Time: Mar
24 , 2023

**1. Spectral Theory of Graph 1-Laplacian (CHANG KungChing)**

**2. Self-similar Solutions of Curve Shortening Problem (JIANG Meiyue )**

The generalized curve-shortening flow has been extensively studied in the literature. Self-similar solutions of the flow are solutions whose shapes change homothetically during the evolution. Such solutions are important in understanding the long time behaviors and the structure of singularities. A case which is called the affine curve shortening problem, has special interest in geometry. In terms of the support function, self-similar solutions reduce to an important 2-th order equation which also appears in image processing, 2-dimensional L^p-Minkowski problem, etc. Such equation with special parameter function is invariant with respect to affine transformations of R^2, which leads to some special features of the problem, for instance, there are obstructions for the solvability, and there is no a priori estimates for the solutions without additional assumption. This problem has been studied for the pi-periodic function. However, the method does not work for 2pi-periodic functions, which is more natural from the geometric point of view.Meiyue Jiang and his collaborators developed a new approach for the a priori estimate and existence of solutions, and proved the existence of solutions of the anisotropic affine curve shortening problem under some basic conditions [Jiang- Wang-Wei, Calc. Var. Partial Differential Equations (2011); Jiang-Wei, Discrete Contin. Dyn. Syst. (2016)].

**3. Harmonic Analysis and Its Applications in PDE (TANG Lin)**

Lin Tang obtained some important results on pseudo-differential operators, Schrodinger equations and parabolic Monge-Ampère equations.Tang [Tang-Zhang, J. Funct. Anal. (2016)] obtained weighted norm inequalities for pseudo- differential operators with smooth symbols and their commutators and introduced a new non-double weighted class, which generalizes essentially the classical Munckenhoupt weighted class, which also give a direction for studying pseudo- differential operators in the future.Lin Tang and his collaborators [Pan-Tang, J. Funct. Anal. (2016)] established the L^p boundedness of Schrodinger equations with discontinuous coefficients by using new method.Parabolic Monge-Ampère equations have some important applications in Ricci flow. Tang [Tang, J. Differential Equations (2013)] obtained the interior regularity for two type parabolic Monge-Ampère equations with VMO type data. In addition, Tang [Tang, J. Differential Equations (2015)] also solved the boundary regularity problem on parabolic Monge-Ampère equations.

**4. Solutions of L2 Extension Problem and Strong Openness Conjecture (GUAN Qi-an )**