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Winter Quarter
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9 January
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Speaker: Pierre Albin (University of Illinois)
Title: Hodge cohomology on singular spaces (Note the change)
Abstract: The cohomology of a singular space is not as well-behaved as that of a closed manifold, so for `stratified spaces' (such as an algebraic variety, the orbit space of a group action, and many moduli spaces) Goresky and MacPherson introduced a variation, `intersection cohomology'. At the same time, Cheeger was studying the de Rham cohomology of differential forms associated to metrics with iterated edge singularities, and discovered that this yields an analytic approach to the same theory. I will report on joint work with Markus Banagl, Eric Leichtnam, Rafe Mazzeo, and Paolo Piazza extending and refining both the topological and analytic approaches to cohomology on stratified spaces.
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16 January
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Speaker: TBA
Title: TBA
Abstract:
TBA
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23 January
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Speaker: TBA
Title: TBA
Abstract:
TBA
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30 January
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Speaker: Christian Baer (Potsdam)
Title: Geometrically formal 4-manifolds with nonnegative sectional curvature
Abstract: A Riemannian manifold is called geometrically formal if the wedge product of any two harmonic forms is again harmonic. We classify geometrically formal compact 4-manifolds with nonnegative sectional curvature. If the sectional curvature is strictly positive, the manifold must be homeomorphic to S^4 or diffeomorphic to CP^2. This conclusion stills holds true if the sectional curvature is strictly positive and we relax the condition of geometric formality to the requirement that the length of harmonic 2-forms is not "too nonconstant". In particular, the Hopf conjecture on S^2 x S^2 holds in this class of manifolds.
TBA
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6 February
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Speaker: Robert Haslhofer (NYU)
Title: Quantitative stratification and the regularity of mean curvature flow
Abstract:
We introduce techniques for turning estimates on the infinitesimal
behavior of geometric flows (statements about tangent flows and limit
flows) into more quantitative estimates. In the present talk, we focus
on weak solutions of the mean curvature flow and explain how these
techniques can be used to obtain a quantitative version of the
regularity theory of Brian White. In the k-convex case we prove in
particular that the singular set has parabolic Minkowski dimension at
most k-1 and that the second fundamental form lies in L^p for any p < n
+1-k. In fact, our results yield even stronger control including volume
estimates for tubular neighborhoods of the quantitative singular strata
and L^p bounds for the inverse regularity scale. This is joint work with
Jeff Cheeger and Aaron Naber.
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13 February
3pm (special time)
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Speaker: Raphael Ponge (Seoul National University)
Title: The logarithmic singularities of the Green functions of the conformal powers of the Laplacian
Abstract: Green functions play an important role in conformal geometry. In this talk, we shall explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the Laplacian. These operators the Yamabe and Paneitz operators, as well as the conformal fractional powers of the Laplacian arising from scattering theory for Poincare-Einstein metrics. The results are formulated in terms of Weyl conformal invariants defined via the ambient metric of Fefferman-Graham.
This talk is part of the Analysis and PDE Seminar
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13 February
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Speaker: Guofang Wei (UCSB)
Title: Various Covering Spectra
Abstract: Joint with C. Sormani, we introduced covering spectrum
which measures the sizes of holes of a metric space or a Riemannian
manifold. For compact metric spaces, the covering spectrum is a subset
of the length spectrum and continues under Gromov-Hausdorff
convergence. These are not true for non-compact spaces. Instead we
introduce cut-off covering spectrum, rescaled covering spectrum and
study their properties. We also analyze these spectra on Riemannian manifolds with nonnegative sectional
and Ricci curvature.
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20 February
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Speaker: Israel Michael Sigal (Toronto)
Title: Singularity formation under the mean-curvature flow
Abstract: l will review some recent results, joint with my former students, Wenbin Kong and Zhou Gang, and with Dan Knopf, on collapse and neckpinching of hypersurfaces under the mean-curvature flow.
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27 February
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Speaker: TBA
Title: TBA
Abstract:
TBA
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6 March
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Speaker: Adrian Butscher (MPII Saarbrücken / Stanford)
Title: TBA
Abstract:
TBA
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13 March
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Speaker: TBA
Title: TBA
Abstract:
TBA
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