Weyl laws relate the asymptotic behaviors of the eigenvalues of certain geometric operators with the geometric/dynamical properties of the underlying space. In this talk I will briefly describe these connections, with an emphasis on the relation between the eigenvalue counting problem for special planar domains with integrable billiard flows and the classical lattice point counting problem.

We will introduce the S-closed conformal transformation in Finsler geometry. We show some properties of such transformation. Using some results, we show how to refine a rigidity theorem of Matsumoto problem about conformally equivalent between two non-Riemannian Berwald manifolds.

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.