Krasnoselskii's theorem in product spaces of operators and applications
Authorship
L.M.F.P.
Master in Mathematics
L.M.F.P.
Master in Mathematics
Defense date
07.18.2025 09:00
07.18.2025 09:00
Summary
Fixed point theory has proven to be a branch of mathematics with great potential for solving multiple problems in nonlinear analysis, such as proving the existence, uniqueness, or multiplicity of solutions to both integral and differential equations. Specifically, Krasnoselskii’s fixed point theorem in cones of expansion-compression has been widely used in numerous research works to obtain nontrivial solutions to such problems. With the aim of obtaining existence results for systems of differential and integral equations with all components being nontrivial, different versions of this result have been developed, adapted to operators defined on product spaces. In this work, we introduce the fixed point index, a powerful tool with which we will prove the main results. Additionally, we propose a new version of Krasnoselskii’s theorem, in which the conditions on the operator are expressed in terms of the norms of the spaces forming the product space under consideration. This original result has motivated minor improvements to existing versions, which we also describe. Finally, we apply the most novel among these results to establish conditions that guarantee the existence of solutions with positive components for various systems of differential and integral equations.
Fixed point theory has proven to be a branch of mathematics with great potential for solving multiple problems in nonlinear analysis, such as proving the existence, uniqueness, or multiplicity of solutions to both integral and differential equations. Specifically, Krasnoselskii’s fixed point theorem in cones of expansion-compression has been widely used in numerous research works to obtain nontrivial solutions to such problems. With the aim of obtaining existence results for systems of differential and integral equations with all components being nontrivial, different versions of this result have been developed, adapted to operators defined on product spaces. In this work, we introduce the fixed point index, a powerful tool with which we will prove the main results. Additionally, we propose a new version of Krasnoselskii’s theorem, in which the conditions on the operator are expressed in terms of the norms of the spaces forming the product space under consideration. This original result has motivated minor improvements to existing versions, which we also describe. Finally, we apply the most novel among these results to establish conditions that guarantee the existence of solutions with positive components for various systems of differential and integral equations.
Direction
Rodríguez López, Jorge (Tutorships)
Rodríguez López, Jorge (Tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
Valuation of the exotic option Callable Reverse Floater under different stochastic models
Authorship
S.G.C.
Master in Industrial Mathematics
S.G.C.
Master in Industrial Mathematics
Defense date
07.16.2025 11:00
07.16.2025 11:00
Summary
This paper explores the valuation of the exotic derivative product Callable Reverse Floater, a combination of a reverse floater and a call option. It establishes a theoretical foundation in stochastic calculus and financial theory. It introduces the SABR model for volatility and analyzes various stochastic models (Hull-White, Black-Karasinski, Linear Gaussian, and CIR) for valuation, with an emphasis on their implementation and calibration to market conditions. Finally, the valuation results obtained using the different models are presented and compared, analyzing their behavior and specific features.
This paper explores the valuation of the exotic derivative product Callable Reverse Floater, a combination of a reverse floater and a call option. It establishes a theoretical foundation in stochastic calculus and financial theory. It introduces the SABR model for volatility and analyzes various stochastic models (Hull-White, Black-Karasinski, Linear Gaussian, and CIR) for valuation, with an emphasis on their implementation and calibration to market conditions. Finally, the valuation results obtained using the different models are presented and compared, analyzing their behavior and specific features.
Direction
Vázquez Cendón, Carlos (Tutorships)
Vázquez Cendón, Carlos (Tutorships)
Court
VAZQUEZ CENDON, MARIA ELENA (Coordinator)
Varas Mérida, Fernando (Chairman)
Terragni , Filippo (Secretary)
López Pouso, Óscar (Member)
VAZQUEZ CENDON, MARIA ELENA (Coordinator)
Varas Mérida, Fernando (Chairman)
Terragni , Filippo (Secretary)
López Pouso, Óscar (Member)
The isoperimetric problem.
Authorship
D.G.D.R.
Master in Mathematics
D.G.D.R.
Master in Mathematics
Defense date
07.18.2025 09:40
07.18.2025 09:40
Summary
The present work addresses the study of the isoperimetric problem, tracing its development from its most classical roots to its formulation in more advanced geometric contexts. This problem seeks to understand which regions within a given ambient space minimize the area of their boundary under a fixed volume constraint. In its classical formulation in the Euclidean plane R2, it is rigorously shown that among all Jordan curves, the one that encloses the greatest area is the circle a fact formalized through the isoperimetric inequality in the plane. Moreover, the isoperimetric problem is generalized to higher dimensions, showing that in Rn, the sphere is the unique compact and connected hypersurface that minimizes surface area for a given volume. This will be demonstrated by proving Alexandrov’s theorem and exploring the variational properties of hypersurfaces with constant mean curvature. This work further extends the study of the isoperimetric problem to the setting of Riemannian manifolds, where solving the isoperimetric problem proves to be significantly more challenging. In this context, the Cheeger isoperimetric constant is introduced, defined as the infimum of the ratios between boundary area and volume of regular domains. This constant also possesses deep analytical properties, as it provides a lower bound for the first eigenvalue of the Laplace Beltrami operator with Dirichlet boundary conditions. Furthermore, the present work provides a detailed study of the explicit computation of the Cheeger isoperimetric constant in a certain family of geometrically significant spaces, such as solvable, simply connected Lie groups equipped with a left invariant metric, where the constant can be expressed in terms of the trace of the adjoint representation of the Lie algebra. Finally, a particularly interesting case is analyzed: symmetric spaces of non compact type. Each of these spaces turns out to be isometric to a solvable, simply connected Lie group endowed with a left invariant metric, allowing for the computation of the Cheeger constant using struc tural tools such as the root space decomposition and the Iwasawa decomposition. Additionally, the explicit calculation of the Cheeger constant is presented for the specific cases of real and complex hyperbolic spaces.
The present work addresses the study of the isoperimetric problem, tracing its development from its most classical roots to its formulation in more advanced geometric contexts. This problem seeks to understand which regions within a given ambient space minimize the area of their boundary under a fixed volume constraint. In its classical formulation in the Euclidean plane R2, it is rigorously shown that among all Jordan curves, the one that encloses the greatest area is the circle a fact formalized through the isoperimetric inequality in the plane. Moreover, the isoperimetric problem is generalized to higher dimensions, showing that in Rn, the sphere is the unique compact and connected hypersurface that minimizes surface area for a given volume. This will be demonstrated by proving Alexandrov’s theorem and exploring the variational properties of hypersurfaces with constant mean curvature. This work further extends the study of the isoperimetric problem to the setting of Riemannian manifolds, where solving the isoperimetric problem proves to be significantly more challenging. In this context, the Cheeger isoperimetric constant is introduced, defined as the infimum of the ratios between boundary area and volume of regular domains. This constant also possesses deep analytical properties, as it provides a lower bound for the first eigenvalue of the Laplace Beltrami operator with Dirichlet boundary conditions. Furthermore, the present work provides a detailed study of the explicit computation of the Cheeger isoperimetric constant in a certain family of geometrically significant spaces, such as solvable, simply connected Lie groups equipped with a left invariant metric, where the constant can be expressed in terms of the trace of the adjoint representation of the Lie algebra. Finally, a particularly interesting case is analyzed: symmetric spaces of non compact type. Each of these spaces turns out to be isometric to a solvable, simply connected Lie group endowed with a left invariant metric, allowing for the computation of the Cheeger constant using struc tural tools such as the root space decomposition and the Iwasawa decomposition. Additionally, the explicit calculation of the Cheeger constant is presented for the specific cases of real and complex hyperbolic spaces.
Direction
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
RODRIGUEZ VAZQUEZ, ALBERTO (Co-tutorships)
DOMINGUEZ VAZQUEZ, MIGUEL (Tutorships)
RODRIGUEZ VAZQUEZ, ALBERTO (Co-tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
Model Categories
Authorship
L.M.S.
Master in Mathematics
L.M.S.
Master in Mathematics
Defense date
07.18.2025 10:00
07.18.2025 10:00
Summary
The main goal of this dissertation is to introduce model categories and the homotopy category of a model category. In order to do that, we study two examples. The first one is in the context of topological spaces. The second one is purely algebraic, we study chain complexes. To accomplish our goal, we need also to introduce Quillen's small object argument.
The main goal of this dissertation is to introduce model categories and the homotopy category of a model category. In order to do that, we study two examples. The first one is in the context of topological spaces. The second one is purely algebraic, we study chain complexes. To accomplish our goal, we need also to introduce Quillen's small object argument.
Direction
Gómez Tato, Antonio M. (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Gómez Tato, Antonio M. (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
Algebraic and combinatoric methods in topological robotics
Authorship
A.M.V.
Master in Mathematics
A.M.V.
Master in Mathematics
Defense date
07.18.2025 10:20
07.18.2025 10:20
Summary
In this dissertation, we develop algebraic and combinatorial methods within the framework of topological robotics. More specifically, we study homotopy invariants related to the motion planning problem, such as the Lusternik-Schnirelmann category and topological complexity, through a unifying notion: homotopic distance. The techniques we use combine classical tools from algebraic topology, such as homotopy groups and (co)homology, with ideas from combinatorial and computational topology via simplicial complexes. This allows us to define new original invariants that strictly improve existing bounds and results in the literature, as well as to design algorithms which, through the implementation of symbolic computation programs, calculate these invariants for any triangulable space.
In this dissertation, we develop algebraic and combinatorial methods within the framework of topological robotics. More specifically, we study homotopy invariants related to the motion planning problem, such as the Lusternik-Schnirelmann category and topological complexity, through a unifying notion: homotopic distance. The techniques we use combine classical tools from algebraic topology, such as homotopy groups and (co)homology, with ideas from combinatorial and computational topology via simplicial complexes. This allows us to define new original invariants that strictly improve existing bounds and results in the literature, as well as to design algorithms which, through the implementation of symbolic computation programs, calculate these invariants for any triangulable space.
Direction
Gómez Tato, Antonio M. (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Gómez Tato, Antonio M. (Tutorships)
MOSQUERA LOIS, DAVID (Co-tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
Failure analysis of lithium-ion batteries due to nail penetration
Authorship
J.N.M.C.
Master in Industrial Mathematics
J.N.M.C.
Master in Industrial Mathematics
Defense date
07.16.2025 12:00
07.16.2025 12:00
Summary
This work focuses on the simulation of nail penetration into a lithium-ion cell, a critical issue for the safety of lithium-ion batteries. Penetration of a metallic object can cause internal short circuits and local generation of excessive heat, leading to fire or explosion risks. To address this problem, a 2D axisymmetric model was developed that simulates the electrochemical and thermal interactions during nail penetration. Gmsh software was used for mesh generation, and FEniCSx together with Multiphenicsx was used to solve the differential equations describing the phenomena involved. In particular, special attention was paid to simulating different convection coefficients and cell states of charge to assess their effects.
This work focuses on the simulation of nail penetration into a lithium-ion cell, a critical issue for the safety of lithium-ion batteries. Penetration of a metallic object can cause internal short circuits and local generation of excessive heat, leading to fire or explosion risks. To address this problem, a 2D axisymmetric model was developed that simulates the electrochemical and thermal interactions during nail penetration. Gmsh software was used for mesh generation, and FEniCSx together with Multiphenicsx was used to solve the differential equations describing the phenomena involved. In particular, special attention was paid to simulating different convection coefficients and cell states of charge to assess their effects.
Direction
Varas Mérida, Fernando (Tutorships)
Varas Mérida, Fernando (Tutorships)
Court
VAZQUEZ CENDON, MARIA ELENA (Coordinator)
Varas Mérida, Fernando (Chairman)
Terragni , Filippo (Secretary)
López Pouso, Óscar (Member)
VAZQUEZ CENDON, MARIA ELENA (Coordinator)
Varas Mérida, Fernando (Chairman)
Terragni , Filippo (Secretary)
López Pouso, Óscar (Member)
Homotopy of spheres
Authorship
V.E.P.B.
Master in Mathematics
V.E.P.B.
Master in Mathematics
Defense date
07.18.2025 11:00
07.18.2025 11:00
Summary
This Master's Thesis addresses the study of the homotopy groups of spheres, one of the classic and fundamental topics in Algebraic Topology. First, some basic topological constructions are introduced, such as pointwise sum, suspensions (of spheres), and smash product, along with a brief introduction to category theory. Subsequently, we will develop the higher-order homotopy groups. We will analyze their definition through homotopy classes of maps of the type $Sn \longrightarrow X$, some of their main properties, and the notion of relative homotopy groups, which allows us to study a long exact sequence between these homotopy groups. Key tools such as the Excision Theorem and the Freudenthal Suspension Theorem are also presented, useful for calculating $\pi_n(Sn)$, among others. Finally, the concept of fibration is introduced, especially the Hopf fibration given by $S1 \longrightarrow S3 \overset{p}{\longrightarrow}S2$, which allows the study of the homotopy group $\pi_3(S2) = \mathbb{Z}$.
This Master's Thesis addresses the study of the homotopy groups of spheres, one of the classic and fundamental topics in Algebraic Topology. First, some basic topological constructions are introduced, such as pointwise sum, suspensions (of spheres), and smash product, along with a brief introduction to category theory. Subsequently, we will develop the higher-order homotopy groups. We will analyze their definition through homotopy classes of maps of the type $Sn \longrightarrow X$, some of their main properties, and the notion of relative homotopy groups, which allows us to study a long exact sequence between these homotopy groups. Key tools such as the Excision Theorem and the Freudenthal Suspension Theorem are also presented, useful for calculating $\pi_n(Sn)$, among others. Finally, the concept of fibration is introduced, especially the Hopf fibration given by $S1 \longrightarrow S3 \overset{p}{\longrightarrow}S2$, which allows the study of the homotopy group $\pi_3(S2) = \mathbb{Z}$.
Direction
Gómez Tato, Antonio M. (Tutorships)
Gómez Tato, Antonio M. (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
Periodic solutions for singular differential equations
Authorship
M.P.A.
Master in Mathematics
M.P.A.
Master in Mathematics
Defense date
07.18.2025 12:00
07.18.2025 12:00
Summary
Singular differential equations present some term tending to infinity at some point in their domain, which often occurs when dealing with problems in electromagnetism or mechanics. In this work, after presenting some useful techniques for the study of periodic boundary value problems, we will introduce the concept of singular differential equations through examples and we will comment the two most common classifications of singularities found in the literature. Afterwards, we will apply the mentioned techniques to prove the existence of positive periodic solutions for different families of second-order singular differential equations, distinguishing between those without damping, that is, those in which the first derivative does not appear, and those with damping.
Singular differential equations present some term tending to infinity at some point in their domain, which often occurs when dealing with problems in electromagnetism or mechanics. In this work, after presenting some useful techniques for the study of periodic boundary value problems, we will introduce the concept of singular differential equations through examples and we will comment the two most common classifications of singularities found in the literature. Afterwards, we will apply the mentioned techniques to prove the existence of positive periodic solutions for different families of second-order singular differential equations, distinguishing between those without damping, that is, those in which the first derivative does not appear, and those with damping.
Direction
Rodríguez López, Rosana (Tutorships)
Rodríguez López, Rosana (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
The p-adic L-function of a modular form
Authorship
J.P.N.
Master in Mathematics
J.P.N.
Master in Mathematics
Defense date
07.18.2025 11:40
07.18.2025 11:40
Summary
This work aims to construct and study the p-adic L-function attached to a modular form. It begins by introducing the concepts of modular forms and modular symbols and their basic properties. Next, the construction of the L p-adic function is made, for which the L functions are introduced and some of the related properties and concepts are studied, proving the algebraicity of some of its concrete values and using Stevens' control theorem (whose proof is addressed later) to finally construct the p-adic L-function associated to a modular form and prove its interpolation property, under certain hypotheses. The last part focuses on studying what happens when the weight of the modular form varies and analyzing the critical and Eisenstein cases.
This work aims to construct and study the p-adic L-function attached to a modular form. It begins by introducing the concepts of modular forms and modular symbols and their basic properties. Next, the construction of the L p-adic function is made, for which the L functions are introduced and some of the related properties and concepts are studied, proving the algebraicity of some of its concrete values and using Stevens' control theorem (whose proof is addressed later) to finally construct the p-adic L-function associated to a modular form and prove its interpolation property, under certain hypotheses. The last part focuses on studying what happens when the weight of the modular form varies and analyzing the critical and Eisenstein cases.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
RIVERO SALGADO, OSCAR (Tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
An introduction to Iwasawa's theory
Authorship
B.Q.C.
Master in Mathematics
B.Q.C.
Master in Mathematics
Defense date
07.18.2025 13:00
07.18.2025 13:00
Summary
This work aims to provide an introduction to Iwasawa theory, beginning with the study of the structure of finitely generated modules over Zp[[T]], which can be understood as a generalization of the classification theorem for modules over a principal ideal domain. This, in turn, allows us to establish Iwasawa’s control theorem for the class number of cyclotomic fields. In the second part, we present more recent results that provide a better understanding of the algebraic structure of Zp-extensions, formulating Iwasawa’s main conjecture, which relates this algebraic structure to a purely analytic object: the p-adic zeta function. Finally, we explain the general strategy used in the proof of this result, and we conclude by illustrating some current research trends in the area.
This work aims to provide an introduction to Iwasawa theory, beginning with the study of the structure of finitely generated modules over Zp[[T]], which can be understood as a generalization of the classification theorem for modules over a principal ideal domain. This, in turn, allows us to establish Iwasawa’s control theorem for the class number of cyclotomic fields. In the second part, we present more recent results that provide a better understanding of the algebraic structure of Zp-extensions, formulating Iwasawa’s main conjecture, which relates this algebraic structure to a purely analytic object: the p-adic zeta function. Finally, we explain the general strategy used in the proof of this result, and we conclude by illustrating some current research trends in the area.
Direction
RIVERO SALGADO, OSCAR (Tutorships)
RIVERO SALGADO, OSCAR (Tutorships)
Court
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
OTERO ESPINAR, MARIA VICTORIA (Chairman)
DOMINGUEZ VAZQUEZ, MIGUEL (Secretary)
ALONSO TARRIO, LEOVIGILDO (Member)
The power series method and its implementation in Python with automatic differentiation.
Authorship
P.V.G.
Master in Mathematics
P.V.G.
Master in Mathematics
Defense date
07.18.2025 12:20
07.18.2025 12:20
Summary
In this work, we delve into the theoretical foundations of the well-known power series method and the lower and upper solutions method, subsequently combining them with Automatic Differentiation in order to to develop two programs in Python. The first program enables the construction of the Taylor polynomial of the desired degree for the solution of an initial value problem associated with an ordinary differential equation, under the assumption that the given function is analytic - a case in which the effectiveness of the power series method is ensured. The second program addresses a boundary value problem of Dirichlet type associated with an ODE with a continuous given function. This program does not aim to approximate the solution, but rather to prove its existence by constructing both a lower and an upper solution for the problem.
In this work, we delve into the theoretical foundations of the well-known power series method and the lower and upper solutions method, subsequently combining them with Automatic Differentiation in order to to develop two programs in Python. The first program enables the construction of the Taylor polynomial of the desired degree for the solution of an initial value problem associated with an ordinary differential equation, under the assumption that the given function is analytic - a case in which the effectiveness of the power series method is ensured. The second program addresses a boundary value problem of Dirichlet type associated with an ODE with a continuous given function. This program does not aim to approximate the solution, but rather to prove its existence by constructing both a lower and an upper solution for the problem.
Direction
LOPEZ POUSO, RODRIGO (Tutorships)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Co-tutorships)
LOPEZ POUSO, RODRIGO (Tutorships)
FERNANDEZ FERNANDEZ, FRANCISCO JAVIER (Co-tutorships)
Court
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)
LADRA GONZALEZ, MANUEL EULOGIO (Chairman)
FERNANDEZ TOJO, FERNANDO ADRIAN (Secretary)
GARCIA RIO, EDUARDO (Member)