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Motives with exceptional Galois groups and the inverse Galois problem
We construct motivic ℓ-adic representations of $\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ into exceptional groups of type E7,E8 and G2 whose image is Zariski dense. This answers a question of
Global Springer theory
On Koszul duality for Kac-Moody groups
For any Kac-Moody group $G$ with Borel $B$, we give a monoidal equivalence between the derived category of $B$-equivariant mixed complexes on the flag variety $G/B$ and (a certain completion of) the
The fundamental lemma of Jacquet and Rallis
We prove both the group version and the Lie algebra version of the Fundamental Lemma appearing in a relative trace formula of Jacquet-Rallis in the function field case when the characteristic is
Kloosterman sheaves for reductive groups
Deligne constructed a remarkable local system on P 1 f 0;1g attached to a family of Kloosterman sums. Katz calculated its monodromy and asked whether there are Kloosterman sheaves for general
Shtukas and the Taylor expansion of L-functions (II)
For arithmetic applications, we extend and refine our results in \cite{YZ} to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension $F'/F$ of function fields
The fundamental lemma of Jacquet-Rallis in positive characteristics
We prove both the group version and the Lie algebra version of the Fundamental Lemma appearing in a relative trace formula of Jacquet-Rallis in the function field case when the characteristic is
Lectures on Springer theories and orbital integrals
These are the expanded lecture notes from the author's mini-course during the graduate summer school of the Park City Math Institute in 2015. The main topics covered are: geometry of Springer fibers,
Macdonald formula for curves with planar singularities
We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to
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