We study the Walsh model of a certain maximal truncation of Carleson's operator related to the Return Times Theorem.

A subgroup $H$ of a group $G$ is said to be \textit{permutable in $G$}, if $HK = KH$ for every subgroup $K$ of $G$. A result due to Stonehewer asserts that every permutable subgroup is ascendant although the converse is false. In this paper we study some infinite groups whose ascendant subgroups are permutable ($AP$--groups). We show that the structure of radical hyperfinite $AP$--groups behave as that of finite soluble groups in which the relation \textit{to be a permutable subgroup} is transitive ($PT$--groups).

Given a holomorphic self-map $\varphi$ of the ball of $\mathbb{C}^N$, we study whether there exists a map $\sigma$ and a linear fractional transformation $A$ such that $\sigma\circ\varphi=A\circ\sigma$. This is an important result when $N=1$ with a great number of applications. We extend this result to the multi-dimensional setting for a large class of maps. Applications to commuting holomorphic self-maps are given.

For functions on a metric measure space, we introduce a notion of ``gradient at a given scale''. This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev inequality at a large enough scale is invariant under large-scale equivalence, a metric-measure version of coarse equivalence. We prove that for a Riemmanian manifold satisfying a local Poincaré inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the $L^p$-isoperimetric profile, for every $1\leq p\leq \infty$ is invariant under quasi-isometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space $X$, providing a natural point of view to understand this phenomenon.

A Klein surface with boundary of algebraic genus $\mathfrak{p}\geq 2$, has at most $12(\mathfrak{p}-1)$ automorphisms. The groups attaining this upper bound are called $M^{\ast}$-groups, and the corresponding surfaces are said to have maximal symmetry. The $M^{\ast}$-groups are characterized by a partial presentation by generators and relators. The alternating groups $A_{n}$ were proved to be $M^{\ast}$-groups when $n\geq 168$ by M. Conder. In this work we prove that $A_{n}$ is an $M^{\ast }$-group if and only if $n\geq 13$ or $n=5,10$. In addition, we describe topologically the surfaces with maximal symmetry having $A_{n}$ as automorphism group, in terms of the partial presentation of the group. As an application we determine explicitly all such surfaces for $n\leq 14$. Each finite group $G$ acts as an automorphism group of several Klein surfaces. The minimal genus of these surfaces is called the real genus of the group, $\rho(G)$. If $G$ is an $M^{\ast}$-group then $\rho(G)=\frac{o(G)}{12}+1$. We end our work by calculating the real genus of the alternating groups which are not $M^{\ast}$-groups.

We study in this paper orthogonal projections in a hyperbolic space to hyperhorospheres and hyperplanes. We deal in more details with the case of embedded surfaces $M$ in $H^3_+(-1)$. We study the generic singularities of the projections of $M$ to horospheres and planes. We give geometric characterizations of these singularities and prove duality results concerning the bifurcation sets of the families of projections. We also prove Koenderink type theorems that give the curvature of the surface in terms of the curvatures of the profile and the normal section of the surface.

A compact Riemann surface of genus $g$ is called an M-surface if it admits an anti-conformal involution that fixes $g+1$ simple closed curves, the maximum number by Harnack's Theorem. Underlying every map on an orientable surface there is a Riemann surface and so the conclusions of Harnack's theorem still apply. Here we show that for each genus $g ϯ 1$ there is a unique M-surface of genus $g$ that underlies a regular map, and we prove a similar result for Riemann surfaces admitting anti-conformal involutions that fix $g$ curves.

We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is shown that, for two generic preimages of the curves to an algebraic framework, their intersection projects exactly onto the stable intersection of the curves. It is also given sufficient conditions for such a generality in terms of the residual coefficients of the algebraic coefficients of defining equations of the curves.

We say that a space $X$ admits a \emph{homology exponent} if there exists an exponent for the torsion subgroup of $H^*(X;\mathbb Z)$. Our main result states that if an $H$-space of finite type admits a homology exponent, then either it is, up to $2$-completion, a product of spaces of the form $B\mathbb Z/2^r$, $S^1$, $\mathbb C P^\infty$, and $K(\mathbb Z,3)$, or it has infinitely many non-trivial homotopy groups and $k$-invariants. Relying on recent advances in the theory of $H$-spaces, we then show that simply connected $H$-spaces whose mod $2$ cohomology is finitely generated as an algebra over the Steenrod algebra do not have homology exponents, except products of mod $2$ finite $H$-spaces with copies of $\mathbb C P^\infty$ and $K(\mathbb Z,3)$.

We construct a family of polynomial maps $\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that their images are open semialgebraic sets whose topological exteriors have arbitrarily many connected components, which are parametric semilines.

We study the existence of everywhere differentiable functions which are almost everywhere solutions of quite general Hamilton-Jacobi equations on open subsets of $\mathbb R^d$ or on $d$-dimensional manifolds whenever $d\geq 2$. In particular, when $M$ is a Riemannian manifold, we prove the existence of a differentiable function $u$ on $M$ which satisfies the Eikonal equation $\Vert \nabla u(x) \Vert_{x}=1$ almost everywhere on $M$.

We prove a Harnack inequality for the positive solutions of ultraparabolic equations of the type $$ \mathcal {L}_0 u + \mathcal {V} u = 0, $$ where $\mathcal {L}_0$ is a linear second order hypoelliptic operator and $\mathcal {V}$ belongs to a class of functions of Stummel-Kato type. We also obtain the existence of a Green function and an uniqueness result for the Cauchy-Dirichlet problem.

We study double Hilbert transforms and maximal functions along surfaces of the form $(t_1,t_2,\gamma_1(t_1)\gamma_2(t_2))$. The $L^p(\mathbb{R}^3)$ boundedness of the maximal operator is obtained if each $\gamma_i$ is a convex increasing and $\gamma_i(0)=0$. The double Hilbert transform is bounded in $L^p(\mathbb{R}^3)$ if both $\gamma_i$'s above are extended as even functions. If $\gamma_1$ is odd, then we need an additional comparability condition on $\gamma_2$. This result is extended to higher dimensions and the general hyper-surfaces of the form $(t_1,\dots,t_{n},\Gamma(t_1,\dots,t_{n}))$ on $\mathbb{R}^{n+1}$.

In this article we prove that for any orthonormal system $(\varphi_j)_{j=1}^n \subset L_2$ that is bounded in $L_{\infty}$, and any $1 < k < n$, there exists a subset $I$ of cardinality greater than $n-k$ such that on $\mathrm{span}\{\varphi_i\}_{i \in I}$, the $L_1$ norm and the $L_2$ norm are equivalent up to a factor $\mu (\log \mu)^{5/2}$, where $\mu = \sqrt{n/k} \sqrt{\log k}$. The proof is based on a new estimate of the supremum of an empirical process on the unit ball of a Banach space with a good modulus of convexity, via the use of majorizing measures.