Phong - Organizational Meeting
Jian Song - The Kähler-Ricci flow on complex surfaces - Reference
Valentino Tosatti - Lower boundedness of the K-energy on deformations of the Mukai threefold I - Reference
Valentino Tosatti - Lower boundedness of the K-energy on deformations of the Mukai threefold II - Reference
Gábor Székelyhidi - Holomorphic discs and the homogeneous complex Monge-Ampère equation
Jacob Sturm - Convergence of Riemannian Manifolds I - Reference
Jacob Sturm - Convergence of Riemannian Manifolds II - Reference
Valentino Tosatti - Quantization of the space of Kähler metrics - References 1 2
Tien-Cuong Dinh - Exponential estimates for plurisubharmonic
functions and applications to dynamics - Reference
Gábor Székelyhidi - K-stability of constant scalar curvature Kähler manifolds - Reference
Valentino Tosatti - Kähler-Einstein metrics on Fano surfaces I - Reference
Valentino Tosatti - Kähler-Einstein metrics on Fano surfaces II - Reference
Gábor Székelyhidi - Okounkov bodies and test configurations - References 1 2
Jian Song - Regularity of plurisubharmonic upper envelopes in big cohomology classes
Jacob Sturm - A variational approach to complex Monge-Ampère equations - Reference
Phong - Convergence properties of the Yang-Mills flow on Kähler surfaces - References 1 2 3 4 5
Gábor Székelyhidi - The Bernstein problem for affine maximal hypersurfaces - Reference
Valentino Tosatti - Harmonic maps and rectifiability - Reference
Ovidiu Munteanu - Regularity of weakly harmonic maps - References 1 2
Nam Q. Le - The affine Plateau problem - Reference
Ben Weinkove (UC San Diego) - Contracting exceptional divisors by the Kähler-Ricci flow - Reference - Abstract
We give a criterion under which a solution g(t) of the Kähler-Ricci flow
contracts exceptional divisors on compact manifold and can be uniquely
continued on a new manifold. This is a joint work with Jian Song.
Vincent Guedj (Marseille) - Viscosity solutions to degenerate complex Monge-Ampère equations - Abstract
Degenerate complex Monge-Ampère equations on compact
Kähler manifolds have been recently intensively studied using tools
from pluripotential theory. We develop an alternative approach based on
the concept of viscosity solutions.
This approach works only for a rather restricted type of degenerate
complex Monge-Ampère equations. Nevertheless, we prove that the local
potentials of the singular Kähler-Einstein metrics constructed
previously by the authors are continuous plurisubharmonic functions.
They were previously known to be locally bounded.
Another application is a lower order construction with a C0-estimate
of the solution to the Calabi conjecture which does not use Yau's
celebrated theorem. This is joint work with P.Eyssidieux and A.Zeriahi.
Sławomir Dinew (Jagiellonian University) - Hölder regularity for complex Monge-Ampère equations - Abstract
In geometric analysis many arguments rely on a suitable regularity theory for the
analyzed differential equations. Similarly to the solution of the Calabi conjecture often
deriving suitable a priori estimates is in fact the heart of the matter.
In the talk a regularity result for the complex Monge-Ampère equation will be presented.
We will prove that any C1,1 smooth plurisubharmonic solution u to the problem
det(uij̄) = f
with f strictly positive and Hölder continuous has in fact Hölder continuous second
derivatives. For smoother f this follows form the classical Evans-Krylov theory yet in
our case it cannot be applied directly. Instead we shall follow closely an idea of Xu-Jia
Finally we shall discus how this particular regularity result can be used to justify an
argument in the proof of uniqueness of metrics of constant scalar curvature by Chen and
Yuan Yuan (Rutgers) - Kähler-Ricci flow on singular Calabi-Yau varieties
Adam Jacob - Nonabelian Hodge Theory - References 1 2 3
Jacob Sturm - The structure of spaces with Ricci curvature bounded below - Part II - References 1 2
Jacob Sturm - The structure of spaces with Ricci curvature bounded below - Part I - References 1 2
Valentino Tosatti - Convergence of Einstein manifolds - References 1
Jacopo Stoppa (Cambridge) - Donaldson-Thomas invariants and hyperkähler metrics - Reference - Abstract
I will briefly introduce Donaldson-Thomas invariants counting semistable sheaves on a Calabi-Yau threefold. I will then try to turn this into a differential geometry seminar, relying on the work of Gaiotto, Moore and Nietzke that connects DT invariants to some special hyperkähler metrics. I aim at pointing out one analogy between their metrics
and some results I wrote down in this preprint.