The seminar usually holds on Wednesday from 9:00-10:00 online. For more details, please visit here.
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Wednesday, September 7, 9:00-10:00, Zoom link
(ID: 849 7393 9213, Code: 081038)
Peter Petersen (University of California, Los Angeles) - Curvature and Betti numbers via the Bochner Technique - Abstract
I will give a brief survey of the most recent results that relate to the classical Bochner Technique. The focus will be on restrictions on Betti numbers and holonomy coming from curvature conditions that are based on (weighted) averages of eigenvalues.
Both the standard curvature operator and the curvature operator of the second kind will be used.
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Wednesday, September 14, 9:00-10:00, Zoom link
(ID: 827 0623 2690, Code: 332559)
Daniel Stern (The University of Chicago) - Existence of harmonic maps in higher dimensions and applications - Abstract
I'll discuss recent progress on the existence theory for harmonic maps on higher-dimensional manifolds, giving harmonic maps of optimal regularity from manifolds of dimension n>2 to every non-aspherical closed manifold containing no stable minimal two-spheres.
As an application, we'll see that every manifold carries a canonical family of sphere-valued harmonic maps, which (for n<6) stabilize at a solution of a spectral isoperimetric problem generalizing the conformal maximization of Laplace eigenvalues on surfaces. Based on joint work with Mikhail Karpukhin.
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Wednesday, September 21, 9:00-10:00, Zoom link
(ID: 842 7005 0537, Code: 851096)
Ryosuke Takahashi (Kyushu University) - On the modified $J$-equation - Abstract
In this talk, we consider the modified $J$-equation introduced by Li-Shi. This is a complex Hessian equation including Hamiltonians of holomorphic vector fields, and coincides with the usual $J$-equation introduced by Donaldson and X. Chen when the vector field vanishes.
We show that the solvability of the modified $J$-equation is equivalent to the coercivity of the modified $J$-functional on general compact K\"ahler manifolds. On toric manifolds, we give a numerical necessary and sufficient condition for the existence of solutions, which extends the results of Collins-Sz\'ekelyhidi.
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Wednesday, September 28, 9:00-10:00, Zoom link
(ID: 882 4639 1234, Code: 807208)
Ao Sun (The University of Chicago) - Generic Mean Curvature Flow with Spherical and Cylindrical Singularities - Abstract
We study the local and global dynamics of mean curvature flow with spherical and cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I.
Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University).
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Wednesday, October 12, 9:00-10:00, Zoom link
(ID: 872 8106 9254, Code: 309573)
Wangjian Jian (Chinese Academy of Sciences) - Tangent flow of the Kahler-Ricci flow - Abstract
In this talk, we will introduce some recent progresses on the tangent flow of the Kahler-Ricci flow. First, we will recall the compactness theory of Ricci flow established by Bamler. Then we will introduce a splitting result of the Kahler-Ricci flow, which states that if a limit of a sequence KRF splits a direction R, then it actually splits a direction C.
Finally, we will show how to obtain the well-known Hamilton-Tian conjecture from the tangent flow point of view.
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Wednesday, October 19, 9:00-10:00, Zoom link
(ID: 847 8876 9730, Code: 634239)
Eric Chen (University of California, Berkeley) - Ricci flow and critical integral curvature pinching - Abstract
Smoothing properties of the Ricci flow have provided pointwise pinching results and extensions to pinching in the supercritical L^p, p>n/2 integral curvature cases. However, both here and in other geometric settings, differences arise at the critical, scale-invariant p=n/2 integral norm.
I will describe some cases in which generalizations to curvature pinching in the critical, scale-invariant L^{n/2} sense can still be obtained using consequences of the monotonicity of Perelman's W-functional, such as in pinching to space forms and the Gromov--Ruh Theorem.
This is joint work with Guofang Wei and Rugang Ye.
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Wednesday, October 26, 15:00-16:00 (Special time), Zoom link
(ID: 875 6797 9185, Code: 624510)
Shih-Kai Chiu (University of Oxford) - Higher regularity for singular Kähler-Einstein metrics - Abstract
Let (Z,p) be a Gromov-Hausdorff limit of a complete, polarized, non-collapsing sequence of Kähler-Einstein manifolds. If the germ at p is biholomorphic to the tangent cone at p, then the singular Kähler-Einstein metric on Z is asymptotic at p to the cone metric on the level of Kähler potential.
When the tangent cone has isolated singularity at the vertex, we obtain polynomial convergence of the metric to its tangent cone, generalizing the work of Hein-Sun. A similar result also holds in certain cases when the germ is not isomorphic to the tangent cone. This is joint work with Gábor Székelyhidi.
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Wednesday, November 2, 9:00-10:00, Zoom link
(ID: 871 8438 1471, Code: 821947)
Xiaolong Li (Wichita State University) - The curvature operator of the second kind and Nishikawa's conjecture - Abstract
In 1986, Nishakawa conjectured that a closed Riemannian manifold with positive (resp. nonnegative) curvature operator of the second is diffeomorphic to a spherical space form (resp. a Riemannian locally symmetric space). In this talk, I will present the recent resolution of this conjecture by Cao-Gursky-Tran and myself,
and its further improvements by Nienhaus-Petersen-Wink and myself, as well as some analogue results for Kahler manifolds obtained by Nienhaus-Petersen-Wink-Wylie and myself. Along the way, I will mention some questions and conjectures.
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Wednesday, November 9, 9:00-10:00, Zoom link
(ID: 884 9032 1377, Code: 762983)
Chengjian Yao (ShanghaiTech University) - Einstein-Bogomol'nyi metrics and their limiting behaviors - Abstract
As a dimensional reduction of Einstein-Maxwell-Higgs equations to Riemann surface, Einstein-Bogomol'nyi equation provides the simplest model of a cosmic string. Physically, this equation describes a spacetime together with a magnetic field defined by a U(1) connection and a Higgs field defined by a holomorphic section.
In this talk, we will discuss the existence of solutions to such equation, illustrating how GIT stability condition would play a role. We will also discuss the two types of limiting behavior of the solutions as the volume tends to the lower bound and infinity respectively.
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Wednesday, November 16, 9:00-10:00, Zoom link
(ID: 884 5502 0893, Code: 744922)
Sven Hirsch (Duke University) - Spacetime harmonic functions and a Hawking mass monotonicity formula for initial data sets - Abstract
One of the central results in mathematical relativity is the positive mass theorem which has first been proven by Schoen-Yau using the Jang equation, and Witten using spinors.
We show how level-sets of spacetime harmonic functions can be used to give a new proof of the spacetime positive mass theorem and demonstrate how this leads to an optimal rigidity statement. This is based on joint work with Demetre Kazaras, Marcus Khuri and Yiyue Zhang.
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Wednesday, November 23, 9:00-10:00, Zoom link
(ID: 882 5435 2344, Code: 592955)
Yashan Zhang (Hunan University) - Curvature behavior of long-time solutions to the Kaehler-Ricci flow - Abstract
The curvature behavior is one of the most important aspects of the Kaehler-Ricci flow. In this talk we shall provide an overview on the curvature behavior of long-time solutions to the Kaehler-Ricci flow of collapsing structure, which intimately relate to the underlying complex structure.
In particular, some recent criteria for type-IIb singularities basing on certain analytic or algebraic properties of the underlying manifolds will be introduced.
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Wednesday, November 30, 9:00-10:00, Zoom link
(ID: 820 1324 4896, Code: 275008)
Yong Wei (University of Science and Technology of China) - Curvature measures and volume preserving curvature flows - Abstract
Volume preserving mean curvature flow was introduced by Huisken in 1987 and it was proved that the flow deforms convex initial hypersurface smoothly to a round sphere. This was generalized later by McCoy in 2005 and 2017 to volume preserving flows driven by a large class of 1-homogeneous symmetric curvature functions.
In this talk, we will discuss the flows with higher homogeneity and describe the convergence result for volume preserving curvature flows in Euclidean space by arbitrary positive powers of k-th mean curvature for all k=1,...,n.
As key ingredients, the monotonicity of a generalized isoperimetric ratio will be used to control the geometry of the evolving hypersurfaces and the curvature measure theory will be used to prove the Hausdorff convergence to a sphere.
We also discuss some generalizations including the flows in the anisotropic setting, and the flows in the hyperbolic setting. The talk is based on joint work with Ben Andrews (ANU), Yitao Lei (ANU), Changwei Xiong (Sichuan Univ.), Bo Yang (CAS) and Tailong Zhou (USTC).
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Wednesday, December 7, 16:00-17:00 (Special time), Zoom link
(ID: 823 1558 6297, Code: 782306)
Robert Berman (Chalmers University of Technology) - Sharp bounds on the height of Fano varieties - Abstract
According to a result of Fujita-Liu the n-dimensional complex projective space may be characterized as the Fano variety which has the largest algebraic degree among all K-semistable Fano varieties. In this talk - which is a non-technical introduction to a joint work with Rolf Andreasson - an arithmetic analog of this result is discussed, where the role of degree is played by Falting's height.
The height is formulated in terms of arithmetic intersection theory and depends on the choice of a metric on the anti-canonical line bundle of the corresponding Fano variety X. Our main result establishes an arithmetic analog of Fujita-Liu's result for toric Fano varieties of dimension n \leq 6.
The conjectural extension to any dimension n hinges on a conjectural gap-hypothesis for the algebraic degree.
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Wednesday, December 14, 9:00-10:00, Zoom link
(ID: 816 5199 9786, Code: 092396)
Jiayin Pan (University of California, Santa Cruz) - Ricci curvature meets sub-Riemannian geometry - Abstract
I had a short conversation with Richard Montgomery. It led to a surprising connection between Ricci curvature and sub-Riemannian geometry: Pan-Wei's example of Ricci limit space is isometric to half of the Grushin plane, a classical example in sub-Riemannian geometry.
Inspired by this connection, we construct the Grushin hemisphere as a Ricci limit space with curvature >=1.