# Geometry and Topology Seminar 2019-2020: Difference between revisions

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| [[#Jonathan Zhu| "Entropy and self-shrinkers of the mean curvature flow"]] | | [[#Jonathan Zhu| "Entropy and self-shrinkers of the mean curvature flow"]] | ||

| Lu Wang | | Lu Wang | ||

|- | |- | ||

|November 11 | |November 11 |

## Revision as of 15:23, 4 November 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 .

## Fall 2016

## Fall Abstracts

### Ronan Conlon

*New examples of gradient expanding K\"ahler-Ricci solitons*

A complete K\"ahler metric $g$ on a K\"ahler manifold $M$ is a \emph{gradient expanding K\"ahler-Ricci soliton} if there exists a smooth real-valued function $f:M\to\mathbb{R}$ with $\nabla^{g}f$ holomorphic such that $\operatorname{Ric}(g)-\operatorname{Hess}(f)+g=0$. I will present new examples of such metrics on the total space of certain holomorphic vector bundles. This is joint work with Alix Deruelle (Universit\'e Paris-Sud).

### 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*

### Richard Kent

*Analytic functions from hyperbolic manifolds*

Thurston's Geometrization Conjecture, now a celebrated theorem of Perelman, tells us that most 3-manifolds are naturally geometric in nature. In fact, most 3-manifolds admit hyperbolic metrics. In the 1970s, Thurston proved the Geometrization conjecture in the case of Haken manifolds, and the proof revolutionized 3-dimensional topology, hyperbolic geometry, Teichmüller theory, and dynamics. Thurston's proof is by induction, constructing a hyperbolic structure from simpler pieces. At the heart of the proof is an analytic function called the *skinning map* that one must understand in order to glue hyperbolic structures together. A better understanding of this map would more brightly illuminate the interaction between topology and geometry in dimension three. I will discuss what is currently known about this map.

### 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