The seminar usually holds on Wednesday. For more details, please visit
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Wednesday, September 13, 9:00-10:00, Zoom link
(ID: 822 9523 6179, Code: 960801)
Shiguang Ma (Nankai University) - Potential theory and conformal geometry - Abstract
Partial differential equations (PDEs) are usually efficient tools in studying conformal geometry. Different types of PDEs arise when one studies different type curvatures. Potential theory is a powerful tool to study certain kinds of PDEs.
Especially it is convenient to use potential theory when one studies singular solutions (supersolutions). In this talk, I will mention our series of works, using potential theory to study conformal geometry.
In particular, I will talk about our recent work about the relationship between p-Laplace operator and conformal geometry.
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Wednesday, September 20, 10:00-11:00 (Special time), Zoom link
(ID: 857 6539 0760, Code: 577466)
Jiakun Liu (University of Wollongong) - Free boundary problems in optimal transportation - Abstract
In this talk, we introduce some recent regularity results of free boundary in optimal transportation. Particularly for higher order regularity, when densities are Hölder continuous and domains are C^2, uniformly convex, we obtain the free boundary is C^{2,alpha} smooth.
We also consider another model case that the target consists of two disjoint convex sets, in which singularities of optimal transport mapping arise. Under similar assumptions, we show that the singular set of the optimal mapping is an (n-1)-dimensional C^{2,alpha} regular sub-manifold of R^n.
These are based on a series of joint work with Shibing Chen and Xu-Jia Wang.
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Wednesday, September 27, 8:00-9:00 (Special time), Zoom link
(ID: 830 8530 4937, Code: 782471)
Paula Burkhardt-Guim (Courant Institute of Mathematical Sciences) - ADM mass for $C^0$ metrics and distortion under Ricci-DeTurck flow - Abstract
We show that there exists a quantity, depending only on $C^0$ data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$ sense and has nonnegative scalar curvature in the sense of Ricci flow.
Moreover, the $C^0$ mass at infinity is independent of choice of $C^0$-asymptotically flat coordinate chart, and the $C^0$ local mass has controlled distortion under Ricci-DeTurck flow when coupled with a suitably evolving test function.
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Wednesday, October 11, 9:00-10:00, Zoom link
(ID: 857 0208 4447, Code: 166323)
Otis Chodosh (Stanford University) - The p-widths of a surface - Abstract
The p-widths are a geometric invariant of a Riemannian manifold closely related to the spectrum of the Laplacian. I will describe some properties of the p-widths on a surface (joint with C. Mantoulidis).
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Wednesday, October 25, 9:00-10:00, Zoom link
(ID: 847 8137 1046, Code: 066678)
Jeffrey Streets (University of California, Irvine) - Formal structure of scalar curvature in generalized Kahler geometry - Abstract
The Fujiki-Donaldson moment map formulation of scalar curvature, and the attendant Mabuchi-Semmes-Donaldson geometry of a Kahler class, play a central role in addressing the existence and uniqueness of constant scalar curvature Kahler metrics.
Generalized Kahler (GK) geometry is a natural extension of Kahler geometry arising from Hitchins generalized geometry program and mathematical physics, and forms a particularly well-structured extension of Kahler geometry.
Recently Goto defined a notion of scalar curvature in GK geometry as the moment map of a particular Hamiltonian action on the space of generalized Kahler structures.
In this talk I will describe joint work with Vestislav Apostolov and Yury Ustinovskiy where we give an explicit description of the scalar curvature, and define a natural generalization of the Mabuchi-Semmes-Donaldson metric,
leading to a Calabi-Lichnerowicz-Matsushima obstruction, generalizations of Futaki's invariants, and a conditional uniqueness result.
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Wednesday, November 1, 9:00-10:00, Zoom link
(ID: 876 9745 2824, Code: 060811)
Yangyang Li (The University of Chicago) - Min-max widths of the unit 3-sphere and a strong multiplicity one theorem - Abstract
The recent advancements in Almgren-Pitts min-max theory have revealed the abundance of minimal hypersurfaces in closed manifolds. Specifically, each min-max width can be represented by a disjoint union of minimal hypersurfaces.
However, obtaining precise estimates on these widths remains a significant challenge, even in the case of the unit 3-sphere.
In this talk, I will discuss a recent joint work with Adrian Chu, in which we show that the min-max 10-width to 13-width of the unit 3-sphere lie strictly between 2π^2 (the area of a Clifford torus) and 8π (twice the area of an equator).
This partially answers a question posed by F. Marques and A. Neves. Our result is achieved via a strong multiplicity one theorem.
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Wednesday, November 8, 9:00-10:00, Zoom link
(ID: 836 3199 0429, Code: 072570)
Ravi Shankar (Princeton University) - The sigma-2 equation in dimension four - Abstract
The sigma-2 equation is the remaining equation in the Monge-Ampere / sigma-n family to be understood. It unclear whether solutions are smooth inside their domains.
With Yu Yuan, we confirm the interior regularity in dimension four. In higher dimensions, we find an interior estimate under a weak condition. The dimension two case is by Heinz in the 1950's, and dimension three is by Warren and Yuan in the 2000's.
Our method is pointwise and combines several overlooked ingredients from the past two decades. The idea is to propagate partial regularity using a three-sphere inequality. The method also gives new pointwise proofs of the Monge-Ampere and special Lagrangian equation results.
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Wednesday, November 15, 9:00-10:00, Zoom link
(ID: 892 7039 8293, Code: 511067)
Chao-Ming Lin (The Ohio State University) - On the Solvability of General Inverse $\sigma_k$ Equations - Abstract
In this talk, first, I will introduce general inverse $\sigma_k$ equations in Kähler geometry. Some classical examples are the complex Monge-Ampère equation, the J-equation, the complex Hessian equation, and the deformed Hermitian-Yang-Mills equation.
Second, by introducing some new real algebraic geometry techniques, we can consider more complicated general inverse $\sigma_k$ equations. Last, analytically, we study the solvability of these complicated general inverse $\sigma_k$ equations.
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Wednesday, November 29, 9:00-10:00, Zoom link
(ID: 839 6098 6853, Code: 104348)
Yiyue Zhang (University of California, Irvine) - The stability of Llarull's theorem - Abstract
Llarull's Theorem states that on an n-sphere, a Riemannian metric dominating the standard one must have scalar curvature less than n(n-1) at some point. Utilizing Llarull's proof, we have obtained a stability result in the intrinsic flat sense, assuming a uniformly bounded Poincaré constant.
Additionally, I will share the proof of the Incomplete Llarull's Theorem for dimension three, using level set techniques. These works are joint efforts with Sven Hirsch, Demetre Kazaras, and Marcus Khuri.
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Wednesday, December 6, 10:00-11:00 (Special time), Zoom link
(ID: 844 3814 6451, Code: 150000)
Ngoc Cuong Nguyen (Korea Advanced Institute of Science and Technology) - Regularity of the Siciak-Zaharjuta extremal function on compact Kahler manifolds - Abstract
We prove that the regularity of the extremal function of a compact subset of a compact Kahler manifold is a local property, and that the
continuity and Holder continuity are equivalent to classical notions of the local L-regularity and the locally Holder continuous property in pluripolential theory. As a consequence we give an effective characterization of the (C^{\alpha},C^{\alpha'})-regularity of compact sets, the notion introduced by Dinh, Ma and Nguyen.
Using this criterion all compact fat subanalytic subsets in R^n are shown to be regular in this sense.
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Wednesday, December 13, 9:00-10:00, Zoom link
(ID: 878 0965 2245, Code: 837865)
Wei Wei (Nanjing University) - The $\sigma_2$-curvature equation on a compact manifold with boundary - Abstract
We first establish local $C^2$ estimates of solutions to the $\sigma_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the background metric has nonnegative mean curvature on totally non-umbilic boundary,
for dimensions three and four we prove the existence of a conformal metric with a prescribed positive $\sigma_2$-curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis in the latter existence result. This is a joint work with Xuezhang Chen.
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Wednesday, December 20, 9:00-10:00, Zoom link
(ID: 819 7581 6692, Code: 254959)
Yiqi Huang (Massachusetts Institute of Technology) - Volume Estimates for Singular Set of Elliptic PDEs with Hölder Coefficients - Abstract
Consider the weak solution $u$ to the elliptic equation $\mathcal{L}(u)=\partial_{i}(a^{ij}(x)\partial_{j}u)+b^{i}(x)\partial_{i}u+c(x)u=0$. There has been extensive study about the singular set $\{u(x)=\nabla u(x)=0\}$ for the equation with weakly regular coefficients.
In this talk, I will discuss the recent progress about the volume estimates for singular sets with $a^{ij}$ assumed only to be Hölder continuous. It is sharp as it is the weakest condition in order to define the singular set of $u$ according to elliptic estimates. This talk is based on joint work with Wenshuai Jiang.
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Wednesday, December 27, 9:00-10:00, Zoom link
(ID: 812 0810 6093, Code: 020500)
Wei Sun (ShanghaiTech University) - The weak solutions to complex Hessian equations - Abstract
We shall talk on the weak solutions to complex Hessian equations on compact Hermitian manifolds. With appropriate assumptions, it is possible to obtain weak solutions in different senses. Different than previous works, we shall utilize PDE method.