Group of Mathematics Applied to Biological and Physical Sciences

Department of Applied Mathematics
Faculty of Mathematical Science

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.
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.
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.

Adequate office space, libraries, online resources, and administrative support for researchers and their visitors. Basic computational resources for code development such as laptops, desktops, printers, as well as access to parallel computing through the UCM central servers. Software library including Matlab, Comsol licences, a variety of compilers and standard computer equipment.

Postdoctoral researchers can apply to carry out research within our lines of work. Some possible projects:
1) Nowadays an ever increasing amount of data central to different illnesses is becoming available: measurements of expression of genes and protein synthesis, image recordings, 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. We plan to combine the usage of different distances (Wasserstein, Fermat) with topological data analysis and machine learning tools, quantifying the perfomance of the proposed techniques with bayesian approaches.
2) 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 strategies to reduce the computational cost is essential to face three dimensional applications. We plan to investigate the influence that the choice of mathematical representations for the shapes and fields has on the reconstruction process.
The researchers will have the opportunity to interact with institutions with whom we already have well established collaborations, as well as with the academic and non academic partners belonging to the UNAEUROPA Alliance.