I will present how uniform K-stability translates into a convex geometric problem for polarized spherical varieties. From this, we will derive a combinatorial sufficient condition of existence of constant scalar curvature Kahler metrics on smooth spherical varieties, and a complete solution to the Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

This is a report on joint work with Kyler Siegel that develops new ways to count $J$-holomorphic curves in $4$-dimensions, both in the projective plane with multi-branched tangency constraints, and in noncompact cobordisms between ellipsoids. These curves stabilize, i.e. if they exist in a given four dimensional target manifold $X$ they still exist in the product $X \times {\mathbb R}^{2k}$. This allows us to establish new cases of the stabilized embedding conjecture for symplectic embeddings of an ellipsoid into a ball (or ellipsoid).

Lagrangian fibrations on holomorphic symplectic manifolds and orbifolds are higher-dimensional generalizations of elliptic K3 surfaces. They are fibrations whose general fibres are abelian varieties that are Lagrangian with respect to the symplectic form. Markushevich and Tikhomirov described the first example whose fibres are Prym varieties, and their construction was further developed by Arbarello, Ferretti, and Sacca and by Matteini to yield more examples. In this talk we describe the general framework, and consider a new example. We describe its singularities and show that it is a 'primitive' symplectic variety. We also construct the dual fibration, using ideas of Menet. This is joint work with Chen Shen.

A natural intriguing question is the following: how much the moduli spaces of certain polarized varieties know about the symplectic geometry of the underneath manifold? After giving a general overview, I will discuss work-in-progress with T. Baier, G. Granja and R. Sena-Dias where we investigate some relations between the topology of the moduli spaces of certain varieties, of the symplectomorphism group and of the space of compatible integrable complex structures. In particular, using results of J. Evans, we show that the space of such complex structures for monotone del Pezzo surfaces of degree four and five is weakly homotopically contractible.

Both triangulated categories as well as persistence homology play an important role in symplectic topology. The goal of this talk is to explain how to put the two structures together, leading to the notion of a triangulated persistence category. The guiding principle comes from the theory of Lagrangian cobordism.

The talk is based on ongoing joint work with Octav Cornea and Jun Zhang.

Sasakian manifolds are odd-dimensional counterparts of Kahler manifolds in even dimensions, with K-contact manifolds corresponding to symplectic manifolds. In this talk, we give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki.

For this, we translate the question about K-contact 5-manifolds to constructing symplectic 4-orbifolds with cyclic singularities containing disjoint symplectic surfaces of positive genus. The question on Sasakian 5-manifolds translates to the existence of algebraic surfaces with cyclic singularities containig disjoint complex curves of positive genus. A key step consists on bounding universally the number of singular points of the algebraic surface.

The goal of the talk is to explain a duality theorem between Rabinowitz-Floer homology and cohomology. These are Floer homology groups associated to the contact boundary of a Liouville domain, and the duality isomorphism is compatible with canonically defined product structures. Dual to the cohomological product is a homology coproduct which satisfies a remarkable compatibility relation with the product structure. We will also discuss the relationship to loop spaces and Chas-Sullivan/Goresky-Hingston products.

There is a long history of attacking problems involving nonabelian Lie groups by reducing to a maximal abelian subgroup. It has been understood in the last decade that

1) the exact WKB method for studying linear ODEs, 2) the computation of classical Chern-Simons invariants of flat connections, 3) the study of some link invariants, such as the Jones polynomial,

can all be understood as aspects of this general idea. I will describe this point of view, trying to emphasize the common features of all three problems, and (briefly) their common origin in supersymmetric quantum field theory. Parts of the talk are a report of joint work with Dan Freed, Davide Gaiotto, Greg Moore, Lotte Hollands, and Fei Yan.

Finding the minimal number of fixed points of a periodic flow on a compact manifold is, in general, an open problem. We will consider almost complex manifolds and see how one can obtain lower bounds by retrieving information from a special Chern number.

Projective hyper-Kähler (HK) manifolds are among the building blocks of projective manifolds with trivial first Chern class. Fano manifolds are projective manifolds with positive first Chern class.

Despite the fact that these two classes of algebraic varieties are very different (HK manifolds have a holomorphic symplectic form which governs all of its geometry, Fano manifolds have no holomorphic forms) their geometries have some strong ties. For example, starting from some special Fano manifolds one can sometimes construct HK manifolds as parameter spaces of objects on the Fano. In this talk I will explain this circle of ideas and focus on some recent work exploring the converse: given a projective HK manifold, how to recover a Fano manifold from it?

The talk is based on joint work with Roger Bielawski about twistor constructions of higher dimensional non-compact hyperkähler manifolds with maximal and submaximal volume growth. In the first part of the talk, based on arXiv:2012.14895, I will discuss the case of hyperkähler metrics with maximal volume growth: in the same way that ALE spaces are closely related to the deformation theory of Kleinian singularities, we produce large families of hyperkähler metrics asymptotic to cones exploiting the theory of Poisson deformations of affine symplectic singularities. In the second part of the talk, I will report on work in progress about the construction of hyperkähler metrics generalising to higher dimensions the geometry of ALF spaces of dihedral type. We produce candidate holomorphic symplectic manifolds and twistor spaces from Hilbert schemes of hypertoric manifolds with an action of a Weyl group. The spaces we define are closely related to Coulomb branches of 3-dimensional supersymmetric gauge theories.

A closed connected contact manifold is called Besse when all of its Reeb orbits are closed (the terminology comes from Arthur Besse's monograph "Manifolds all of whose geodesics are closed", which deals indeed with Besse unit tangent bundles). In recent years, a few intriguing properties of Besse contact manifolds have been established: in particular, their spectral and systolic characterizations. In this talk, I will focus on Besse contact spheres. In dimension 3, it turns out that such spheres are strictly contactomorphic to rational ellipsoids. In higher dimensions, an analogous result is unknown and seems out of reach. Nevertheless, I will show that at least those contact spheres that are convex still "resemble" a contact ellipsoid: any stratum of the stratification defined by their Reeb flow is an integral homology sphere, and the sequence of their Ekeland-Hofer capacities coincides with the full sequence of action values, each one repeated according to a suitable multiplicity. This is joint work with Marco Radeschi.

The moduli spaces of Higgs bundles and holomorphic connections both have important affine holomorphic Lagrangian subvarieties, these are the Hitchin section and the space of opers, respectively. Both of these spaces arise from the same Lie theoretic mechanism, namely a regular nilpotent element of a Lie algebra. In this talk we will generalize these parameterizations to other nilpotents. The resulting objects are not related by the nonabelian Hodge correspondence, but by an operation called the conformal limit. Time permitting, we will also discuss their relation to Higher Teichmuller spaces.

The category of totally real (TR) submanifolds was traditionally of interest mainly to complex analysts. We will present a survey of recent work towards a "TR geometry" and explain its relevance to the study of minimal Lagrangians and of convexity properties of the volume functional.

In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system.

Tropical geometry is a useful tool to study the Gromov-Witten type invariants, which count the number of holomorphic curves with incidence conditions. On the other hand, holomorphic discs with boundaries on the Lagrangian fibration of a Calabi-Yau manifold play an important role in the quantum correction of the mirror complex structure. In this talk, I will introduce a version of open Gromov-Witten invariants counting such discs and the corresponding tropical geometry on (log) Calabi-Yau surfaces. Using Lagrangian Floer theory, we will establish the equivalence between the open Gromov-Witten invariants with weighted count of tropical discs. In particular, the correspondence theorem implies the folklore conjecture that certain open Gromov-Witten invariants coincide with the log Gromov-Witten invariants with maximal tangency for the projective plane.