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−  == Fall Abstracts ==
 
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−  === Ronan Conlon ===
 
−  ''New examples of gradient expanding K\"ahlerRicci solitons''
 
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−  A complete K\"ahler metric $g$ on a K\"ahler manifold $M$ is a \emph{gradient expanding K\"ahlerRicci soliton} if there exists a smooth realvalued 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 ParisSud).
 
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−  === Jiyuan Han ===
 
−  ''Deformation theory of scalarflat ALE Kahler surfaces''
 
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−  We prove a Kuranishitype theorem for deformations of complex structures on ALE Kahler surfaces. This is used to prove that for any scalarflat Kahler ALE surfaces, all small deformations of complex structure also admit scalarflat Kahler ALE metrics. A local moduli space of scalarflat 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 scalarflat 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.
 
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−  === Sean Howe ===
 
−  ''Representation stability and hypersurface sections''
 
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−  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 pointcounting 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}^{n1}!
 
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−  === Nan Li ===
 
−  ''Quantitative estimates on the singular sets of Alexandrov spaces''
 
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−  The definition of quantitative singular sets was initiated by Cheeger and Naber. They proved some volume estimates on such singular sets in noncollapsed 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 krectifiable and such structure is sharp in some sense. This is a joint work with Aaron Naber.
 
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−  === Yu Li ===
 
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−  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 3manifolds with nonnegative scalar curvature.
 
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−  === Peyman Morteza ===
 
−  ''We develop a procedure to construct Einstein metrics by gluing the Calabi metric to an Einstein orbifold. We show that our gluing problem is obstructed and we calculate the obstruction explicitly. When our obstruction does not vanish, we obtain a nonexistence result in the case that the base orbifold is compact. When our obstruction vanishes and the base orbifold is nondegenerate and asymptotically hyperbolic we prove an existence result. This is a joint work with Jeff Viaclovsky. ''
 
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−  === Caglar Uyanik ===
 
−  ''Geometry and dynamics of free group automorphisms''
 
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−  A common theme in geometric group theory is to obtain structural results about infinite groups by analyzing their action on metric spaces. In this talk, I will focus on two geometrically significant groups; mapping class groups and outer automorphism groups of free groups.We will describe a particular instance of how the dynamics and geometry of their actions on various spaces provide deeper information about the groups.
 
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−  === Bing Wang ===
 
−  ''The extension problem of the mean curvature flow''
 
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−  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 twosided pseudolocality property of the mean curvature flow, whenever the mean curvature is bounded.
 
−  This is a joint work with Haozhao Li.
 
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−  === Ben Weinkove ===
 
−  ''Gauduchon metrics with prescribed volume form''
 
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−  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 MongeAmpere type equation. This is a joint work with Gabor Szekelyhidi and Valentino Tosatti.
 
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−  === Jonathan Zhu ===
 
−  ''Entropy and selfshrinkers of the mean curvature flow''
 
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−  The ColdingMinicozzi 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 entropystable selfshrinkers that may have a small singular set.
 
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−  ===Yu Zeng===
 
−  ''Short time existence of the Calabi flow with rough initial data''
 
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−  Calabi flow was introduced by Calabi back in 1950’s as a geometric flow approach to the existence of extremal metrics. Analytically it is a fourth order nonlinear parabolic equation on the Kaehler potentials which deforms the Kaehler potential along its scalar curvature. In this talk, we will show that the Calabi flow admits short time solution for any continuous initial Kaehler metric. This is a joint work with Weiyong He.
 
   
 == Spring Abstracts ==   == Spring Abstracts == 
Revision as of 10:55, 2 February 2017
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 .
Spring 2017
Spring Abstracts
Lucas Ambrozio
"TBA"
Rafael Montezuma
"Metrics of positive scalar curvature and unbounded minmax widths"
In this talk, I will construct a sequence of Riemannian metrics on the threedimensional sphere with scalar curvature greater than or equal to 6, and arbitrarily large minmax widths. The search for such metrics is motivated by a rigidity result of minmax minimal spheres in threemanifolds obtained by Marques and Neves.
Carmen Rovi
The mod 8 signature of a fiber bundle
In this talk we shall be concerned with the residues modulo 4 and modulo 8 of the signature of a 4kdimensional geometric Poincare complex. I will explain the relation between the signature modulo 8 and two other invariants: the BrownKervaire invariant and the Arf invariant. In my thesis I applied the relation between these invariants to the study of the signature modulo 8 of a fiber bundle. In 1973 Werner Meyer used group cohomology to show that a surface bundle has signature divisible by 4. I will discuss current work with David Benson, Caterina Campagnolo and Andrew Ranicki where we are using group cohomology and representation theory of finite groups to detect nontrivial signatures modulo 8 of surface bundles.
Yair Hartman
"Intersectional Invariant Random Subgroups and Furstenberg Entropy."
In this talk I'll present a joint work with Ariel Yadin, in which we solve the Furstenberg Entropy Realization Problem for finitely supported random walks (finite range jumps) on free groups and lamplighter groups. This generalizes a previous result of Bowen. The proof consists of several reductions which have geometric and probabilistic flavors of independent interests.
All notions will be explained in the talk, no prior knowledge of Invariant Random Subgroups or Furstenberg Entropy is assumed.
Bena Tshishiku
"TBA"
Mark Powell
Stable classification of 4manifolds
A stabilisation of a 4manifold M is a connected sum of M with some number of copies of S^2 x S^2.
Two 4manifolds are said to be stably diffeomorphic if they admit diffeomorphic stabilisations.
Since a necessary condition is that the fundamental groups be isomorphic, we study this equivalence relation for a fixed group. I will discuss recent progress in classifying 4manifolds up to stable diffeomorphism for certain families of groups, arising from work with Daniel Kasprowski, Markus Land and Peter Teichner.
As a byproduct we also obtained a result on the analogous question with the complex projective plane CP^2 replacing S^2 x S^2.
Autumn Kent
Analytic functions from hyperbolic manifolds
At the heart of Thurston's proof of Geometrization for Haken manifolds is a family of analytic functions between Teichmuller spaces called "skinning maps." These maps carry geometric information about their associated hyperbolic manifolds, and I'll discuss what is presently known about their behavior. The ideas involved form a mix of geometry, algebra, and analysis.
Xiangwen Zhang
"TBA"
Archive of past Geometry seminars
20162017 Geometry_and_Topology_Seminar_20162017
20152016: Geometry_and_Topology_Seminar_20152016
20142015: Geometry_and_Topology_Seminar_20142015
20132014: Geometry_and_Topology_Seminar_20132014
20122013: Geometry_and_Topology_Seminar_20122013
20112012: Geometry_and_Topology_Seminar_20112012
2010: Fall2010GeometryTopology