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

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)

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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)

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Wednesday, Mar 25, 2020

Stephen Humphries (Brigham Young University)

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Wednesday, Apr 1, 2020

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Wednesday, Apr 8, 2020

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Wednesday, Apr 15, 2020

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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)

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

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Winter 2020

Seminars will be on Wednesdays at 11:00 - 1150 am in 294 TMCB, unless otherwise noted.