16:45 – 18:00
Subcentric linking systems
Paula González Sáez (UB)
Linking systems are crucial tools for studying the homotopy theory of fusion systems, while also being of deep interest from a purely algebraic perspective. However, the classical axioms defining these categories restrict the objects to quasicentric subgroups, which limits the structural flexibility of the theory.
To address this limitation, E. Henke proposed extending the set of objects to a broader class of subgroups: subcentric subgroups. Using the framework of localities, Henke established that every saturated fusion system admits a unique associated subcentric linking system, up to isomorphism. In this thesis, we translate Henke's results into the classical categorical language, thereby avoiding the machinery of partial groups and localities. Furthermore, we offer an explicit categorical construction of a subcentric linking system for realizable fusion systems (that is, fusion systems of finite groups).
Mathematical theory of the magneto-optical properties of two-dimensional van der Waals and topological matter
Joan Pérez Alemany (UV-UPV)
This thesis formulates a comprehensive mathematical theory governing the macroscopic magnetic response of excitons in low-dimensional topological materials, addressing the limitations of the independent-particle Bloch framework. The main original contribution is a rigorous and gauge-invariant derivation of the generalized excitonic magnetic moment, obtained directly from the Hamiltonian of the Bethe-Salpeter equation (BSE) for many-body systems with interactions. By introducing mass-weighting coefficients in a periodic-field regularization scheme, this work successfully resolves the distributional singularities of the position operator, formally decoupling the relative internal motion of the electron-hole pair from the macroscopic dynamics of its center of mass. Furthermore, the thesis establishes completely novel geometric upper bounds and topological lower bounds for these excitonic magnetic moments, demonstrating that they are strictly constrained by the quantum geometric tensor of the constituent bands; specifically, it links the limits of the magnetic response to the fundamental quantum metric and the topological Chern number. These fundamental bounds, analytically validated through a solvable two-dimensional massive Dirac model, provide crucial theoretical references for bounding numerical truncation errors in computational methodologies and offer predictive mathematical pathways for the design of next-generation optoelectronic and valleytronic quantum materials.
KAM theory for periodic orbits
Laura Cara Sala (UB)
The classical approach to analyzing periodic orbits and their behavior is Floquet theory. This allows linearizing the system around the periodic orbit to reduce the study of local stability to a linear system with periodic coefficients.
Following Moore's idea of using Floquet theory to study periodic orbits, in my master's thesis we developed a theoretical framework for the convergence of a Newton-type convergence method in a system with a periodic orbit, demonstrating the existence of the latter with an explicitly computed error. In this work, we adapt analytical techniques from KAM theory by applying a small KAM iterative scheme in terms of losing part of the analytic domain of definition in exchange for finding the periodic orbit
As the main result of the work, we present a convergence theorem that proves the existence of periodic orbits when the initial error is sufficiently small. This theorem can be applied to the so-called computer-assisted proofs (computer-assisted proofs) to demonstrate the existence of periodic orbits.