Màster Oficial - Matemàtica Avançada

URI permanent per a aquesta col·leccióhttps://hdl.handle.net/2445/42661

Treballs finals del Màster en Matemàtica Avançada de la Facultat de Matemàtiques i Informàtica de la Universitat de Barcelona

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Finite group actions on surfaces: A Riemann surface approach to classification by genus
(2026-06-10) Vicente Sastre, Andrea Isabel; Mundet i Riera, Ignasi
Actions of finite groups on compact surfaces can be studied by fixing the surface and asking which groups are able to act on it. The present work addresses this question by classifying the effective actions of finite groups on compact surfaces according to their genus. The central idea is to pass from the differentiable to the holomorphic setting: averaging an arbitrary Riemannian metric yields a G-invariant one, which determines an almost-complex structure on the surface. A central result establishes that every such structure arises from a unique Riemann surface structure so then the elements of the group act as holomorphic or antiholomorphic automorphisms. The required theory of Riemann surfaces is developed, with special attention to uniformization and the Riemann-Hurwitz formula. In this way, the problem is reduced to understanding automorphisms. This is carried out in each of the three cases determined by the genus, where the geometry of the surface dictates how much symmetry is possible. In genus zero the surface admits a large amount of symmetry, whereas from genus one onwards the admissible groups become increasingly rigid, culminating in the bound that governs the high genus case.
Causality, Singularities, and the Topology of MOTS
(2026-06-06) Ujaldon Garcia, Victor; Porti, Joan
This Master's Thesis explores the global causal structure of Lorentzian manifolds, utilizing techniques from differential geometry and geometric analysis to study spacetime singularities and apparent horizons. First, we establish the foundational topological properties of causal curves and the critical role of global hyperbolicity. Through the kinematics of timelike congruences and the Raychaudhuri equation, we formally present Hawking's Singularity Theorem. This result proves that, under the strong energy condition, timelike geodesics generically focalize and collapse into conjugate points, rendering the spacetime geodesically incomplete. Subsequently, the framework is adapted to the degenerate regime of null geometry to study the causal boundaries defined by trapped surfaces. This work highlights a profound geometric dichotomy: while the non-degenerate timelike geometry inevitably succumbs to pathological singularities, the degenerate null geometry framing these singularities exhibits remarkable topological rigidity. By analyzing the spectral properties of the stability operator associated with Marginally Outer Trapped Surfaces (MOTS), we prove the Galloway-Schoen theorem. In particular for the four-dimensional spacetime satisfying the dominant energy condition, the topology of a compact apparent horizon is strictly constrained to the sphere $S^2$, or the flat torus $S^1 \times S^1$ in the strictly Ricci-flat limit.
An equivariant cohomological approach to Smith theory and its converse
(2026-06-12) Solaguren Anaut, Mikel; Castellana i Vila, Natàlia
The present work is devoted to the study of the topological properties of the fixed point set of a group action. In particular, we will first set the foundational machinery needed to study G-spaces using equivariant cohomology. Then, we will apply this machinery to generalize the classical result of Smith theory that relates the cohomology of a space with the cohomology of the fixed points of a p-group action following a proof by P. May. Finally, we will address the converse problem studied by L.Jones: when can a space be realized as the fixed point set of a semi-free group action on a contractible space.
Group C*-Algebras: amenability and nuclearity
(2026-06-12) Sales Cabrera, Miguel; Perera Domènech, Francesc
The goal of this work is to study group $C^*$-algebras, with particular emphasis on the notions of amenability and nuclearity. We develop the necessary background on representations of groups and Banach $*$-algebras, in order to construct two $C^*$-algebras associated to a locally compact group. This objects will allow us to investigate the interplay between algebraic properties of groups and analytic properties of operator algebras. The concepts of amenability and nuclearity are then introduced. The first describes a fundamental regularity property of groups, admitting several equivalent characterizations. The second is a regularity property of $C^*$-algebras, expressed through the behavior of tensor products and finite-dimensional approximations. These two notions, originating in seemingly distinct mathematical settings, are shown to be equivalent in the culminating theorem of this work.
Handwriting Generation Using Spiking Neural Networks
(2026-06-10) Pomar Pallares, Marc; Ruiz Cirera, Albert
Spiking Neural Networks (SNNs) provide a biologically inspired framework for processing temporal information through discrete spike events. Their event-driven nature makes them especially suitable for sequence-generation tasks, where the timing of the signal is central to the problem. This Master’s Final Project studies the generation of handwritten characters using recurrent SNNs trained with eligibility propagation (e-prop) in the NEST simulator. The project is based on the idea that handwriting should not only be treated as a static image, but as a temporal movement trajectory. A handwritten character is produced by a sequence of pen displacements, stroke transitions, pressure variations, and pen lifts. Therefore, the model developed in this thesis generates handwriting through three temporal readout signals: horizontal displacement, vertical displacement, and pen pressure. The spatial readouts are represented as derivatives, $\Delta x(t)$ and $\Delta y(t)$, and the visible trajectory is reconstructed only after simulation by cumulative summation. The work develops a controlled pipeline for training, executing, and decoding recurrent SNN handwriting models. The system receives frozen spike-train inputs associated with characters, processes them through a recurrent spiking population, and generates the corresponding trajectory through a small readout layer. The pressure readout allows the model to represent multi-stroke symbols without drawing artificial connections between strokes, while also providing a more natural interpretation of stroke visibility and intensity. The thesis focuses on improving control over the generated output. This includes the use of derivative targets to obtain a clearer movement representation, pressure-based decoding for multi-stroke generation, input spike perturbations to produce natural variation. The results show that e-prop trained SNNs can reproduce legible handwritten symbols from very limited data, while also showing that decoding consistency, input timing, and training duration strongly affect the quality of the generated trajectory. Overall, this work presents a compact and data-efficient approach to handwriting generation with recurrent SNNs. Its main contribution is not only to generate visible characters, but to define a more interpretable and controlled framework for online handwriting generation, where movement, pressure, variability, and stability are treated as separate components of the system.
A pathway to triangulated categories
(2026-06-12) Pineda Pascual, Arnau; Castellana i Vila, Natàlia
Rough volatility: from empirical evidence to the Rbergomi model
(2026-06-12) Estévez Lengua, Francisco; Vives i Santa Eulàlia, Josep, 1963-
The Black–Scholes–Merton model assumes constant volatility, contradicting the implied volatility surface of equity markets. A key feature of this surface is the at-the-money skew, which empirically explodes as a power law $\Psi(\tau) \sim \tau^{-0.4}$ as the maturity $\tau \to 0$. Classical stochastic volatility models such as Heston and Bergomi cannot reproduce this: driven by Brownian motion, their short-maturity skew stays bounded. Following Gatheral, Jaisson and Rosenbaum, we show that modelling log-volatility as a fractional Brownian motion with Hurst exponent $H \approx 0.1$ resolves this, and we develop the rough Bergomi model, whose Volterra kernel $(t-s)^{H-1/2}$ generates the empirical scaling $\Psi(\tau) \sim \tau^{H-1/2}$. We add three numerical contributions: a replication of the Hurst estimation on eight equity indices, a Monte Carlo validation of the method, and a Bergomi vs. rBergomi simulation comparison.
Gröbner Bases on $\Sigma$-Algebras
(2026-06-12) Pérez Arrese-Igor, Iker; de Felipe Paramio, Ana Belén; D'Andrea, Carlos, 1973-
Copilot said: In this text, we develop a theory of Gröbner bases for $\Sigma$-algebras. To do so, we first give introductory notions on Gröbner bases, and on $\Sigma$-algebras. Moreover, we propose a method to represent certain monomial orderings of $\Sigma$-polynomial algebras via infinite matrices. Within this setting, and despite the non-Noetherianity, we present an algorithm to compute Gröbner bases whenever feasible. Finally, we illustrate these concepts with an example and provide implementations of the proposed algorithms.
Homotopy Tiny Objects in Enriched Diagram Categories
(2026-06-12) Otero Escobar, Agustín; Castellana i Vila, Natàlia
The reconstruction theorem of \[Mon22], which gives a homotopical rendering of the density theorem of the nerve-realization adjunction \[Kel82], provides the left adjoint given by the subcategory generated under weak equivalences and homotopy weighted colimits of what are called *homotopy tiny objects* in order to identify a model category enriched over a closed symmetric monoidal model category $\mathcal{V}$ with a category of enriched presheaves generated by the small full subcategory of the homotopy tiny objects by Quillen equivalence. We extend the theorem to the case where the base of enrichment is a diagram category $\mathcal{V}^{\mathcal{D}}$, for $\mathcal{D}$ a small $\mathcal{V}$-category, equipped with the projective model structure. Such diagram bases arise naturally in equivariant operad or infinite loop space theory \[Bar22, BH15, GMMO19], global homotopy theory \[Sch18, Sch20], and related forms of indexed homotopy theory \[MS06, BT14], where homotopical information is organized simultaneously across families of subgroups, isotropy types, or other indexing data. We define $\mathcal{V}^{\mathcal{D}}$-homotopy tiny objects and prove the extended reconstruction theorem, from which two applications follow: a reproof of the classical Elmendorf theorem using these methods which recovers its modern formulation due to Stephan \[Ste16], as well as a generalization of the Schwede–Shipley theorem, first given in \[SS03], identifying a stable $\mathcal{V}^{\mathcal{D}}$-model category with compact generators $\mathcal{G}$ with $(Sp^{\Sigma})^{\mathcal{G}^{op}\times\mathcal{D}}$ for $\mathcal{V}=Sp^{\Sigma}$.
On Critical Integrability of PDE Solutions via Convex Integration
(2026-06-12) Molina Bakhos, David A.; Clop, Albert
On the mordell - Weil rank of hyperelliptic curves
(2026-06-12) Mata Carmona, Guillem; Masdeu Sabaté, Marc
The Jacobian of an algebraic curve C defned over a number feld K is an abelian variety whose group of K-rational points has finite rank, by the Mordell-Weil theorem. We study the proof of this theorem and the problem of computing the rank. We give some bounds on this rank in a certain family of curves by studying the Selmer group, which is a cohomological device that can be explicitly computed.
Avoiding collisions: an introduction to the topology of configuration spaces
(2026-06-12) Martínez Sant, Jordi; Castellana i Vila, Natàlia
This thesis provides a comprehensive study of configuration spaces, tracing their topological properties from classical homogeneous manifolds to the discrete setting of topological graphs. Configuration spaces serve as a fundamental bridge between algebraic topology, geometric group theory, and robotics, capturing the collision-free motion of multiple particles. We begin by establishing the foundational properties of ordered configuration spaces on connected manifolds, with a rigorous focus on the forgetting map and the Fadell–Neuwirth fibration theorem. This geometric framework transitions naturally into the study of braid groups, where we explore their algebraic presentations and characterisations. Following this, we analyze the specific topology of the planar configuration space $\mathcal{F}_n(\mathbb{R}^2)$ by examining the classical Fox-Neuwirth theorem. Additionally, we introduce the complexities that arise when extending these techniques to compact connected surfaces, where the splitting behavior of the Fadell–Neuwirth sequence is no longer guaranteed. Finally, we shift our focus to the distinct combinatorial nature of configuration spaces on graphs. We investigate the local structure by analyzing the configuration space of the star of a vertex, and scale these techniques to analyze the global space of a whole graph. To illustrate the concrete applications of this discrete framework, we present a fully worked example modeling a segment graph with three points.
Analytic and Signed Riesz Capacities in BMO, Lip(s) and L∞ Spaces
(2026-06-12) Martín Agüera, Àlex; Prat Baiget, Laura
This thesis develops a unified distributional framework for the study of analytic capacity and its generalisation to higher dimensions via the $\alpha$-Riesz transform. The work is organised around four main themes. In the first part, the Cauchy transform is extended to compactly supported distributions, yielding a flexible reformulation of analytic capacity. Within this framework, the classical $L^\infty$ Hausdorff content estimate for analytic capacity, due to Painlevé, the BMO and $\mathrm{Lip}(s)$ variants due to Verdera [Ver86] and Melnikov [Mel69], are presented with unified proofs written in distributional language. These arguments serve as an explicit template for the subsequent chapters. In the second part, we introduce the $\alpha$-Riesz transform and study criteria for its $L^2(\mu)$-boundedness with respect to non-doubling Radon measures. For $0 < \alpha \leq 1$, we present a proof of a $T1$ theorem by combining the Good Lambda method [Tol14] with a Menger-type curvature $c_\alpha$ arising from symmetrisation of the kernel. A $Tb$ theorem, which relaxes the testing condition to a suitable function $b \in L^\infty(\mu)$, is also stated and will play a key role in the final chapter. The third part introduces the $\alpha$-Riesz capacity $\gamma_\alpha$ and presents, following Prat [Pra04b], its Hausdorff content characterisation in the $L^\infty$ setting. We then construct a family of self-similar Cantor sets with $0 < \mathcal{H}^{\alpha}(E) < \infty$ but $\gamma_\alpha(E) = 0$, providing a negative answer to the question of whether positive $\mathcal{H}^{\alpha}$ measure implies positive capacity. Building on this distributional framework, we prove complete characterisations of the BMO and $\mathrm{Lip}(s)$ variants, $\gamma_{\alpha,\mathrm{BMO}}$ and $\gamma_{\alpha,\mathrm{Lip}(s)}$, in terms of Hausdorff content via purely real-variable methods. Finally, following Prat [Pra04b], the case $0 < \alpha < 1$ is studied in depth. Using the $Tb$ theorem together with curvature divergence and kernel regularisation arguments, it is shown that every compact set $E \subset \mathbb{R}^d$ with $\mathcal{H}^{\alpha}(E) < \infty$ satisfies $\gamma_\alpha(E) = 0$.
Kummer’s Lemma and Fermat’s Last Theorem
(2026-06-12) Huélamo Longás, David; Dieulefait, L. V. (Luis Victor)
The present thesis is devoted to the study and proof of Fermat’s Last Theorem (FLT) for regular primes, with a specific focus on the more intricate Case II. Following the historical and mathematical framework established by Ernst Kummer, the core of this proof relies on a profound intermediate result known as Kummer’s Lemma. While the statement of this lemma is deceptively simple, its rigorous justification demands a considerable amount of preliminary mathematical machinery. There is not a single way to prove Kummer’s Lemma. One well-known approach relies on the advanced architecture of class field theory; however, developing that theory from the ground up would require an extensive detour, occupying many pages of dense discussion. Consequently, we have chosen an alternative, more analytical path. This route leans heavily on the elegant framework of p-adic analysis, specifically p-adic L-functions and p-adic regulators, drawing significant inspiration from Lawrence C. Washington’s foundational text, Introduction to Cyclotomic Fields. Because Washington’s book is a comprehensive treatise on the broader theory of cyclotomic fields rather than a specialized text on Fermat’s Last Theorem, extracting the exact sequence of results needed for Kummer’s Lemma can be a daunting task, and there is a lack of pedagogical literature dedicated specifically to unpacking this proof in detail. The primary objective of this project, therefore, is to streamline and clarify this theory. By isolating and presenting strictly the necessary analytical and algebraic tools, this thesis aims to provide a self-contained and direct pathway to the proof of Kummer’s Lemma and, ultimately, the second case of Fermat’s Last Theorem for regular primes.
Option Pricing under Black–Scholes and Jump-Diffusion Models: A Partial Differential Equation Approach
(2026-06-12) Honey, Jenna; Vives i Santa Eulàlia, Josep, 1963-
This thesis studies the pricing of European options using partial differential equation methods. The starting point is the Black–Scholes model, where the underlying asset is assumed to follow a geometric Brownian motion. Under this assumption, the option price satisfies a parabolic partial differential equation, and for European call and put options this equation can be solved explicitly to obtain the classical Black–Scholes formula. The first part of the thesis develops this model from the basic financial and probabilistic ideas. We introduce options, the no-arbitrage principle, Brownian motion, geometric Brownian motion and Itô’s lemma, before deriving the Black–Scholes equation by a delta-hedging argument. The analytical solution is then obtained by transforming the Black–Scholes PDE into the heat equation. Since closed-form solutions are not available in many more general situations, we also study finite difference methods for parabolic PDEs, including the explicit, implicit, Crank–Nicolson and $(\theta)$-schemes, together with the ideas of consistency, stability and convergence. The second part considers models where the asset price is allowed to jump. This is motivated by the fact that real financial markets can experience sudden movements which are not well described by continuous Brownian paths. We introduce Poisson and compound Poisson processes, and then study the Merton and Kou jump-diffusion models. In this setting, the option price satisfies a partial integro-differential equation, where the integral term represents the effect of jumps. We derive this equation and discuss its numerical solution using an explicit-implicit scheme, treating the differential part implicitly and the integral part explicitly. The thesis finishes with numerical studies implemented in MATLAB. These are used to compare finite difference prices with the Black–Scholes formula, to test the IMEX method for the Merton model against the corresponding series representation, and to compare the implied volatility smiles generated by the Black–Scholes, Merton and Kou models. The overall aim is to show how the PDE approach to option pricing extends from the classical continuous-path model to jump-diffusion models, and how numerical methods are used when explicit formulae are no longer available.
Sandwiched Volterra Volatility models: theory and applications
(2026-06-07) He, Yuanxi; Vives i Santa Eulàlia, Josep, 1963-; Márquez, David (Márquez Carreras)
This thesis begins with an introduction to mathematical finance, reviewing the historical development of asset-price models and several stylised facts observed in financial markets. We then motivate the study of fractional and rough volatility models, which are driven by fractional Brownian motion and provide a more realistic description of volatility dynamics. The necessary definitions and theoretical results concerning fractional Brownian motion are also presented. Building on the work of Di Nunno et al. [15], we study the Sandwiched Volterra Volatility (SVV) model. In particular, we reformulate the general SVV framework in a one-dimensional setting, which allows for a more transparent exposition of the model and more detailed proofs of several key results. We further illustrate the model through a number of examples. In addition, we investigate the Malliavin differentiability of both the volatility and asset-price processes within the SVV framework and provide rigorous proofs of the corresponding results. Finally, we analyse classical rough volatility models in order to characterise those that can be represented within the Sandwiched Volterra Volatility (SVV) framework. We show that most of the models studied fall into this class, with the exception of the Rough Fractional Stochastic Volatility model (for Hurst parameter $H \neq \frac{1}{2}$), the rough Heston model, and the quadratic rough Heston model.
Subcentric Linking Systems
(2026-06-12) González Sáez, Paula; Broto, Carles, 1959-
Linking systems are crucial tools for studying the homotopy theory of fusion systems, while also being of profound independent interest from a purely algebraic perspective. However, the classical axioms defining these categories restrict their objects to quasicentric subgroups, which limits the structural flexibility of the theory. To address this limitation, E. Henke [Hen19] proposed expanding the object set to a broader class of subgroups: subcentric subgroups. Utilizing the framework of localities, Henke established that every saturated fusion system admits a unique, up to isomorphism, associated subcentric linking system. In this thesis, we translate Henke’s results into classical categorical language, thereby bypassing the machinery of partial groups and localities. Furthermore, we provide an explicit categorical construction of a subcentric linking system for realizable fusion systems (the fusion systems of finite groups).
Introduction to Symplectic and Contact Geometry and the Study of Periodic Orbits
(2026-06-12) Garriga Sànchez, Enric; Cardona Aguilar, Robert
The objective during the first two sections of the present Master Final Thesis is to provide an introduction as self-contained as possible to the fields of Symplectic and Contact Geometry and, ultimately, to devote the last section to prove a main theorem due to Hofer regarding existence of periodic orbits of the Reeb vector field on overtwisted contact $3$-manifolds. This theorem proves a particular case of the much more general Weinstein conjecture, which still remains open at the moment of writing this Thesis. Some more concepts studied in this Thesis include Darboux’s Theorem, an overview of the interplay between contact and Symplectic Geometry and a theorem on the existence of periodic orbits for Hamiltonian vector fields along convex energy surfaces inside $\mathbb{R}^{2n}$, another instance where the Weinstein conjecture is resolved positively. In the final section, special attention is given to the study of pseudoholomorphic curves and their properties, since they constitute the main tool to prove Hofer’s Theorem, and also to related ideas such as Bishop Families and pseudoconvexity. The proof of Hofer’s Theorem is finally given by combining the machinery developed throughout the work.
The real interpolation method
(2026-06-12) Garcia Muñoz, Marçal; Martín i Pedret, Joaquim
[eng] This Master’s Final Project provides a systematic study of the real interpolation method, tracing its development from the classical theorems of Riesz-Thorin and Marcinkiewicz. The abstract framework is constructed using Peetre’s $K$-functional and the associated $J$-method, establishing essential structural properties such as the Density, Reiteration, and Wolff’s theorems. The project also investigates the behavior of operator compactness under interpolation. To address this, we present the approach relying on the ball measure of non-compactness to demonstrate how compactness is preserved between interpolated spaces. Furthermore, the core theory is extended to the broader category of quasi-normed Abelian groups. This generalization is a necessary mathematical step to provide a proof of the general Marcinkiewicz Interpolation Theorem. Finally, the work concludes by briefly illustrating these abstract concepts through concrete mathematical examples, such as the computation of interpolation spaces for Sobolev couples or the classical endpoints in harmonic analysis $H^1$ and $BMO$. [cat] Aquest Treball Final de Màster proporciona un estudi sistemàtic del mètode d’interpolació real, resseguint el seu desenvolupament des dels teoremes clàssics de Riesz-Thorin i Marcinkiewicz. El marc abstracte es construeix utilitzant el $K$-funcional de Peetre i el $J$-mètode associat, establint propietats estructurals essencials com els teoremes de Densitat, Reiteració i Wolff. El projecte també investiga el comportament de la compacitat dels operadors sota interpolació. Per abordar-ho, presentem l’enfocament basat en la mesura de no-compacitat de bola per demostrar com es preserva la compacitat entre espais interpolats. A més, la teoria central s’estén a la categoria més àmplia dels grups Abelians quasi-normats. Aquesta generalització és un pas matemàtic necessari per proporcionar una demostració del Teorema d’Interpolació de Marcinkiewicz general. Finalment, el treball conclou il·lustrant breument aquests conceptes abstractes a través d’exemples matemàtics concrets, com ara el càlcul d’espais d’interpolació per a parelles de Sobolev o per als típics extrems en anàlisi harmònica $H^1$ i $BMO$.
The approximation theorem and topological hochschild homology
(2026-06-21) Garcia Comas, Pau; Bayındır, Haldun Özgür
This work is divided in two halves. In the first half we introduce the basic notions of the theory of operads, including free algebras over an operad, a study of the little $n$-cube operads, and their action on iterated loop spaces. This leads to the proof of the approximation theorem by Peter May. In the second half we introduce the basic definitions and properties of spectra and its tensor product. We then define topological Hochschild homology and give the proof of Bökstedt’s periodicity theorem on the description of $\mathrm{THH}(\mathbb{F}_p)$ by Nikolaus and Krause, which relates it to a theorem by Hopkins and Mahowald using the approximation theorem of May.

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