List of (confirmed) speakers
Minicourses :

Bo Berndtsson , Chalmers University

Hans Joachim Hein , Fordham University

Valentino Tosatti , Northwestern University

Jeff Viaclovsky , Wisconsin University
Talks :

Thibaut Delcroix, ENS Paris

Eleonora Di Nezza , Imperial College

Jakob Hultgren , Chalmers University

Zakarias Sjostrom Dyrefelt , Université de Toulouse
Titles and abstracts
 Bo Berndtsson (Chalmers University) : Direct image bundles and variations of complex structures
Given a smooth proper fibration $p:\mathcal X\to B$ and $L$ a line bundle over $\mathcal X$, the direct image $$ E:= p_*(L) $$ is in many cases a holomorphic vector bundle over $B$. Its fibers are the spaces of holomorphic sections of $L$ over the fibers of $p$, $X_t=p^{1}(t)$, and they can be given various $L^2$metrics. In case the fibration is of relative dimension $n$ so that the fibers are compact Riemann surfaces, special cases of this situation can be used to study the variation of complex structures on the fibers $X_t$. (The fibers are all diffeomorphic, but their complex structure varies with $t$, so we can view the family $X_t$ as a family of variations of complex structures on one fixed smooth manifold.) When the relative dimension is higher than one the situation is more complicated and one needs to consider also higher direct images. I will discuss the problems that arise in this connection, with previous work of Siu, Schumacher and ToYeung and some recent joint work with Xu Wang and Mihai Paun.
 HansJoachim Hein (Fordham University) : Tangent cones of CalabiYau varieties
It has been known for about 10 years that the classical CalabiYau theorem on the existence and uniqueness of Ricciflat Kahler metrics on smooth complex manifolds with zero first Chern class can be extended to a natural setting of weak Kahler metrics on singular complex varieties. However, until relatively recently nothing was known  even in the simplest nontrivial examples  about the precise asymptotic behavior of these weak Ricciflat metrics at the singularities of the underlying varieties. I will explain work of DonaldsonSun, HNaber and HSun that resolves this question in certain cases.
 Valentino Tosatti (Northwestern University) : Metric Limits of CalabiYau Manifolds
In this minicourse I will give an introduction to the study of limits of Ricciflat Kahler metrics on a compact CalabiYau manifold when the Kahler class degenerates to the boundary of the Kahler cone. Analytically, the problem is to prove suitable uniform a priori estimates for solutions of a degenerating family complex MongeAmpère equations, away from some singular set. Geometrically, this can be used to understand the GromovHausdorff limit of these metrics. And if the manifold is projective algebraic and the limiting class is rational, the limits possess an algebraic structure and are obtained from the initial manifold via contraction morphisms from Mori theory.
 Jeff Viaclovsky (Wisconsin University) : The geometry of SFK ALE metrics
I will discuss some of the basics of scalarflat Kaehler (SFK) metrics, and focus on the geometry of SFK metrics which are asymptotically locally Euclidean (ALE). These space arise as "bubbles" in the compactness theory of Calabi's extremal Kaehler metrics. I will also present some of the deformation theory of SFK ALE metrics.
 Thibaut Delcroix (ENS Paris) : Kähler geometry of horospherical manifolds
Horospherical manifolds form a class of almost homogeneous manifolds whose Kähler geometry is very close to that of toric manifolds. They strictly contain homogeneous toric bundles, to which a lot of results holding for toric manifolds have been extended. I will present horospherical manifolds, trying to convince you that they are not much harder to deal with, and in particular I will present the criterion for Kstability in the Fano case that follows either from my work on spherical varieties, or from a direct, WangZhu type, approach.
 Eleonora Di Nezza (Imperial College) : MongeAmpère energy and weak geodesic rays
The recent proof of Demailly's conjecture by Witt Nyström gives another evidence that pluripotential theory play a key role when working with complex MongeAmpère equations in order to solve problems in differential and algebraic geometry. In this talk we investigate pluripotential tools: we characterise MongeAmpère energy classes in terms of envelopes. And in order to do that, we develop the theory of weak geodesic rays in a big cohomogy class. We also give a positive answer to an open problem in pluripotential theory. This is a joint work with Tamas Darvas and Chinh Lu.
 Jakob Hultgren (Chalmers University) : Coupled KählerEinstein Metrics
A central theme in complex geometry is to study various types of canonical metrics, for example KählerEinstein metrics and cscK metrics. In this talk we will introduce the notion of coupled KählerEinstein (cKE) metrics which are ktuples of Kähler metrics that satisfy certain coupled KählerEinstein equations. We will discuss existence and uniqueness properties and elaborate on related algebraic stability conditions. (Joint work with David Witt Nyström)
 Zakarias Sjostrom Dyrefelt (Université de Toulouse) : Kstability of constant scalar curvature Kähler manifolds
In this talk we introduce a variational/pluripotential approach to the study of Kstability of Kähler manifolds with transcendental cohomology class, extending a classical picture for polarised manifolds. Our approach is based on establishing a formula for the asymptotic slope of the Kenergy along certain geodesic rays, from which we deduce that cscK manifolds are Ksemistable. Combined with a recent properness result of R. Berman, T. Darvas and C. Lu we further deduce uniform Kstability of cscK manifolds with discrete automorphism group, thus confirming one direction of the YTD conjecture in this setting. If time permits we also discuss possible extensions of these results to the case of compact Kähler manifolds admitting holomorphic vector fields.