We show that a Riemannian foliation on a topological $n$–sphere has leaf dimension $1$ or $3$ unless $n=15$ and the Riemannian foliation is given by the fibers of a Riemannian submersion to an $8$–dimensional sphere. This allows us to classify Riemannian foliations on round spheres up to metric congruence.

We prove the $K$– and the $L$–theoretic Farrell–Jones conjectures with coefficients in additive categories and with finite wreath products for arbitrary lattices in virtually connected Lie groups.

We find many tight codes in compact spaces, in other words, optimal codes whose optimality follows from linear programming bounds. In particular, we show the existence (and abundance) of several hitherto unknown families of simplices in quaternionic projective spaces and the octonionic projective plane. The most noteworthy cases are $15$–point simplices in $ℍ‘{\mathbb{P}}^2$ and $27$–point simplices in $\mathbb{O}‘{\mathbb{P}}^2$, both of which are the largest simplices and the smallest $2$–designs possible in their respective spaces. These codes are all universally optimal, by a theorem of Cohn and Kumar. We also show the existence of several positive-dimensional families of simplices in the Grassmannians of subspaces of ${ℝ}^n$ with $n\le 8$; close numerical approximations to these families had been found by Conway, Hardin and Sloane, but no proof of existence was known. Our existence proofs are computer-assisted, and the main tool is a variant of the Newton–Kantorovich theorem. This effective implicit function theorem shows, in favorable conditions, that every approximate solution to a set of polynomial equations has a nearby exact solution. Finally, we also exhibit a few explicit codes, including a configuration of $39$ points in $\mathbb{O}‘{\mathbb{P}}^2$ that form a maximal system of mutually unbiased bases. This is the last tight code in $\mathbb{O}‘{\mathbb{P}}^2$ whose existence had been previously conjectured but not resolved.

This paper is devoted to the classification of ${GL}^{+}\left(2,ℝ\right)$–orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of abelian differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus.

The proof uses several topological properties of Prym eigenforms. In particular, the tools and the proof are independent of the recent results of Eskin and Mirzakhani and Eskin, Mirzakhani and Mohammadi.

As an application we obtain a finiteness result for the number of closed ${GL}^{+}\left(2,ℝ\right)$–orbits (not necessarily primitive) in the Prym eigenform locus $\Omega E_D\left(2,2\right)$ for any fixed $D$ that is not a square.

The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an application, the first nontrivial lower bounds on the $C\left(K\right)$–distortion of important classes of separable Banach spaces, where $K$ is a countable compact space in the family $\left\{\left[0,\omega \right],\left[0,\omega \cdot 2\right],\dots ,\left[0,{\omega }^2\right],\dots ,\left[0,{\omega }^k\cdot n\right],\dots ,\left[0,{\omega }^{\omega }\right]\right\}$ are obtained.

Let $F$ be a homotopy functor with values in the category of spectra. We show that partially stabilized cross-effects of $F$ have an action of a certain operad. For functors from based spaces to spectra, it is the Koszul dual of the little discs operad. For functors from spectra to spectra it is a desuspension of the commutative operad. It follows that the Goodwillie derivatives of $F$ are a right module over a certain “pro-operad”. For functors from spaces to spectra, the pro-operad is a resolution of the topological Lie operad. For functors from spectra to spectra, it is a resolution of the trivial operad. We show that the Taylor tower of the functor $F$ can be reconstructed from this structure on the derivatives.

We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker stability. Under a boundedness assumption which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, we prove that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $\omega \in N^1{\left(X\right)}_{ℝ}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus flips.

Using a Toda bracket computation $\langle {\theta }_4,2,{\sigma }^2\rangle$ due to Daniel C Isaksen, we investigate the $45$–stem more thoroughly. We prove that ${\theta }_4^2=0$ using a $4$–fold Toda bracket. By work of Barratt, Jones and Mahowald, this implies that ${\theta }_5$ exists and there exists a ${\theta }_5$ such that $2{\theta }_5=0$. Based on ${\theta }_4^2=0$, we simplify significantly their $9$–cell complex construction to a $4$–cell complex, which leads to another proof that ${\theta }_5$ exists.

This paper gives sharp linear bounds on the genus of a normal surface in a triangulated compact orientable 3–manifold in terms of the quadrilaterals in its cell decomposition — different bounds arise from varying hypotheses on the surface or triangulation. Two applications of these bounds are given. First, the minimal triangulations of the product of a closed surface and the closed interval are determined. Second, an alternative approach to the realisation problem using normal surface theory is shown to be less powerful than its dual method using subcomplexes of polytopes.

We give a Thurston-like definition for laminations on higher Teichmüller spaces associated to a surface $S$ and a semi-simple group $G$ for $G={SL}_m$ or ${PGL}_m$. The case $G={SL}_2$ or ${PGL}_2$ corresponds to the classical theory of laminations on a hyperbolic surface. Our construction involves positive configurations of points in the affine building. We show that these laminations are parametrized by the tropical points of the spaces ${\mathcal{X}}_{G,S}$ and ${\mathcal{A}}_{G,S}$ of Fock and Goncharov. Finally, we explain how the space of projective laminations gives a compactification of higher Teichmüller space.

We study geometrical properties of translation surfaces: the finite blocking property, bounded blocking property, and illumination properties. These are elementary properties which can be fruitfully studied using the dynamical behavior of the $SL\left(2,ℝ\right)$–action on the moduli space of translation surfaces. We characterize surfaces with the finite blocking property and bounded blocking property, completing work of the second-named author. Concerning the illumination problem, we also extend results of Hubert, Schmoll and Troubetzkoy, removing the hypothesis that the surface in question is a lattice surface, thus settling a conjecture of theirs. Our results crucially rely on the recent breakthrough results of Eskin and Mirzakhani and of Eskin, Mirzakhani and Mohammadi, and on related results of Wright.

We use the famous knot-theoretic consequence of Freedman’s disc theorem — knots with trivial Alexander polynomial bound a locally flat disc in the $4$–ball — to prove the following generalization: the degree of the Alexander polynomial of a knot is an upper bound for twice its topological slice genus. We provide examples of knots where this determines the topological slice genus.

There are three main components to this article:

1. A formula for the $\eta$–invariant of the signature complex for any finite subgroup of $SO\left(4\right)$ acting freely on $S^3$ is given. An application of this is a nonexistence result for Ricci-flat ALE metrics on certain spaces.

2. A formula for the orbifold correction term that arises in the index of the self-dual deformation complex is proved for all finite subgroups of $SO\left(4\right)$ which act freely on $S^3$. Some applications of this formula to the realm of self-dual and scalar-flat Kähler metrics are also discussed.

3. Two infinite families of scalar-flat anti-self-dual ALE spaces with groups at infinity not contained in $U\left(2\right)$ are constructed. Using these spaces, examples of self-dual metrics on $n\#{ℂ\mathbb{P}}^2$ are obtained for $n\ge 3$. These examples admit an $S^1$–action, but are not of LeBrun type.