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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).

Upcoming Talks

Date Speaker Title Abstract
June 23 Lisa Piccirillo (MIT) The trace embedding lemma and PL surfaces 4-manifold topologists have long been interested in understanding smooth (resp. topological) embedded surfaces in smooth (resp. topological) 4-manifolds, and as such have developed rich suites of tools for obstructing the existence of smooth (resp. topological) surfaces. Understanding PL surfaces in smooth 4-manifolds has historically garnered less interest, but several problems about PL surfaces have recently arisen in modern lines of questioning. Presently there are far fewer tools available to obstruct PL surfaces. In this talk, I’ll discuss how to use a classical observation, called the trace embedding lemma, to repurpose smooth surface obstructions as PL surface obstructions. I’ll discuss applications of these retooled obstructions to problems about spinelessness, exotica, and geometrically simply connectedness. This is joint work with Kyle Hayden.
June 30 Kathryn Mann (Cornell) Groups that recognize homeomorphisms Given a sufficiently rich group of homeomorphisms of a space, it is sometimes possible to reconstruct the action of a homeomorphism from purely algebraic information from the group. This is an old (if vague!) principle, but in this talk I will discuss a new precise version from joint work with Maxime Wolff, and how we use it to give a new short proof of a recent result of Kim and Koberda on diffeomorphism groups of 1-manifolds: one can distinguish the diffeomorphism groups of different regularities (Cr versus Cs versus smooth...) by their finitely generated subgroups.
July 7 Mikhail Khovanov (Columbia University) Introduction to foam evaluation and its uses tba
July 14 Claudius Zibrowius (UBC Vancouver) tba tba
July 21 Marc Lackenby (University of Oxford) tba tba

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


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.

Past Talks (with slides and videos for download)

Date Speaker Title Downloads Abstract
May 5 Peter Kronheimer (Harvard) Genus versus double-points for immersed surfaces Video If X is a simply-connected closed 4-manifold containing an oriented embedded surface S of genus g, is there always an immersed sphere S' which represents the same homology class and has only g transverse double-points? Colloquially, can we "trade handles for double points"? This is an open question, though a "relative" version of the question (concerning surfaces in the 4-ball bounding a given knot in the 3-sphere) is known to have a negative answer. For closed surfaces in closed 4-manifolds, a particularly interesting class of examples comes from algebraic geometry, and includes the question of whether two smooth quintic surfaces can intersect in a singular rational curve. We will explore whether gauge theory might be a tool that can be used to explore these questions.
May 12 Danny Calegari (University of Chicago) Taut foliations leafwise branch cover the 2-sphere Video, Slides A co-oriented foliation of an oriented 3-manifold is taut if and only if there is a map from the 3-manifold to the 2-sphere which is a branched cover when restricted to each leaf. I shall give two proofs of this theorem and explain its relation to theorems of Ghys and Donaldson.
May 19 Richard Webb (University of Manchester) Quasimorphisms on diffeomorphism groups Video, Slides I will explain how to construct an unbounded quasimorphism on the group of isotopically-trivial diffeomorphisms of a surface of positive genus. As a corollary the commutator length and fragmentation norm are both (stably) unbounded, which solves a problem of Burago--Ivanov--Polterovich. The proof uses a new hyperbolic graph on which these groups act by isometries, which is inspired by techniques from mapping class groups. This is joint work with Jonathan Bowden and Sebastian Hensel.
May 26 Zhenkun Li (MIT) Instanton Floer homology and the depth of taut foliations Video, Slides Sutured manifold hierarchy is a powerful tool introduced by Gabai to study the topology of 3-manifolds. The length of a sutured manifold hierarchy gives us a measurement of how complicated the sutured manifold is. Also, using this tool, Gabai proved the existence of finite depth taut foliations. However, he didn’t discuss how finite the depth could be.

Sutured Instanton Floer homology was introduced by Kronheimer and Mrowka and is defined on balanced sutured manifolds. In this talk, I will explain how sutured instanton Floer homology could offer us bounds for the minimal length of a sutured hierarchy and the minimal depth of a foliation on a fixed balanced sutured manifold.

June 2 no talk (pentecoste)
June 9 Boyu Zhang (Princeton) Instanton Floer homology for tangles and applications in Khovanov homology Video, Slides In this talk, I will present an excision formula for singular instanton Floer homology where the excision surfaces intersect the singular locus. This formula allows us to define an instanton Floer homology theory for sutured manifolds with tangles. Similar to the non-singular case, the instant Floer homology for tangles is non-vanishing on taut manifolds with tangles and detects trivial products. As applications, we prove several detection results for Khovanov homology. This is joint work with Yi Xie.
June 16 Gordana Matic (Univ. of Georgia/MPIM Bonn) Spectral order invariant and obstruction to Stein fillability Video, Slides In this joint work with Cagatay Kutluhan, Jeremy Van Horn-Morris and Andy Wand, we define an invariant of contact structures in dimension three arising from introducing a filtration on the boundary operator in Heegaard Floer homology. This invariant takes values in the set of positive integers union infinity. It is zero for overtwisted contact structures, ∞ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. I will give the definition and discuss computability of the invariant. As an application, we give an easily computable obstruction to Stein fillability on closed contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.


Jonathan Bowden, Lukas Lewark, Raphael Zentner.