# Warwick-Seoul National University Symplectic Geometry Workshop

Tue 18th—Wed 19th Feb 2014

## Organisers: Miles Reid, Weiyi Zhang

This is current planning. There is a REGISTRATION PAGE, and it makes our planning and administration much easier if EVERYONE registers as soon as possible.

PARK Jongil of Seoul National University will be at Warwick Mon 17th-Sat 22nd Feb. The workshop will involve UK and some European specialists in Symplectic geometry and 4-manifold topology

## Participants

Paolo Cascini (Imperial), Jacqui Espina (UCL), Jonny Evans (UCL), Luis Haug (ETH), Jarek Kedra (Aberdeen), YankI Lekili (King's, London), LIM Seonhee (SNU), PARK Jongil (SNU), Miles Reid (Warwick), Antonio Rieser (Technion), Dmitriy Rumynin (Warwick), Dmitry Tonkonog (Cambridge), Peter Topping (Warwick), Henry Wilton (UCL), ZHANG Weiyi (Warwick)

## Schedule (subject to revision)

Warwick-Seoul National University Symplectic Geometry

Workshop, Tue 18th-Wed 19th Feb 2014

Tue 18th Feb

15:00 Jarek Kedra (Aberdeen) *Braids and concordance classes of knots *(Room **A1.01**)

16:30 Dmitry Tonkonog (Cambridge) *Commuting symplectomorphisms and Dehn twists in divisors* (Room **B3.01**)

18:00 wine party and dinner in Mathematics Institute Common Room

Wed 19th Feb

10:00 Paolo Cascini (Imperial) *Topological bounds in birational geometry* (Room **A1.01**)

11:30 Luis Haug (ETH) *The Lagrangian cobordism group of the 2-torus* (Room **D1.07**)

13:00 lunch in Mathematics Institute Common Room

14:00 YankI Lekili (King's, London) *Floer cohomology and Platonic Solids* (Room **B3.01**)

15:30 Antonio Rieser (Technion) *Coisotropic Hofer-Zehnder capacities, non-squeezing for relative embeddings, and energy-capacity inequalities* (Room **B3.01**)

17:00 PARK Jongil (SNU)* On Knot surgery 4-manifolds* (Room **B3.01**)

**Abstract**

**Jarek Kedra: Braids and concordance classes of knots.**

I will construct a map (essentially by closing braids)

F: B_n ---> Conc(S^3)

from the braid group to the concordance group of knots in the three dimensional sphere and prove that this map has good geometric properties. Namely, it is Lipschitz with respect to the biinvariant word metric on the braid group and the four ball genus on the concordance group.

As an application I will construct infinite families of knots (and concodrance classes) with uniformly bounded four ball genus and infinite sequences of concordance classes with growing

four ball genus.

Joint work with Michael Brandenbursky.

**Dmitry Tonkonog: Commuting symplectomorphisms and Dehn twists in divisors.**

Let f, g two commuting symplectomorphisms of a (weakly monotone) symplectic manifold X. We define an action of f on the Floer homology HF(g) and an action of g on HF(f), and prove the supertraces of these actions are equal. Using this, we obtain a lower bound on dim HF(g) if g is a symplectomorphism of X commuting with a symplectic involution.

We apply this to the following. Let X be a (weakly monotone) smooth divisor in Gr(k,n). Take a Lagrangian sphere in X which is a vanishing cycle for an algebraic degeneration of X. It defines a symplectomorphism of X called the Dehn twist. We prove this Dehn twist has infinite order in the symplectic mapping class group of X.

**Paolo Cascini: Topological bounds in birational geometry.**

Many birational invariants of a smooth projective three-fold, such as the number and the singularities of its minimal models, are related with the topology of the underlying manifold. Using these methods, I will discuss some recent progress towards a question by Hirzebruch on the Chern numbers of a smooth projective threefold.

**Luis Haug: The Lagrangian cobordism group of the 2-torus.**

The derived Fukaya category DFuk(M) of a symplectic manifold M is a triangulated category whose objects are the Lagrangian submanifolds in M. Recent work of Biran--Cornea provides a way of understanding cone decompositions in DFuk(M) via Lagrangian cobordisms. One consequence of their work is the existence of a canonical homomorphism from a naturally defined Lagrangian cobordism group to the Grothendieck group of DFuk(M). After explaining some parts of the general theory, I will explain the proof that this map is an isomorphism when M is a 2-torus.

**YankI Lekili:** **Floer cohomology and Platonic Solids. **

We consider Fano threefolds on which SL(2,C) acts with a dense open orbit. This is a finite list of threefolds whose classification follows from the classical work of Mukai-Umemura and Nakano. Inside these threefolds, there sits a Lagrangian space form given as an orbit of SU(2). I will discuss the interesting case of a Lagrangian SU(2)/D_6 in CP^3. We prove this Lagrangian is non-displaceable by Hamiltonian isotopies via computing its Floer cohomology over a field of characteristic 5 and that it (strongly) generates the Fukaya category of CP^3 (which in particular implies that it is non-displaceable from any other object of the Fukaya category, such as the Clifford torus). The computation depends on certain counts of holomorphic disks with boundary on the Lagrangian, which we explicitly identify.

**Anothony Rieser: Coisotropic Hofer-Zehnder capacities, non-squeezing for relative embeddings, and energy-capacity inequalities.**

We introduce a notion of symplectic capacity relative to a coisotropic submanifold of a symplectic manifold, and we define a modification of the Hofer-Zehnder capacity and show it provides an example. As a corollary, we obtain a non-squeezing theorem for symplectic embeddings relative to coisotropic constraints. We further show an several cases where an energy-capacity inequality may be obtained, giving an upper bound on the capacity. Joint work with Samuel Lisi.

**PARK Jongil: On Knot surgery 4-manifolds**

See also:

Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

Past Events

Past Symposia

**Internet Access at Warwick**:

Where possible, visitors should obtain an EDUROAM account from their own university to enable internet access whilst at Warwick.

**WiFi**whilst at Warwick, click here for instructions (upon arrival at Warwick)

**Registration**:

You can register for any of the symposia or workshops online. To see which registrations are currently open and to submit a registration, please click here.

**Contact:**

Mathematics Research Centre

Zeeman Building

University of Warwick

Coventry CV4 7AL - UK

**E-mail:**

mrc@maths.warwick.ac.uk