Difference between revisions of "Geometry and Topology Seminar"
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|November 4 | |November 4 | ||
| Jonathan Zhu (Harvard University) | | Jonathan Zhu (Harvard University) | ||
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| Lu Wang | | Lu Wang | ||
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=== Jonathan Zhu === | === Jonathan Zhu === | ||
− | '' | + | ''Entropy and self-shrinkers of the mean curvature flow'' |
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+ | The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set. | ||
== Archive of past Geometry seminars == | == Archive of past Geometry seminars == |
Revision as of 18:29, 18 October 2016
The Geometry and Topology seminar meets in room 901 of Van Vleck Hall on Fridays from 1:20pm - 2:10pm.
For more information, contact Alexandra Kjuchukova or Lu Wang .
Contents
Fall 2016
date | speaker | title | host(s) |
---|---|---|---|
September 9 | Bing Wang (UW Madison) | "The extension problem of the mean curvature flow" | (Local) |
September 16 | Ben Weinkove (Northwestern University) | "Gauduchon metrics with prescribed volume form" | Lu Wang |
September 23 | Jiyuan Han (UW Madison) | "Deformation theory of scalar-flat ALE Kahler surfaces" | (Local) |
September 30 | |||
October 7 | Yu Li (UW Madison) | "Ricci flow on asymptotically Euclidean manifolds" | (Local) |
October 14 | Sean Howe (University of Chicago) | "Representation stability and hypersurface sections" | Melanie Matchett Wood |
October 21 | Nan Li (CUNY) | "Quantitative estimates on the singular Sets of Alexandrov spaces" | Lu Wang |
October 28 | Ronan Conlon | "TBA" | Bing Wang |
November 4 | Jonathan Zhu (Harvard University) | "Entropy and self-shrinkers of the mean curvature flow" | Lu Wang |
November 7 | Gaven Martin (University of New Zealand) | "TBA" | Simon Marshall |
November 11 | |||
November 18 | Caglar Uyanik (Illinois) | "TBA" | Kent |
Thanksgiving Recess | |||
December 2 | Peyman Morteza (UW Madison) | "TBA" | (Local) |
December 9 | |||
December 16 | |||
Fall Abstracts
Ronan Conlon
TBA
Jiyuan Han
Deformation theory of scalar-flat ALE Kahler surfaces
We prove a Kuranishi-type theorem for deformations of complex structures on ALE Kahler surfaces. This is used to prove that for any scalar-flat Kahler ALE surfaces, all small deformations of complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics is then constructed, which is shown to be universal up to small diffeomorphisms (that is, diffeomorphisms which are close to the identity in a suitable sense). A formula for the dimension of the local moduli space is proved in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of \C^2/\Gamma, where \Gamma is a finite subgroup of U(2) without complex reflections. This is a joint work with Jeff Viaclovsky.
Sean Howe
Representation stability and hypersurface sections
We give stability results for the cohomology of natural local systems on spaces of smooth hypersurface sections as the degree goes to \infty. These results give new geometric examples of a weak version of representation stability for symmetric, symplectic, and orthogonal groups. The stabilization occurs in point-counting and in the Grothendieck ring of Hodge structures, and we give explicit formulas for the limits using a probabilistic interpretation. These results have natural geometric analogs -- for example, we show that the "average" smooth hypersurface in \mathbb{P}^n is \mathbb{P}^{n-1}!
Nan Li
Quantitative estimates on the singular sets of Alexandrov spaces
The definition of quantitative singular sets was initiated by Cheeger and Naber. They proved some volume estimates on such singular sets in non-collapsed manifolds with lower Ricci curvature bounds and their limit spaces. On the quantitative singular sets in Alexandrov spaces, we obtain stronger estimates in a collapsing fashion. We also show that the (k,\epsilon)-singular sets are k-rectifiable and such structure is sharp in some sense. This is a joint work with Aaron Naber.
Yu Li
In this talk, we prove that if an asymptotically Euclidean (AE) manifold with nonnegative scalar curvature has long time existence of Ricci flow, it converges to the Euclidean space in the strong sense. By convergence, the mass will drop to zero as time tends to infinity. Moreover, in three dimensional case, we use Ricci flow with surgery to give an independent proof of positive mass theorem. A classification of diffeomorphism types is also given for all AE 3-manifolds with nonnegative scalar curvature.
Gaven Marin
TBA
Peyman Morteza
TBA
Caglar Uyanik
TBA
Bing Wang
The extension problem of the mean curvature flow
We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3. A key ingredient of the proof is to show a two-sided pseudo-locality property of the mean curvature flow, whenever the mean curvature is bounded. This is a joint work with Haozhao Li.
Ben Weinkove
Gauduchon metrics with prescribed volume form
Every compact complex manifold admits a Gauduchon metric in each conformal class of Hermitian metrics. In 1984 Gauduchon conjectured that one can prescribe the volume form of such a metric. I will discuss the proof of this conjecture, which amounts to solving a nonlinear Monge-Ampere type equation. This is a joint work with Gabor Szekelyhidi and Valentino Tosatti.
Jonathan Zhu
Entropy and self-shrinkers of the mean curvature flow
The Colding-Minicozzi entropy is an important tool for understanding the mean curvature flow (MCF), and is a measure of the complexity of a submanifold. Together with Ilmanen and White, they conjectured that the round sphere minimises entropy amongst all closed hypersurfaces. We will review the basics of MCF and their theory of generic MCF, then describe the resolution of the above conjecture, due to J. Bernstein and L. Wang for dimensions up to six and recently claimed by the speaker for all remaining dimensions. A key ingredient in the latter is the classification of entropy-stable self-shrinkers that may have a small singular set.
Archive of past Geometry seminars
2015-2016: Geometry_and_Topology_Seminar_2015-2016
2014-2015: Geometry_and_Topology_Seminar_2014-2015
2013-2014: Geometry_and_Topology_Seminar_2013-2014
2012-2013: Geometry_and_Topology_Seminar_2012-2013
2011-2012: Geometry_and_Topology_Seminar_2011-2012
2010: Fall-2010-Geometry-Topology