For Fall/Spring tutorials, see this page.
Welcome Message
You can sign up for a tutorial only by emailing Prof. Gaitsgory,
gaitsgde@math.harvard.edu. 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 (cindy@math.harvard.edu)
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:
- the basic theory of elliptic curves
- modular curves as moduli spaces
- various interpretations of modular forms
- Hecke operators
- Galois representations and L-functions
- The Modularity Theorem
Prerequisites: Number theory. Familiarity with basic algebraic geometry
is useful, but not required.