In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first- and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik, Kerov, Logan, and Shepp (for the first-order asymptotics) and Kerov (for the second-order asymptotics). This gives more evidence for the connection with the Gaussian $\beta$-ensemble, already suggested by a work of Matsumoto.

Our main tool is a polynomiality result for the structure constants of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interesting in itself and we give several other applications of it.

We prove a version of the quantum geometric Langlands conjecture in characteristic $p$. Namely, we construct an equivalence of certain localizations of derived categories of twisted crystalline $\mathcal{D}$-modules on the stack of rank $N$ vector bundles on an algebraic curve $C$ in characteristic $p$. The twisting parameters are related in the way predicted by the conjecture and are assumed to be irrational (i.e., not in ${\mathbb{F}}_p$). We thus extend some previous results Braverman and Bezrukavnikov concerning a similar problem for the usual (nonquantum) geometric Langlands. In the course of the proof, we introduce a generalization of $p$-curvature for line bundles with nonflat connections, define quantum analogues of Hecke functors in characteristic $p$, and construct a Liouville vector field on the space of de Rham local systems on $C$.

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a nontrivial element of the kth homology group of the space of cycles and such that the mass of each cycle is bounded above by $Cmax\hspace{0.17em}\{\sqrt{k},\sqrt{g}\}\sqrt{Area\left(M\right)}$. This result is optimal up to a multiplicative constant.

We prove in this article that, for any prime $p$ and tame level $N$, the projection from the eigencurve to the weight space satisfies a rigid analytic version of the valuative criterion for properness introduced by Buzzard and Calegari. This gives a negative answer to a question of Coleman and Mazur.