The seminar usually holds on Wednesday from 9:00-10:00 online. For more details, please visit
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Wednesday, March 4, 9:00-10:00, Zoom link
(ID: 825 3248 0199, Code: 931144)
Xinrui Zhao (Yale University) - On the Rate of Convergence of Cylindrical Singularity in Mean Curvature Flow - Abstract
We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges at a super-exponential rate, then it must coincide with the cylinder itself. We also show that this result is sharp by constructing local graphical counterexamples with arbitrarily fast super-exponential convergence and rapidly expanding domains.
These examples form infinite-dimensional families of Tikhonov-type solutions and show that unique continuation fails for local graphical solutions. Our constructions apply to a broad class of nonlinear equations. This talk is based on joint work with Yiqi Huang.
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Wednesday, March 11, 16:00-17:00 (Special time), Zoom link
(ID: 889 3611 0023, Code: 928471)
Ling Wang (Bocconi University) - A Revisit to the De Giorgi Conjecture: Savin's Proof and Applications - Abstract
In this talk, I will first introduce the Allen-Cahn equation and the related De Giorgi conjecture. I will then present the key ideas and a detailed sketch of Savin's groundbreaking proof in dimensions up to 8. Building on Savin's framework, I will discuss a half-space version of De Giorgi's conjecture, based on joint work with Wenkui Du and Yang Yang.
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Thursday (Special day), March 19, 10:00-11:00 (Special time), Zoom link
(ID: 823 4782 9349, Code: 362787)
Ruobing Zhang (University of California, San Diego) - Regularity and collapsing geometry of metric spaces with curvature bounds - Abstract
In this talk, we will exhibit a series of regularity results and structure theory for collapsing metric spaces with lower Ricci curvature and mixed curvature bounds. These results are new even in the smooth setting.
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Wednesday, March 25, 9:00-10:00, Zoom link
(ID: 885 7626 6237, Code: 293861)
Ying Wang (University of Michigan) - Non-Archimedean Perspectives on Affine Calabi-Yau Metrics - Abstract
Non-Archimedean geometry can be viewed as a degeneration of complex geometry. In recent years, it has found applications in the existence problem for canonical metrics via K-stability, particularly for Fano varieties. On the other hand, while the existence of canonical metrics on smooth projective Calabi-Yau varieties is classical, much less is understood in the affine setting.
In this talk, we discuss some recent progress on the existence problem for canonical metrics on affine Calabi-Yau varieties, focusing on a non-Archimedean perspective. In particular, we present a solution to a corresponding non-Archimedean existence problem in certain cases using a kind of non-Archimedean spaces called Berkovich spaces. We also explain how this non-Archimedean solution relates to the original complex analytic problem.
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Wednesday, April 1, 9:00-10:00, Zoom link
(ID: 820 9062 6605, Code: 117123)
Hongyi Liu (Princeton University) - Poincaré-Einstein 4-manifolds with conformally Kähler geometry - Abstract
Poincaré-Einstein metrics play an important role in geometric analysis and mathematical physics, yet constructing new examples beyond the perturbative regime is difficult. In this talk, I will describe a class of four-dimensional Poincaré-Einstein manifolds that are conformal to Kähler metrics.
These metrics admit a natural symmetry generated by a Killing field, which reduces the Einstein equations to a Toda-type system. This approach leads to existence and uniqueness results in the case of complex line bundles over surfaces of genus at least one.
The construction produces large-scale, infinit-dimensional families of new Poincaré-Einstein metrics with conformal infinities of non-positive Yamabe type. This is joint work with Mingyang Li.
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Wednesday, April 8, 9:00-10:00, Zoom link
(ID: 851 3392 2600, Code: 574264)
Adrian Chun-Pong Chu (Cornell University) - Enumerative problems for minimal surfaces with prescribed genus - Abstract
We will present the enumerative min-max theory, which relates the number of genus g minimal surfaces in 3-manifolds to topological properties of the set of all embedded surfaces of genus ≤g.
As a consequence, we can show that in every 3-sphere of positive Ricci curvature, there exist ≥5 minimal tori (confirming a conjecture by B. White (1989) in the Ricci-positive case), ≥4 minimal surfaces of genus 2, and ≥1 minimal surface of genus g for all g.
This is based on a joint work with Yangyang Li and Zhihan Wang.
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Wednesday, April 15, 9:00-10:00, Zoom link
(ID: 897 4475 0018, Code: 682168)
Ming Xiao (University of California, San Diego) - Obstruction flatness and Bergman logarithmic flatness of circle bundles - Abstract
Let $\Omega \subset \mathbb{C}^n$ be a smoothly bounded strictly pseudoconvex domain. The boundary $\partial\Omega$ is said to be obstruction flat if the log singularity (the obstruction function) of the log-potential of the complete Kähler-Einstein metric on $\Omega$ vanishes.
It is called Bergman logarithmically flat if the log singularity in the Fefferman expansion of the Bergman kernel vanishes. Both notions of flatness depend only on the local CR geometry of the boundary.
In this talk, we consider real hypersurfaces arising as unit circle bundles of negative Hermitian line bundles over a complex manifold $M$. We study the relationship between obstruction flatness and Bergman logarithmic flatness of these circle bundles and the Kähler geometry of the induced metric on $M$.
The talk is based on joint work with Peter Ebenfelt and Hang Xu.
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Wednesday, April 22, 16:00-17:00 (Special time), Zoom link
(ID: 864 1606 9736, Code: 183145)
Shengxuan Zhou (Institut de Mathématiques de Toulouse) - An Explicit Uniform Bound for Rational Points on Curves - Abstract
The celebrated Mordell conjecture, proved by Faltings, asserts that a curve of genus greater than one over a number field has only finitely many rational points. A deep uniform upper bound on the number of rational points follows from Vojta's inequality and the recent works of Dimitrov-Gao-Habegger and Kühne.
In this talk, I will introduce an explicit version of this uniform bound. Our approach relies on analyzing Arakelov Kähler forms via localization of Bergman kernels. This is joint work with Jiawei Yu and Xinyi Yuan.
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Wednesday, April 29, 16:00-17:00 (Special time), Zoom link
(ID: 871 5029 4872, Code: 237909)
Shujing Pan (Goethe-University Frankfurt) - Quermassintegral Inequalities and horo-convexity - Abstract
In this talk, I will introduce a new notion of convexity in the unit sphere called horo-convexity, inspired by its analogue in hyperbolic space. For horo-convex hypersurfaces, we prove the smooth convergence of the Guan/Li inverse curvature flow and, as a consequence, establish the full set of quermassintegral inequalities on the sphere.
If time permits, I will also discuss parallel results for the hypersurface with capillary boundary lying in the Euclidean unit ball. This talk is based on joint works with Julian Scheuer.
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Wednesday, May 13, 16:00-17:00 (Special time), Zoom link
(ID: 893 4998 4451, Code: 514608)
Yuxin Ge (University of toulouse) - A Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature - Abstract
We consider a new Yamabe problem for the quotient between the $Q$ curvature and the scalar curvature $R$: Find a conformal metric $g$ in a given conformal class $[g_0]$ with $Q_g\slash R_g=const$. When the dimension $n\geq5$, it relates to a new Sobolev inequality between the total $Q$-curvature and the total scalar curvature on $\mathbb{S}^n$ ($n\geq5$).
With this inequality we introduce a new Yamabe constant $Y_{4,2}(M,[g_0])$ and prove the existence of the above problem provided that $Y_{4,2}(M,[g_0])
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Wednesday, May 20, 9:00-10:00, Zoom link
(ID: 862 1107 7021, Code: 727974)
Wenrui Kong (Courant Institute of Mathematical Sciences, New York University) - Curvature Estimates along Immortal Kähler-Ricci Flows - Abstract
On a compact Kähler manifold X, a Kähler-Ricci flow (KRF) is immortal when the canonical bundle of X is numerically effective. In this case, assuming the abundance conjecture and intermediate Kodaira dimension, the collapsing behavior of the normalized KRF, as already known, poses difficulty on uniform curvature estimates.
We extend known C^0 estimates on the scalar curvature (by Song-Tian) and Ricci curvature (recently by Hein-Lee-Tosatti) to orders up to 2, and explain the failure for higher orders in general.
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Wednesday, June 3, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Xingyu Zhu (Michigan State University) - TBA - Abstract
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Wednesday, June 10, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Yaoting Gui (Xiamen University) - TBA - Abstract