The seminar usually holds on Wednesday from 9:00-10:00 online. For more details, please visit
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Wednesday, September 9, 9:00-10:00, Zoom link
(ID: 830 0216 4749, Code: 752097)
Antoine Song (California Institute of Technology) - On the geometry of large sets - Abstract
I will talk about what happened behind the resolution of Talagand's convexity conjecture (with Dongming Merrick Hua and Stefan Tudose) and propose new directions, in particular a Riemannian version of the conjecture for manifolds with nonnegative Ricci curvature.
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Wednesday, September 16, 9:00-10:00, Zoom link
(ID: 834 6727 7014, Code: 440423)
Biao Ma (East China Normal University) - Gårding polynomials and applications in PDEs and Combinatorics - Abstract
Motivated by the classical hyperbolic polynomials and Caffarelli-Nirenberg-Spruck theory, we introduce Gårding and ideal Gårding polynomials to capture the ellipticity and convexity features that appear in fully nonlinear PDEs.
I will discuss some interesting properties and their connections to real stable and Lorentzian polynomials. Applications in fully nonlinear PDEs and combinatorics will be discussed, including a new Liouville type Theorem,
Rayleigh inequalities and log-concavity for matroids and graphs. This talk is based on joint works with Hao Fang and Jinyang Wu.
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Wednesday, September 23, 15:00-16:00 (Special time), Zoom link
(ID: 838 5510 7596, Code: 544440)
Harish Seshadri (Indian Institute of Science) - Holomorphic sectional curvature and volume - Abstract
I will outline the proof of a sharp volume comparison theorem and the corresponding rigidity result for compact Kahler manifolds with positive holomorphic sectional curvature. This approach also gives Kewei Zhang's sharp volume estimate for compact Kahler manifolds with positive Ricci curvature.
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Wednesday, September 30, 9:00-10:00, Zoom link
(ID: 841 2962 6664, Code: 373132)
Bo Yang (Xiamen University) - Function Theory on Several Classes of Noncompact Complex Manifolds - Abstract
In this talk, I will present some results on holomorphic functions on noncompact Kahler or Hermitian manifolds, including growth estimates, dimension estimates for holomorphic functions of polynomial growth, and other rigidity results under suitable curvature assumptions.
The manifolds we study include total spaces of certain vector bundles over compact Kahler manifolds and complete Hermitian manifolds with flat Chern connection.
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Wednesday, October 14, 9:00-10:00, Zoom link
(ID: 827 5799 0746, Code: 790479)
Shrey Aryan (Massachusetts Institute of Technology) - Contracting Transport Maps on Curved Spaces - Abstract
Caffarelli's contraction theorem states that the Brenier optimal transport map from the standard Gaussian measure to a more log-concave probability measure is 1-Lipschitz. In view of its applications to geometric and functional inequalities, Villani asked whether this theorem extends to Riemannian manifolds.
Motivated by results in comparison geometry, E. Milman formulated several conjectures on spheres and on weighted manifolds satisfying the $\mathrm{CD}(K,\infty)$ condition with $K>0$. Such a contracting transport map implies a corresponding spectral comparison.
In the spherical setting, this comparison was also conjectured by Colding and Minicozzi in 1998 for closed connected manifolds with a positive Ricci curvature lower bound. In this talk, I will present counterexamples to the spherical conjectures in dimensions $d\geq4$, followed by joint work with Bang-Xian Han and Zhuo-Nan Zhu proving the spherical transport conjecture when $d=2$.
Together with a recent counterexample by Shengjie Lin, Haibin Wang, and Guoyi Xu in dimension $d=3$, these results settle the spherical transport and spectral comparison conjectures in every dimension $d\geq2$. If time permits, I will discuss some open problems.
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Wednesday, October 21, 9:00-10:00, Zoom link
(ID: 832 9276 8120, Code: 652352)
Xuwen Zhu (Northeastern University) - Geometric microlocal analysis of gravitational instantons - Abstract
Gravitational instantons are non-compact Calabi-Yau metrics with L^2 bounded curvature and are categorized into six types. I will describe several projects on gravitational instantons including: Fredholm theory and deformation, non-collapsing degeneration limits, and existence of stable non-holomorphic minimal spheres in certain types.
These projects utilize geometric microlocal analysis in different settings, where the analysis of singular operators is obtained through geometric resolutions. Based on works jointly with Rafe Mazzeo, Yu-Shen Lin, Sidharth Soundararajan, and Alia Yusaini.
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Wednesday, November 4, 15:00-16:00 (Special time), Zoom link
(ID: 857 7894 7365, Code: 875291)
Jiaxiang Wang (Nankai University) - When Harmonic Functions Meet the Complex Monge-Ampère Equation - Abstract
In 2024, we constructed a singular solution to the complex Monge-Ampère equation with constant right-hand side. In this talk, we explain the mechanism underlying that construction and show how it yields new examples with more severe singularities.
For two symmetry classes in complex dimension two, a partial Legendre transform relates the complex Monge-Ampère equation to the Laplace equation. This correspondence allows us to construct plurisubharmonic solutions of \det(u_{i\bar j})=1 in the Bedford-Taylor sense from superpositions of harmonic functions.
In every complex dimension n\geq2, these include:
1. A bounded solution on the unit polydisk \mathbb D^n that is discontinuous almost everywhere and nowhere classically differentiable.
2. A continuous exhaustion function on \mathbb C^n that is not in W^{2,1}_{\mathrm{loc}}(\mathbb C^n). This example disproves a conjecture of Błocki on regularity in hyperconvex domains and a conjecture of B. Chen on the completeness of metrics induced by proper entire solutions.
3. A solution on \mathbb D^n, continuous up to the boundary, whose distributional Laplacian has a nonzero singular part.
We also revisit the Taub-NUT metric, a classical gravitational instanton, and discuss its relation to the quadratic polynomial via the partial Legendre transform.
This is joint work with Wenlong Wang.
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Wednesday, November 11, 9:00-10:00, Zoom link
(ID: 882 7049 2281, Code: 509184)
Junbang Liu (The Hong Kong University of Science and Technology) - The J-Equation: singular solutions and numerical obstructions - Abstract
The smooth solvability of the J-equation on a compact Kähler manifold is closely related to numerical positivity of pairs of Kähler classes. When these numerical conditions degenerate or fail, a natural question is whether the equation admits a canonical singular solution.
In this talk, I will discuss recent progress on degenerate J-equations, focusing on the existence, uniqueness, and partial regularity of solutions at the birational minimal slope. I will also describe related results in the J-nef setting, including an analytic characterization of the J-null locus and the finiteness of J-null subvarieties.
These results clarify the relationship between numerical obstructions and analytic degeneration.
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Wednesday, November 18, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Yao Wan (The Chinese University of Hong Kong) - TBA - Abstract
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Wednesday, November 25, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Hongyi Sheng (Westlake University) - TBA - Abstract
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Wednesday, December 9, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Zetian Yan (The Chinese University of Hong Kong) - TBA - Abstract
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Wednesday, December 23, 9:00-10:00, Zoom link
(ID: TBA, Code: TBA)
Yaxiong Liu (Chongqing University of Technology) - TBA - Abstract