Informal Geometric Analysis Seminar

Academic Year 2017-18

Organizers: G. Liu, V. Tosatti, B. Weinkove, S. Zelditch

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The seminar met on Thursdays from 3.00pm to 4.00pm in Lunt 107.

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Schedule

• October 5
  Ben Weinkove - Gauss curvature flow: the fate of the rolling stones

• October 12
  Jian Xiao - The M-ellipsoid, Symplectic Capacities and Volume

• October 19
  Greg Edwards - Geometric estimates for complex Monge-Ampere equations

• October 26
  Nick McCleerey - The minimum sets and free boundaries of strictly plurisubharmonic functions

• November 2
  Gang Liu - Positivity of holomorphic sectional curvature and a conjecture of Yau - Abstract

  A conjecture of Yau states that a compact Kähler manifolds with positive holomorphic sectional curvature is projective and rationally connected. We present a proof of Xiaokui Yang on this conjecture.

• November 16
  Jian Xiao - The Hormander proof of the Bourgain-Milman theorem - Abstract

  We present Nazarov's proof of the Bourgain-Milman theorem based on Hormander¡¯s existence theorem for solutions of the dbar-problem.

• January 18
  Yu Wang - Long time existence for Yang-Mills flow

• January 25
  Robin Neumayer - Some rigidity theorems for Riemannian manifolds with lower Ricci curvature bounds - Abstract

  We will discuss rigidity theorems involving certain optimal Sobolev and log-Sobolev constants for Riemannian manifolds with Ricci curvature bounded below, along with some applications.

• February 1
  Greg Edwards - Direct images of fiberwise Ricci-flat metrics on Calabi-Yau fibrations

• February 8
  Jian Xiao - Effective very ampleness - References 1 2 3

• February 15
  Jian Xiao - Effective very ampleness - References 1 2 3

• February 22
  Albert Chau (UBC) - The cusp Kähler Ricci flow on Quasi projective Varieties - Abstract

  In this talk I will discuss the existence of cusp-like solutions g(t) to the Kähler Ricci flow on quasi projective varieties M (in other words, M=N\D for a compact complex manifold N with divisor D). A Kähler metric g is called cusp-like on M if it is equivalent to the standard hyperbolic punctured disc model in complex directions approaching D. The discussion will include complete solutions to the flow on M emerging from complete initial metrics with unbounded curvature, and also complete solutions to the flow on M emerging from smooth initial metrics on N (thus becoming ``instantaneously complete" on M). The talk will be based on joint work with Ka Fai Li and Liangming Shen.

• March 8
  Robin Neumayer - A brief overview of the Yamabe problem - Abstract

  The Yamabe problem asserts that every closed Riemannian manifold is conformally equivalent to one of constant scalar curvature. In this talk, we sketch the main ideas of the proof.

• April 12
  Jian Xiao - Invariance of plurigenera - Abstract

  A fundamental result in complex geometry is the invariance of plurigenera. This problem was solved for varieties of general type by Siu in 1998, and general case by Siu in 2000. For the general case, Paun also gave a very nice and simplified proof in 2007, which we will present in the talk. All the proofs are analytical, based on the construction of singular metrics and extension theorem.

• April 19
  Gang Liu - Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below - Abstract

  A famous result of Donaldson-Sun states that non-collapsed Gromov-Hausdorff limits of polarized Kähler manifolds, with 2-side Ricci curvature bounds are homeomorphic to normal projective varieties. Following their approach, we remove the upper bound of the Ricci curvature. More precisely, we prove that non-collapsed Gromov-Hausdroff limits of polarized Kähler manifolds, with Ricci curvature bounded below are normal projective varieties, and the metric singularities are precisely given by a countable union of analytic subvarieties. The work is joint with Gabor Székelyhidi.

• May 3
  Nick McCleerey - On the singularity type of full mass currents in big cohomology classes

• May 10
  Jian Xiao - Compact Kähler threefolds without nontrivial subvarieties

• May 17
  Ben Weinkove - Constant scalar curvature Kähler metrics - existence results

• May 24
  Greg Edwards - Relative volume comparison along the Käahler-Ricci flow

• May 31
  Robin Neumayer - An epsilon regularity theorem for the Ricci flow

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Schedule for the past years:

2012-13 - 2013-14 - 2014-15 - 2015-16