Research Seminar: Symplectic Geometry (Winter Semester 2021/22)

This is a working group seminar run by Klaus Mohnke, Chris Wendl, and Thomas Walpuski on recent developments in symplectic geometry and related areas. Participants are expected to be familiar with the basics of symplectic geometry, including some knowledge of holomorphic curves and/or Floer-type theories. The seminar is conducted in English.

Time: Mondays 13-15 (starting on 2021-10-25)
Location: BMS Seminar Room
Moodle page (key: gromov)

Doug Schultz (HU Berlin)
Multiple covers of holomorphic curves with Lagrangian boundary conditions

For a holomorphic curve in symplectic manifold with Lagrangian boundary conditions, it is not always guaranteed that the curve factors through a branched cover of a simple curve. This is much unlike the case for closed holomorphic curves. Lazzarini, and later Zehmisch, showed that in the case of holomorphic discs in dimension at least 4, one can perturb J so that such a disc does factor through a branched cover of a simple disk. We will talk about these results, and make some observations about the case for higher genus holomorphic curves with Lagrangian boundary conditions.

Marc Kegel (HU Berlin)
Stein traces
Every Legendrian knot leaves a traces in the 4-dimensional symplectic world. In this talk we will investigate whether a 4-dimensional tracker (with the necessary mathematical education) can determine the 3-dimensional creature that left the trace. This is based on joint work with Roger Casals and John Etnyre.
Fabio Gironella (HU Berlin)
Exact orbifold fillings of contact manifolds
The topic of the talk will be Floer theories on exact symplectic orbifolds with smooth contact boundary. More precisely, I will first describe the construction, which only uses classical transversality techniques, of a symplectic cohomology group on such symplectic orbifolds. Then, I will give a geometrical application to the smooth setting, namely the existence, in any odd dimension at least 5, of a pair of contact manifolds with no exact symplectic (smooth) cobordisms in either direction. This is joint work with Zhengyi Zhou.
Chris Wendl (HU Berlin)
Taubes's Gromov invariant
In the mid-1990's Taubes discovered a deep relationship between the Seiberg-Witten invariants on symplectic 4-manifolds and a counting invariant for embedded holomorphic curves, called the Gromov invariant. It is related to the Gromov-Witten invariants, but it packages its curve counts differently so that they are always integers, and its construction does not require any virtual techniques, but relies instead on bifurcation analysis. In this talk, I will say nothing about Seiberg-Witten theory, but will give a basic sketch of how the Gromov invariant is defined and how the underlying bifurcation theory works. The tricky part is to understand a pitchfork-type bifurcation that can emerge from doubly covered holomorphic tori, which forces the invariant to include terms counting these double covers.
Michael Rothgang (HU Berlin)
Bai-Swaminathan, I
It has long been conjectured that in symplectic Calabi-Yau 6-manifolds, there should exist an analogue of the Gromov invariant that gives integer-valued counts of embedded holomorphic curves, together with contributions from their multiple covers that are determined by bifurcation analysis. It is still an open question how to define such an invariant in general, but a recent preprint of Bai and Swaminathan explains how to do it in situations where the multiple covers making contributions have degree at most 2. This talk will give a general overview of the construction.
Paramjit Singh (HU Berlin)
Bai-Swaminathan, II
Continuing on the topic from the previous week, this talk will focus on one of the main technical tools that Bai and Swaminathan use in their construction: it is a new result that gives a necessary condition for a sequence of embedded holomorphic curves (with a converging sequence of almost complex structures) to converge to a nodal multiple cover. This can be used to rule out bifurcations involving nodal curves with ghost bubbles, so that the proof of invariance of curve counts does not require obstruction bundle gluing.
Naageswaran Manikandan (HU Berlin)
Dominik Gutwein, Gorapada Bera (HU Berlin)

Note for new students: If you think you might be interested in this seminar but have neither attended before nor spoken with Profs. Mohnke, Wendl, or Walpuski about it, it is a good idea to get in touch with one of us ahead of time!