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This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

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This is a sample blog post. Lorem ipsum I can’t remember the rest of lorem ipsum and don’t have an internet connection right now. Testing testing testing this blog post. Blog posts are cool.

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Published in *Proc. Amer. Math. Soc. 148 (2020), 4967-4982*, 2020

This paper is about rigidity of round spheres as self-conformal solutions to inverse curvature flows.

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Published in *arxiv preprint*, 2022

This paper is about the physical-space Morawetz estimates for axisymmetric waves on extremal Kerr spacetime which based on the vector field method.

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Published in *arxiv preprint*, 2023

This paper is about the rigidity of map between compact manifolds with positive curvature, which is based on harmonic map heat flow approach.

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Published in *The Journal of Geometric Analysis*, 2023

This paper is about the natural boundary effects/conditions for the positive intermediate curvature condition introduced by Brendle-Hirsch-Johne.

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Published in *arxiv preprint*, 2023

This paper is about the rigidity of area-nonincreasing map between compact manifolds with positive curvature.

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We give the basic ingredients for Stern’s Bochner formula and apply this formula to harmonic maps $M^3\rightarrow \mathbb{S}^1$ where $M$ is a compact three-manifold with or without boundary. This gives a beautiful inequality relating the average Euler characteristics of harmonic map’s level sets and the scalar curvature of $M$.

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For an asymptotically flat initial data set (three dimensional), with the mass density large on a large region, Schoen and Yau showed that there is an apparent horizon in the initial data. (SY83’) Their proof is based on a contradiction argument where assuming no apparent horizon in the region will give a global solution to Jang’s equation over the region (SY81’). Furthermore, positivity of mass density and this global solvability of Jang’s equation will give rise to positivity of a certain operator on this region. Finally, the argument is completed by showing such positivity of certain operator on the region will assert that the region can not be too large in a certain sense. Aaron Chow and the speaker used a slicing technique that was introduced in a recent paper of S. Brendle, S. Hirsch, and F. Johne (BHJ22’) to show that in a $n+1$-dimensional torical band $T^n \times [0,1]$ where $n+1\leq 7$, positivity of a similar operator will assert that the band cannot be too long. If time permits, we will also discuss the possibility of generalising Schoen-Yau existence result of Black hole to higher dimensions, with such torical band width estimate.

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Hawking’s theorem on the topology of black holes asserts that cross sections of the event horizon in 4-dimensional asymptotically ﬂat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres. Geometrically and very roughly, this is analogous to the topological restriction of a stable minimal surface in positively curved 4-manifold. In this reading seminar talk, the speaker is going to talk about Galloway and Schoen’s generalization of Hawking’s theorem to any dimensional Spacetime satisfying the dominant energy condition, asserting that outer apparent horizon is Yamabe positive, except some very special cases.

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In this reading seminar talk, the speaker will present Hamilton’s Maximum Principle for Ricci flow and discuss applications for convergence of Ricci flow under certain positive curvature conditions. Then, as a comparison, the speaker will present a version of Bony’s strict maximum principle for degenerate elliptic equations and discuss its application on rigidity results (where we change the previous positive curvature conditions to non-negative curvature conditions). If time permits, the speaker will present and discuss some results where a version of Hamilton’s Maximum Principle or Bony’s strict maximum principle was applied.

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Dirac Operator is a powerful tool to study positive scalar curvature. Concerning positive scalar curvature on a manifold with boundary, it’s natural to ask how to formulate a valid boundary value problem for Dirac operator (1st order elliptic). In fact, Dirichlet boundary condition, which is natural for Laplacian operator (2nd order elliptic), turns out to be too strong for 1st order elliptic operators. In this talk, we focus on Dirac-type operators, with principal symbols capturing the Clifford relation just like the usual Dirac operator, and discuss some basic materials to get ready for elliptic boundary value problems.

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Last time, we defined the differential operator on the boundary associated to the given Dirac type operator using the information of the symbols. Recall that such associated boundary operator is a self-adjoint elliptic first order operator defined on a closed manifold (the boundary), so we can make use of its spectrum (L^2 decomposition of sections defined on the boundary) to investigate what kind of boundary condition is natural for the Dirac type operator. Statements will be given and the ideas of the proof will be sketched, and some important examples will be discussed.

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Last time, we studied the hybrid Sobolev spaces using the spectrum of a boundary adapted operator. In particular, the hybrid Sobolev space $\check{H}(A)$ is the image of (extended) trace map on the Dirac-maximal domain, and any boundary condition we will consider is a closed subspace of $\check{H}(A)$. Among all the boundary conditions, a special class called D-elliptic boundary conditions will be the main subject we are discussing. To understand this D-elliptic boundary condition, we will start with the famous APS condition, and try to explore based on that. Under these D-elliptic boundary conditions, nice boundary regularity result is obtained. And we will also discuss many other examples which belong to the class of D-elliptic boundary conditions.

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In this talk, we will discuss the isoperimetric surface technique that was developed by Professor Hubert Bray in his PhD thesis. In particular, we are going to use such technique to study volume comparison theorems for positively curved manifolds, which includes a new proof for the Bishop’ volume comparison theorem and also the “Bray’s football theorem”- a volume comparison theorem for scalar curvature in the case that the metric is close to the standard S^3 metric. We remark that the rigid case of Bray’s football theorem can be found in a beautiful survey paper written by Professor S.Brendle. (But we might not have the time cover it)

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In this talk, we will continue to discuss the isoperimetric surface technique that was developed by Professor Hubert Bray in his PhD thesis. This time, we are going to study the effect of non-negativity of scalar curvature on a complete asymptotically flat three-manifold. In particular, we will use the isoperimetric surface technique to prove two conditional Penrose inequality for single/multiple horizons cases. In contrast to last time, we are dealing with a minimizing process in non-compact manifolds, so we need to address the existence of area minimizers with its volume constraint.

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In this talk, I will review the vector field method/Energy method towards scalar wave equations on Schwarzschild or Kerr spacetime. To do so, we will emphasize some geometric/physical phenomena of Schwarzschild and Kerr involving the ergoregion, the photon sphere and also the Red-shift effect. Remarkably, these phenomena represent either obstructions or stabilizing mechanism for decay estimates of the wave equations. During the Energy estimate of wave equations, I intend to explain which phenomenon causes what obstruction and which phenomenon can help us fix some of the obstructions. At the end, I will also try to explain why the study of scalar wave equations is an essential step towards the (non-)linear stability of Blackholes.

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In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

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Lecture series on 11/8, 18/8, 25/8, 1/9(cancelled due to Typhoon)

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In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

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In this talk, we are going to consider the rigidity of map between possibly curved closed manifolds, which is motivated by the recent work of Tsai-Tsui-Wang. We show that distance non-increasing map between complex projective spaces is either an isometry or homotopically trivial. The rigidity result also holds on a wider class of manifolds with positive curvature and weaker contracting property on the map in between distance non-increasing and area non-increasing. This is based on the harmonic map heat flow and it partially solves a Conjecture of Tsai-Tsui-Wang.

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In this talk, we’ll discuss a paper by Brendle-Hung-Wang that applies a monotonicity formula of inverse mean curvature flow to prove a Minkowski inequality in AdS-Schwarzschild manifolds. Despite the lack of full convergence in this context (IMCF on Asymptotically hyperbolic manifold), we establish a roundedness estimate that helps estimate the lower bound of a monotone quantity along the flow. Besides IMCF, Professor Brendle’s geometric inequality and a sharp Sobolev inequality on the standard sphere are key components of the proof.

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Lagrangian Mean Curvature Flow (LMCF) provides a natural way to find special Lagrangian. In this talk, we discuss papers by Lotay-Oliveira that investigates the LMCF in very special ambient spaces, which are HyperKahler 4-manifolds generated by Gibbon-Hawking Ansatz. Notably, these spaces include the Multi-Eguchi-Hanson (ALE) and Multi-Taub-NUT (ALF). Moreover, HyperKahler 4-manifold in Gibbon-Hawking Ansatz admits a U(1)-symmetry which makes studying the U(1)-invariant Lagrangian surfaces and U(1)-invariant special Lagrangian surfaces very accessible. In particular, a U(1)-invariant LMCF in Gibbon-Hawking Ansatz will reduce to modified curve shortening flow on a plane by the HyperKahler moment map. This reduction provides us opportunities to understand various statements in the field of LMCF such as the notion of stability in Thomas-Yau conjecture and neck-pinch phenomenon in Joyce conjecture.

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We continue the discussion about IMCF and a Minkowksi inequality in AdS-Schwarzschild manifold. Last time, we went over the basic a priori estimates for IMCF using both the parametric and non-parametric versions of the flow. In particular, short time existence and long time existence were clarified. In this talk, we will delve into an improved roundness estimate for the flow and the monotonicity formula along the flow. It’s worth noting that IMCF doesn’t fully converge in asymptotically hyperbolic manifolds. To overcome this, we will rely on Beckner’s sharp Sobolev inequality on standard spheres and a geometric inequality by Professor Brendle to estimate the lower bound of the monotone quantity as we approach the limit. References: A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

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We continue the discussion on LMCF in Hyperkahler 4-manifolds with circle symmetry. In this talk, we will discuss the hyperkahler moment map reduction of the LMCF where special Lagrangians will be mapped to certain line segments in the image and the LMCF will be reduced to a modified curve shortening flow. With all these understanding, we will give a heuristic outline of the Thomas conjecture and Thomas-Yau conjecture in this setting.

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In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case, which was first proved by Huisken-Ilmanen using the Inverse mean curvature flow in dimension three. As the first part of this series of talks, we are going to discuss some basic notions and the Geroch monotonicity of Hawking mass under smooth inverse mean curvature flow. After this brief introduction, we will focus on the Huisken-Ilmanen’s weak formalism of inverse mean curvature flow, which is the essential ingredient in the proof of Riemannian Penrose inequality.

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Last time, we discussed the Geroch monotonicity of Hawking mass under smooth inverse mean curvature flow that can lead to Riemannian Penrose Inequality. We also gave a brief overview of Huisken-Ilmanen’s argument. In this talk, we aim to delve deeper into the intricacies of Huisken-Ilmanen’s weak formalism concerning the inverse mean curvature flow. To provide a more comprehensive understanding, we will focus on the freezing and variation approach, essential components in defining a weak solution for the level set inverse mean curvature flow. This approach not only grants us analytical advantages but also plays a crucial role in demonstrating the existence of weak solutions. To enhance our geometric understanding of weak IMCF, we will also cover an equivalence formulation utilizing sublevel sets. With both the analytic and geometric understanding of the weak IMCF, we will briefly talk about the heuristic reason for Geroch monotonicity along weak IMCF.

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This talk explores an alternative method for establishing the Riemannian Penrose inequality through Bray’s conformal flow. The conformal flow preserves the non-negativity of scalar curvature and the outermost minimizing property of the inner boundary. By using the positive mass theorem, one can demonstrate that the ADM mass is non-increasing under Bray’s conformal flow, eventually converging to half of Schwarzschild space. This convergence implies the desired Riemannian Penrose inequality. Notably, Bray’s approach offers advantages over IMCF approach, as it can be generalized to dimensions less than eight due to the absence of Gauss-Bonnet and enhances the inequality by replacing the area of the largest component of the horizon with the total area of the horizon, owing to fewer topology restrictions on the flow.

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Within this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form $F(D^2u)=0$, which is invariant under translation. We will introduce a useful approximation of the viscosity solution, called Jenson Approximation, which effectively captures the translation invariance of our PDE. Using Jenson Approximation, we establish a variation of Jenson’s uniqueness theorem for Dirichlet problem, which concerns the difference of two viscosity solutions. And again, due to the translation invariance of the PDE, we can apply the same idea to study the difference quotient of a viscosity solution, leading to interior $C^{1,\alpha}$ estimate. If time permits, we will also cover the application to concave $F$, in which we basically apply the same idea to study second order difference quotient (hence second derivatives) of a solution. (Notice: the additional concavity condition is crucial for discussing second order difference quotient.) References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]

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In this talk, we restrict ourselves to the fully nonlinear uniformly elliptic equation of the form $F(D^2u)=0$ with the additional condition that $F$ is concave as a function on symmetric matrices. In particular, we will make use of weak Harnack inequality and local maximum principle in chapter 4 of Caffarelli-Cabre as ingredients, demonstrating their application in proving the Evans-Krylov theorem and interior $C^{2,\alpha}$ regularity for concave equations. During the proof, some results implied by Jensen approximation we discussed last time (Chapter 5 of Caffarelli-Cabre) would also be used, leveraging translation invariance of the PDE we are interested in. References: [Caffarelli-Cabre Fully Nonlinear Elliptic Equations]

Undergraduate course, *University 1, Department*, 2014

This is a description of a teaching experience. You can use markdown like any other post.

Workshop, *University 1, Department*, 2015

This is a description of a teaching experience. You can use markdown like any other post.