28 September
4:15pm-5:15pm |
Speaker: Bo'az Klartag (Tel Aviv)
Title: Poincaré Inequalities and Moment Maps (Joint with MRC)
Abstract:
We will discuss emerging connections between Poincar\'e-type
inequalities on convex bodies and K\"ahler geometry. Most of the
volume of a high-dimensional convex body is concentrated near the
surface of its Legendre ellipsoid of inertia. This unexpected effect
lies at the heart of the analysis related to the central limit
theorem for convex bodies. In order to prove such ``thin shell
estimates'', we are led to the study of Poincar\'e-type inequalities
on convex bodies in high dimension. Excellent estimates are known in
the presence of symmetries, but not in the general case. For an
arbitrary convex body, our idea is to introduce additional symmetries to the
problem by considering a certain transportation of measure from a
space of twice or thrice the dimension. In this talk we will analyze
the method via a few examples which demonstrate its potential.
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| 5 October |
Speaker: TBA
Title: TBA
Abstract:
TBA
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| 12 October |
Speaker: Thomas Mettler (MSRI)
Title: Weyl metrisability for projective surfaces
Abstract:
The existence problem for Riemannian metrics on a surface with prescribed unparametrised geodesics was first studied by R. Liouville. He observed that the problem can be formulated as a linear first order PDE system which in general will not admit solutions. The necessary and sufficient conditions for local existence of solutions were found only recently by Bryant, Dunajski and Eastwood. Surprisingly the conditions are rather complicated. However if one looks for Weyl structures on surfaces with prescribed unparametrised geodesics the situation is different. In this talk I will use techniques from complex geometry to show that the corresponding PDE system always admits local solutions. I will also show that the Weyl structures on the 2-sphere whose geodesics are the great circles, are in one-to-one correspondence with the smooth quadrics without real points in the complex projective plane. If time permits, I will explain that the Weyl metrisability problem for projective surfaces has a natural analogue in all even dimensions.
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| 19 October |
Speaker: TBA
Title: TBA
Abstract:
TBA
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26 October
(3pm in 383N) |
Speaker: Robert Haslhofer (ETH)
Title: Singularities in 4d Ricci flow
Abstract:
In this talk, we discuss the formation of singularities in higher-dimensional Ricci flow without pointwise curvature assumptions. We show that the space of singularity models with bounded entropy and locally bounded energy is orbifold-compact in arbitrary dimensions. In dimension four, a delicate localized Gauss-Bonnet estimate even allows us to drop the assumption on energy in favor of (essentially) an upper bound for the Euler characteristic. We will also see how these results are part of a larger project exploring high curvature regions in 4d Ricci flow. This is all joint work with Reto Mueller.
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26 October
(4pm in 383N) |
Speaker: Fernando Marques (IMPA)
Title: Deforming four-manifolds with positive isotropic curvature
Abstract:
We have recently been able to prove that the moduli space of metrics
with positive scalar curvature of an orientable compact 3-manifold is
path-connected. The proof uses the Ricci flow with surgery, the
conformal method,
and the connected sum construction of Gromov and Lawson. The work of
Perelman on Hamilton's Ricci flow was fundamental.
In this talk we will review the proof of that result and discuss the
necessary modifications to prove similar theorems in the context of
4-manifolds with positive isotropic curvature.
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| 2 November |
Speaker: Mu-Tao Wang (Columbia)
Title: Construction of soliton solutions of curvature flows
Abstract:
I shall discuss new constructions of self-similar solutions of
Lagrangian mean curvature flows and K\"ahler-Ricci flows based on a
separation of variable ansatz.
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| 9 November |
Speaker: Dmitry Jakobson (McGill)
Title: Curvature of random metrics
Abstract:
We study Gauss curvature for random Riemannian metrics on a
compact surface, lying in a fixed conformal class; our questions are
motivated by comparison geometry. Next, analogous questions are
considered for the scalar curvature in dimension n>2, and for the
Q-curvature of random Riemannian metrics. This is joint work with I.
Wigman and Y. Canzani.
|
| 16 November |
Speaker: Lu Wang (Johns Hopkins/MSRI)
Title: Uniqueness of Self-shrinkers of Mean Curvature Flow
Abstract:
Recently, using the desingularization technique, a new family of
complete properly embedded self-shrinkers (of mean curvature flow)
asymptotic to cones in three dimensional Euclidean space have been
constructed by Kapouleas-Kleene-Moeller and independently by Nguyen.
In this talk, we present the uniqueness of self-shrinking ends asymptotic
to any given cone in general Euclidean space. The feature of our uniqueness
result is that we do not require the control on the boundaries of
self-shrinking ends or the rate of convergence to cones at infinity. As
applications, we show that, there do not exist complete properly embedded
self-shrinkers other than hyperplanes having ends asymptotic to
rotationally symmetric cones.
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| 30 November |
Speaker: David Fisher (Indiana/MSRI)
Title: TBA
Abstract:
TBA
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