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September 8
Organizational Meeting
Valentino Tosatti - Nonsmooth geodesics
in the space of Kähler metrics
- Reference
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September 15
J.J. Kohn (Princeton) - Hypoellipticity and loss of derivatives - Reference
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September 22
Jian Song - Bounding scalar curvature for global solutions of
the Kähler-Ricci flow -
References 1 2 3
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September 29
Donovan McFeron - The Mabuchi metric and the Kähler-Ricci flow - Reference
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October 6
Sławomir Dinew - Moser-Trudinger type inequalities for complex Monge-Ampère operators and a conjecture of Aubin - Reference
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October 13
Valentino Tosatti - Volume estimates for Kähler-Einstein manifolds
- Reference
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October 27
Xiaowei Wang - b-stability and blowups - Reference
November 3
Ovidiu Munteanu - Bounds on volume growth of geodesic balls under Ricci flow
- References 1 2
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November 10
Tristan Collins - The transverse entropy functional and the Sasaki-Ricci flow
- Reference
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November 17
Adam Jacob - The Yang-Mills flow and the Atiyah-Bott formula on compact Kähler manifolds - Reference
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December 1
Xiaowei Wang - b-stability and blowups II - Reference
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December 8
Tristan Collins - The ACC Conjecture for log canonical thresholds - References 1
2
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January 26
Organizational Meeting 2:40pm-3:00pm
Frederick T.-H. Fong (Stanford) - Collapsing Behavior of the Kähler-Ricci flow and its Singularity Analysis
- References 1 2 - Abstract
In this talk, I will discuss my recent works on the collapsing behavior of the Kähler-Ricci flow. The first work studies the Kähler-Ricci flow on P1-bundles over Kähler-Einstein manifolds. We proved that if the initial Kähler metric is constructed by the Calabi's Ansatz and is in the suitable Kähler class, the flow must develop Type I singularity and the singularity model is P1 X Cn. It is an extension of Song-Weinkove's work on Hirzebruch surfaces. The second work discusses the collapsing behavior in a more general setting without any symmetry assumption. We showed that if
the limiting Kähler class of the flow is given by a holomorphic submersion and the Ricci curvature is uniformly bounded from above with respect to the initial metric, then the fibers will collapse in an optimal rate, i.e. diam~(T-t)1/2. It gives a partial affirmative answer to a conjecture stated in Song-Szekelyhidi-Weinkove's work on projective bundles.
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February 2
Ben Weinkove (UCSD) - The evolution of a Hermitian metric by its Chern-Ricci form
- Reference - Abstract
I will discuss the evolution of a Hermitian metric on a
compact complex manifold by its Chern-Ricci form. This is an evolution
equation first studied by M. Gill, and coincides with the Kähler-Ricci
flow if the initial metric is Kähler. I will describe the maximal
existence time for the flow in terms of the initial data. I will
discuss the behavior of the flow on complex surfaces when the initial
metric is Gauduchon, on complex manifolds with negative first Chern
class, and on some Hopf manifolds. This is a joint work with
Valentino Tosatti.
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February 9
Yu Wang - C2,α regularity for the complex Monge-Ampère equation - Reference
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February 16
Tristan Collins - Affine K-stability
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February 23
Sławomir Dinew - Hölder continuous solutions to Monge-Ampère equations - Reference
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March 1
Valentino Tosatti - Strominger's system - Reference
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March 8
Valentino Tosatti - Strominger's system II - Reference
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March 22
Jian Song - Form-type Calabi-Yau equations - References 1 2
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March 29
Xiaowei Wang - Kähler metrics with cone singularities along a divisor - Reference
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April 5
Yu Wang - Integrability exponents of plurisubharmonic functions - Reference
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April 12
Michael Siepmann (ETH Zürich) - Ricci Flat Cones and expanding (Kähler) Ricci Solitons - Abstract
We consider the question whether a Ricci flat cone admits a smooth Ricci flow coming out of it.
After some general observations about such Ricci flows we study the Monge-Ampère equation associated to the (expanding) soliton equation on Kähler manifolds with one asymptotically Ricci flat conical end. Solutions will provide examples of possibly non-rotationally symmetric expanding Kähler Ricci solitons. We will also sketch an approach to construct explicit non-rotationally symmetric expanders which flow out of Ricci flat cones.
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April 16 - Special Date - 4.30pm to 5.30pm in Math 507
Martine Klughertz (Toulouse) - The holonomy group at infinity of the Painlevé VI Equation - Abstract
It will be proven that the holonomy group at infinity of the Painlevé VI equation is virtually commutative.
It is is related to differential Galois theory and motivated by the study of non-integrability of
Hamiltonian systems by Ziglin-Morales-Ramis-Simo. (joint work with Bassem ben Hamed and Lubomir Gavrilov).
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April 20
Special Day on Complex Geometry and PDE
- Details
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May 3
Dan Rubin - The Willmore flow with small initial energy - Reference