DMS Graduate Student Seminar

Title: What Is Differential Geometry? From Curves and Surfaces to Modern Geometry

Abstract: Differential geometry is an important area of mathematics that combines ideas from calculus, linear algebra, and topology to study curved spaces. Starting from the geometry of curves and surfaces, I will introduce basic notions such as curvature, geodesics, and metrics, and explain how these ideas extend to general manifolds. I will also discuss the relation between curvature and topology, for example, through the Gauss–Bonnet theorem. In the second part, I will give a short introduction to geometric analysis, which uses partial differential equations to understand geometric and topological properties of spaces. Examples include mean curvature flow and the Ricci flow.