Costante Bellettini | Stable CMC hypersurfaces

In a joint work with N. Wickramasekera (Cambridge) we develop a regularity and compactness theory for a class of codimension-1 integral $n$-varifolds with generalised mean curvature in $L^{p}_{loc}$ for some $p\gt n$. Subject to suitable variational hypotheses on the regular part (namely criticality and stability for the area functional with respect to variations that preserve the "enclosed volume") and two necessary structural assumptions, we show that the varifolds under consideration are "smooth" (and have constant mean curvature in the classical sense) away from a closed singular set of codimension 7. In the case that the mean curvature is non-zero, the smoothness is to be understood in a generalised sense, i.e. also allowing the tangential touching of two smooth CMC hypersurfaces (e.g. two spheres touching).

Simon Donaldson | Singularities of Kahler-Einstein metrics (after Hein and Sun)

It many cases it is known that singular algebraic varieties admit Kahler-Einstein metrics, for example as the limit of such metrics on smoothings of the variety. Then there is a question of relating the metric structure at the singularities to algebraic geometry and in particular describing the metric tangent cones. This is a central and long-standing problem in complex differential geometry and there has been decisive recent progress by H-J. Hein and S. Sun. The lecture will be an introduction to this circle of ideas.

Kota Hattori | The nonuniqueness of tangent cones at infinity of Ricci-flat manifolds

Colding and Minicozzi have shown the uniqueness of the tangent cones at infinity of Ricci-flat manifolds with Euclidean volume growth which has a tangent cone at infinity with a smooth cross section. In this talk I will show an example of the Ricci-flat manifold implying that the assumption for the volume growth in the above result is essential. More precisely, I will show that some of the hyperKähler manifolds constructed by Anderson, Kronheimer and LeBrun have infinitely many tangent cones at infinity and one of them has a smooth cross section.

Hiroshi Iriyeh | On Mahler's conjecture in the three dimensional case

Mahler's conjecture states that for a centrally symmetric convex body $K$ in the $n$-dimensional Euclidean space, the product of the volume of $K$ and that of the polar body is greater than or equal to $4^n/n!$. The two dimensional case is solved by Mahler in 1939. In spite of the simpleness of the problem, this conjecture is still open for $n>2$. We prove Mahler's conjecture for three dimensional centrally symmetric convex bodies with a hyperplane symmetry. We also discuss the relation between Mahler's conjecture and Viterbo's isoperimetric-type conjecture for symplectic capacities. The talk is based on a joint work with Masataka Shibata.

Dominic Joyce | Lagrangian Mean Curvature Flow in Calabi-Yau manifolds and the Thomas-Yau conjecture I, II.

Let $X$ be a Ricci-flat Calabi-Yau manifold and $L_0$ a compact Lagrangian submanifold in $X$. The Lagrangian Mean Curvature Flow $L_t : t\in [0,T)$ flows $L_0$ in the direction of its mean curvature in $X$. If $X$ is a Ricci-flat Calabi-Yau manifold the flow stays within Lagrangian submanifolds, and if $L_0$ is graded (Maslov zero) then the flow stays within the Hamiltonian isotopy class of $L_0$. The fixed points of the flow are special Lagrangian submanifolds.

In 2001, motivated by Mirror Symmetry, Thomas and Yau conjectured that if $L_0$ satisfies a "stability condition", then the Lagrangian MCF of $L_0$ should exist for $t\in[0,\infty)$, and should converge as $t\to\infty$ to a special Lagrangian submanifold $L_\infty$ in $X$.

The Thomas-Yau conjecture cannot be true in its exact original form. Their definition of stability condition is dubious, and Neves proved that in any Hamiltonian isotopy class we can find Lagrangians for which singularities of Lagrangian MCF develop in finite time.

Nonetheless, the author believes that a modified, and rather more complex, version of the Thomas-Yau conjecture may hold. These lectures will explain the author’s attempt in arXiv:1401.4949 to update the Thomas-Yau conjecture in the light of subsequent developments in Mirror Symmetry, Fukaya categories, Bridgeland stability conditions, and Lagrangian MCF.

The main features of our picture are:

• We should work in the derived Fukaya category $D^bF(X)$ of (immersed) graded Lagrangians in X. A graded Lagrangian $L$ in $X$ is an object of $D^bF(X)$ if it has unobstructed Lagrangian Floer cohomology, and a choice of bounding cochain $b$.
• We allow finite time singularities in Lagrangian MCF, but aim to continue the flow past the singularity after doing a surgery, in a similar way to Perelman’s proof of the Poincaré Conjecture using Ricci flow. These surgeries may change the topology of $L_t$, but we insist that they should stay in the same isomorphism class in $D^bF(X)$.
• We expect there should be a Bridgeland stability condition on the triangulated category $D^bF(X)$, whose semistable objects are (roughly) represented by special Lagrangians.
• Starting from a Lagrangian $L_0$ with unobstructed Lagrangian Floer cohomology, the Lagrangian MCF with surgeries starting from $L_0$ should exist uniquely for all time, and converge to a finite union of special Lagrangians with different phases, which corresponds to the Harder-Narasimhan type decomposition of $[L_0]$ into semistable objects in $D^bF(X)$ coming from the Bridgeland stability condition.
• If $L_0$ has obstructed Lagrangian Floer cohomology, then finite time singularities of Lagrangian MCF may occur after which you cannot continue the flow by doing a surgery.

Ailsa Keating | A symplectic geometer's introduction to hypersurface singularities

This mini-course will aim to give an introduction to some of the symplectic properties of isolated hypersurface singularities and their Milnor fibres, targeted at graduate students in geometry. A particular emphasis will be put on ideas from Picard-Lefschetz theory, some low-dimensional techniques, and examples, in the hope that these will prove useful to researchers with a variety of geometric interests. A tentative list of topics for each lecture is as follows:

1. Basic concepts: Milnor fibres and Milnor fibrations, symplectic parallel transport, vanishing cycles. A'Campo's algorithm for the Milnor fibre of a two-variable singularity, and examples.
2. Dehn twists: different descriptions, basic properties, mutations. Notions from singularity theory: Milnor's theorem, intersection forms, modality, adjacency. Lefschetz fibration techniques for describing Milnor fibres in higher dimensions (start).
3. Lefschetz fibration techniques for describing Milnor fibres in higher dimensions (continued). Examples of Fukaya categories associated to singularities.

Kirill Krasnov | Gravity and 3-forms

I will describe how gravity in three and four dimensions can be obtained as the dimensional reduction on $S^3$ (in one case on $\mathbf{R}^3$) of certain theories of 3-forms in six and seven dimensions.

From a more mathematical perspective, I will describe how solutions of certain second order PDE's in three and four dimensions can be lifted to complex structures in six, metrics of holonomy $G_2$ in seven, and metrics of holonomy $Spin(7)$ in eight dimensions. This generalises the known constructions by Bryant and Salamon.

Yankı Lekili | Duality between Legendrian and Lagrangian invariants

Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(\Lambda)$, with coefficients in chains of the based loop space of $\Lambda$ and study its relation to the Floer cohomology $CF^{\ast}(L)$ of $L$. We show that under connectivity and locally finiteness assumptions on $CE^*(\Lambda)$, there is a duality between $CE^{\ast}(\Lambda)$ and $CF^{\ast}(L)$. In some cases, this gives explicit computations of symplectic cohomology of X. Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. This is a joint work with Tobias Ekholm.

Reiko Miyaoka | Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces

This is a joint work with H. Iriyeh, H. Ma and Y. Ohnita (Bull. London Math. Soc. 2016). To compute the Floer homology of certain Lagrangian submanifold, the first step is to show the Hamiltonian non-displaceability (HND for short) , since the intersection with its Hamiltonian deformation is the generator of the Floer homology. We have a rich class of Lagrangian minimal submanifolds in the complex hyperquadric obtained as the Gauss images of isoparametric hypersurfaces in spheres. The latter is a class of hypersurfaces having constant principal curvatures, including infinitely many homogeneous and non-homogeneous examples. For such hypersurfaces, we show that the Gauss images are HND if the multiplicities of the principal curvatures are not less than $2$. Under this assumption, when the number $g$ of distinct principal curvatures is $3$, the Floer homology coincides with the singular homology with $\mathbf{Z}_2$ coefficients, and when $g=4$ and $6$, we show they are HND. For the proof, we use Damian's lifted Floer complex and spectral sequence.

Takeo Nishinou | Periodic plane tropical curves and holomorphic curves on tori

Tropical curves are combinatorial object satisfying certain harmonicity condition. They reflect properties of holomorphic curves, and a few precise correspondence is known between tropical curves in real affine spaces and holomorphic curves in toric varieties. A natural question is whether there is a correspondence between periodic tropical curves and holomorphic curves on complex tori. The two dimensional case can be solved in a satisfactory manner, but the situation is rather different from the non-periodic case. This is a joint work with Tony Yue Yu.

Kaoru Ono | Lagrangian Floer theory - around generating criterion for Fukaya category

I will explain some construction in Lagrangian Floer theory and present Generating Criterion for Fukaya category. I would also like to mention some related topics. The talk is based on a joint work with M. Abouzaid, K. Fukaya, Y.-G. Oh and H. Ohta as well as other joint works with Fukaya, Oh and Ohta.

Alexander Ritter | The cohomological McKay correspondence via Floer theory

The goal of my talk is to present work in progress, jointly with Mark McLean (Stony Brook, NY), which proves the cohomological McKay correspondence using symplectic topology techniques. This correspondence states that given a crepant resolution $Y$ of the singularity $\mathbf{C}^n / G$, where $G$ is a finite subgroup of $SL(n,\mathbf{C})$, the conjugacy classes of $G$ are in 1-1 correspondence with generators of the cohomology of $Y$. This statement was proved by Batyrev (1999) and Denef-Loeser (2002) using algebraic geometry techniques. We instead construct a certain symplectic cohomology group of Y whose generators are Hamiltonian orbits in Y to which one can naturally associate a conjugacy class in G. We then show that this symplectic cohomology recovers the classical cohomology of Y. This work is part of a large-scale project which aims to study the symplectic topology of resolutions of singularities also outside of the Calabi-Yau setup.

Kazushi Ueda | Mirror symmetry and Grassmannians

The Grassmannian $Gr(r,n)$ of $r$-spaces in an $n$-space is a classical object appearing in various aspects of mathematics. In the talk, we will discuss the relation between mirror symmetry and Grassmannians.

To study classical mirror symmetry for $Gr(r,n)$, it is convenient to use the description as the GIT quotient of the space $Mat(r,n)$ of $n$ times $r$ matrices by the natural action of $GL(r)$. This leads to the quasimap theory, which is a mathematical formulation of A-twisted non-abelian gauged linear sigma model, and related to the vortex theory by the Hitchin-Kobayashi correspondence. One of the central themes is the abelian/non-abelian correspondence.

To study homological mirror symmetry, one can use the Gelfand-Cetlin system, which is based on the description as a coadjoint orbit. The potential function, which is a Floer-theoretic invariant which "counts" the number of Maslov index 2 disks, gives the Landau-Ginzburg potential of the mirror. In the case of $Gr(2,n)$, one can associate an integrable systems to a triangulation of an $n$-gon, and the corresponding potential functions are related by cluster transformations.

One relation between classical mirror symmetry and homological mirror symmetry is given by Dubrovin's conjecture and its refinement called the gamma conjecture.