Seminars and Colloquia by Series

Invariance of Knot Lattice Homology

Series
Geometry Topology Seminar
Time
Monday, September 27, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Seppo Niemi-ColvinDuke University

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.

Mitsumatsu's Liouville domains are stably Weinstein

Series
Geometry Topology Seminar
Time
Monday, September 20, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Austin ChristianGeorgia Tech

In 1995, Mitsumatsu constructed a large family of Liouville domains whose topology obstructs the existence of a Weinstein structure.  Stabilizing these domains yields Liouville domains for which the topological obstruction is no longer in effect, and in 2019 Huang asked whether Mitsumatsu's Liouville domains were stably homotopic to Weinstein domains.  We answer this question in the affirmative.  This is joint work-in-progress with J. Breen.

A curve graph for Artin groups

Series
Geometry Topology Seminar
Time
Monday, September 13, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
online
Speaker
Rose Morris-WrightUCLA

Please Note: Meeting URL https://bluejeans.com/770198652/3456?src=join_info Meeting ID 770 198 652 Participant Passcode 3456 Want to dial in from a phone? Dial one of the following numbers: +1.408.419.1715 (United States (San Jose)) +1.408.915.6290 (United States (San Jose)) (see all numbers - https://www.bluejeans.com/numbers) Enter the meeting ID and passcode followed by # Connecting from a room system? Dial: bjn.vc or 199.48.152.152 and enter your meeting ID & passcode

Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. Braid groups are can be thought of as the mapping class groups of a punctured disc. The combinatorial and geometric structure of the mapping class group is reflected in a Gromov-hyperbolic space called the curve graph of the mapping class group. Using the curve graph of the mapping class group of a punctured disc, we can define a graph associated to a given braid group. In this talk, I will discuss how to generalize this construction to more general classes of Artin groups. I will also discuss the current known properties of this graph and further open questions about what properties of the curve graph carry over to this new graph. 

Chi-slice 3-braid links

Series
Geometry Topology Seminar
Time
Monday, August 30, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan SimoneGeorgia Tech

A link L in the 3-sphere is called chi-slice if it bounds a properly embedded surface F in the 4-ball with Euler characteristic 1. If L is a knot, then this definition coincides with the usual definition of sliceness. One feature of such a link L is that if the determinant of L is nonzero, then the double cover of the 3-sphere branched over L bounds a rational homology ball. In this talk, we will explore the chi-sliceness of 3-braid links. In particular, we will construct explicit families of chi-slice quasi-alternating 3-braids using band moves and we will obstruct the chi-sliceness of almost all other quasi-alternating 3-braid links by showing that their double branched covers do not bound rational homology 4-balls. This is a work in progress joint with Vitaly Brejevs.

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.

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