Abstract:
Gastric cancer is a disease with high incidence and mortality in our population. The prognosis of patients with this disease is closely related to the neoplasm stage at diagnosis, including the following characteristics of the tumor: extension into the gastric wall thickness, spread to locoregional lymph nodes and the ability to generate distant metastases, as described by the TNM classification. For localized tumors characterized only by invasion of mucosa or submucosa at diagnosis, survival at 5 years is between 70 and 95% with exclusive surgical management; however, when extension into the gastric wall is higher and/or there is locoregional nodal involvement, survival decreases to 20-30% at 5 years. Currently, at high-volume centers, the extent of gastrectomy is individualized based on several parameters, which in an increasing number of cases allows a total gastrectomy with D2 lymphadenectomy and preservation of the spleen and pancreas. This improved procedure increases the chance of R0 surgery and improves the relationship between resected and affected lymph nodes, resulting in a decreased risk of the long-term locoregional recurrence. To improve these results, different therapeutic strategies combining chemotherapy or chemoradiotherapy with surgery have been tested. Previously, the Intergroup 0116 clinical trial, published in 2001, which changed clinical practice in the United States, showed that adjuvant chemoradiotherapy improved survival (from 26 to 37 months overall survival) of these patients. In Europe, perioperative chemotherapy has been considered the standard treatment, since the publication of two randomized phase III trials showed an increase at 5 years survival in the group treated with chemotherapy. El cáncer gástrico es un tumor de alta incidencia y mortalidad en nuestro medio, y su pronóstico está íntimamente relacionado con la situación neoplásica al diagnóstico, que incluye su extensión en el grosor de la pared gástrica, sobre los ganglios linfáticos locorregionales y su capacidad de generar metástasis a distancia, extensión basada en la clasificación TNM. En aquellos tumores localizados al diagnóstico, caracterizados por la invasión únicamente de mucosa-submucosa, la supervivencia a 5 a os se establece entre el 70 y el 95% con manejo quirúrgico exclusivo, sin embargo, cuando la extensión en la pared es mayor y/o existe afectación ganglionar locorregional, la supervivencia disminuye al 20-30% a 5 a os. Actualmente en centros con alto volumen de pacientes, la extensión de la gastrectomía se individualiza en función de varios parámet

Abstract:
gastric cancer is a disease with high incidence and mortality in our population. the prognosis of patients with this disease is closely related to the neoplasm stage at diagnosis, including the following characteristics of the tumor: extension into the gastric wall thickness, spread to locoregional lymph nodes and the ability to generate distant metastases, as described by the tnm classification. for localized tumors characterized only by invasion of mucosa or submucosa at diagnosis, survival at 5 years is between 70 and 95% with exclusive surgical management; however, when extension into the gastric wall is higher and/or there is locoregional nodal involvement, survival decreases to 20-30% at 5 years. currently, at high-volume centers, the extent of gastrectomy is individualized based on several parameters, which in an increasing number of cases allows a total gastrectomy with d2 lymphadenectomy and preservation of the spleen and pancreas. this improved procedure increases the chance of r0 surgery and improves the relationship between resected and affected lymph nodes, resulting in a decreased risk of the long-term locoregional recurrence. to improve these results, different therapeutic strategies combining chemotherapy or chemoradiotherapy with surgery have been tested. previously, the intergroup 0116 clinical trial, published in 2001, which changed clinical practice in the united states, showed that adjuvant chemoradiotherapy improved survival (from 26 to 37 months overall survival) of these patients. in europe, perioperative chemotherapy has been considered the standard treatment, since the publication of two randomized phase iii trials showed an increase at 5 years survival in the group treated with chemotherapy.

Abstract:
A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.

Abstract:
We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about the zeta function. But it gives a new perspective to many known results. Also it may prove useful to show to students of Complex Variables or Analytic Number Theory. (This was my initial motivation to make the drawings).

Abstract:
Let us consider an abelian variety defined over $\mathbb{Q_{\ell}}$ with good supersingular reduction. In this paper we give explicit conditions that ensure that the action of the wild inertia group on the $\ell$-torsion points of the variety is trivial. Furthermore we give a family of curves of genus 2 such that their Jacobian surfaces have good supersingular reduction and satisfy these conditions. We address this question by means of a detailed study of the formal group law attached to abelian varieties.

Abstract:
We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical technique (maximal slope principle) for proving inequalities between elementary functions. The second proof is fully analytical. Finally we prove a general rearrangement theorem and apply it to the remaining case 0 < r < 1. Some of these inequalities are needed for obtaining general sharp bounds for the errors committed when applying the Riemann-Siegel expansion of Riemann's zeta function.

Abstract:
In this paper we prove the Geyer-Jarden conjecture on the torsion part of the Mordell-Weil group for a large class of abelian varieties defined over finitely generated fields of arbitrary characteristic. The class consists of all abelian varieties with big monodromy, i.e., such that the image of Galois representation on l-torsion points, for almost all primes l, contains the full symplectic group.

Abstract:
In this paper we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic $\ell>3$ as the Galois group of a tamely ramified Galois extension of $\mathbb{Q}$. The strategy is to consider the Galois representation $\rho_{\ell}$ attached to the Tate module at $\ell$ of a suitable abelian surface. We need to choose the abelian varieties carefully in order to ensure that the image of $\rho_{\ell}$ is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such that the Galois representation attached to the $\ell$-torsion points of their Jacobian varieties provide tame Galois realizations of the desired symplectic groups.

Abstract:
This article is the second part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part is concerned with symplectic Galois representations having a huge residual image, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. A key ingredient is a classification of symplectic representations whose image contains a nontrivial transvection: these fall into three very simply describable classes, the reducible ones, the induced ones and those with huge image. Using the idea of an (n,p)-group of Khare, Larsen and Savin we give simple conditions under which a symplectic Galois representation with coefficients in a finite field has a huge image. Finally, we combine this classification result with the main result of the first part to obtain a strenghtened application to the inverse Galois problem.

Abstract:
This article is the first part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. In this first part, we determine the smallest field over which the projectivisation of a given symplectic group representation satisfying some natural conditions can be defined. The answer only depends on inner twists. We apply this to the residual representations of a compatible system of symplectic Galois representations satisfying some mild hypothesis and obtain precise information on their projective images for almost all members of the system, under the assumption of huge residual images, by which we mean that a symplectic group of full dimension over the prime field is contained up to conjugation. Finally, we obtain an application to the inverse Galois problem.