Q-curvature is 4th-order analog of scalar curvature, which is an important geometric object in the study of conformal geometry. In this talk, after reviewing some of our previous work about deformations of Q-curvature, I will talk about some new progresses in the study of the volume comparison result of Einstein manifolds with respect to Q-curvature. This series of works are joint with Yueh-Ju Lin in Wichita State University.

Let Ω be a domain of a closed manifold $(M, g_0)$ with dim M>2. Let $g=u^\frac{4}{n-2}g_0$ be a complete metric defined on Ω. We will show that $M\setminus Ω$ is a finite set when $\int_Ω|Ric(g)|^\frac{n}{2}dV_g<+∞. Such a result is not true if we replace Ricci curvature with Scalar curvature. We will discuss the properties of conformal metrics with $\|R\|_{L^\frac{n}{2}}<+∞$ on a punctured ball of a Riemannian manifold , and give some geometric obstacles for Huber's theorem in this case.

As is well known, potential theory is a useful tool to study PDE. In this talk, I will briefly show that how to apply it to study conformal geometry and hypersurfaces in hyperbolic spaces. There are joint works with Professor Jie Qing at UCSC.

As a curvature of riemannian manifold, like the sectional curvature and the Ricci curvature, there are topology and geometry results about the scalar curvature. In this talk, we talk the rigid/extremal property of scalar curvature which is proposed by Gromov in his “Four lectures on scalar curvature“.

We study the positive solutions to a class of general semilinear elliptic equations ∆u(x) + uh(ln u) = 0 defined on a complete Riemannian manifold (M, g) with Ric(g) ≥ −Kg, and obtain the Li-Yau type gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). We also obtain some Liouville-type theorems for these equations when (M, g) is noncompact and Ric(g) ≥ 0 and establish some Harnack inequalities as consequences. Then, as applications of main theorem we extend our techniques to the Lichnerowicz-type equations

We develop a new class of distribution–free multiple testing rules for FDR control. I will mainly illustrate the idea via multiple testing with general dependence. A key element in our proposal is a symmetrized data aggregation (SDA) approach to incorporating the dependence structure via sample splitting, data screening and information pooling. The SDA substantially outperforms the knockoff method in power under moderate to strong dependence, and is more robust than existing methods based on asymptotic p-values. I will also talk about some other applications, such as the selection of the number of change-points and threshold selection in feature screening.

The theory of Gromov-Witten invariants is a curve counting theory defined by integration on the moduli of stable maps. Varying the stability condition gives alternative compactifications of the moduli space and defines similar invariants. One example is epsilon-stable quasimaps, defined for a large class of GIT quotients. When epsilon tends to infinity, one recovers Gromov-Witten invariants. When epsilon tends to zero, the invariants are closely related to the B-model in physics. The space of epsilon's has a wall-and-chamber structure. In this talk, I will explain how wall-crossing helps to compute the Gromov-Witten invariants and sketch a proof of the wall-crossing formula.

Combustion or ignition equation is a special reaction diffusion equation which is used to describe the propagation of flames. In this talk we consider the Cauchy problem and the Stefan free boundary problem for such equations. We will specify the asymptotic behavior for the solutions, in particular, the spreading phenomena and its feature in these problems.