Exploring D-modules and Riemann-Hilbert correspondence from relative and logarithmic perspectives and applications

The theory of D-modules plays a central rule in various fields of contemporary mathematics. It provides tools to solve Hilbert's 21st problem and leads to the development of the theory of Riemann-Hilbert correspondence. The primary purpose of the research project is to explore Riemann-Hilbert correspondence from relative and logarithmic perspectives and to apply relative and logarithmic D-modules and the correspondence in the study of singularities in algebraic geometry and other related research areas in pure mathematics. "Relative" means to study D-modules relatively over one or more independent (analytic or algebraic) parameters, while "logarithmic" means that over logarithmic spaces in the  category of logarithmic geometry. Based on results in the literature and more importantly from my previous research, the project aims to: (1) study and establish Riemann-Hilbert correspondence over algebraic spaces of higher dimension, (2) study and establish Riemann-Hilbert correspondence over certain smooth logarithmic spaces, (3) apply relative and logarithmic D-modules and the correspondence to study Bernstein-Sato polynomials/ideals (for instance for hyperplane arrangements) and the topological zeta functions. The major potential impacts of the project include that it will increase people’s knowledge of the categories where Riemann-Hilbert correspondence applies, and that it will provides brand new applications of D-modules in algebraic geometry.