Speaker: Kartick Ghosh
Date: November 16, 2023
Affiliation: TIFR, Mumbai

Abstract: I will talk about two PDEs: Coupled K$\ddot{\mathrm{a}}$hler-Einstein and Hermitian-Yang-Mills equations and vortex-type equations. <p> Firstly, I will introduce the coupled K$\ddot{\mathrm{a}}$hler-Einstein and Hermitian-Yang-Mills equations. I will then talk about two obstructions to the existence of solutions of these equations: Matsushima-Lichnerowicz type obstruction and Futaki inavariant type obstruction. We will solve the equations is some cases using deformation. Using Calabi ansatz, we will solve the equations on some projective bundles. <p> In the second part, I will introduce vortex-type equation and derive a priori estimates for it. Then I will give some applications.