Harvard Mathematics Department : Summer Tutorials 2020

You can sign up for a tutorial only by emailing Prof. Gaitsgory, [email protected]. Please email me by noon (EST), Tuesday, May 12, if you wish to sign up. If more than 10 students (per tutorial) have e-mailed by that time, there will be a lottery for places. If there are still open spots after that deadline, places will be filled on a first-come, first-served basis. In the past, some tutorials have filled up quickly. Please email Dennis Gaitsgory with your choice of tutorial and your Math experience and classes to determine if you meet the requirements. Please email Cindy ([email protected]) for any questions.

The topics and leaders of the tutorials this summer are:

Differential forms in algebraic topology

Taught by Joshua Wang Some people say you like spectral sequences when you first learn them if and only if you learn them from Bott and Tu's book "Differential Forms in Algebraic Topology". In this tutorial, we'll explore algebraic topology through the lens of differential forms. We will begin with differential forms on Euclidean space, and eventually we will see Poincaré duality, characteristic classes of vector bundles, and spectral sequences all in a concrete hands-on way. Armed with these concepts, we'll continue into homotopy theory and, among other things, compute some homotopy groups of spheres. Prerequisites include first courses in algebra, topology, and analysis. Some familiarity with smooth manifolds is expected, but I'll include a crash course on the basics. Those who have studied algebraic topology or smooth manifold theory will particularly benefit from seeing how the two complement each other. The purpose of this course is not just to understand algebraic topology more concretely; it also serves as a gateway into geometric and low-dimensional topology, more advanced topics in algebraic topology, and even topics in analysis like Hodge theory and index theory. Course website is here.

Modular forms

Taught by Samuel Marks Modular forms, of Fermat's Last Theorem fame, are ubiquitous in modern number theory. Yet after students' first glance at modular forms, they often come away with two questions: "What do modular forms have to do with number theory?" and "What is the connection between modular forms and elliptic curves?" The short answers are "Galois representations" and "modular curves," respectively. The long answers are the focus of this course. More specifically, this course will cover: