Title: Diophantine tuples

Abstract: A tuple of numbers $(a,b,c)$ is a Diophantine tuple if the product of two distinct elements added by $1$ is a perfect square. This was introduced by Diophantus and he gave an example of such a quadruple of rational numbers. Fermat came up with such a triple with integer entries. Many generalizations of this notion came up over the years.

The aim is to talk about these tuples and their generalizations. In the way I will touch upon some of my works in this direction.

Title: Crossed or paired type problems of meromorphic functions.

Abstract: There are many important research problems in the value distribution of meromorphic functions and complex differential equations, such as Picard exceptional values of meromorphic functions, Hayman conjecture on the zeros of complex differential polynomials, the periodicity and the parity of complex differential polynomials, Malmquist or Riccati differential equations. In this talk,  their crossed or paired types problems will be established and the latest results on these problems will be given.

Title: Some Liouville properties in geometric analysis.

Abstract: In my talk, I will mention several types of vanishing phenomenons in geometric analysis. I will start with the gradient estimate method to derive Liouville results for positive solutions for non-linear equations on curved manifolds. Then I will introduce the nonlinear potential method and Moser iteration to investigate some vanishing results on Riemannian manifolds. Some geometric applications are also presented.