Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity.

We characterize the LC property for d-spheres (“the sphere minus a facet collapses to a (d−2)-complex”) and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are:

– Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.)

There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.)

– All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.)

– Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)

We present the third in the series of papers describing Poisson properties of planar directed networks in the disk or in the annulus. In this paper we concentrate on special networks N_u,v in the disk that correspond to the choice of a pair (u, v) of Coxeter elements in the symmetric group S_n and the corresponding networks $N_{u,v}^\circ$ in the annulus. Boundary measurements for N_u,v represent elements of the Coxeter double Bruhat cell G^u,v⊂GL_n. The Cartan subgroup H acts on G^u,v by conjugation. The standard Poisson structure on the space of weights of N_u,v induces a Poisson structure on G^u,v, and hence on the quotient G^u,v/H, which makes the latter into the phase space for an appropriate Coxeter–Toda lattice. The boundary measurement for $N_{u,v}^\circ$ is a rational function that coincides up to a non-zero factor with the Weyl function for the boundary measurement for N_u,v. The corresponding Poisson bracket on the space of weights of $N_{u,v}^\circ$ induces a Poisson bracket on the certain space $ {\mathcal{R}_n} $ of rational functions, which appeared previously in the context of Toda flows.

Following the ideas developed in our previous papers, we introduce a cluster algebra $ \mathcal{A} $ on $ {\mathcal{R}_n} $ compatible with the obtained Poisson bracket. Generalized Bäcklund–Darboux transformations map solutions of one Coxeter–Toda lattice to solutions of another preserving the corresponding Weyl function. Using network representation, we construct generalized Bäcklund–Darboux transformations as appropriate sequences of cluster transformations in $ \mathcal{A} $.

We prove that there are no stable intersections of regular Cantor sets in the C¹ topology: given any pair (K, K′) of regular Cantor sets, we can find, arbitrarily close to it in the C¹ topology, pairs $ \left( {\tilde{K},\tilde{K}'} \right) $ of regular Cantor sets with $ \tilde{K} \cap \tilde{K}' = \emptyset $.

Moreover, for generic pairs (K, K′) of C¹-regular Cantor sets, the arithmetic difference $ K - K' = \left\{ {x - y:x \in K,y \in K'} \right\} = \left\{ {t \in \mathbb{R}:K \cap \left( {K' + t} \right) \ne \emptyset } \right\} $ has empty interior (and so is a Cantor set). This is very different from the situation in the C² topology: according to a theorem by Moreira and Yoccoz, typical pairs (K, K′) of C²-regular Cantor sets whose sum of Hausdorff dimensions is larger than 1 are such that their arithmetic difference K, K′ is the closure of its interior.

We also show that there is a generic set $ \mathcal{R} $ of C¹ diffeomorphisms of M such that, for every $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\psi }} \in \mathcal{R} $, there are no tangencies between leaves of the stable and unstable foliations of Λ₁, Λ₂, for any horseshoes Λ₁, Λ₂ of $ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\psi }} $.

Let $ \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) $ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant c > 0 with the property that, for arbitrarily large values of X, the inequalities$ \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} $admit no non-zero solution $ \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} $. Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each X ≥ 1. Such extremal numbers are transcendental and their set is stable under the action of $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ on $ \mathbb{R}\backslash \mathbb{Q} $ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf_q→∞q||qξ|| is $ \frac{1}{3} $, the largest possible value for a non-quadratic number, and that there is a natural bijection between the $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $-equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.

We consider the Stokes conjecture concerning the shape of extreme 2-dimensional water waves. By new geometric methods including a non-linear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.