Title: Index Theory for Spaces with Signularities and its Applications
Prerequisites: The prerequisite of the course is basic knowledge in functional analysis.
Familiarity with basic differential geometry is encouraged, but we will cover some basics at the beginning of the course.
Course Description: The main goal of this course is to discuss a new index theory for spaces with singularities and its
applications to geometry. Some of the main applications include positive solutions to some conjectures of Gromov on scalar
curvature (such as the dihedral rigidity extremality/rigidity conjecture and the flat corner domination conjecture) and a
positive solution to Stokerâ€™s conjecture (a long-standing open question in discrete geometry). First we plan to cover basics
of the following: Fredholm theory, manifolds, differential operators on manifolds, the classical index theory for closed manifolds
(and compact manifolds with smooth boundary) and analysis on conic metrics. Then we give a self-contained account of a new index
theory for spaces with singularities and its applications.
Avg. Amount of Time Dedicated per Week: There will be no homework or exams. The students may need to several hours weekly to
review the course notes and to do some further reading.