It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of $\mathrm{GL}_2$-type over $\mathbb{Q}$ by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.

Analogously to extraspecial $p$-groups, we define the class of small Frattini $p$-loops and investigate some properties of small Frattini Bol 2-loops. We show that they are related to certain invariant transversals in extraspecial 2-groups. This fact enables us to describe the automorphisms and isomorphisms of small Frattini Bol 2-loops by linear algebraic tools. We use this description to obtain a lower bound for the number of non-isomorphic small Frattini Bol 2-loops of given order.

In this paper we study periodic functions of one and two variables that are invariant under a subgroup of the Euclidean group. Starting with a function $f$ defined on the plane we obtain a function of one variable by two methods: (1) we project the values on a strip into its edge, by integrating $f$ along the width; and (2) we restrict $f$ to a line. If the functions had been obtained by solving a partial differential equation equivariant under the Euclidean group, how do the symmetries of the function of one variable compare to those of solutions of equations formulated directly in one dimension?

Some of the symmetries of projected and of restricted functions may be obtained knowing only the symmetries of the original functions. Extra symmetries arise at special widths of the strip and at some special positions of the line used for restriction. We obtain a general description of the two types of symmetries and discuss how they arise in the wallpaper groups (crystallographic groups of the plane). We show that the projections and restrictions of solutions of differential equations in the plane may have symmetry groups larger than those of solutions of problems formulated in one dimension.

We give a constructive proof of a Theorem of Izmestiev and Joswig. Namely, given $(L,\omega)$ where $L$ is a link in $S^{3}$ and $\omega$ a simple (not necessarily transitive) representation of $\pi_{1}(S^{3}\backslash L)$ onto the symmetric group $\Sigma_{4}$ of four elements $\{1,2,3,4\}$ we construct a triangulation of $S^{3}$ giving rise to $(L,\omega)$ in a natural way.

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\mathbb{Z}_n$ covering spaces, $M_n$, in which $\beta_1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop, by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and Katz).

We show that the Nielsen–Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum $SU(n)$ representations, for any fixed $n\geq 2$. In the Pseudo–Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) $SU(2)$-TQFT representation matrices. It follows that at big enough levels, Pseudo–Anosov mapping classes are represented by matrices of infinite order.

On a smooth Banach manifold $M$, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle $T^{2}M$ of $M$. This is a vector bundle in the presence of a linear connection $\nabla$ on $M$ and the corresponding local structure is heavily dependent on the choice of $\nabla$. In this paper we study the extent of this dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on $T^{2}M$ remains invariant under conjugate connections with respect to diffeomorphisms of $M$.

We consider Exel's new construction of a crossed product of a $C^*$-algebra $A$ by an endomorphism $\alpha$. We prove that this crossed product is universal for an appropriate family of covariant representations, and we show that it can be realised as a relative Cuntz–Pimsner algbera. We describe a necessary and sufficient condition for the canonical map from $A$ into the crossed product to be injective, and present several examples to demonstrate the scope of this result. We also prove a gauge-invariant uniqueness theorem for the crossed product.

Suppose that $P(D)$ is a linear differential operator with complex coefficients and $\{f_k\}_{k \in \mathbb{Z}}$ is a two-sided sequence of complex-valued functions on $\mathbb{R}$ such that $f_{k+1} \,{=}\, P(D)f_k$ with the following growth condition: there exist $a\,{\in}\,[0,1)$ and $N\,{\geq}\,0$ such that $|f_k(x)|\,{\leq}\,M_{|k|}\exp (N|x|^a)$, where the sequence $\{M_k\}_{k=0}^{\infty}$ satisfies that for any $\varepsilon\,{>}\,0$, the sequence $\{{M_k}/{(1+\varepsilon)^{k}}\}_{k=0}^{\infty}$ has a bounded subsequence. Then $f_0$ is an entire function of an exponential growth. This result is a generalization of those of Roe, Burkill and Howard.

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.

Global existence results in the past time direction of cosmological models with collisionless matter and a massless scalar field are presented. It is shown that the singularity is crushing and that the Kretschmann scalar diverges uniformly as the singularity is approached. In the case without Vlasov matter, the singularity is velocity dominated and the generalized Kasner exponents converge at each spatial point as the singularity is approached.