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September 22
Ben Weinkove - The Fu-Yau equation
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September 29
Greg Edwards - Conical Kähler Ricci flow on Hirzebruch surfaces
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October 6
Valentino Tosatti - Geodesics in the space of Kähler metrics
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October 20
Gang Liu - On the contractibility of Kähler manifolds which are close to metric cones - Abstract
We prove that, if a geodesic ball in a complete Kähler manifold
is sufficiently close to a noncollapsed metric cone and the bisectional
curvature has a lower bound close to zero, then a smaller (fixed size)
geodesic ball is contractible in a slightly larger ball.
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October 27
Guillaume Roy-Fortin - Polynomial upper bound for the Hausdorff measure of nodal sets of
Laplace eigenfunctions - Reference
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November 3
Xavier Garcia - On the expected number of critical points of a random holomorphic section of CPm - Reference
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November 10
Greg Edwards - The Continuity Method on Elliptic Surfaces - Reference
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November 17
Valentino Tosatti - Geodesics in the space of Kähler metrics II
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January 5
Valentino Tosatti - Entropy of holomorphic maps - Reference
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January 12
Peng Zhou - Complex Laurent polynomials and Lagrangian thimbles - Abstract
Given a complex Laurent polynomial f in n variables, f:(C*)n->C,
we may study its critical points and vanishing cycles, as in complex Morse theory or Picard-Lefschetz theory.
I will explain how to turn the problem into a combinatorial one, by allowing the coefficients in f to depend
on a large parameter R.
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January 19
Henri Guenancia (Stony Brook) - Singular varieties with trivial canonical bundle - Abstract
If X is a smooth projective variety (or compact Kähler manifold) with trivial first Chern class, then a famous result of Beauville and Bogomolov asserts that up to a finite étale cover, X is a product of varieties of three possible type: abelians varieties (or tori), Calabi-Yau varieties or Hyperkähler varieties. These last two classes are defined using properties of the algebra of global holomorphic forms.
If X is singular though (say with torsion canonical bundle and klt singularities) this result is not known and presumably very difficult. In this talk, I will explain that if in addition X is assumed to be strongly stable (which is an infinitesimal version of irreducibility) then X falls into one of the singular analogues of the two categories above (Calabi-Yau and Hyperkähler varieties).
This is ongoing joint work with Stefan Kebekus and Daniel Greb.
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January 26
Gang Liu - A Liouville theorem for the complex Monge-Ampère equation on product manifolds - Reference
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February 2
Greg Edwards - Schauder estimates for equations with cone metrics - Reference
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February 9
Xin Jin - Dehn twist and the symplectomorphism group of the cotangent bundle of flag varieties - Reference
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February 16
Jian Xiao - Hyperbolicity of compactifications of certain quotients of bounded symmetric domains - Reference
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February 23
Yu Wang - Quantitative stratification of stationary Yang-Mills fields - Reference
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March 2
Valentino Tosatti - Regularity of envelopes on Kähler manifolds - Reference
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March 9
Yashan Zhang - Holomorphic sectional curvature and the canonical bundle - References 1 2 3
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March 30
Sławomir Kołodziej (Jagiellonian University) - L∞ estimates for the Monge-Ampère equation on Hermitian manifolds
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April 5 - Special date: 3.00pm, Lunt 107
Nguyen-Bac Dang (École Polytechnique) - Degrees of iterates of dominant rational maps - Abstract
In this talk, I will explain how the study of positivity properties of algebraic cycles allows us to understand the behavior of the sequence of degrees of iterates of a dominant rational map.
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April 13
Gang Liu - On the compactification of certain Kähler manifolds with nonnegative Ricci curvature - Abstract
Let M be a complete noncompact Kähler manifold with nonnegative Ricci curvature and maximal volume growth.
1. If M is Ricci flat with quadratic curvature decay, then M is the crepant resolution of an affine algebraic variety. Moreover, there exists two step degeneration from the affine variety to the unique tangent cone of M at infinity.
2. If M has positive Ricci with quadratic curvature decay, then M is quasi-projective.
3. If the bisectional curvature lower bound decay faster than quadratic, then M can be compactified as a Moishezon manifold.
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April 20
Bogdan Georgiev (Max Planck Bonn) - Some recent estimates on nodal geometry in the high-energy limit - Abstract
Given a closed manifold, the Laplace operator is known to
possess a discrete spectrum of eigenvalues converging to infinity. We are
interested in properties of the corresponding eigenfunctions as the eigenvalue becomes
large (i.e. the high-energy limit). From a physical point of view, the eigenfunctions represent
stationary states of a free quantum particle - when properly normalized,
they may be interpreted as the probability density of a particle in the manifold.
Various questions about the geometry of Laplace-eigenfunctions have been studied thoroughly
- for example, distribution and measure of the vanishing (nodal) set; localization;
shape and inner radius of nodal domains, etc.
We present some recent results along these lines - this also includes joint
work with M. Mukherjee.
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April 27
Valentino Tosatti - Seshadri Constants
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May 4
Gang Liu - On Yau's uniformization conjecture - Abstract
Let Mn be a complete noncompact Kähler manifold with nonnegative bisectional curvature and maximal volume growth, we prove M is biholomorphic to Cn. This confirms Yau's uniformization conjecture when M has maximal volume growth.
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May 11
Nick McCleerey - Strong openness conjecture - Reference
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May 18
Chi Li (Purdue) - On metric tangent cones at klt singularities - Abstract
Motivated by the study of metric tangent cones in Kähler
geometry, I will introduce a normalized volume functional defined on
spaces of real centered valuations for Kawamata log terminal (klt)
singularities. Klt singularities form an important class of
singularities in algebraic geometry and appear naturally on
Gromov-Hausdorff limits of Kähler-Einstein manifolds. We will discuss
the minimization problem of the normalized volume functional whose
minimal value and the associated minimizer are new invariants for any
klt singularity. These algebraic invariants should uniquely determine
metric tangent cones on singular Kähler-Einstein varieties. As a model
example, which generalizes a result of Martelli-Sparks-Yau, we show that
the real valuation associated to the Reeb vector field of a
Sasaki-Einstein metric uniquely minimizes the normalized volume among
quasi-monomial real valuations. Further application of this minimization
problem to the study metric tangent cones on singular Kähler-Einstein
metrics will be discussed. Part of this work is joint with Yuchen Liu
and Chenyang Xu.
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May 25
Greg Edwards - Recent developments on the Ricci flow with marked points - Reference