Research Team / Research Group Name (if any)
Mathematics applied to biological and physical systems
Brief description of the Research Team / Research Group / Department
Currently, we have two major interests: 1) the development of adequate mathematical and computational techniques for biomedical problems, including data analysis and agent based models for cellular aggregates, 2) inverse problems and imaging, including optimization with differential constraints, bayesian approaches and Markov Chain Monte Carlo sampling.<br />In previous years, we have made contributions to the modeling, numerical simulation and mathematical analysis of topics in biological sciences (pattern formation in cellular aggregates, protein folding and unfolding, nerve impulse propagation, angiogenesis, analysis of cancer data), imaging (holographic techniques, inverse scattering in acoustics, photodetection and electromagnetism), materials science (defects and ripples in graphene, dislocation nucleation and motion in crystals, nucleation and aggregation processes, oscillations in semiconductors, nonlinear waves in oscillator networks), as well as to the analysis of fluid mechanics equations, nonlinear wave equations, and integrodifferential kinetic models.<br />We have projects, contacts and publications with researchers from Harvard University, Stanford University, MIT, UC Santa Barbara, UC Berkeley, Duke University, New York University, Oxford University, Paris-Sorbonne Universités, Technische Universität Berlin, and also from institutions such as the Sloan Kettering Cancer Center, the Institut Curie, the National Center of Biotechnlogy, and Hospitals of the Madrid area.
Research lines / projects proposed
Postdoctoral researchers can apply to carry out research within our lines of work. Some possible projects:<br />1) Mathematical methods for medical data. Nowadays an ever increasing amount of data central to different illnesses is becoming available: measurements of expression of genes and protein synthesis, image recordings, patients records, metadata regarding social and psicological factors. Mathematical and computational tools to extract meaningful information from such data need to be identified and developed. The usage of standard machine learning tools faces a number of problems, for instance, available data may not be big enough, show gaps or just not be comparable, also the outcome may fail to have a medical interpretation, moreover, uncertainty should be quantified.<br />2) Inverse Problems. Inverse problems arising in multiple frameworks (medicine, public security, geophysics) involve finding large sets of parameters defining unknown shapes and fields, by either optimizing costs which compare an output to measured data or by determining maximum likelihood options. These schemes require solving large amounts of auxiliary problems, often formulated in terms of partial differential equations. Devising efficient strategies to reduce the computational cost is essential to face three dimensional applications. <br />3) Modeling and simulation in biomedicine. Developing descriptions of cellular aggregates, for instance, requires the study of mechanisms at the microcospic and macroscopic level as well as their interaction. Mathematical approaches for cell behavior often have a stochastic basis, informed by the dynamics of continuum variables (concentrations, flows...). Developing hybrid models combining agent based models for cell behavior and continuum models is a challenging task.