Let $(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present talk is concerned with the set $\Red(z)$ of all reduced expressions for any $z\in W$. By associating each bc-expression to a certain symbol, we describe the set $\Red(z)$ and compute its cardinal $|\Red(z)|$ in terms of symbols. An explicit formula for $|\Red(z)|$ is deduced, where the Fibonacci numbers play a crucial role.

We are concerned with a system of equations governing the evolution of isothermal, viscous and capillary compressible fluids, which is used as the phase transition model. In the case of zero sound speed, it is found that the linearized system admits a purely parabolic structure. Consequently, one can establish the global-in-time existence and Gevrey analyticity of Lp solutions in hybrid Besov spaces, which improves the prior L2 bounds on the low frequencies of density and velocity due to acoustic waves. The proof mainly relies on new Besov (-Gevrey) estimates for product and composition of functions.

Recently, the deformed Hermitian Yang-Mills equation has been extensively studied. In this talk, we introduce a deformed Hermitian Yang-Mills flow in the supercritical case on a compact Kähler manifold. Under a suitable condition on the subsolution, we show the longtime existence of the flow and we prove that the solution converges exponentially to the solution of the elliptic deformed Hermitian Yang-Mills equation which has been solved by Collins-Jacob-Yau by the method of continuity. This is a joint work with Professor Jixiang Fu.

We would like to talk about the Minkowski problem for Gaussian surface area measure. Both the uniqueness and existence results are investigated. This is a joint work with Huang Yong and Zhao Yiming.

In this work, we transform the equation in the upper half space first studied by Caffarelli and Silvestre to an equation in the Euclidean unit ball $\mathbb{B}^n$. We identify the Poisson kernel for the equation in the unit ball. Using the Poisson kernel, we define the extension operator. We prove an extension inequality in the limit case and prove the uniqueness of the extremal functions in the limit case using the method of moving spheres. In addition we offer an interpretation of the limit case inequality as a conformally invariant generalization of Carleman's inequality.

Let $(W,S)$ be a Coxeter system with a strictly complete Coxeter graph. The present talk is concerned with the set $\Red(z)$ of all reduced expressions for any $z\in W$. By associating each bc-expression to a certain symbol, we describe the set $\Red(z)$ and compute its cardinal $|\Red(z)|$ in terms of symbols. An explicit formula for $|\Red(z)|$ is deduced, where the Fibonacci numbers play a crucial role.

We investigate the fully nonlinear elliptic equations F(D^2u,Du,u,x) = 0, which satisfy the structural condition previously posed by Bianchini-Longinetti-Salani in 2009. By establishing a novel differential inequality, we prove a weak Harnack inequality for the principal curvatures of the level surfaces of the solutions. This result is indeed a quantitative version of the constant rank theorem showed by Guan-Xu in 2013.

In this talk, we will discuss stability and index estimates for compact and noncompact capillary surfaces. A classical result in minimal surface theory says that a stable complete minimal surface in R^3 must be a plane. We show that, under certain curvature assumptions, a strongly stable capillary surface in a 3-manifold with boundary has only three possible topological configurations. In particular, we prove that a strongly stable capillary surface in a half-space of R^3 which is minimal or has the contact angle less than or equal to $\pi/2$ must be a half-plane. We also give index estimates for compact capillary surfaces in 3-manifolds by using harmonic one-forms.