We consider equidistribution of angles for certain hyperbolic lattice points in the upper half-plane. Extending work of Friedlander and Iwaniec, we show that for the full modular group equidistribution persists for matrices with a^{2}+b^{2}+c^{2}+d^{2}=p with p prime; at least if we assume sufficiently good lower bounds in the hyperbolic prime number theorem by Friedlander and Iwaniec. We also investigate related questions for a specific arithmetic co-compact group and its double cosets by hyperbolic subgroups. The general equidistribution problem was studied by Good, and in this case, we show, that equidistribution holds unconditionally when restricting to primes.

We describe the dynamics of a group \Gamma generated by Dehn twists along two filling multi-curves or a family of filling curves on the \mathsf{SU}(2) -representation variety of closed surfaces. Consequently, we provide explicit \Gamma -invariant rational functions on the representation variety of the genus two closed surface S_{2} for some pair of multi-curves. We establish a similar result for the \mathsf{SU}(2) -character variety of genus four non-orientable surfaces N_{4} for some family of filling curves.

We show that the family of systoles of hyperbolic surfaces associated with congruence lattices in \operatorname{SL}_{2}(\mathbb{Z}) have asymptotically minimal crossing number.

We show that the problem of deciding whether a closed three-manifold admits an elliptic structure lies in NP. Furthermore, determining the homeomorphism type of an elliptic manifold lies in the complexity class FNP. These are both consequences of the following result. Suppose that M is a lens space which is neither \mathbb{RP}^{3} nor a prism manifold. Suppose that \mathcal{T} is a triangulation of M . Then there is a loop, in the one-skeleton of the 86 th iterated barycentric subdivision of \mathcal{T} , whose simplicial neighbourhood is a Heegaard solid torus for M .

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations \rho \colon \Gamma \to \mathrm{SO}(p,q+1) of closed p -manifold groups. In particular: We provide a uniform lower bound of the product entropy times volume that depends only on the geometry of the abstract group \Gamma . We prove that the entropy is bounded from above by p-1 with equality if and only if \rho is conjugate to a representation inside \mathrm{S}(\mathrm{O}(p,1)\times\mathrm{O}(q)) , which answers affirmatively to a question of Glorieux and Monclair. Lastly, we prove finiteness and compactness results for groups admitting convex cocompact representations with bounded diameter.

Morita [Osaka J. Math. 21 (1984), 545–563] showed that for each integer k \geq 1 , there are examples of flat \mathbb{S}^{1} -bundles for which the k -th power of the Euler class does not vanish. Haefliger [Enseign. Math. (2) 24 (1978), 154] asked if the same holds for flat odd-dimensional sphere bundles. In this paper, for a manifold M with a free torus action, we prove that certain M -bundles are cobordant to a flat M -bundle and as a consequence, we answer Haefliger’s question. We show that all monomials in the Euler class and Pontryagin classes p_{i} for i\leq n-1 are non-trivial in H^{*}(\operatorname{BDiff}^{\delta}_{+}(\mathbb{S}^{2n-1});\mathbb{Q}) .

We consider an area-minimizing integral current T of codimension higher than 1 in a smooth Riemannian manifold \Sigma . In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as the ‘singularity degree’. In this paper, we show that the set of points for which the singularity degree is strictly larger than 1 is (m-2) -rectifiable. In a subsequent work, we prove that the remaining flat singular points form a \mathcal{H}^{m-2} -null set, thus concluding that the singular set of T is (m-2) -rectifiable.

Wandering Fatou components were recently constructed by Astorg et al. (2016) for higher-dimensional holomorphic maps on projective spaces. Their examples are polynomial skew products with a parabolic invariant line. In this paper we study this wandering domain problem for polynomial skew product f with an attracting invariant line L (which is the more common case). We show that if f is unicritical (in the sense that the critical curve has a unique transversal intersection with L ), then every Fatou component of f in the basin of L is an extension of a one-dimensional Fatou component of f|_{L} . As a corollary there is no wandering Fatou component. We will also discuss the multicritical case under additional assumptions.

In a setting of a complex manifold with a positive line bundle and a submanifold, we consider the optimal Ohsawa–Takegoshi extension operator, sending a holomorphic section of the line bundle on the submanifold to the holomorphic extension of it on the ambient manifold with the minimal L^{2} -norm. We show that for a tower of submanifolds and large tensor powers of the line bundle, the extension operators act transitively modulo some small defect, which is a Toeplitz type operator. We calculate the first significant term in the asymptotic expansion of this “transitivity defect”. As a byproduct, we deduce composition rules for Toeplitz type operators, the extension and restriction operators and calculate the second term in the asymptotic expansion of the optimal constant in the semi-classical version of the extension theorem.

Let H_{D}(T) denote the Hilbert class polynomial of the imaginary quadratic order of discriminant D . We study the rate of growth of the greatest common divisor of H_{D}(a) and H_{D}(b) as |D| \to \infty for a and b belonging to various Dedekind domains. We also study the modular support problem: if for all but finitely many D every prime ideal dividing H_{D}(a) also divides H_{D}(b) , what can we say about a and b ? If we replace H_{D}(T) by T^{n}-1 and the Dedekind domain is a ring of S -integers in some number field, then these are classical questions that have been investigated by Bugeaud–Corvaja–Zannier, Corvaja–Zannier, and Corrales-Rodrigáñez–Schoof.