We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean’s spectral sequence converging to the Floer cohomology of the $m$‑th iterate of the monodromy, when the polynomial has an isolated singularity. Inspired by this connection, we conjecture that if two germs of holomorphic functions are embedded topologically equivalent, then the Milnor fibers of their tangent cones are homotopy equivalent.

We adapt the method of Simon [26] to prove a $C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone $\mathbf{C}^2_0$ over an equiangular geodesic net. For varifold classes admitting a “no-hole” condition on the singular set, we additionally establish $C^{1,\alpha}$-regularity near the cone $\mathbf{C}^2_0 \times \mathbb{R}^m$. Combined with work of Allard [2], Simon [26], Taylor [29], and Naber–Valtorta [21], our result implies a $C^{1,\alpha}$-structure for the top three strata of minimizing clusters and size-minimizing currents, and a Lipschitz structure on the $(n-3)$-stratum.

A subject of recent interest in inverse problems is whether a corner must diffract fixed frequency waves. We study the related question of which cones $[0,\infty) \times Y$ do not diffract high frequency waves. We prove that if $Y$ is analytic and does not diffract waves at high frequency then every geodesic on $Y$ is closed with period $2\pi$. Moreover, we show that if $\operatorname{dim} Y=2$, then $Y$ is isometric to either the sphere of radius $1$ or its $\mathbb{Z}^2$ quotient, $\mathbb{RP}^2$.

We prove that the conclusion of Selberg’s Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds.

In this paper, we improve a result by Chodosh and Ketover [2]. We prove that, in an asymptotically flat $3$-manifold $M$ that contains no closed minimal surfaces, fixing $q \in M$ and $V$ a $2$-plane in $T_q M$ there is a properly embedded minimal plane $\Sigma$ in $M$ such that $q \in \Sigma$ and $T_q \Sigma = V$. We also prove that fixing three points in $M$ there is a properly embedded minimal plane passing through these three points.

We study the asymptotics of the number $N(t)$ of geometrically distinct closed geodesics of length $\leq t$ of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups, and apply the results to the prime decomposition of a three-manifold. In particular we show that the function $N(t)$ grows at least like the prime numbers on a compact $3$-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact $3$-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simplyconnected manifold which is not homeomorphic to a sphere.