In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n (n \geq 2)$ is a level set of a noncritical holomorphic function on $\mathbb{B}_n$ all of whose level sets are complete. This shows that $\mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $\mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required.

Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $\mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $q \in \lbrace 1, \dotsc , n-1 \rbrace$.

We analyse cusp excursions of random geodesics for Weil–Petersson type incomplete metrics on orientable surfaces of finite type: in particular, we give bounds for maximal excursions.

We also give similar bounds for cusp excursions of random Weil–Petersson geodesics on non-exceptional moduli spaces of Riemann surfaces conditional on the assumption that the Weil–Petersson flow is polynomially mixing.

Moreover, we explain how our methods can be adapted to understand almost greasing collisions of typical trajectories in certain slowly mixing billiards.

Uniform measures have played a fundamental role in geometric measure theory since they naturally appear as tangent objects. For instance, they were essential in the groundbreaking work of Preiss on the rectifiability of Radon measures. However, relatively little is understood about the structure of general uniform measures. Indeed, the question of whether there exist any non-flat uniform measures beside the one supported on the Kowalski–Preiss cone has been open for 30 years, ever since Kowalski and Preiss classified $n$‑uniform measures in $\mathbb{R}^{n+1}$.

In this paper, we answer the question and construct an infinite family of $3$‑uniform measures in arbitrary codimension. We define a notion of distance symmetry for points and prove that every collection of $2$‑spheres whose centers are distance symmetric gives rise to a $3$‑uniform measure. We then develop a combinatorial method to systematically produce distance symmetric points. We also classify conical $3$‑uniform measures in $\mathbb{R}^5$ by proving that they all arise from distance symmetric spheres.

We study knots in $S^3$ with infinitely many $SU(2)$-cyclic surgeries, which are Dehn surgeries such that every representation of the resulting fundamental group into $SU(2)$ has cyclic image. We show that for every such nontrivial knot $K$, its set of $SU(2)$-cyclic slopes is bounded and has a unique limit point, which is both a rational number and a boundary slope for $K$. We also show that such knots are prime and have infinitely many instanton L‑space surgeries. Our methods include the application of holonomy perturbation techniques to instanton knot homology, using a strengthening of recent work by the second author.