__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, Mar 7, 2018**

*Seungwon** Kim *(National Institute for Mathematical Sciences)

Title:
Representativity and waist of cable knots

Abstract:
We study the essential surfaces in the exterior of a cable knot to compute the representativity and waist of most cable knots. Our
computation answers Ozawa’s question about the relationship between the representativity and the waist of a knot in the negative.

**Wednesday, Mar 14, 2018**

*Nick
Callor *(Brigham Young University)

Title:
An
Introduction to Multidimensional Persistent Homology

Abstract:
We discuss
the work of Gunnar Carlsson and Afra
Zomorodian to extend persistent homology to multiparameter filtrations.

**Wednesday, Mar 21, 2018**

*Martha
Kilpack *(Brigham Young University)

Title:
Algebraic structure for closure operator lattice

Abstract:
Algebraic structures come in many forms: from groups and rings to unnamed
types. Given an algebraic structure A,
the subalgebras form a lattice under the partial
order of set inclusion. When starting
with a lattice L that is generated from compact
elements, there exists an algebraic structure, which has a subalgebra
lattice isomorphic to L. How do we find
these structures?

We
will look at some ways to find the desired structures. We will also consider special lattices called
closure operator lattices. We will consider two structures: groups and
loops. We will begin to answer the
question, when is a closure operator lattice isomorphic to a subgroup or subloop lattice?

**Wednesday, Mar 28, 2018**

*Greg
Conner *(Brigham Young University)

Title:
TBD

Abstract:
TBD

**Wednesday, Apr 4, 2018**

Title:
TBD

Abstract:
TBD

**Wednesday, Apr 11, 2018**

Title:
TBD

Abstract:
TBD

**Wednesday, Apr 18, 2018**

Title:
TBD

Abstract:
TBD