Joontae Kim (KIAS, Seoul): Classification of real Lagrangian manifolds in S2 x S2

A Lagrangian submanifold in a symplectic manifold is called real if it is the fixed point set of an antisymplectic involution. In this talk, we give the classification of Hamiltonian isotopy classes of real Lagrangians in S2 x S2, namely there are exactly two classes from the antidiagonal sphere and the Clifford torus. In particular, the uniqueness of real Lagrangian tori shows a real symplectic phenomenon.

Diego Matessi (Università degli Studi di Milano): On the topology of the fixed point set of some anti-symplectic involutions

I will discuss some results concerning the mod 2 cohomology groups of a Lagrangian submanifold which is the fixed point set of a fibre-preserving involution of some symplectic Calabi-Yau manifolds with a Lagrangian fibration. Some years ago, together with Castano-Bernard and Solomon we constructed these involutions. A bit later, with Castano-Bernard we found an exact sequence relating the mod 2 Betti numbers of the Lagrangian with some mod 2 Hodge numbers of the Calabi-Yau. Recently Arguz and Prince have found a very interesting way to compute this sequence using mirror symmetry and squaring of divisors.


Jake Solomon (Hebrew University, Jerusalem): Real bounding chains

I will compare the general problem of counting J-holomorphic disks with boundary in a Lagrangian submanifold with the particular case of a real Lagrangian. In both cases, under appropriate conditions, one can define invariants in arbitrary dimension with both boundary and interior constraints, and an open version of the WDVV equation holds. However, in the real case, one can treat Lagrangian submanifolds with more complicated topology. The special behavior of the real case stems from the notion of a real bounding chain in a Fukaya A-infinity algebra. The same notion underlies the unobstructedness of the fibers of an SYZ fibration. This talk is based on joint work with S. Tukachinsky.

Gleb Smirnov (ETH, Zürich): Infinitely many real symplectic forms on S2 x S2

We study the space of monotone anti-invariant symplectic forms on a complex quadric endowed with an anti-holomorphic involution. We show that this space is disconnected. Also, during the course of the proof, we will produce a diffeomorphism of the grassmannian (2,4) which induces the identity map on all homology and homotopy groups, but which is not homotopic to the identity.

Joé Brendel (Université de Neuchâtel): Real Lagrangian Tori in toric symplectic manifolds

It is a well-known fact that fixed point sets of anti-symplectic involutions are either empty or Lagrangian. On the other hand, one can ask if a given Lagrangian can be realized as the fixed point set of some anti-symplectic involution. In this talk, we will present an obstruction to this property in terms of the displacement energies of nearby Lagrangians. In particular, we apply this obstruction to toric fibres and Chekanov tori in toric symplectic manifolds. For toric fibres, we obtain a complete answer to the above question (partially joint work with J. Kim and J. Moon) and for Chekanov tori the answer is always no, which generalizes a result in S2 x S2 obtained by J. Kim.

Grigory Mikhalkin (Université de Genève): Mutations by patchworking

Some 140 years ago A.A. Markov has arrived to the Diophantine equation a2 + b2 + c2 = 3abc, and to mutations propagating its solutions (now known as Markov triples). In 2010 S. Galkin and A. Usnich have upgraded each Markov triple to a lattice triangle with a canonical integer function on its lattice points and conjectured a connection to symplectic geometry. Several important advances in the last decade have confirmed this conjecture. In particular, in the works of R. Vianna and R. Casals - R. Vianna, a connection was made in the realm of almost toric geometry through toric mutations (in place of Markov's mutations). In this talk we consider another connection, which stays in the realm of conventional toric geometry as it appears in Viro's patchworking technique. The corresponding patchworking mutations are obtained simply as duals to the original Galkin-Usnich mutations. Based on a joint work with Sergey Galkin.