The seminar usually holds on Wednesday. For more details, please visit
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Wednesday, September 11, 9:00-10:00, Zoom link
(ID: 815 0182 4414, Code: 803005)
Tamás Darvas (University of Maryland) - The trace operator of quasi-plurisubharmonic functions on compact Kähler manifolds - Abstract
We introduce the trace operator for quasi-plurisubharmonic functions on compact Kähler manifolds, allowing us to study the singularities of such functions along submanifolds where their generic Lelong numbers vanish.
Using this construction we obtain novel Ohsawa-Takegoshi extension theorems and give applications to restricted volumes of big line bundles (joint work with Mingchen Xia).
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Wednesday, September 18, 9:00-10:00, Zoom link
(ID: 870 9960 9401, Code: 546758)
Shiyu Zhang (University of Science and Technology of China) - On structure of compact Kähler manifolds with nonnegative holomorphic sectional curvature - Abstract
By establishing a Bochner-type result on compact Kähler manifolds with nonnegative holomorphic sectional curvature (HSC), we proved that any nontrivial holomorphic p-form induces a decomposition of the tangent bundle, with one component being flat.
In this talk, we will explain why this result is crucial for the development of structure theorems of nonnegative HSC. As a corollary, we generalized Yau's conjecture to the quasi-positive case.
Additionally, we classified all non-projective Kähler 3-folds with nonnegative HSC, which must be either a 3-torus or a P^1-bundle over a 2-torus. This is joint work with Professor Xi Zhang.
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Wednesday, October 9, 9:00-10:00, Zoom link
(ID: 842 8607 6700, Code: 216116)
Douglas Stryker (Princeton University) - Stable minimal hypersurfaces in R^5 - Abstract
I will discuss why every complete two-sided stable minimal hypersurface in R^5 is flat, based on joint work with Otis Chodosh, Chao Li, and Paul Minter.
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Wednesday, October 16, 9:00-10:00, Zoom link
(ID: 871 3102 7073, Code: 772553)
Yujie Wu (Stanford University) - The $\mu$-bubble Construction of Capillary Surfaces - Abstract
We introduce a method of constructing (generalized) capillary surfaces via Gromov's "$\mu$-bubble" method. Using this, we study low-dimensional manifolds with nonnegative scalar curvature and strictly mean convex boundary.
We prove a fill-in question of Gromov, a band-width estimate, and a compactness conjecture of M. Li in the case of surfaces.
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Wednesday, October 23, 16:00-17:00 (Special time), Zoom link
(ID: 879 1300 9727, Code: 434180)
Liangjun Weng (Università di Pisa) - Asymptotic behavior of volume-preserving mean curvature flow and its application - Abstract
In this talk, we discuss the volume-preserving mean curvature flows (VPMCF), where a hypersurface evolves with a velocity determined by its mean curvature, along with an additional constraint that ensures constant enclosed volume.
Such flow and its variations have been extensively studied over the past half-century, with significant contributions from M. Gage, G. Huisken, B. Andrews, P. Guan, and many others.
VPMCF is closely related to the isoperimetric problem and various optimal geometric inequalities. For instance, it provides an effective "path" for finding the minimizing set of the perimeter functional under a volume constraint.
The constrained term in velocity presents distinct challenges depending on their nature and formulation of singularities. For example, the avoidance principle fails when the velocity involves a non-local term of VPMCF.
Beyond closed hypersurfaces, we will focus on compact hypersurfaces with free or capillary boundaries, which naturally arise in variational problems and fluid mechanics.
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Wednesday, October 30, 9:00-10:00, Zoom link
(ID: 889 9514 6006, Code: 896307)
Michael Albanese (The University of Adelaide) - Aspherical 4-Manifolds, Complex Structures, and Einstein Metrics - Abstract
Using results from the theory of harmonic maps, Kotschick proved that a closed hyperbolic four-manifold cannot admit a complex structure. We give a new proof which instead relies on properties of Einstein metrics in dimension four.
The benefit of this new approach is that it generalizes to prove that another class of aspherical four-manifolds (graph manifolds with positive Euler characteristic) also fail to admit complex structures. This is joint work with Luca Di Cerbo.
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Wednesday, November 6, 15:00-16:00 (Special time), Zoom link
(ID: 849 9160 7322, Code: 674508)
Eva Kopfer (Universität Bonn) - Ricci curvature, optimal transport and functional inequalities - Abstract
We review characterization of lower Ricci curvature bounds. Of particular interest for us are characterization which generalize to nonsmooth spaces. We further investigate in Ricci bounds, i.e. we combine lower with upper curvature bounds.
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Wednesday, November 13, 9:00-10:00, Zoom link
(ID: 852 8581 1801, Code: 142236)
Liam Mazurowski (Lehigh University) - Infinitely many constant mean curvature surfaces splitting a manifold in half - Abstract
A constant mean curvature surface (CMC) is a critical point of the area functional subject to a volume constraint. Let M be a closed, three dimensional Riemannian manifold. The solution to the isoperimetric problem implies that, for each v between 0 and the volume of M, there is a constant mean curvature surface in M enclosing volume v.
In this talk, we focus on the case where v is half the volume of M. We show that, when the metric on M is generic, there actually exist infinitely many distinct constant mean curvature surfaces cutting M into two pieces of equal volume. This solves a natural CMC version of Yau's conjecture for generic metrics.
This is joint work with Xin Zhou.
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Wednesday, November 20, 16:00-17:00 (Special time), Zoom link
(ID: 890 6551 6489, Code: 447089)
Daniil Mamaev (London School of Geometry and Number Theory) - On Banach's isometric subspaces problem - Abstract
Let V be a real normed vector space such that for a fixed 2 <= k < dim V any two k-dimensional subspaces of V are isometric. Is the norm on V necessarily induced by an inner product?
This question of Banach is currently known to have an affirmative answer unless k + 1 = dim V = 4m >= 8 or k + 1 = dim V = 134, in which cases the question is open.
After an overview of earlier results I will sketch a proof for the cases k = 2 and k = 3, obtained in a joint work with Sergei Ivanov and Anya Nordskova. In particular, we handle the case k = 3, dim V = 4 which was out of reach of the known global topological methods since the 3-sphere is parallelisable.
Our proof is based on a differential-geometric analysis in a neighbourhood of a single k-plane, which also allows us to solve a stronger, local version of the problem.
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Wednesday, November 27, 9:00-10:00, Zoom link
(ID: 872 4642 0257, Code: 760606)
Yueqiao Wu (Johns Hopkins University) - K-semistability of log Fano cone singularities - Abstract
Local K-stability is the algebro-geometric criterion for the existence of a Ricci-flat K\"ahler cone metric on a log Fano cone singularity, which itself is a generalization of affine cones over Fano varieties.
By a result of Li-Xu, to test K-stability of Fano varieties, it suffices to test the so-called 'special' test configurations. In this talk, we will talk about a local version of this result for log Fano cone singularities.
Our method relies on a non-Archimedean characterization of local K-stability. This is joint work with Yuchen Liu.
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Wednesday, December 4, 16:00-17:00 (Special time), Zoom link
(ID: 875 3791 2581, Code: 694900)
Pietro Mesquita-Piccione (Sorbonne Université and Université Paris Cité) - A non-Archimedean approach to the Yau-Tian-Donaldson Conjecture - Abstract
In Kähler Geometry, the Yau-Tian-Donaldson Conjecture relates the differential geometry of compact Kähler manifold with an algebro-geometric notion called K-stability.
I will start with a brief overview of the topic, and then I will discuss a possible non-Archimedean approach to solve this conjecture, generalizing a result of Chi Li to the transcendental setting.
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Wednesday, December 11, 9:00-10:00, Zoom link
(ID: 870 9231 8508, Code: 865566)
Qi Yao (Stony Brook University) - The Asymptotic behavior of the Solution to Homogeneous Complex Monge-Ampere Equation on ALE Kähler manifolds - Abstract
Initiated by Mabuchi, Semmes, and Donaldson, homogeneous complex Monge-Ampere (HCMA) equations become a central topic in understanding the uniqueness and existence of canonical metrics in Kähler classes. Under the setting of ALE Kähler manifolds, one of the main difficulties is to understand the asymptotic behaviors of solutions to HCMA equations.
In this talk, I will give an introduction to canonical metric problems under the setting of ALE Kähler manifolds and discuss the recent progress on this problem. I will present a new result on the asymptotic behavior of HCMA solutions and outline the proof of the result. (The paper will be available on arXiv very shortly.)
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Wednesday, December 18, 9:00-10:00, Zoom link
(ID: 857 3909 2367, Code: 323846)
Mathew George (Purdue University) - Complex Monge-Ampère equation for positive (p,p) forms - Abstract
A complex Monge-Ampère equation for differential (p,p) forms is introduced on compact Kähler manifolds. For any 1≤p<n, we show the existence of smooth solutions unique up to adding constants.
For p=1, this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for p=n-1, this gives the Monge-Ampère equation for (n-1) plurisubharmonic functions solved by Tosatti-Weinkove.
For other p values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. In this talk, we will give an overview of this theory and discuss the main ideas involved in the proof of existence of solutions.