Why Stacks
What is a moduli problem?

Motivation from topology: classifying spaces for vector bundles over a fixed space.

What are some common moduli problems?

Vector bundles on a fixed space/curve.

Complex structures on a Riemann surface \(\Sigma\) up to homeos isotopic to the identity.

Equivalently, marked hyperbolic structures on \(\Sigma\) where \(m\in {\operatorname{Homeo}}(\Sigma)\) mod isotopy.

DonaldsonThomas theory: flat connections on a vector bundle up to gauge equivalence.

GromovWitten theory: all curves of a fixed genus, e.g. moduli of elliptic curves.

Floer theory: pseudoholomorphic discs, e.g. with Lagrangian boundary conditions
What is a stack?

What are some examples?

\({ \mathcal{M}_{g, n} }(X)\)

\({\mathbf{B}}G = K(G, 1)\)

\(\operatorname{Pic}^0(X)\).

\({\mathsf{Bun}}_G(X)\)

\({ \mathsf{Vect} }_n(X) = {\mathsf{Bun}}_{\operatorname{GL}_n}(X)\)

\({\mathbf{P}}^n = [{\mathbf{A}}^{n+1}/ {\mathbf{G}}_m]\).

\({\mathsf{Coh}}(X)\) and \({\mathsf{QCoh}}(X)\).

\(\mathsf{Loc}\Sys(X) \cong {\mathsf{Rep}}(\pi_1 X) \cong { \mathsf{Vect} }_n^\flat(X)\).
Issues with schemes and sets

Why not just schemes?

Existence of universal families

Quotients of schemes by group actions (orbit spaces)

Representability of moduli functors.

Why groupoids?
Why Infinity Categories

What is an infinity groupoid?

What is the homotopy hypothesis?

What is a simplicial set?

What is the nonabelian derived category?

What is the nerve?

What is an infinity category?

What is an equivalence of infinity categories?
Why “derived”/higher stacks

What does “derived” mean?

When is a derived stack strictly necessary?

What are some examples of higher stacks?
What problems does this solve?

Geometric Langlands

What tools are available in the theory?