuous map; that is, what is the image of the $\pi_1$ functor on homomorphisms?

Wednesday, Oct 18, 2017

Sam Corson (University of Basque Country)

Title: Prime root extraction in one-relator groups and slenderness

Abstract:  One-relator groups have been studied since the 1930s and furnish a supply of good test examples for conjectures.  I will give some background and examples of these groups, as well as sketch a proof of a new theorem concerning prime extraction in such groups.

Wednesday, Oct 25, 2017

Martha Kilpack (Brigham Young University)

Title: Loop representation for closure operator lattices.

Abstract:  Will loops be more effective than groups as a algebraic representative for closure operator lattices?

A theorem of Birkhoff and Frink we know every algebraic lattice can be represented as a subalgebra lattice of a finite algebraic structure. In work by the presenter and Magidin we know for a finite lattice L, the generated closure operator lattice is isomorphic to the lattice of subgroups of a group if and only if L is a chain.

What about loops? As loops are groups without the associative there are many lattices which are subloop but not subgroup lattices. We will look at some preliminary finding showing many types of  closure operator lattices are isomorphic to a subloop lattices. For example, the closure operator lattice generated from the non-chain 4 element lattices is a subloop lattice but not a subgroup lattice.

Wednesday, Nov 1, 2017

Eric Swenson (Brigham Young University)

Title: Normal forms and wicks forms.

Abstract:  Abnormal correlations on the mapping class group of a once punctured surface.

Wednesday, Nov 8, 2017

Curt Kent (Brigham Young University)

Title: Asymptotic cones and topological groups

Abstract: We will discuss when asymptotic cones of groups are topological groups.

Wednesday, Nov 29, 2017

Nick Castro (University of California, Davis)

Title: Interactions Between 3- and 4-manifolds via Trisections

Abstract: A trisection of a smooth 4-manifold X is the 4-dimensional analog of a Heegaard splitting of a 3-manifold, decomposing X into three diffeomorphic, codimension 0 submanifolds whose intersections encode the complexity of X. In the case that X has non-empty boundary, a trisection induces a fiber bundle over the circle on the bounding 3-manifold(s) known as an open book decomposition.
In this talk, I will focus on the diagrammatic version of trisections which allows us to describe any smooth, compact, oriented 4-manifold as three collections of curves on a surface. I will give an algorithm that explicitly determines the open book decomposition induced by a trisection (diagram). This is joint work with David Gay and Juanita Pinzón-Caicedo. If time permits, I will discuss the gluing theorem, which highlights the importance of open book decompositions, and the algorithm, in the theory of relative trisections.

Wednesday, Dec 6, 2017

Alex Shumway (Brigham Young University)

Title: Bass-Serre Theory

Abstract: In this talk, I will give an overview of Bass-Serre theory and touch on my work towards expanding the range of groups it can analyze. We will discuss HNN extensions and free products with amalgamation both algebraically and topologically, and we will see the outcome of using these tools to create graphs of groups. We will also discuss using Bass-Serre Theory to analyze complexes of groups more complicated than graphs.

Wednesday, Dec 13, 2017

Nick Larsen (Brigham Young University)

Title: TBD

Abstract: TBD

Winter 2018

Seminars will be on Wednesdays at 2:00 - 2:50 pm in 294 TMCB, unless otherwise noted.

Wednesday, Jan 17, 2018

Curt Kent (Brigham Young University)

Title: Fibrations of Peano continua

Abstract: We will discuss a sufficient condition for unique path lifting fibrations to be path connected.

Wednesday, Jan 24, 2018

Mark Hughes (Brigham Young University)

Title: The immersed cross-cap number of a knot

Abstract: The immersed Seifert genus of a knot in 3-space can be defined as the minimal genus of an orientable immersed surface whose boundary is the given knot.  By a result of Gabai, this value is always equal to the (embedded) Seifert genus of the knot.  In this talk, I will discuss the embedded and immersed cross-cap numbers of a knot, which are the non-orientable versions of these invariants.  Unlike their orientable counterparts these values do not always coincide, and can in fact differ by an arbitrarily large amount.  In further contrast to the orientable case, there are families of knots with arbitrarily high embedded 4-ball cross-cap numbers, but which are easily seen to have immersed cross-cap number 1.  After describing these examples, I will discuss a classification of knots with immersed cross-cap number 1.  This is joint work with Seungwon Kim.

Wednesday, Jan 31, 2018

Steve Humphries (Brigham Young University)

Title: Character varieties

Abstract: Surfaces with hyperbolic structure can be studied all at once by considering representations of their fundamental groups G into SL(2,C). Each such representation determines a character, X:G -> C, given by the trace. These trace functions are really polynomial functions in a finite number of variables. The group G may be a free group, in which case there is an action of Aut(G), that 'gives' an action on these polynomials.  We indicate certain results relating to the dynamics of this action, including fixed points.

Wednesday, Feb 7, 2018

Wolfgang Herfort (Vienna University of Technology)

Title: The Abelianization of Certain Topologist's Products

Abstract: Reviewing the notion of “Topologist's Product”, as introduced in the 50s by Griffiths and Higman. The most prominent example is the Hawaiian Earring Group (HEG). For the latter the “free factors” are all isomorphic to the group Z of integers. K Eda and K. Kawamura 2000 determined the structure of the commutator quotient (the first homology group) of the HEG. Their proof relies on elaborate concepts of infinite words developed and considered by K. Eda, and independently, by J. Cannon & G. Conner, as well as by A. Zastrow.

During this talk we shall consider the case of topologist's product when all “free factors” are isomorphic to a group of p elements, p a fixed prime. We shall determine the algebraic structure of the commutator quotient (abelianization). Our method of proof relies on classical facts about cotorsion groups and the concept of “Higman completeness”, as introduced earlier by the same authors, but already present as a technical device in Higman's seminal paper from 1956.

Wednesday, Feb 14, 2018

Petar Pavešić (University of Ljubljana)

Title: Triangulations with small links are combinatorial

Abstract: We study lower bounds for the number of vertices in combinatorial triangulations of closed manifolds. In particular we show that a combinatorial triangulation of a d-dimensional manifold with non-free fundamental group has at least 3d+1 vertices. As a corollary, every d-dimensional homology sphere that admits a combinatorial triangulation with at most 3d vertices is homeomorphic to S^d.  Another interesting consequence is that every triangulation with small links of a closed manifold is necessarily combinatorial.

Wednesday, Feb 21, 2018

Michael Andersen (Brigham Young University)

Title: TBD

Abstract: TBD

Wednesday, Feb 28, 2018

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Wednesday, Mar 7, 2018

Seungwon Kim (National Institute for Mathematical Sciences)

Title: TBD

Abstract: TBD

Wednesday, Mar 14, 2018

Nick Callor (Brigham Young University)

Title: TBD

Abstract: TBD

Wednesday, Mar 21, 2018

Martha Kilpack (Brigham Young University)

Title: TBD

Abstract: TBD

Wednesday, Mar 28, 2018

Greg Conner (Brigham Young University)

Title: TBD

Abstract: TBD

Wednesday, Apr 4, 2018

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Wednesday, Apr 11, 2018

Title: TBD

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Wednesday, Apr 18, 2018

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