Seminars and Colloquia by Series

Branched cyclic covers and L-spaces

Series
Geometry Topology Seminar
Time
Wednesday, July 7, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hannah TurnerUniversity of Texas, Austin

A 3-manifold is called an L-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" 3-manifolds is known. Manifolds obtained as the double-branched cover of alternating knots in the 3-sphere give examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are L-spaces for every index n. I will also discuss an application to "visibility" of certain periodic symmetries of a knot. Some of this work is joint with Ahmad Issa.
 

Normal surface theory and colored Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, May 3, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Christine Ruey Shan LeeUniversity of South Alabama

The colored Jones polynomial is a generalization of the Jones polynomial from the finite-dimensional representations of Uq(sl2). One motivating question in quantum topology is to understand how the polynomial relates to other knot invariants. An interesting example is the strong slope conjecture, which relates the asymptotics of the degree of the polynomial to the slopes of essential surfaces of a knot. Motivated by the recent progress on the conjecture, we discuss a connection from the colored Jones polynomial of a knot to the normal surface theory of its complement. We give a map relating generators of a state-sum expansion of the polynomial to normal subsets of a triangulation of the knot complement. Besides direct applications to the slope conjecture, we will also discuss applications to colored Khovanov homology.

On the length of the shortest closed geodesic on positively curved 2-spheres.

Series
Geometry Topology Seminar
Time
Monday, April 26, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/579155918
Speaker
Franco Vargas PalleteYale University

Following the approach of Nabutovsky and Rotman for the curve-shortening flow on geodesic nets, we'll show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. On the pinched curvature setting, we prove a bound on the first eigenvalue of the Laplacian and use it to prove a new isoperimetric inequality for pinched 2-spheres sufficiently close to being round. This allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting. This is joint work with Ian Adelstein.

3-manifolds that bound no definite 4-manifold

Series
Geometry Topology Seminar
Time
Monday, April 19, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Marco GollaUniversité de Nantes

All 3-manifolds bound 4-manifolds, and many construction of 3-manifolds automatically come with a 4-manifold bounding it. Often times these 4-manifolds have definite intersection form. Using Heegaard Floer correction terms and an analysis of short characteristic covectors in bimodular lattices, we give an obstruction for a 3-manifold to bound a definite 4-manifold, and produce some concrete examples. This is joint work with Kyle Larson.

Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins

Series
Geometry Topology Seminar
Time
Monday, April 12, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
On line
Speaker
David GayUGA

I'm interested in the smooth mapping class group of S^4, i.e. pi_0(Diff^+(S^4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S^4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory. Most of what works in Cerf theory works in dimension 5 and higher, but with a little digging one discovers statements that work in dimension 4 as well. Putting all this stuff together we can show that there is a surjective homomorphism from (a certain limit of) fundamental groups of spaces of embeddings of 2-spheres in connected sums of S^2XS^2 onto this smooth mapping class group of S^4. Furthermore, we can identify two natural, and in some sense complementary, subgroups of this fundamental group, one in the kernel of this homomorphism and one whose image we can understand explicitly in terms of Dehn twist-like diffeomorphisms supported near pairs of embedded S^2's in S^4 (Montesinos twins).

Obstructions to embeddings in 4-manifolds

Series
Geometry Topology Seminar
Time
Friday, April 9, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
On line
Speaker
Anubhav MukherjeeGeorgia Tech

Please Note: Note special day and time.

In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk)  in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.

Right-veering open books and the Upsilon invariant

Series
Geometry Topology Seminar
Time
Monday, April 5, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Diana HubbardBrooklyn College, CUNY

Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon.  I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.

Topology of the Shift Locus via Big Mapping Class Groups by Yan Mary He

Series
Geometry Topology Seminar
Time
Monday, March 29, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Yan Mary HeUniversity of Toronto

The shift locus of (monic and centered) complex polynomials of degree d > 1 is the set of polynomials whose filled-in Julia set contains no critical points. Traversing a loop in the shift locus gives rise to a holomorphic motion of Cantor Julia sets, which can be extended to a homeomorphism of the plane minus a Cantor set up to isotopy. Therefore there is a well-defined monodromy representation from the fundamental group of the shift locus to the mapping class group of the plane minus a Cantor set. In this talk, I will discuss the image and the kernel of this map as well as a combinatorial model for the shift locus. This is joint work with J. Bavard, D. Calegari, S. Koch and A. Walker.

Infinite-type surfaces and the omnipresent arcs by Tyrone Ghaswala

Series
Geometry Topology Seminar
Time
Monday, March 22, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Tyrone GhaswalaCIRGET, Université du Québec à Montréal

Please Note: A pre-talk will be given at 1 and office hours will be held at 3 (following the seminar talk).

In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and $\delta$-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.

This is joint work with Federica Fanoni and Alan McLeay

Big mapping class groups and rigidity of the simple circle by Lvzhou Chen

Series
Geometry Topology Seminar
Time
Monday, March 15, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Lvzhou ChenUT Austin

Please Note: Office hours will be held 3-4 pm.

Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.

Pages