Introduction to Shtukas and their Moduli
Let $k$ be a finite field $F_q$. For two k-stacks $\mathcal{X}$ and $\mathcal{Y}$, we use $\mathcal{X}$x $\mathcal{Y}$ to denote their fibred product over $spec$ $k$.
Let $X$ be geometrically connected smooth projective curve over $k$. Let $|X|$ set of closed points of $X$. For $x\in |X|$, let $\mathcal{O}_x$, $\mathcal{m}_x$, $F_x$, $k_x$ denote completed ring of $|X|$ at $x$, its maximal ideal, fraction field and residue field.
Let $k$ be an field. An algebraic $k$-scheme is a scheme $X$ over $k$ such that the structure morphism $X\rightarrow Spec(k)$ is of finite type.
Given a topological group $G$, a $G$-set is a set $X$ equipped with a continuous group action $\mu:G$x$X\rightarrow X$, where $X$ is given discreter topology.
An equivariant function is a homomorphism between two $G$-sets:
$$f:X\rightarrow Y$$ $f(g\cdot x)=g\cdot f(x)$ for all $g\in G$ and $x\in X$.
A groupoid $G$ consists of two sets: a set of objects $G_0$ and set of arrows $G_1$ together with five structure maps$(s,t,u,i,c).$

$G$-$bundles:$ Let $G$ be a split reductive group over $k$.
A $G$-$bundle$ over a $k$-$scheme$ $Y$ is a scheme $\mathcal{E}$ with right action of $G$ and a $G$-invariant map
$\pi:\mathcal{E}\rightarrow {Y}$ such that $\acute{e}tale$ locally over $Y$, $\mathcal{E}$ is $G$-invariantly isomorphic to $G$x$Y$ with right translation action of $G$.
Let Bun$_G$:Sch$_k\rightarrow$Gpd be the functor that assigns a $k$-scheme $S$ the groupoid of $G$-bundles over $X$x$S$.

Derived Aspects of the Langlands Correspondence

Talk by Tony Feng and Michael Harris

If the object is abelian group, then derived object is chain complex of $\mathbb{Z}- modules.$

Similarly, if the object is commutative ring, then derived object is simplicial commutative rings.

Characterization of Simplicial Commutative Rings (SCR)

For a topological space $T$ there are two algebraic invariants: homotopy group and homology

In similar fashion, for SCR $\mathcal{R}$ there are two algebraic invariants: homotopy group which is graded commutative ring and cotangent complex $\coprod_R.$

Notation:

For classical case: $A, B, R, X, Y$
For derived case: $\mathcal{A}, \mathcal{B}, \mathcal{R}, \mathcal{X}, \mathcal{Y}$

Brief review of Cotangent Complex

Consider the morphism of schemes $f:X\longrightarrow Y$ then $\coprod_f:=\coprod_{X/Y}\in$ Derived category of quasi coherent sheaves of $X$. It governs the deformation theory of $f$.

Result:
A map $f:\mathcal{R}\longrightarrow\mathcal{R'}$ in SCR is an isomorphism if and only if $\pi_*(\mathcal{R}) \xrightarrow{\sim} \pi_*(\mathcal{R'})$.

Definition: A map of derived schemes/stacks
$f:\mathcal{X}\longrightarrow \mathcal{Y}$ is quasi smooth if $\coprod_f$ is perfect in degree $\geq-1$.

When not to derive moduli space
If give moduli space $M$ is local complete intersection of correct dimension then no need to derive it.