The Effect of Gender in the Publication Patterns in Mathematics

Despite the increasing number of women graduating in mathematics, a systemic gender imbalance persists and is signified by a pronounced gender gap in the distribution of active researchers and professors. Especially at the level of university faculty, women mathematicians continue being drastically underrepresented, decades after the first affirmative action measures have been put into place. A solid publication record is of paramount importance for securing permanent positions. Thus, the question arises whether the publication patterns of men and women mathematicians differ in a significant way. Making use of the zbMATH database, one of the most comprehensive metadata sources on mathematical publications, we analyze the scholarly output of ~150,000 mathematicians from the past four decades whose gender we algorithmically inferred. We focus on development over time, collaboration through coautorships, presumed journal quality and distribution of research topics -- factors known to have a strong impact on job perspectives. We report significant differences between genders which may put women at a disadvantage when pursuing an academic career in mathematics.Comment: 24 pages, 12 figure

Extended objects in quantum field theory in three dimensions and applications

In this thesis the systematic study of Quantum Field Theories (QFT) in various dimensions is proposed from the point of view of mathematical and theoretical physics, paying special attention to systems of one and three spatial dimensions (in addition to the temporal dimension in both cases) under the influence of some particular external conditions. These conditions vary from local interactions with other external classical fields to ideal boundary conditions in confining geometries. More specifically, the main objective of this work is the study of the spectrum of quantum fluctuations of the fields in the vacuum state subject to the external conditions indicated. This study will be applied to the calculation of several relevant parameters in three-dimensional and one-dimensional extended structures. These systems have recently received increasing interest in material physics (in micro-electromechanical devices based on the Casimir effect or topological defects in metamaterials and nanotubes) and in fundamental physics (quantum effects in modern cosmology and topological defects such as domain walls, monopoles and skyrmions). Different configurations of quantum fields both in compact domains and in open ones with boundaries will be studied: -A scalar field confined between plates mimicked by the most general type of lossless and frequently independent boundary conditions. -Scalar fields propagating at finite temperature under the influence of generalised Dirac delta lattices and Pöschl-Teller combs. -Scalar fields between two parallel plates mimicked by Dirac delta potentials in a curved background of a topological Pöschl-Teller kink. -Relativistic fermionic particles propagating in the real space under the influence of either a single and a double Dirac delta potential. Only effective theories will be considered. Here effective means that the microscopic degrees of freedom relative to the atoms and quarks of the matter composing the plates or objects between which the vacuum quantum interaction energy will be studied are not going to be taken into account. The methodology developed for the project is the following. Firstly, the spectrum of the non-relativistic Schrödinger operator or the relativistic Dirac one that will give rise to the set of one-particle states of the corresponding QFT will be characterised. Secondly, analytical and numerical results of the vacuum interaction energy between extended objects at zero temperature will be obtained. Finally, the study will be generalised to other thermodynamic magnitudes of interest such as the one loop quantum corrections to the Helmholtz free energy, the entropy and the Casimir force between objects at finite non zero temperature. Furthermore, graphical representations obtained numerically with the software Mathematica will be added. The thesis is structured in such a way that Chapter 1 gives an introduction to the work as a whole and the following chapters present the concrete results of each of the systems listed above. Finally, Chapter 6 summarises the main conclusions to give an overall view of the work carried out.El objetivo de esta tesis es el estudio, bajo el punto de vista de la física matemática, de teorías cuánticas de campos (TCC) en una y en tres dimensiones espaciales (aparte de la temporal) bajo la influencia de diversas condiciones externas. Estas condiciones comprenden tanto la interacción con otros campos clásicos externos así como condiciones de borde en geometrías confinantes. En particular, el principal interés de este trabajo es el estudio del espectro de las fluctuaciones cuánticas de los campos en el estado de vacío sujeto a las condiciones externas anteriormente indicadas. Este estudio permitirá obtener parámetros relevantes en algunas estructuras extensas en una y tres dimensiones. Este tipo de sistemas han suscitado recientemente un gran interés en la física de materiales (por ejemplo en dispositivos microelectromecánicos basados en el efecto Casimir, nanotubos y defectos topológicos en metamateriales) y en física fundamental (defectos topológicos como paredes de dominio, cuerdas cósmicas, monopolos y skyrmiones). A lo largo de la tesis se van a estudiar las diferentes configuraciones de campos cuánticos, tanto en dominios compactos como en dominios abiertos con bordes, que se enumeran a continuación: -Campos escalares confinados entre placas representadas por las condiciones de contorno independientes de las frecuencias más generales posibles. -Campos escalares que se propagan a temperatura finita bajo la influencia de redes de tipo delta de Dirac generalizadas y peines formados con potenciales Pöschl-Teller. -Campos escalares en un sistema formado por dos placas paralelas modelizadas por potenciales delta de Dirac introducidas en un potencial de fondo curvo dado por un kink topológico de tipo sine-Gordon. -Partículas fermiónicas relativistas que se propagan en el espacio real bajo la influencia de tanto uno como varios potenciales delta de Dirac. Es importante destacar que en esta tesis se van a considerar únicamente teorías ”efectivas”, en el sentido de que no se van a tener en cuenta los grados de libertad microscópicos relativos a los átomos y los quarks de la materia que compone las placas o objetos entre los cuales se va a estudiar la energía de interacción cuántica de vacío. La metodología general empleada para la obtención de estos objetivos es la siguiente: primero se caracterizará el espectro del operador de Schrödinger no relativista que dará lugar al conjunto de estados de una partícula de la teoría cuántica de campos correspondiente, después se obtendrán fórmulas analíticas para el cálculo de la energía de vacío de interacción entre los objetos extensos considerados a temperatura cero y finalmente se generalizará el estudio a otras magnitudes termodinámicas de interés como las correcciones cuánticas a un lazo de la energía libre de Helmholtz, la entropía y la fuerza de Casimir entre los objetos a temperatura finita no nula. Se obtendrán resultados analíticos cuando sea posible y además, todo ello irá acompañado de representaciones gráficas obtenidas numéricamente con ayuda del software Mathematica. La tesis está organizada de forma que el primer capítulo es una introducción bibliográfica al trabajo en su conjunto y los siguientes capítulos presentan los resultados concretos de cada uno de los sistemas anteriormente enumerados. Finalmente, el capítulo 6 resume las principales conclusiones de la tesis para dar una visión global del trabajo realizado.Escuela de DoctoradoDoctorado en Físic

Casimir effect and TGTG-formula for curved backgrounds

The quantum vacuum interaction energy between a pair of semitransparent two-dimensional plates in the topological background of a sine-Gordon kink is studied. Quantum vacuum oscillations around the sine-Gordon kink solution can be interpreted as a quantum scalar field theory in the spacetime of a domain wall. An extension of the TGTG-formula, firstly discovered by O. Kenneth and I. Klich, to weak curved backgrounds is obtained.Comment: 5 pages, 1 figur

One-dimensional scattering of fermions in double Dirac delta potentials

The spectrum of bound and scattering states of the one dimensional Dirac Hamiltonian describing fermions distorted by a static background built from two Dirac delta potentials is studied. A distinction will be made between mass-spike and electrostatic Dirac delta-potentials. The second quantisation is then performed to promote the relativistic quantum mechanical problem to a relativistic quantum field theory and study the quantum vacuum interaction energy for fermions confined between opaque plates. The work presented here is a continuation of [Guilarte et al 2019 Front. Phys.7 109].Comment: 30 pages, 29 figure

Casimir energy through transfer operators for weak curved backgrounds

The Casimir energy between a pair of two-dimensional plates represented by Dirac delta potentials and embedded in the topological background of a sine-Gordon kink is studied in [L. Santamaria-Sanz, letter (2023)] through an extension of the TGTG-formula, firstly discovered by O. Kenneth and I. Klich, to weak curved backgrounds. More details of the calculations are provided here, not only regarding the spectrum of the corresponding associated non-relativistic quantum mechanical problem but also concerning the Green's function and the transfer operators of the corresponding Quantum Field Theory. These details allow a better understanding of the issue. In fact, a more general potential consisting of a Dirac delta as well as its first derivative would be used to represent each plate. Moreover, the relation between the phase shift and the density of states (the well-known Dashen-Hasslacher-Neveu formula) is also exploited to characterize the quantum vacuum energy.Comment: 15 pages, 3 figure

Revisiting the Casimir Energy with General Boundary Conditions, and applications in 1D Crystals

Producción CientíficaWe obtain new expressions for the Casimir energy between plates that are mimicked by the most general possible boundary conditions allowed by the principles of quantum field theory. This result enables to provide the quantum vacuum energy for scalar fields propagating under the influence of a one-dimensional crystal represented by a periodic potential formed by an infinite array of identical potentials with compact support.MINECO (MTM2014-57129-C2-1-P) and Junta de Castilla y Leon (BU229P18 and VA137G18)

One-Dimensional Scattering of Fermions on δ-Impurities

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Que los problemas no sean un problema

Las clases de matemáticas no han de destinarse a resolver algoritmos (en la mayoría de las ocasiones, rígidos, mecanizados e impuestos) durante la mayor parte del tiempo de forma aislada. Las operaciones han de estar contextualizadas, ya que la resolución de problemas ha de ser nuestro eje vertebrador en nuestras aulas. Debemos trabajar todo tipos de problemas. El material manipulativo es fundamental para entenderlos y modelizarlos. Se debe aumentar el trabajo con cantidades (que aportan un elemento cualitativo y significatividad) frente al trabajo de cifras (números) “vacías” que no ofrecen un trabajo competencial y de aplicación a corto plazo. A modo de ejemplo podríamos decir que 19:5 en una calculadora (o en papel del alumno) son 3,80 ( la calculadora es fundamental, pero calcula, no razona). Sin embargo ¿y si son 19 pelotas a dividir en 5 redes? ¿sigue siendo la respuesta 3,80? ¿importa contextualizar la división? ¿cuesta mucho añadir 19 pelotas: 5 redes o 19 € :5 personas? ¿Son conscientes los alumnos que dicha operación tiene 2 soluciones