We construct analogues of the Hecke operators for the moduli space of G-bundles on a curve X over a local field F with parabolic structures at finitely many points. We conjecture that they define commuting compact normal operators on the Hilbert space of half-densities on this moduli space. In the case $F=\mathbb{C}$, we also conjecture that their joint spectrum is in a natural bijection with the set of ${}^{L}G$-opers on X with real monodromy. This may be viewed as an analytic version of the Langlands correspondence for complex curves. Furthermore, we conjecture an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators. Assuming the compactness conjecture, this formula follows from a certain system of differential equations satisfied by the Hecke operators, which we prove here for $G={\mathrm{PGL}}_n$.

We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let $(M^{n+1},g)$ be a closed Riemannian manifold, and let $\mathrm{\Sigma }\subset M$ be a closed embedded minimal hypersurface with area at most $A>0$ and with a singular set of Hausdorff dimension at most $n-7$. We show the following bounds: there is $C_A>0$ depending only on n, g, and A so that

$$\begin{array}{rcl}\sum _{i=0}^n{b}^i(\mathrm{\Sigma })& \le & C_A(1+index(\mathrm{\Sigma }))\text{if}3\le n+1\le 7,\\ {\mathcal{H}}^{n-7}(Sing(\mathrm{\Sigma }))& \le & C_A{(1+index(\mathrm{\Sigma }))}^{7∕n}\text{if}n+1\ge 8,\end{array}$$

where $b^i$ denote the Betti numbers over any field, ${\mathcal{H}}^{n-7}$ is the $(n-7)$-dimensional Hausdorff measure, and $Sing(\mathrm{\Sigma })$ is the singular set of Σ. In fact, in dimension $n+1=3$, $C_A$ depends linearly on A. We list some open problems at the end of the paper.

Assuming a lower bound on the dimension, we prove a long-standing conjecture concerning the classification of global solutions of the obstacle problem with unbounded coincidence sets.

The purpose of this article is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by $\{M(r)\mid r\in \mathbb{Q}\}$, which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol manifold $M(0)$ due to Plante, we have:

As a consequence of the second result, we get infinitely many closed orientable hyperbolic 3-manifolds that carry taut foliations but do not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by Schwider, which is used to carry essential laminations on $M(r)$.