# Motivation: Intersection Theory

# 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 $m\in \Homeo(\Sigma)$ mod isotopy.
	- Donaldson-Thomas theory: flat connections on a vector bundle up to gauge equivalence. 
	- Gromov-Witten theory: all curves of a fixed genus, e.g. moduli of elliptic curves.
	- Floer theory: pseudo-holomorphic discs, e.g. with Lagrangian boundary conditions

## What is a stack?
- What are some examples?
	- $\mgn(X)$ 
	- $\B G = K(G, 1)$
	- $\Pic^0(X)$.
	- $\Bun_G(X)$
	- $\Vect_n(X) = \Bun_{\GL_n}(X)$
	- $\PP^n = [\AA^{n+1}/ \GG_m]$.
	- $\Coh(X)$ and $\QCoh(X)$.
	- $\Loc\Sys(X) \cong \Rep(\pi_1 X) \cong \Vect_n^\flat(X)$.

## 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?