Cannon Seminar: Fall
2019/Winter 2020
Winter 2020
Seminars will be on Wednesdays at 12:00 – 1:00 am in
294 TMCB, unless otherwise noted in green.
Tuesday, Jan 14, 2020
David
Constantine (Wesleyan University)
Title: Entropy
rigidity for Hilbert geometries
Abstract: Hilbert geometries, or convex projective geometries,
generalize hyperbolic geometry in a non-Riemannian direction. Though they
generally do not satisfy the metric conditions to be negatively curved, they
nonetheless possess many properties similar to negatively curved spaces. In
this talk I'll discuss some recent work joint with Illesanmi
Adeboye and Harrison Bray proving an entropy rigidity
statement for these spaces: If a compact manifold of dimension at least 3
supports a Hilbert geometry and a hyperbolic metric, then the hyperbolic metric
uniquely minimizes a certain functional relating entropy and volume. To prove
this, we use techniques developed by Besson-Courtois-Gallot in the Riemannian setting. I'll also discuss some
applications of this result and, if time permits, an extension to the
finite-volume case joint with Bray.
Wednesday, Jan 22, 2020
Curtis
Kent (Brigham Young University)
Title:
Topologies on the fundamental group, covering spaces, and fibrations
Abstract:
We will discuss two distinct topologies on the set of homotopy
classes of based paths and their relationships to covering spaces and unique
path lifting fibrations.
Tuesday, Jan 28, 2020 at 10 am in 294 TMCB
Khadim War (University
of Chicago)
Title:
Margulis estimates for geodesic flows on
surfaces without conjugate points of genius at least two
Abstract:
In this talk we first discuss the construction of the measure of maximal
entropy for the geodesic flow on surfaces
without conjugate points via Patterson-Sullivan measures. We then use the
geometric properties of this construction to
give an exact asymptotic growth rate of closed geodesics which is known
as Margulis estimates in the case of
negative curvature, i.e. we prove that the number of free homotopy
classes of closed geodesics of length less than T>0 is of order of exp(hT)/(hT)
where h is the topological entropy of the system. This is based on a joint work
with Vaughn Climenhaga and Gerhard Knieper.
Wednesday, Feb 5, 2020
Eric
Swenson (Brigham Young University)
Title:
Separators of Topological spaces and pretreess
Abstract:
Given a collection of closed separators $\mathcal A$
of a connected topological space $Z$ satisfying some reasonable properties, we
show how to construct a pretree encoding the
separation properties of $\mathcal A$.
Wednesday, Mar 4, 2020
Mark
Hughes (Brigham Young University)
Title:
Abstract:
Wednesday, Mar 11, 2020
Boštjan Lemež (University
of Ljubljana)
Title:
Generalized inverse limits indexed by the integers
Abstract: We introduce the generalized inverse limits
indexed by integers and compare them to the generalized inverse limits indexed
by positive integers.
We
also construct the upper semi-continuous function $f$ such that the inverse
limit of the closed unit intervals $[0,1]$ with $f$ as
the only bonding function is a 3-cell.
Wednesday, Mar 18, 2020
Garbriel Islambouli
(University of Waterloo)
Title:
Abstract:
Wednesday, Mar 25, 2020
Stephen
Humphries (Brigham Young University)
Title:
Abstract:
Wednesday, Apr 1, 2020
TBD
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Wednesday, Apr 8, 2020
TBD
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Wednesday, Apr 15, 2020
TBD
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Fall 2019
Seminars will be on Wednesdays at 11:00 am - 11:50 am
in 294 TMCB, unless otherwise noted.
Wednesday, Sept 11, 2019
Tyler
Hills (Brigham Young University)
Title:
An introduction to shape theory
Abstract:
We explore foundational topics of shape theory beginning with directed sets,
inverse
systems, and inverse limits in an arbitrary category. We then discuss nerves of
open covers of
topological spaces which will enable us to associate a sequence of simplicial
complexes which
approximate the space. We end with the definition of the shape group of a
topological space.
Wednesday, Sept 18, 2019
No
seminar
Wednesday, Sept 25, 2019
Discussion
on Roller boundaries of CAT(0) cube complex groups,
directed by Curt Kent.
Wednesday, Oct 2, 2019
Mark
Hughes (Brigham Young University)
Title:
Braid rank and genera of knots
Abstract: We describe a new technique to recast geometric problems
involving the 4-ball and ribbon genera of a link in terms of algebraic
properties of a braid representative. These techniques make use of braided cobordisms, and require the study of certain shortest word
problems in the braid group described by Rudolph
Wednesday, Oct 9, 2019
Eric Swenson (Brigham Young University)
Title:
CAT(0) spaces with isolated flats
Abstract: We give a new proof of the result of Kleiner
and Hurska that a CAT(0)
group with isolated flats is relatively hyperbolic wrt
maximal abelian subgroups.
Wednesday, Oct 16, 2019 at
3:00 pm
Emily
Stark (University of Utah)
Title:
Action rigidity for free products of hyperbolic manifold groups
Abstract: The relationship between the large-scale geometry of a
group and its algebraic structure can be studied via three notions: a group's
quasi-isometry class, a group's abstract commensurability class, and geometric
actions on proper geodesic metric spaces. A common model geometry for groups G
and G' is a proper geodesic metric space on which G and G' act geometrically. A
group G is action rigid if every group G' that has a common model geometry with
G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold
group is not action rigid for all n at least three. In contrast, we show that
free products of closed hyperbolic manifold groups are action rigid.
Consequently, we obtain the first examples of Gromov
hyperbolic groups that are quasi-isometric but do not virtually have a common
model geometry. This is joint work with Daniel Woodhouse.
Wednesday, Oct 23, 2019
Jacob
Pichelmeyer
Title:
Genera of knots in the complex projective plane
Abstract: Let $K : S^1\to S^3$ be a knot and $M$ be a smooth, closed
four-dimensional manifold. The $M$-genus of K is the least genus among all
smooth, orientable surfaces $\Sigma$ smoothly and properly embedded in $M \
B^4$ such that $\partial\Sigma= K$. In the cases where $M$ is the four-sphere
$S^4$ or $S^2\times S^2$, the $M$-genus has been computed for all 2,977 prime
knots up to 12 crossings. The case of the complex projective plane $\mathbb CP^2$ stands in contrast. There are 2,000+ prime
knots up to 12 crossings along with their mirrors for which computation of the
$\mathbb CP^2$-genus is non-trivial. Of these, the $\mathbb CP^2$-genus was known for only 8 such knots before
our work. We have obtained both obstruction and construction results, allowing
for the computation of 146 more prime knots of 12 crossings or less for which
computation of the $\mathbb CP^2$-genus is
nontrivial, along with an infinite family. We present background on this topic,
explanation of how the constructions and obstructions were obtained, and how computations
were made using these results.
Wednesday, Oct 30, 2019
Curt
Kent (Brigham Young University)
Title:
Lacunary properties of CAT(0)
groups
Abstract:
A group is hyperbolic if and only if all of its asymptotic cones are $\mathbb R$-trees. A group is lacunary
hyperbolic if one of its asymptotic cones is an $\mathbb R$-tree. Thus we will say a
group is lacunary $P$ if one of its asymptotic
cones has property $P$. I will present some properties of groups that can
be characterised by their asymptotic cones and some lacunary properties. Specifically we will show that a
CAT(0) group is a product if and only if it a lacunary product and that a group CAT(0) group has isolated
flats if and only if it one of its asymptotic cones is tree-graded with respect
to flats.
Wednesday, Nov 13, 2019
Maggie Miller (Princeton)
Title:
Dehn surgery on links and the Thurston norm
Abstract:
Let $L$ be a multiple-component link in a rational homology sphere $Y$ with
nonzero linking numbers. Let $S$ be a surface properly embedded in $Y-N(L)$ which is minimum-genus in its relative homology class.
We may obtain a closed surface $\hat{S}$ by Dehn-surgering each component of $L$ according to $\partial
S$ and capping off. After some exposition on sutured manifolds, foliations, and
some interesting moves one can do on each, I will show that for
"most" choices of $[S]$ (this “most” can be stated precisely),
$\hat{S}$ is also minimum-genus, and discuss some examples where it is not. The
proof is constructive; we find a taut foliation in the surgered
manifold achieving $\hat{S}$ as a leaf. This is
analogous to Gabai’s proof of the property R
conjecture, and is inspired by his methods.
Wednesday, Nov 20, 2019
Seungwon Kim (Center for Geometry and Physics - Institute for Basic Science Korea)
Title:
Non-orientable surfaces in 4-space
Abstract:
In this talk, we will consider isotopy problems of non-orientable surfaces
using banded unlink diagrams and knot group.
Wednesday, Dec 4, 2019
Nick
Callor (Brigham Young University)
Title:
TBD
Abstract:
TBD
Winter 2020
Seminars will be on Wednesdays at 11:00 - 1150 am in
294 TMCB, unless otherwise noted.