All the notes linked here are preparatory in nature, and we as the organizer did not proofread them at all. As such, there may be mistakes that the speakers did not realize and/or did not bother to correct. Readers should read at their own risk.
[Alp24] Jarod Alper. Stacks and Moduli. Aug. 2024. url: https://sites.math.washington.
edu/~jarod/moduli.pdf.
[Fu11] L. Fu. Etale Cohomology Theory. Nankai tracts in mathematics. World Scien-
tific, 2011. isbn: 9789814307727. url: https://books.google.ca/books?id=
WhaY64UEVXAC.
[Har77] Robin Hartshorne. Algebraic geometry. eng. Graduate texts in mathematics ; 52.
New York: Springer-Verlag, 1977. isbn: 0387902449.
[Ols16] Martin Olsson. Algebraic Spaces and Stacks. en. Vol. 62. Colloquium Publications.
Providence, Rhode Island: American Mathematical Society, 2016. isbn: 978-1-4704-
2798-6. doi: 10.1090/coll/062. url: http://www.ams.org/coll/062.
[Vak25] R. Vakil. The Rising Sea: Foundations of Algebraic Geometry. Princeton University
Press, 2025. isbn: 9780691268675. url: https://books.google.ca/books?id=
N2Xx0AEACAAJ.
[Vis04] Angelo Vistoli. “Notes on Grothendieck topologies, fibered categories and descent
theory”. en. In: (Dec. 2004). url: https://arxiv.org/abs/math/0412512v4.
Date: December 6, 2024
Speaker: Dingchang Zhou
Title: Projectivity of \bar{M_{g}}
Abstract: We will briefly cover different approaches towards proving projectivity, and then focus on Kollár's method. This uses sheaf theory, criterion of positivity, and a genius observation from Kollár.
Date: November 22, 2024
Speaker: Connor Lehmacher
Title: Sheaves on Stacks
Abstract: Today we’ll define sheaves on stacks and provide some standard propositions. This will be in preparation for proving the projectivity of Mg.
Date: November 15, 2024
Speaker: Dingchang Zhou
Title: Irreducibility of Mg
Abstract: Today we'll prove the irreducibility of Mg. We'll use degeneration to the boundary to give an induction argument. If time permits, we'll also introduce other approaches.
Date: November 1, 2024
Speaker: Matthew Huynh
Title: Stable reduction
Disclaimer by the Speaker: There are mistakes in the above note and that the reader should read at their own risk.
Abstract: We start by showing that \bar{M}_{g,n} is finite type. Then we will state the Valuative Criterion of Properness for nice morphisms of algebraic stacks. We'll sketch the proof of stable reduction, which implies that \bar{M}_{g,n} --> Spec(Z) satisfies the existence part of the Valuative Criterion for Properness. Time permitting, we will also show that it satisfies the uniqueness part.
Date: October 25, 2024
Speaker: Shuo Gao
Title: \mathcal{M}_g,n are Deligne-Mumford Stack
Abstract: We start with an overview of Hilbert schemes, whose theory turns out to be at heart at the algebraicity of many stacks, especially \mathcal{M}_g,n. After this, we work through the proof of \mathcal{M}_g,n being a Deligne-Mumford stack, piecing everything we have covered so far together.
Date: October 18, 2024
Speaker: Shuo Gao
Title: Nodal Curves, Stable Curves and Their Deformation
Abstract: As the first talk in a series of talks about M_{g,n}, we focus our attention to the object/point of this extremely useful Moduli space. We start by reviewing an important toolbox from curve theory including Riemann-Roch and Serre duality. We will use these tools to find both the basic properties and the deformation theory of nodal curves and stable curves.
Date: October 11, 2024
Speaker: Dingchang Zhou
Title: Calculating Deformation-Obstruction Theories
Abstract: In this talk we'll calculate various deformation-obstruction theories of common objects. We'll not go through the abstract definition of the deformation-obstruction theory of a stack, but hopefully we can give an idea of how deformation-obstruction theory works through these examples.
Date: October 4, 2024
Speaker: Connor Lehmacher
Title: Examples of Stacks, Isom, 2-Fibre Products
Abstract: Today we will discuss some examples of stacks, the sheaf Isom(E,E') and how it relates to the stack conditions, and fiber products of sheafs and what makes them 2-limits rather than limits.
Date: September 27, 2024
Speaker: Matthew Huynh
Title: Descent Theory
Abstract: We will discuss some general descent theory, and phrase the condition for a prestack to be a stack in terms of descent. After that, I’ll show that the prestack of quasicoherent sheaves is a stack. In the remaining time, we will talk about properties of schemes and morphisms that descend under fppf/smooth/etale morphisms.
Date: September 20, 2024
Speaker: Ze Yun
Title: Commutative Algebra Background for Stack theory
Abstract: I’ll talk about flat morphisms and faithfully flat morphisms. They are some algebraic conditions for “nice” families. We also prove for faithfully flat morphisms a gluing theorem remained from last time. Then I’ll talk about Kähler differentials, and unramified morphisms. I’ll introduce étale morphisms and smooth morphisms, which are analogues of covering maps and submersions of manifolds.
Date: September 13, 2024
Speaker: Shuo Gao
Title: Categorical Background for Stack theory
Abstract: The notion of a stack is an abstraction and unifying generalization of classifying spaces, moduli, algebraic spaces and many more. In this talk, we will discuss all the relevant definitions from category theory towards stacks. Specifically, we introduce the notion and theory of Grothendieck topology and sheaves over such topology, in analogy with the theory of usual topology and sheaves; we also introduce the definition of stacks and their theories of cohomology, in analogy to examples such as \”Cech cohomology with which we are familiar.