Programme

09h30 - 10h00 Welcoming & Opening Session (in the presence of Vice-Rector Artur Silva)

10h00 - 11h00 Ivan Pombo (Research Engineer at Inductiva Research Labs, and former PDM student) - The PhD Life - Tackling the Calderón Problem

11h00 - 11h30 Rui Martins (PDMat) - A C++ implementation of discrete adjoint sensitivity analysis for systems of ODEs

11h30 - 12h00 Marco Brito (MAP-Fis) - Stability and physical properties of spherical excited scalar boson star 

12h00 - 14h00 Lunch-Break

14h00 - 14h30 Etevaldo Costa (MAP-Fis) - Proca-Higgs model in a UV completion for Proca self-interactions

14h30 - 15h00 Franco Madriaga (PDMA) - Study of some Diophantine Equations 

15h00 - 15h30 João Costa (PDMat) - Schatten Classes on Banach Quaternionic Modules 

15h30 - 16h00 Juan Díaz (PDMA) - Multiple orthogonality and applications

Abstract: We present a new C++ library for sensitivity analysis of optimization problems involving ordinary differential equations (ODEs). The discrete adjoint sensitivity analysis method is implemented for adaptive explicit Runge-Kutta (ERK) methods available in the C++ boost library using automatic adjoint differentiation (AAD). Update expressions for the adjoint variables at each time step are derived and AAD is employed for efficient evaluations of products between vectors and the Jacobian of the right hand side of the ODE. This approach avoids the low-level drawbacks of the black box approach of employing AAD on the entire ODE solver and opens the possibility to leverage parallelization. We study the performance of other methods and implementations of sensitivity analysis and we find that our algorithm is competitive to equivalent existing implementations.

Abstract: We consider a Proca-Higgs model wherein a complex vector field gains mass via spontaneous symmetry breaking, by coupling to a real scalar field with a Higgs-type potential. This vector version of the Friedberg-Lee-Sirlin model can be considered a UV completion of a complex self-interacting Proca model. We study the flat spacetime and self-gravitating solitons of the model, as well as hairy black holes, exploring the domain of solutions and describing some of their mathematical and physical properties. Under certain limits, the model reduces to the pure Proca scenario; moreover, we show that it is free of the hyperbolicity problems that plague the self-interacting Proca models, thus making it an exciting arena for exploring the dynamics of such Proca-Higgs solitons in a self-consistent manner. We also prove a no-hair theorem for static, spherically symmetric BH. Hence, within this model, BH with Proca-Higgs hair requires rotation and no static limit.

Abstract: In this presentation we introduce an axiomatic approach to the theory of s-numbers in quaternionic analysis aiming to extend the notion of Schatten classes to the quaternionic framework while shedding some light on the theory of quaternionic operator ideals. To this end,  A. Pietsch's approach to s-number theory is adapted to the quaternionic algebra, following the works of F. Colombo and I. Sabadini on quaternionic spectral theory. One of the main results of this research is the uniqueness of s-numbers over quaternionic Hilbert modules. Moreover, examples will be given in the quaternionic framework together with the introduction of nuclear numbers. A consequence of the presented theory is a basis independent definition of the Schatten classes over quaternionic Hilbert and Banach spaces.