Vincent Lafforgue's work span many topics and contain many striking results but the most probable recent work to be described in the spirit of the question is *Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale*. In slogan form, this work proves the Langlands correspondence in the "automorphic to Galois" direction for reductive group over global function field of positive characteristic.

More precisely, let $F$ be the function field of a smooth, projective, geometrically irreducible curve $X$ over $\mathbb F_{p}$, let $\mathbb A$ be its adele ring and let $G$ be a connected, reductive group over $F$ assumed split for simplicity. For $N$ a finite sub-scheme of $X$, denote by $K_N$ the compact, open subgroup of $G(\mathbb A)$ equal to the kernel of $G(\mathbb O)\rightarrow G(\mathcal O_N)$ where $\mathbb O$ is the product of the unit balls of the local fields $F_v$ and $\mathcal O_N$ is the ring of functions on $N$. Denote by $Z$ the center of $G$ and fix $\Xi$ a lattice inside $Z(F)\backslash Z(\mathbb A)$. Finally fix $E$ a finite extension of $\mathbb Q_\ell$ where $\ell\nmid p$.

Then the finite-dimensional $E$ vector space
$$C^{\operatorname{cusp}}_c(G(F)\backslash G(\mathbb A)/K_N\Xi,E)$$
of cuspidal functions admits a direct-sum decomposition indexed by global Langlands parameter
$$\sigma:\operatorname{Gal}(\bar{F}/F)\longrightarrow \widehat{G}(E)$$
with values in the Langlands dual group. This decomposition is compatible with the Satake isomorphism.

In addition to the result itself, the method he introduced (the so-called excursion operators) looks very promising even in the characteristic zero case.

Let's say something about what this method entails. The standard strategy, going back at least to Eichler-Shimura, for proving statements of this type is to study the cohomology groups of some space (in the classical case, a modular curve or Shimura variety, and in the function field case, a moduli space of Shtukas) and show that it admits a Galois action and an action of Hecke operators, and then show that the Hecke eigenspace is a Galois representation arising from a Langlands parameter $\sigma:\operatorname{Gal}(\bar{F}/F)\longrightarrow \widehat{G}(E)$ composed with some fixed representation of $\widehat{G}(E)$.

This works quite well when $G = GL_n$, and was used by Vincent's brother Laurent Lafforgue to great effect in that case. However, there are some difficulties for other groups. If we pick out the subspace of the cohomology corresponding to a particular automorphic form, it may be difficult to show that the Galois action factors through the group $\hat{G}$, or it might factor through the group $\hat{G}$ in multiple indistinguishable ways.

V. Lafforgue solves this by working simultaneously with the cohomology of a huge array of spaces, in which Langlands parameters are expected to appear via different representations. He defines several different maps between the cohomology of these different spaces. Composing these maps appropriately, he defines excursion operators on the original space $C^{cusp}_c$ of automorphic forms. These operators include the Hecke operators, but are not limited to them. They satisfy some relations, forming a ring, whose characters he checks by abstract group theory correspond to Langlands parameters. The compatibility with Satake can be established by comparing these operators to the classical Hecke operators.

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