May 02, 2016
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Fully nonlinear flows with surgery
Simon Brendle (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We will present joint work with Gerhard Huisken on a fully nonlinear flow for hypersurfaces in Riemannian manifolds. Unlike mean curvature flow, this flow preserves two-convexity in a general ambient manifold. For this fully nonlinear flow, we establish a convexity estimate, a cylindrical estimate, and a pointwise curvature derivative estimate. These estimates allow us to extend the flow beyond singularities by a surgery procedure, similar to the ones developed by Hamilton and Perelman for the Ricci flow and by Huisken and Sinestrari for mean curvature flow
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Ricci flow from metrics with isolated conical singularities
Felix Schulze (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Let $(M,g_0)$ be a compact n-dimensional Riemannian manifold with a finite number of singular points, where at each singular point the metric is asymptotic to a cone over a compact (n-1)-dimensional manifold with curvature operator greater or equal to one. We show that there exists a smooth Ricci flow starting from such a metric with curvature decaying like C/t. The initial metric is attained in Gromov-Hausdorff distance and smoothly away from the singular points. To construct this solution, we desingularize the initial metric by glueing in expanding solitons with positive curvature operator, each asymptotic to the cone at the singular point, at a small scale s. Localizing a recent stability result of Deruelle-Lamm for such expanding solutions, we show that there exists a solution from the desingularized initial metric for a uniform time T>0, independent of the glueing scale s. The solution is then obtained by letting s->0. We also show that the so obtained limiting solution has the corresponding expanding soliton as a forward tangent flow at each initial singular point. This is joint work with P. Gianniotis
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Singularities of mean curvature flow
Tom Ilmanen (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Manifolds with almost nonnegative curvature operator
Burkhard Wilking (Westfälische Wilhelms-Universität Münster)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We show that n-manifolds with a lower volume bound v and upper diameter D bound whose curvature operator is bounded below by −ε(n,v,D) also admit metrics with nonnegative curvature operator. The proof relies on heat kernel estimates for the Ricci flow and shows that various smoothing properties of the Ricci flow remain valid if an upper curvature bound is replaced by a lower volume bound
- Supplements
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May 03, 2016
Tuesday
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09:30 AM - 10:30 AM
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Hermitian curvature flows
Gang Tian (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In this talk, I will discuss Hermitian curvature flows introduced by Streets and myself a few years ago and recent progress on these flows
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Asymptotic rigidity of noncompact shrinking gradient Ricci solitons
Brett Kotschwar (Arizona State University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Shrinking gradient Ricci solitons are models for the local geometry about a developing singularity under the
Ricci flow. At present, all known examples of complete noncompact shrinkers are either locally reducible as products
or possess conical structures at infinity. I will survey some recent results related to the problem of their classification
including some joint work with Lu Wang in which we study the uniqueness of such asymptotic structures as a problem of parabolic unique continuation
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Flow by the Gauss curvature and its power
Lei Ni (University of California, San Diego)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Hypersurfaces of Low Entropy
Lu Wang (Yale University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The entropy is a natural geometric functional introduced by Colding-Minicozzi to study the singularities of mean curvature flow, and it roughly measures the complexity of a hypersurface of Euclidean space.
In this talk, I will survey some recent progress with Jacob Bernstein on understanding the geometry and topology of hypersurfaces with low entropy
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
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- Video
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May 04, 2016
Wednesday
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09:30 AM - 10:30 AM
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Ricci flow through singularities
Bruce Kleiner (New York University, Courant Institute)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
It has been a long-standing problem in geometric analysis to find a good definition of generalized solutions to the Ricci flow equation that would formalize the heuristic idea of flowing through singularities. I will discuss a notion in the 3-d case that has good analytical properties, enabling one to prove existence and compactness of solutions, as well as a number of structural results. It may also be used to partly address a question of Perelman concerning the convergence of Ricci flow with surgery to a canonical flow through singularities. This is joint work with John Lott
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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The Ricci flow on the sphere with marked points
Jian Song (Rutgers University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We study the limiting behavior of the Ricci flow on the 2-sphere with marked points. We show that the normalized Ricci flow will always converge to a unique constant curvature metric or a shrinking gradient soliton metric. In the semi-stable and unstable cases of the 2-sphere with more than two marked points, the limiting metric space carries a different conical and the complex structure from the initial structure. We also study the blow-up behavior of the flow in the semi-stable and unstable cases. This is a joint work with Phong, Sturm and Wang
- Supplements
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May 05, 2016
Thursday
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09:30 AM - 10:30 AM
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Convergence of weak Kaehler-Ricci flows on minimal models of positive Kodaira dimension
Vincent Guedj (Institut de Mathématiques de Toulouse)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Studying the behavior of the Kaehler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampere equations. I will explain how viscosity methods allow one to define and study the long term behavior of the normalized Kaehler-Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models. This is joint work with P.Eyssidieux and A.Zeriahi
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Convergence of Ricci flows with bounded scalar curvature
Richard Bamler (University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
It is a basic fact that the Riemannian curvature becomes unbounded at every finite-time singularity of the Ricci flow. Sesum showed that the same is true for the Ricci curvature. It has since remained a conjecture whether also the scalar curvature becomes unbounded at any singular time.
In this talk I will show that, given a uniform scalar curvature bound, the Ricci flow can only degenerate on a set of codimension bigger or equal to 4, if at all. This result is a consequence of a structure theory for such Ricci flows, which relies on and generalizes recent work of Cheeger and Naber
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Ancient solutions, their asymptotics and uniqueness
Natasa Sesum (Rutgers University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We will discuss construction of new ancient solutions in the Yamabe flow. alwe will also address the uniqueness question of ancient solutions in the mean curvature flow
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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The Chern-Ricci flow
Ben Weinkove (Northwestern University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The Chern-Ricci flow is a geometric flow on complex manifolds. It can be regarded as a generalization of the Kahler-Ricci flow to the non-Kahler setting. In this talk, I will give an overview of results on the Chern-Ricci flow and describe some open problems
- Supplements
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May 06, 2016
Friday
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09:30 AM - 10:30 AM
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Global solutions of the Teichmueller harmonic map flow
Peter Topping (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The Teichmueller harmonic map flow is a gradient flow of the classical harmonic map energy, in which both a map from a surface and the metric on that surface are allowed to evolve. In principle, the flow wants to find minimal immersions. However, in general, the domain metric might degenerate in finite time. In this talk we show how to flow beyond finite time singularities, and this allows us to decompose a general map into a collection of minimal immersions.
This is forthcoming work joint with Melanie Rupflin
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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A gap theorem and some uniform estimates for Ricci flows on homogeneous spaces
Miles Simon (Otto-von-Guericke-Universität Magdeburg)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We prove a gap theorem for homogeneous spaces : If the norm of the Riemannian curvature is one, then the norm of the Ricci curvature is larger than $\ep(n)$, where $\ep(n)$ is a positive constant depending only on the dimension $n$ of the homogeneous space. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$.
This is joint work with Christoph Böhm (University of Münster), Ramiro Lafuente (University of Münster)
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Smoothing properties and uniqueness of the weak Kaehler-Ricci flow
Eleonora Di Nezza (Institut de Mathématiques de Jussieu; École Normale Supérieure)
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- Location
- SLMath:
- Video
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- Abstract
Let X be a compact Kaehler manifold. I will show that the Kaehler-Ricci flow can be run from a degenerate initial data, (more precisely, from an arbitrary positive closed current) and that it is immediately smooth in a Zariski open subset of X. Moreover, if the initial data has positive Lelong number we indeed have propagation of singularities for short time. Finally, I will prove a uniqueness result in the case of zero Lelong numbers.
(This is a joint work with Chinh Lu)
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
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- Video
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- Abstract
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- Supplements
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