Neurological disorders such as epilepsy (affecting about 5 MILL people worldwide) are associated with deviations from the optimal network structure and dynamics. Epilepsy surgery (costing 1000-3000 EUR per surgery in EU countries) consists in the removal of the minimum amount of tissue needed to stop seizure propagation, but it is only successful in aprox. 2/3 of the cases. Analyses of brain network structure, and computational studies simulating seizure dynamics over it, have related aspects of the patients' brain network (e.g. hub distribution, excitability) with surgery outcome. However, results are not always replicable, and have not been able to surpass the current clinical standard. In the MENTE (which stands for "mind" in Spanish) project, I will explicitly consider the previously neglected effect of high-order (HO) brain structure on seizure dynamics and epilepsy surgery. I will characterize the HO brain structure of epilepsy patients (via Topological Data Analysis), in relation to surgery outcome. I will then create a HO computational model of seizure dynamics, by extending pairwise multiscale (MS) neuronal models of ictal activity. I will study the emergent behavior of the system via theoretico-computational analysis to characterize the healthy-to-ictal transition. Finally, I will fit the model to intracranial EEG data of ictal activity, and use it to model epilepsy surgery, with the goal of predicting surgery outcome and finding alternative resection strategies (smaller of with better outcome), to reduce the societal and economical burden of epilepsy. MENTE will validate the role of HO coupling on brain activity and promote the theoretico-computational modeling, analysis and application to neuroscience of network dynamics including HO and MS coupling. The fellowship will strengthen mine and the host's expertise these topics, and allow me develop my academic career to the point of scientific maturity.

Public key cryptography is the backbone of internet security. Yet it is very likely that within the next few decades some government or corporate entity will succeed in building a general-purpose quantum computer that is capable of breaking all of today's public key protocols. Lattice cryptography, which appears to be resilient to quantum attacks, is currently viewed as the most promising candidate to take over as the basis for cryptography in the future. Recent theoretical breakthroughs have additionally shown that lattice cryptography may even allow for constructions of primitives with novel capabilities. But even though the progress in this latter area has been considerable, the resulting schemes are still extremely impractical. The central objective of the FELICITY project is to substantially expand the boundaries of efficient lattice-based cryptography. This includes improving on the most crucial cryptographic protocols, some of which are already considered practical, as well as pushing towards efficiency in areas that currently seem out of reach. The methodology that we propose to use differs from the bulk of the research being done today. Rather than directly working on advanced primitives in which practical considerations are ignored, the focus of the project will be on finding novel ways in which to break the most fundamental barriers that are standing in the way of practicality. For this, I believe it is productive to concentrate on building schemes that stand at the frontier of what is considered efficient -- because it is there that the most critical barriers are most apparent. And since cryptographic techniques usually propagate themselves from simple to advanced primitives, improved solutions for the fundamental ones will eventually serve as building blocks for practical constructions of schemes having advanced capabilities.

Waves propagating through a complex medium provide a non-invasive way to probe its interior structures. In ambient noise imaging, the input data are the cross-correlation of the stochastic wavefields. To reconstruct the properties of the medium, the waveform inversion is formulated as an optimization problem involving a misfit function whose convexity plays a critical role in the achievable spatial resolution of the inversion results, especially in the absence of a priori information about the medium. Current inversions are often limited by computational cost, cross-talk between the physical quantities, and the use of single-scattering approximations. Project INCORWAVE proposes to create a new mathematical and computational framework for nonlinear inversion of full waveform cross-correlation. Two specific problems are considered: first, for the reconstruction of geophysical visco-elasticity tensors with applications to Earth's subsurface monitoring; secondly, for the reconstruction of three-dimensional flows in the Sun to characterize the poorly understood properties of deep solar convection. To improve the convexity of misfit functions, the inversion procedure of project INCORWAVE will follow a hierarchical progression which is established by selecting subsets of input data, unknown parameters, and frequencies. The choice of each of these subsets, as well as the associated misfit function, is controlled by criteria in form of convergence estimates. Indispensable to meaningful inversion is accurate modeling operators that describe the physics under consideration and that are adapted to the treatment of real data. For the reconstruction of the elasticity tensor, the project will develop a solver in terms of P- and S-potentials for heterogeneous media. A 3D global Sun vector-wave solver is created for the inversion of the convection component of the solar flow that does not bear symmetry.