MATH 99R: Tutorial

Math 99r courses are small group tutorial courses with limited enrollment. There is just one Math 99r tutorial for Fall 2022**; it is titled Moduli problems and stacks and it will be led by Taeuk Nam. What follows momentarily is a description of the tutorial. There will also be a Zoom meeting to describe this tutorial on Monday, August 22 from 7-8pm (Boston time). The meeting url is here.

Moduli problems and stacks: This course is an introduction to stacks, the modern framework in which to think about moduli problems. A moduli problem is the study of families of objects that vary "nicely" along a base.

During the first part of the course, we will build up the theory of moduli problems and stacks in the general setting, with our main focus being the topological setting; in other words, the bases along which our families of objects live over will be topological spaces. We will start by working towards defining the notion of a stack, then introduce concepts including symmetry groupoids of families, fine and coarse moduli spaces, versal families, and quotient stacks. The necessary category theory will be intertwined with the other material and introduced when the need arises.

The second part of the course will be about the algebraic setting of the theory, where we instead consider families of objects over algebraic varieties and schemes. After considering the concepts introduced in the first section and specializing them to our new setting, we will introduce concepts specific to the algebraic context, such as algebraic spaces, algebraic stacks, and Delign-Mumford stacks, with the overarching goal of working towards the Keel-Mori Theorem. The necessary algebraic geometry will also be introduced between the other material as the need arises.

Some other possible topics, depending on time and participant interest, include sheaves/bundles on stacks, higher/homotopy stacks, applications to geometric representation theory, orbifolds, etc. One possibility is to have participants give presentations on some of these topics.

This tutorial is inspired and influenced by a course on algebraic stacks, taught by Kai Behrend at the University of British Columbia, that the instructor took as an undergraduate.

Notes: Math 99r counts for Mathematics concentration credit except if taken previously in which case it only counts for Harvard credit.  In any event, the final paper for Math 99r can be used to fill the Mathematics concentration third year paper requirement.

**The class number for Math 99r is 21799.