Seminars will be on Wednesdays at 12:00 - 12:50 pm in
292 TMCB, unless otherwise noted.
Wednesday, Jan 18, 2017
Martha
Kilpack (Brigham Young University)
Title:
A First Lecture in Universal Algebras.
Abstract:
Universal Algebra started as a way to generalized ideas that are
seen in different areas of algebra. In this talk, we see how algebraic
structures are defined and some areas of study in
Universal Algebras.
Wednesday, Jan 25, 2017
Seungwon Kim
(City University of New York)
Title:
A topological characterization of toroidally
alternating knots
Abstract:
We extend Howie’s characterization of alternating knots to give a topological
characterization of toroidally alternating knots,
which were defined by Adams. We provide necessary and
sufficient conditions for a knot to be toroidally
alternating. We also give a topological characterization of almost-alternating knots which is different from Ito’s recent characterization.
Wednesday, Feb 1, 2017
Martha
Kilpack (Brigham Young University)
Title:
The lattice of algebraic closure operators on finite lattices and infinite
subgroup lattices.
Abstract:
For a finite lattice L the closure operators which act on L form an algebraic
lattice c.o.(L). We will show for any finite lattice L
that c.o.(L) is a subgroup lattice if and only if L is
a chain (totally ordered set).
Finally
we investigate the case where L isomorphic to sub(G),
a subgroup lattice where G is an infinite group. We settle the case in
which G has torsion, and the torsionfree case in
which G has a nontrivial abelian normal subgroup (including the case when G is
itself abelian). We conjecture that for infinite groups G, a.c.o(subgrps(G)),
the algebraic closer operator lattice, is isomorphic to a subgroup lattice if
and only if G is isomorphic to the Prufer p-group for
some prime p.
Wednesday, Feb 8, 2017
Alex
Safsten (Brigham
Young University)
Title: Busemann geodesic
spaces: the role of curvature and the infinite dimensional case
Abstract: Busemann
Geodesic spaces (g-spaces) are metric spaces endowed with a several properties
that together guarantee the existence of geodesics. A long-standing conjecture
suggests that all g-spaces are manifolds. We show that g-spaces with locally
bounded Alexandrov curvature do indeed possess
manifold properties such as the existence of a local projection to a geodesic.
We will present a standard definition for an infinite-dimensional metric space,
and outline an unfinished argument for the non-existence of an infinite
dimensional g-space.
Wednesday, Feb 22, 2017
Curt
Kent (Brigham Young University)
Title: Coarse geometry of CAT(0)
groups
Abstract: CAT(0)
spaces are a class of geodesic metric spaces that have a non-positive curvature
condition. We will consider the coarse
geometric structure of groups with act geometrically on CAT(0) spaces by
studying the relationships between their canonical boundaries and their
asymptotic cones.
Wednesday, Mar 1, 2017
Kathryn Bryant (Colorado
College)
Title: Classical Results in 4-Manifold Topology
Abstract: One of the most interesting aspects
of 4-dimensional topology is the fact that the category of topological
manifolds is not equivalent to the category of smooth manifolds, and dimension
four is the first dimension in which we see this distinction. As such, huge
amounts of effort are put in to better understand the
delineations between these two categories. One of the earliest and most
relevant tools (still) for accomplishing this task is the intersection form. In
this talk, I will discuss a way to describe smooth 4-manifolds, I will define
the intersection form, and then I will discuss some of the foundational results
in 4-dimensional topology that are based on the intersection form such as
Freedman's Theorem and Donaldson's Theorem.
Wednesday,
March 8, 2017
Curt
Kent (Brigham Young University)
Title: Asymptotic cones and boundaries of CAT(0) groups
Abstract: It is well known that the Tits
boundary of a CAT(0) space X embeds in any asymptotic
cone of X. We will demonstrate how to endow the set of asymptotic cones
of a CAT(0) space with a limit structure which
respects this canonical embedding of the Tits boundary into the asymptotic
cone. We will then show how this limit structure illustrates the relationship
between the visual boundary and the Tits boundary of a CAT(0)
group. In particular, we will prove that the direct limit of the asymptotic
cones of a CAT(0) space is isometric to the Euclidean
cone on its Tits boundary and the connected maps are determined by the topology
on the visual boundary.
Wednesday,
March 15, 2017
Nick
Callor (Brigham Young University)
Title: Topological complexity of fibrational kinematic maps (Part I)
Abstract: We review the classic definition of
topological complexity and some properties of it. We then give a generalization
to kinematic maps. We will then prove several properties about this value when
the kinematic map happens to be a fibration.
Wednesday,
March 22, 2017
Nick
Callor (Brigham Young University)
Title: Topological complexity of fibrational kinematic maps (Part II)
Abstract: We review the classic definition of topological
complexity and some properties of it. We then give a generalization to
kinematic maps. We will then prove several properties about this value when the
kinematic map happens to be a fibration.
Wednesday, April
5, 2017
Tyler Hills (Brigham Young University)
Title: The Solenoid and Warsawanoid
are Sharkovskii Spaces
Abstract: We extend Sharkovskii’s theorem
concerning orbit lengths of endomorphisms of the real
line to endomorphisms of a path component of the
solenoid and certain subspaces of the War-
sawanoid. In particular, Sharkovskii
showed that if there exists an orbit of length 3 then there exist orbits of all
lengths. The solenoid is the inverse limit of double covers over the circle,
and the Warsawanoid is the inverse limit of double
covers over the Warsaw circle. We show Sharkovskii’s result is true for path
components of the solenoid and certain subspaces of the Warsawanoid.
Cannon Seminar: Fall 2016
Seminars will be on Wednesdays at 3:00 -3:50 pm in 292
TMCB, unless otherwise noted.
Wednesday,
August 31, 2016
Curt
Kent (Brigham Young University)
Title:
Coarse geometry of groups
Abstract:
We will define asymptotic cones of finitely generated groups and consider some
natural examples.
Wednesday, September 7, 2016
Nick
Callor (Brigham Young University)
Title:
Introduction to Topological Complexity of a Map
Abstract:
Topological complexity of a map is a generalization of topological complexity
of a space, a concept originally introduced by Michal Farber in 2003. We will
first review the classical results from Farber's definition of topological
complexity. We will then introduce the concept of topological complexity of a
map, including the motivation for this generalization and a brief survey of
some of the basic theorems concerning it.
Wednesday, September 14, 2016
Stephen
Humphries (Brigham Young University)
Title:
Braid Groups
Abstract:
We will give a brief introduction to classical Braid
groups.
Wednesday, September 21, 2016
Petar Pavešić (University of
Ljubljana)
Title:
Manipulation of complexity of spidery linkages
Abstract:
We are going to introduce the manipulation problem for a mechanical system with
a given kinematic map. As an illustration, we will compute the manipulation
complexity of a special class of planar linkages.
Wednesday, October 5, 2016
Martha
Kilpack (Brigham Young University)
Title:
Algebraic lattices and the isomorphic subalgebra lattices.
Abstract:
We will look at definitions of algebraic structures, lattice (in at algebraic
sense), and algebraic lattices. We will also look at the big theorems and
open questions in lattice theory. Finally was consider a few algebraic
lattices and find the algebraic structure which have an isomorphic subalgebra lattice.
Wednesday, October 12, 2016
Michael
Hull (University of Florida)
Title:
Highly Transitive Groups
Abstract:
A group action on a set S is called k-transitive
if for any two ordered k-tuples of distinct elements of S, some group element
takes the first k-tuple to the second. The study of k-transitive actions of
finite groups has a long history and is now well-understood using the
classification of finite simple groups. However, much less is
known in the case of infinite groups. We will show that many infinite
groups G which arise in various algebraic and geometric
contexts satisfy the following dichotomy: either G admits no faithful
2-transitive actions, or G is highly transitive, that is G admits a faithful
action that is k-transitive for all k. In addition, we prove that being highly transitive has strong algebraic consequences for a
group. In particular, a highly transitive group G either
contains a normal subgroup isomorphic to the infinite alternating group or G is
mixed identity free. This work is joint with Denis Osin.
Friday, October 14, 2016 (at
2 pm in 292 TMCB)
Ken
Johnson (Penn State Abington)
Title:
Spherical functions and Random walks
Abstract:
The theory of random walks on a group or a set acted on by a group is a part of
harmonic analysis. In the in
nite
case the standard example is the action of \( SO(3) \)
on the 3-sphere. An important part is played by the double coset
algebra \(SO(3)\backslash SO(2)/SO(3) \) of the point
stabilizer, and a basis for the functions on this algebra is the set of
spherical functions. These are constant on the horizontal circles and may be given in terms of Legendre polynmials.
For a random walk on a
nite group G acting transitively on a set X the
corresponding probabilities are constant on the double coset
algebra \( G\backslash H/G \) where H is the point stabilizer, and the
spherical functions become relevant. If the random walk is actually on a
nite group G, the situation is much easier to analize if a probability is given
which is constant on the conjugacy class algebra of G. However
Steve Humphries and I have shown that splittings of
the conjugacy classes of the class algebra of G can be made so that for probablities constant on the split classes the analysis
remains easy. I will discuss generalizing out techniques to arbitrary
walks on a set with a group action, with the possibility that some new
"special functions" may be produced.
Wednesday, October 19, 2016
Sam
Corson (Vanderbilt University)
Title:
New results on the noncommutative Specker phenomenon
Abstract:
We give an overview of the commutative Specker phenomenon,
including motivating examples and theorems. The definition of
n-slenderness is motivated and some of the historic results are
reviewed. We conclude with a sketch of the proof that
torsion-free word hyperbolic groups are n-slender and some open questions.
Wednesday, October 26, 2016
Wolfram
Hojka
Title:
2 is just the worst
Abstract:
We will discuss three constructions in algebra and wild topology where elements
of order 2 present an unexpected obstacle in the proof.
First we use a word length estimate (similar to the one given by Sam Corson
last week) to show inverse limits cannot split nontrivially
as free products. Then we turn to the difficulties created by 2-torsion in
archipelago groups and small loop spaces.
To end on a positive note, we calculate the abelianization
of \((*)_n (\mathbb Z / 2\mathbb Z)\), that is, the first homology group of a
projective Hawaiian earring
Wednesday, November 2, 2016
Rémi Coulon (Université
de Rennes I)
Title:
Automorphisms of lacunary
hyperbolic groups
Abstract:
Asymptotic cone is a tool to capture the large scale
geometry of a group G. Roughly speaking it is a metric space obtained by
viewing G from infinitely far away. A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is
an R-tree. Introduced by Ol'shanskii, Osin and Sapir this class of groups is very broad. It
contains for instance hyperbolic groups, some infinite torsion groups,
some elementary amenable groups, etc. In this talk we will focus on
the outer automorphism group Out(G)
of a lacunary hyperbolic group G. One of the
goal is to understand the structure of Out(G) when
every action of G on an R-tree has a global fixed point. We will also
investigate the Hopf and co-Hopf properties for lacunary hyperbolic groups.
This is a joint work with Vincent Guirardel.
Wednesday, November 9, 2016
Rebeca
Paulsen (Brigham Young University)
Title:
Crystallographic Groups and their Fibrifold Names
Abstract:
John Conway together with collaborators William Thurston and Daniel Huson have proposed a naming convention for the reducible
space groups (three dimensional crystallographic
groups) which is based on the fibrifold of the group.
Each of these topological names are built from the orbifold
name of the two dimensional crystallographic group (wallpaper group) contained
in the space group. The advantage of this naming convention is that the
name contains enough information to construct the group up to isotopy. We
explore some examples of such groups and their names. We will also look
at some irreducible space groups and discuss their fibrifold
names.