The seminar usually holds on Wednesday from 9:00-10:00 online. For more details, please visit
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Wednesday, September 10, 9:00-10:00, Zoom link
(ID: 828 4077 1514, Code: 083690)
Chao Li (Courant Institute of Mathematical Sciences, New York University) - On the topology of stable minimal hypersurfaces in S^4 - Abstract
Given an n-dimensional manifold (with n at least 4), it is generally impossible to control the topology of a homologically minimizing hypersurface M. In this talk, we construct stable (or locally minimizing) hypersurfaces with optimal restrictions on its topology in a 4-manifold X with natural curvature conditions (e.g. positive scalar curvature),
provided that X admits certain embeddings into a homeomorphic S^4. As an application, we obtain black hole topology theorems in such 4-dimensional asymptotically flat manifolds with nonnegative scalar curvature. This is based on joint work with Boyu Zhang.
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Wednesday, September 17, 9:00-10:00, Zoom link
(ID: 859 5741 3376, Code: 521730)
Nicholas McCleerey (Purdue University) - Lines in the space of Kähler metrics - Abstract
We report on joint work with Tamás Darvas, in which we establish a Ross-Witt Nyström correspondence for weak geodesic lines in the (completed) space of Kähler metrics. Using this, we construct a wide range of weak geodesic lines on any projective Kähler manifold which are not generated by holomorphic vector fields, thus disproving a folklore conjecture popularized by Berndtsson.
Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of Kähler metrics.
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Wednesday, October 15, 9:00-10:00, Zoom link
(ID: 820 7268 4790, Code: 191946)
Zilu Ma (University of Tennessee, Knoxville) - Structure of Parabolic Singular Sets - Abstract
We present new estimates for the size and structure of the nodal and singular sets of parabolic equations. In fact, we obtain estimates of the quantitative nodal and singular sets following the works of Naber-Valtorta, Cheeger-Jiang-Naber, and others for elliptic equations.
This is a joint work with Max Hallgren and Robert Koirala.
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Wednesday, October 22, 9:00-10:00, Zoom link
(ID: 897 9777 1251, Code: 153144)
Yipeng Wang (Columbia University) - Fill-in Estimate with Scalar Curvature bounded from Below - Abstract
A central problem in differential geometry is understanding how the geometry of a boundary determines the geometry of its interior. Gromov's fill-in problem suggests that when a closed Riemannian manifold is filled with a region of large curvature, the extrinsic curvature of the boundary must be bounded above in some sense.
The fill-in problem, particularly in the context of scalar curvature, is closely related to certain notions of quasi-local mass in general relativity. In this talk, I will discuss some recent progress on the scalar curvature fill-in problem under the hyperbolic setting.
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Wednesday, October 29, 9:00-10:00, Zoom link
(ID: 854 3929 9342, Code: 197607)
Huiping Pan (South China University of Technology) - Ray structures on Teichmuller space - Abstract
Given an oriented closed surface S of genus at least two, the Teichmuller space of S is the space of equivalence classes of complex structures on S. It is also the space of equivalence classes of hyperbolic structures on S. Deformations of these structures provide several ray structures on the Teichmuller space.
In this talk, we will show a transition between Teichmuller geodesics and Thurston geodesics via harmonic map (dual) rays. As an application, we construct a new family of Thurston geodesics, the harmonic stretch lines, and show the existence and uniqueness of such lines for any two hyperbolic surfaces in the Teichmuller space.
A key ingredient of the proof is a generalized Jenkin-Serrin problem: existence and uniqueness of some tree-valued minimal graphs over hyperbolic domains. This is a joint work with Michael Wolf.
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Wednesday, November 5, 9:00-10:00, Zoom link
(ID: 842 4840 2232, Code: 988512)
Yang Yang (University of Wisconsin-Madison) - The anisotropic Bernstein problem - Abstract
The Bernstein problem asks whether entire minimal graphs in R^{n+1} are necessarily hyperplanes. It is known through spectacular work of Bernstein, Fleming, De Giorgi, Almgren, Simons, and Bombieri-De Giorgi-Giusti that the answer is positive if and only if n < 8.
The anisotropic Bernstein problem asks the same question about minimizers of parametric elliptic functionals, which are natural generalizations of the area functional that both arise in many applications and offer important technical challenges.
We will discuss the recent solution of this problem (the answer is positive if and only if n < 4). This is joint work with C. Mooney.
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Wednesday, November 12, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Mingxiang Li (Chinese University of Hong Kong) - TBA - Abstract
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Wednesday, November 19, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Liam Mazurowski (Lehigh University) - TBA - Abstract
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Wednesday, November 26, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Jikang Wang (University of California, Berkeley) - TBA - Abstract
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Wednesday, December 3, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Zhenhua Liu (Princeton University) - TBA - Abstract
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Wednesday, December 10, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
TBA (TBA) - TBA - Abstract
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Wednesday, December 17, 16:00-17:00 (Special time), Zoom link
(ID: TBA, Code: TBA)
Robert J. Berman (Chalmers University of Technology & University of Gothenburg) - TBA - Abstract
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Wednesday, December 24, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
TBA (TBA) - TBA - Abstract