**Institute of Mathematical Sciences** **(ICMAT)** is a joint research center of the Consejo Superior de Investigaciones Científicas – CSIC (Spanish National Research Council) and three Madrid universities: the Universidad Autónoma de Madrid (UAM), the Universidad Carlos III de Madrid (UC3M), and the Universidad Complutense de Madrid (UCM). The ICMAT is located on the UAM Campus in Madrid, its facilities are exceptional in the field of mathematical research. Its goal is to generate knowledge in all the areas of mathematics, particularly in Algebraic Geometry, Mathematical Physics, Differential Geometry, Geometric Mechanics, Mathematical Analysis, Differential Equations and Applications, Number Theory, Group theory, Statistics, Probability, Operations Research and Mathematics of Quantum Information.

**Algebra and Geometry. **The group conducts research in a broad variety of topics, in the areas of abstract algebra (group theory, commutative algebra), algebraic geometry (arithmetic geometry, number theory, moduli spaces of bundles), differential geometry (geometric analysis, geometric mechanics, dynamical systems and the geometry of PDEs) and topology (topological fluid dynamics, symplectic and contact topology, low-dimensional topology). As such, the research is naturally interdisciplinary, fostering an important level of cross-fertilization between the different areas. In addition, a number of the themes finds their motivation in ideas stemming from physics, such as special metrics, gauge theories and their algebro-geometric counterparts.

**Mathematical Analysis and Differential equations. **These are very active, deeply interrelated fields of research with a preponderant position within the mathematical sciences. This line deals with fundamental problems in the fields of harmonic analysis, partial differential equations, geometric group theory, functional analysis, geometric measure theory, operator algebra, differential geometry and probability, and has been awarded with a total of 7 ERC grants. The topics of interest in this line are classical problems around the Kakeya conjecture and Bochner-Riesz multipliers, the Schrödinger and wave equations, elliptic PDE in rough domains and connections with geometric measure theory, harmonic analysis and geometric group theory for nonamenable groups, classical and abstract Calderón-Zygmund theory and problems around the invariant subspace problem, operator theory, geometry of Banach spaces, complex analysis, quantum probability, and analytic number theory. Also, this group considers differential equations arising in fluid mechanics, spectral theory, mathematical physics and mathematical biology. This is an interdisciplinary subject, with significant applications to engineering, biology and physics.

**Applied Mathematics. **This research group works to develop the mathematical foundations and models needed to deal with the main new societal challenges, with a focus on Data Science, Machine Learning and Quantum Technologies. In particular they consider a wide variety of mathematical problems which are motivated by quantum technologies, such as condense matter and many body systems, quantum control, foundational aspects of quantum mechanics and the theory of operator algebras. In the field of Machine Learning and Data science, they consider Bayesian approaches to treat adversarial risk analysis and adversarial machine learning, and work in the areas of security and cybersecurity, and the tracking of fake news. Finally, other topics of interest are mathematical modelling, simulation, and numerical computation with applications in microfluidics modelling and technological applications, geophysical fluid dynamics, etc.

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