# Solutions to the Monge-Ampere equation with polyhedral and Y-shaped singularities

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The Monge-Ampere equation det(D^2 u) = 1 arises in prescribed

curvature problems and in optimal transport. An interesting feature of the

equation is that it admits singular solutions. We will discuss new examples

of convex functions on R^n that solve the Monge-Ampere equation away from

finitely many points, but contain polyhedral and Y-shaped singular

structures. Along the way we will discuss geometric and applied motivations

for constructing such examples, as well as their connection to a certain

obstacle problem.

# Comparison results for complete noncompact three-dimensional manifolds

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Typical comparison results in Riemannian geometry, such as for

volume or for spectrum of the Laplacian, require Ricci curvature lower

bounds. In dimension three, we can prove several sharp comparison estimates

assuming only a scalar curvature bound. The talk will present these results,

their applications, and describe how dimension three is used in the proofs.

Joint work with Jiaping Wang.

# Symplectic flat bundle

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# Pluriclosed flow on Bismut-flat manifolds

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In this talk, I will present parts of a recent paper written

jointly with Mario Garcia-Fernandez and Jeff Streets ("NonKahler Calabi-Yau

Geometry and Pluriclosed Flow" arxiv:2106.13716). In particular, I intend to

discuss long-time existence and convergence of solutions to pluriclosed flow

on manifolds satisfying a flatness condition. As the pluriclosed flow does

not admit a scalar reduction in general, I will introduce the language of

holomorphic Courant algebroids and a result of J.M. Bismut that will allow

us to derive apriori estimates for an equivalent coupled

Hermitian-Yang-Mills-type flow.

# The hyperbolic positive mass theorem and harmonic level sets

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I will introduce the harmonic level set method developed by Stern in 2019.

This technique has been used to prove the positive mass theorems in various

settings, for example, the Riemannian case, the spacetime case, the

hyperbolic case, and the positive mass theorem with charge. I will focus on

the positive mass theorem for asymptotically hyperbolic manifolds. We give a

lower bound for the mass in the asymptotically hyperbolic setting. In this

setting, we solve the spacetime harmonic equation and give an explicit

expansion for the solution. We also prove some rigidity results as

corollaries. This is joint work with Bray, Hirsch, Kazaras, and Khuri.

# The Type IIA flow and its applications in symplectic geometry

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The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. We prove the wellposedness of this flow and establish the basic estimates. We show that the Type IIA flow can be applied to find optimal almost complex structures on certain symplectic manifolds. It can also be used to prove a stability result about Kahler structures. This is based on joint work with Phong, Picard and Zhang.

# The deformed Hermitian-Yang-Mills equation and the C-subsolution

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The deformed Hermitian-Yang-Mills equation, which will be

abbreviated as dHYM equation, was discovered around the same time in the

year 2000 by Mariño-Minasian-Moore-Strominger and Leung-Yau-Zaslow using

different points of view.

In this talk, first, I will skim through Leung-Yau-Zaslow’s approach in a

simple way. Then I will introduce the C-subsolution which is introduced by

Székelyhidi and Guan, I will go over some known results of the dHYM

equation, and I will bring up my previous results. Last, I will show some of

my recent works which will appear soon.

# Scalar curvature and Riemannian polyhedra

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To characterize scalar curvature, Gromov proposed the dihedral rigidity conjecture which states that a positively curved polyhedron having dihedral angles less than those of a corresponding flat polyhedron should be isometric to a flat one. In this talk, we will discuss some recent progress on this conjecture and its connection with general relativity (ADM mass and quasilocal mass).