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From Higher Invariants


Past talks of the Regensburg low-dimensional geometry and topology seminar

Talks in spring/summer 2020

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.
June 23 Lisa Piccirillo (MIT) The trace embedding lemma and PL surfaces Video, Slides 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) Large-scale geometry of big mapping class groups Video, Slides Mapping class groups of infinite type surfaces are not finitely generated (nor even are they locally compact) groups, but in many cases one can still meaningfully talk about their large scale geometry. I will explain joint work with Kasra Rafi on the problem of which surfaces have mapping class groups with nontrivial coarse geometry, and how this relates to questions of actions on arc and curve complexes.
July 7 Mikhail Khovanov (Columbia University) From SL(2) to GL(N) foam evaluation Video, Slides Foams in 3-space are cobordisms between planar graphs that are heavily used in link homology theories. In this talk we'll explain how foam theory can be built from the ground up starting with an evaluation of closed foams, for SL(2) foams, then GL(2) foams, and finally GL(N) foams for any N. The talk is based on joint work with Louis-Hadrien Robert and on L.-H. Robert and Emmanuel Wagner's work on foam evaluation.
July 14 Claudius Zibrowius (UBC Vancouver) Thin links and Conway spheres Video, Slides When does Dehn surgery along a knot give an L-space? More generally, when does splicing two knot complements give an L-space? Hanselman, Rasmussen and Watson gave very compelling answers to these questions using their technology of immersed curves for three-manifolds with torus boundary. Similar invariants have been developed for four-ended tangles. We use those invariants to study various notions of thinness in both Heegaard Floer and Khovanov homology from the perspective of tangle decompositions along Conway spheres. Interestingly, our results bear strong resemblance to the aforementioned results about L-spaces. Also, we observe strong similarities between Heegaard Floer and Khovanov homology that lead us to ask: What is a thin link? This is joint work with Artem Kotelskiy and Liam Watson.
July 21 Marc Lackenby (University of Oxford) Knot genus in a fixed 3-manifold Video, Slides The genus of a knot is the minimal genus of any of its Seifert surfaces. This is a fundamental measure of a knot's complexity. It generalises naturally to homologically trivial knots in an arbitrary 3-manifold. Agol, Hass and Thurston showed that the problem of determining the genus of a knot in a 3-manifold is hard. More specifically, the problem of showing that the genus is at most some integer g is NP-complete. Hence, the problem of showing that the genus is exactly some integer g is not in NP, assuming a standard conjecture in complexity theory. On the other hand, I proved that the problem of determining the genus of a knot in the 3-sphere is in NP. In my talk, I will discuss the problem of determining knot genus in a fixed 3-manifold. I will outline why this problem is also in NP, which is joint work with Mehdi Yazdi. The proof involves the computation of the Thurston norm ball for knot exteriors.