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From Higher Invariants


Regensburg low-dimensional geometry and topology seminar

The seminar takes place on Zoom (see information below) every Tuesday at 16:00 CET (that is 7am on the US west coast, 10am on the US east coast, 10pm in China and 11pm in Korea and Japan).

Talks are split into a 40 minute and a 20 minute part, with a 15 minute tea break in between (you have to brew your own tea/coffee/beer...).

Talks in spring/summer 2021

The program is also contained in the RLGTS Google Calendar. To receive announcements by email, sign up to our mailing list.

Date Speaker Title Abstract
April 27 Guillem Cazassus (Oxford) The earring correspondence of the pillowcase

Slides Video

Singular instanton homology is a knot invariant introduced by Kronheimer and Mrowka. It is deeply tied to Khovanov homology, and among other things, permits to show that the latter detects the unknot.

In order to compute singular instanton homology, Hedden, Herald and Kirk defined a symplectic (Atiyah-Floer) analogue, called pillowcase homology. This is a Lagrangian Floer homology in the traceless character variety of the four-punctured sphere.

We study the Lagrangian correspondence induced by the earring tangle, an essential ingredient in Kronheimer-Mrowka's construction. Our computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties. This is joint work with Chris Herald, Paul Kirk and Artem Kotelskiy.

May 4 Zoltán Szabó (Princeton) Knot Floer homology constructions and the Pong Algebra

Slides Video

In a joint work with Peter Ozsvath we have developed algebraic invariants for knots using a family of bordered knot algebras. The goal of this lecture is to review these constructions and discuss some of the latest developments
May 11 John Baldwin (Boston College) Instanton L-spaces and splicing

Slides Video

We prove that the 3-manifold obtained by gluing the complements of two nontrivial knots in homology 3-sphere instanton L-spaces, by a map which identifies meridians with Seifert longitudes, cannot be an instanton L-space. This recovers the recent theorem of Lidman–Pinzon-Caicedo–Zentner that the fundamental group of every closed, oriented, toroidal 3-manifold admits a nontrivial SU(2)-representation, and consequently Zentner’s earlier result that the fundamental group of every closed, oriented 3-manifold besides the 3-sphere admits a nontrivial SL(2, C)-representation. This is joint work with Steven Sivek.
May 18 Rostislav Akhmechet (Virginia) Anchored foams and annular homology

Slides Video

We describe equivariant sl(2) and sl(3) homology for links in the solid torus, identified with the thickened annulus, via foam evaluation and universal construction. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams that may intersect the distinguished line. Intersection points, called anchor points, contribute additional terms to the foam evaluation. State spaces and link homology carry additional gradings coming from anchor points. I will describe our evaluation formula and the resulting annular homology theories, as well as relations to other constructions. This is joint work with Mikhail Khovanov.
May 25 no talk (whit tuesday)
June 1 Arunima Ray (MPIM Bonn) A surface embedding theorem

Slides Video

Generalising work of Freedman-Quinn and Stong from 30-40 years ago, we prove a surface embedding theorem for 4-manifolds with good fundamental group, in the presence of potentially unframed dual spheres. The essential obstruction is the Kervaire-Milnor invariant, and I will describe how to compute it and present some useful corollaries of the theorem. The goal of the talk is to be reasonably self-contained, and no prior expert knowledge of 4-manifold topology should be required. This is joint work with Daniel Kasprowski, Mark Powell, and Peter Teichner.
June 8 Cameron Gordon (UT Austin) Toroidal 3-manifolds and the properties in the L-space Conjecture

Slides Video

The L-space Conjecture asserts the equivalence, for prime 3-manifolds, of three properties: not being an L-space (NLS), having left-orderable fundamental group (LO), and supporting a co-orientable taut foliation (CTF). We study these properties for toroidal 3-manifolds. For example, Eftekhary and Hanselman-Rasmussen-Watson have shown that toroidal homology spheres are NLS; we show that they are LO. We also show that the cyclic branched covers of a prime satellite knot are NLS and LO, and CTF if the companion is fibered. A partial extension to links allows us to show that a prime quasi-alternating link is either a (2,q) torus link or hyperbolic, generalizing Menasco's classical result for non-split alternating links. This is joint work with Steve Boyer and Ying Hu.
June 15 Thomas Barthelmé (Queen's University) Hyperbolic-like actions on ℝ and applications to (contact) Anosov flows

Video

Say that an action of a group G on the real line is hyperbolic-like if it commutes with the integer translations and every element either has no fixed points or exactly one pair of attractor/repeller in [0,1).

I will explain how knowing which elements of a hyperbolic-like action admit fixed points is enough to characterize the action up to conjugacy.

Our goal behind this theorem was its application to the problem of classifying (ℝ-covered) Anosov flows up to orbit equivalence, which I will explain. It also turns out that it yields an equivalence between contact Anosov flows up to orbit equivalence and contact structures up to isomorphism, allowing to directly use contact geometry to solve problems on Anosov flows and vice versa.

This is joint work with Kathryn Mann and Jonathan Bowden.

June 22 no talk (ECM)
June 29 Alexandra Kjuchukova (Notre Dame) Covers of S3 bounding covers of B4 Let G be a group, K a knot and ρ a surjective homomorphism π1(S3\K) → G. When does a branched cover of S3 determined by ρ extend over B4, with a smooth branching locus F? Previously, the answer was only known under quite strong assumptions, e.g. when K is slice and G a dihedral group. I will define a new invariant which detects the existence of such an extension for all knots and all metabelian groups. I will give examples of computing the obstruction and constructing the desired surface F. Joint work with Kent Orr.
Monday July 5 (note date change) Jonathan Zung (Princeton) Reeb flows transverse to foliations

Video

Eliashberg and Thurston showed that (roughly) C^2 taut foliations on 3-manifolds can be approximated by tight contact structures. I will explain a new approach to this theorem which allows one to control the resulting Reeb flow and hence produce many hypertight contact structures. Along the way, I will explain how harmonic transverse measures may be used to understand the holonomy of foliations.
Monday July 12 (note date change) Steven Frankel (Washington University in St. Louis) Flows that generalize and organize surfaces

Video

There are two classes of flows - quasigeodesic and pseudo-Anosov - that both generalize and organize aspects of incompressible surfaces. We will discuss potential applications of these flows to the geometry of 3-manifolds, including joint work with Thomas Barthelme and Kathryn Mann on spectral rigidity of transitive pseudo-Anosov flows.
July 20
Regensburg

Past talks

List of past talks in spring/summer 2020 and fall/winter 2020/21, with slides and videos

Zoom information

Zoom Meeting ID: 924 2829 6353

The password is the answer to the following riddle (lower case, singular, no article): As a topologist, what do you typically glue together from charts? (Hint: It's not an orbifold).

We encourage you to enable your video, for a more interactive atmosphere. The talks are recorded and made available on this webpage, including audio and video of questions from the audience.

Organizers

Jonathan Bowden, Lukas Lewark, Raphael Zentner.