Computer Science/Discrete Mathematics Seminar I

Geometric questions for discrete objects (e.g. isoperimetry, integrality gaps, embeddings) are often easier to understand by first considering an appropriate continuous analogue, using additional structure that is available in the continuous setting to prove the desired theorem, and then passing back to a discrete object via an appropriate "discretization" procedure. This last discretization step often relies on ad hoc arguments that are quite involved but technical, yet sometimes it requires conceptually new ideas, and there are instances in which it remains an open problem. In this talk we will discuss attempts to systematize the discretization step in certain important situations. We will present both older and recent theorems along these lines, and describe several basic open questions.