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: 847 7926 9359, Code: 136091)
Mingxiang Li (The Chinese University of Hong Kong) - Higher order Cohn-Vossen inequality and Q-Curvature - Abstract
In this talk, I will discuss some relationships among the positivity of Q-curvature, scalar curvature, and Ricci curvature on conformally flat manifolds. Under positivity conditions on the top-order Q-curvature and the scalar curvature, we establish integral growth estimates for all lower-order Q-curvatures over geodesic balls.
Additionally, we derive growth bounds for Ricci curvature on geodesic balls, which are related to a conjecture of Yau. This is joint work with Juncheng Wei and Xingwang Xu.
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Wednesday, November 19, 9:00-10:00, Zoom link
(ID: 833 8957 8283, Code: 885679)
Liam Mazurowski (Lehigh University) - On the topology of manifolds with positive intermediate curvature - Abstract
I will discuss a conjecture relating the topology of a manifold's universal cover with the existence of metrics of positive intermediate curvature. This conjecture simultaneously generalizes Gromov and Schoen-Yau's K(pi,1) conjecture and Brendle-Hirsch-Johne's generalized Geroch conjecture.
We prove the conjecture in almost all cases for manifolds of dimension up to 6. As an application, we show that a closed, aspherical 6-manifold does not admit a metric of positive 4-intermediate curvature. This is joint work with Tongrui Wang and Xuan Yao.
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Wednesday, November 26, 9:00-10:00, Zoom link
(ID: 854 8955 8991, Code: 345523)
Jikang Wang (University of California, Berkeley) - The collapsing geometry of some RCD spaces - Abstract
We will first overview the collapsing geometry of manifolds with bounded sectional curvature, mainly due to Cheeger, Fukaya and Gromov. Then I shall generalize almost flat manifold theorem and smooth fibration theorem to RCD spaces with non-collapsing (local) universal covers.
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Wednesday, December 3, 9:00-10:00, Zoom link
(ID: 861 6587 3708, Code: 654166)
Zhenhua Liu (Princeton University) - General behavior of area-minimizing subvarieties - Abstract
We will review some recent progress on the general geometric behavior of homologically area-minimizing subvarieties, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as special Lagrangians on a Calabi-Yau, etc.
A fine understanding of the geometric structure of homological area-minimizers can give far-reaching consequences for related problems.
Camillo De Lellis and his collaborators have proven that area-minimizing integral currents have codimension two rectifiable singular sets. A pressing next question is what one can say about the geometric behavior of area-minimizing currents beyond this.
Almost all known examples and results point towards that area-minimizing subvarieties are subanalytic, generically smooth, and calibrated. It is natural to ask if these hold in general. In this direction, we prove that all of these properties thought to be true generally and proven to be true in special cases are totally false in general.
We prove that area-minimizing subvarieties can have fractal singular sets. Smoothable singularities are non-generic. Calibrated area minimizers are non-generic. Consequently, we answer several conjectures of Frederick J. Almgren Jr., Frank Morgan, and Brian White from the 1980s.
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Wednesday, December 10, 9:00-10:00, Zoom link
(ID: 852 4663 8135, Code: 249671)
Ruijia Zhang (Sun Yat-sen University) - Hessian estimates for sigma_k equations and a rigidity theorem - Abstract
We derive a concavity inequality for k-Hessian operators under the semiconvexity condition. As a consequence, we establish interior estimates for semiconvex solutions to the sigma_k equations with vanishing Dirichlet boundary conditions and obtain a Liouville-type result.
This result confirms Chang and Yuan's conjecture under the superquadratic growth condition. Additionally, we present several applications in the global curvature estimates for hypersurfaces with prescribed k-curvature in the space form.
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Wednesday, December 17, 16:00-17:00 (Special time), Zoom link
(ID: 867 4120 5114, Code: 349064)
Robert Berman (Chalmers University of Technology) - Reduced stability thresholds of Fano manifolds and a sharpening of the logarithmic Hardy-Littlewood-Sobolev inequality on the two-sphere - Abstract
This talk provides a non-technical introduction to a conjectural algebro-geometric formula for the reduced analytic stability threshold of a given Fano manifold X, introduced in my recent preprint 'Gibbs polystability of Fano manifolds, stability thresholds and symmetry breaking,' joint with Rolf Andreasson and Ludvig Svensson.
This threshold is central to the existence of canonical metrics, as established by Darvas-Rubinstein's resolution of a conjecture of Tian, which states that X admits a Kähler-Einstein metric if and only if the reduced analytic stability threshold of X is strictly greater than one.
While our broader work is motivated by a probabilistic approach for constructing these metrics, this presentation will focus exclusively on the application to computing reduced analytic stability thresholds.
I will demonstrate how the proof of our conjectural formula in the simplest case-the Riemann sphere-yields a sharp form of the conjecture of Tian and an improvement of the sharp logarithmic Hardy-Littlewood-Sobolev inequality on the two-sphere.
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Wednesday, December 24, 9:00-10:00, Zoom link
(ID: 890 7805 9254, Code: 212868)
Chung-Ming Pan (Université du Québec à Montréal) - Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties - Abstract
Gauduchon metrics are very useful generalizations of Kähler metrics in non-Kähler geometry, as Gauduchon proved that these special metrics always exist on compact complex manifolds. One of their important applications is defining the notion of stability for vector bundles/sheaves on non-Kähler manifolds.
It also leads to studying the existence of Hermite-Einstein metrics and the classification of non-Kähler surfaces. In this talk, I will first introduce a singular version of Gauduchon's theorem and its application to the Hermite-Einstein problem for stable reflexive sheaves on non-Kähler normal varieties.
Then, I will explain one of the main technical points that lies in obtaining uniform Sobolev inequalities for perturbed hermitian metrics on a resolution of singularities.