On a negatively curved surface, we show that the Poincaré series counting geodesic arcs orthogonal to some pair of closed geodesic curves has a meromorphic continuation to the whole complex plane. When both curves are homologically trivial, we prove that the Poincaré series has an explicit rational value at 0 interpreting it in terms of linking number of Legendrian knots. In particular, for any pair of points on the surface, the lengths of all geodesic arcs connecting the two points determine its genus, and, for any pair of homologically trivial closed geodesics, the lengths of all geodesic arcs orthogonal to both geodesics determine the linking number of the two geodesics.

We prove Rietsch’s mirror conjecture that the Dubrovin quantum connection for minuscule flag varieties is isomorphic to the character D-module of the Berenstein–Kazhdan geometric crystal. The idea is to recognize the quantum connection as Galois and the geometric crystal as automorphic. We reveal surprising relations with the works of Frenkel and Gross; Heinloth, Ngô, and Yun; and Zhu on Kloosterman sheaves. The isomorphism comes from global rigidity results where Hecke eigensheaves are determined by their local ramification. As corollaries, we obtain combinatorial identities for counts of rational curves and the Peterson variety presentation of the small quantum cohomology ring.

We study parabolic reductions and Newton points of G-bundles on the Fargues–Fontaine curve and the Newton stratification on the $B_{\mathrm{dR}}^{+}$-Grassmannian for any reductive group G. Let ${\mathrm{Bun}}_G$ be the stack of G-bundles on the Fargues–Fontaine curve. Our first main result is to show that under the identification of the points of ${\mathrm{Bun}}_G$ with Kottwitz’s set $B(G)$, the closure relations on $|{\mathrm{Bun}}_G|$ coincide with the opposite of the usual partial order on $B(G)$. Furthermore, we prove that every non-Hodge–Newton decomposable Newton stratum in a minuscule affine Schubert cell in the $B_{\mathrm{dR}}^{+}$-Grassmannian intersects the weakly admissible locus, proving a conjecture of Chen. On the way, we study several interesting properties of parabolic reductions of G-bundles, and we determine which Newton strata have classical points.