In recent years, much attention has been paid to the description of excitable media, such as the dynamics of the brain and heart. In both cases, the building blocks are excitable cells, neurons, and cardiomyocytes, and a detailed look at the mathematics behind some of their mathematical models provides a good starting point for answering some observed phenomena. In this talk we show how some apparently simple phenomena, such as the spike-adding process, have important consequences in the models and how various elements intervene behind their formation, such as homoclinic bifurcations, fast-slow decompositions, "canards", the completion of the Smale topological template, the formation of Morse surfaces creating geometric bifurcations, etc. Finally, we will illustrate its relevance in insect gait patterns and in the formation of cardiac arrhythmias.

*This talk is based on joint works with Santiago Ibañez, Álvaro Lozano, M.Ángeles Martínez, Lucía Pérez, Esther Pueyo, Marcos Rodríguez and Sergio Serrano.*

The concept of n-width has been introduced by Kolmogorov as a way of measuring the size of compact sets in terms of their approximability by linear spaces. From a numerical perspective it may be thought as a benchmark for the performance of algorithms based on linear approximation. In recent years this concept has proved to be highly meaningful in the analysis of reduced modeling strategies for complex physical problems described by parametric problems. This lecture will first review significant results in this area that concern (i) the practical construction of optimal spaces by greedy algorithms and (ii) the preservation of the rate of decay of widths under certain holomorphic transformation. It will then focus on recent attempts to propose non-linear version of n-widths, how these notions relate to metric entropies, and how they could be relevant to practical applications.

Degenerate differential evolution PDE problems are often characterised by the explicit presence of diffusion/dissipation in some of the spatial directions _only_, yet may still admit decay properties to some long time equilibrium. Classical examples include the inhomogeneous Fokker-Planck equation, Boltzmann equation with various collision kernels, systems of equation arising in micromagnetism or flow vorticity modelling, etc. In the celebrated AMS memoir "Hypocoercivity", Villani introduced the concept of hypocoercivity to describe a framework able to explain decay to equilibrium in the presence of dissipation in some directions only. The key technical idea involved is to exploit certain commutators to overcome the degeneracy of dissipation.

I shall present some results and ideas on the development of numerical methods which preserve the hypocoercivity property upon discretisation. As a result, such numerical methods will be suitable for arbitrarily long-time simulations of complex phenomena modelled by kinetic-type formulations. This will be achieved by addressing the key challenge of lack of commutativity between differentiation and discretisation in the context of mesh-based Galerkin-type numerical methods via the use of carefully constructed non-conforming weak formulations of the underlying evolution problems.Most real world dynamical systems consist of subsystems from different physical domains, modeled by partial-differential equations, ordinary differential equations, algebraic equations, combined with input and output connections. To deal with such complex systems, in recent years the class of dissipative port-Hamiltonian (pH) systems has emerged as a very efficient new modeling methodology. The main reasons are that the network based interconnection of pH systems is again pH, Galerkin projection in PDE discretization and model reduction preserve the pH structure and the physical properties are encoded in the geometric properties of the flow as well as the algebraic properties of the equations. Furthermore, dissipative pH system form a very robust representation under structured perturbations and directly indicate Lyapunov functions for stability analysis. We discuss dissipative pH systems and describe, how many classical models can be formulated in this class. We illustrate some of the nice algebraic properties, including local canonical forms, the formulation of an associated Dirac structure, and the local invariance under space-time dependent diffeomorphisms. The results are illustrated with some real world examples.

We consider proper orthogonal decomposition (POD) methods to approximate the incompressible Navier-Stokes equations. Our aim is to get error bounds with constants independent of inverse powers of the viscosity parameter. This type of error bounds are called robust. In the case of small viscosity coefficients and coarse grids, only robust estimates provide useful information about the behavior of a numerical method on coarse grids. To this end, we compute the snapshots with a full order stabilized method (FOM). We also add stabilization to the POD method. We study a case in which non inf-sup stable elements are used for the FOM and a case in which inf-sup stable elements are used. In the last case to approximate the pressure we use a supremizer pressure recovery method.

We show that in case we have some a priori information about the velocity, a POD data assimilation algorithm converges to the true solution exponentially fast improving the accuracy of the standard POD method.

In practical simulations one can apply some given software to compute the snapshots. It could then be the case that a different discretization for the nonlinear term is used in the FOM and the POD methods. We analyze the influence of using different discretizations for the nonlinear term.

Finally, we also analyze the influence of including snapshots that approach the velocity time derivative. We study the differences between projecting onto $L^2$ and $H^1$ and prove pointwise in time error bounds in both cases.

Since the early 1940s, the field of robotics has evidenced a paradigm shift from conventional hard robotics to soft robotics, through the exploration of machines or components with biomimetic dexterous features capable of superseding the ability of humans whilst safely interacting with them. Soft robots are highly nonlinear systems made of highly deformable materials such as elastomers, polymers and other soft matter, that often exhibit intrinsic uncertainty in their elastic responses under large strains due to microstructural inhomogeneity. As a consequence, control of soft robots, potentially actuated by means of a wide spectrum of complex external stimuli (electric or magnetic field, mechanical pressure, osmotic pressure, etc.) is not a trivial task.

This presention will review on modelling, mathematical analysis, control and design of soft materials. The theoretical analysis and numerical modeling relies on the theory of hyperelasticity. In addition to the constitutive model, another aspect of paramount importance in optimal control of soft materials is the choice of the cost functional to be minimized. Tracking-type cost functionals based on distance functions, such as the Hausdorff distance, have been recently proposed as a natural choice in this field. The numerical treatment of uncertainty quantification is another difficulty due, among others, to the well-known phenomenon of the curse of dimensionality. All of these issues will be illustrated in the talk through several concrete examples. Finally, some mathematical challenges in this emergent field will be described.

The original results included in the presentation have been obtained in collaboration with Jesús Martínez-Frutos (UPCT), Carlos Mora-Corral (UAM), Rogelio Ortigosa-Martínez (UPCT) and Pablo Pedregal (UCM).

*T*he numerical simulation of coupled mechanical deformation and fluid flow in porous media has become of increasing importance in several branches of technology and natural sciences. Geothermal energy extraction, CO2 storage, hydraulic fracturing or cancer research are among typical societal relevant applications of poromechanics. In this talk, we will introduce the Biot's model of poroelasticity, and we will focus on the numerical difficulties that appear in its numerical solution. More concretely, we will treat mathematical and practical aspects of models for poroelasticity, with an emphasis on the analysis of stable numerical discretizations and the efficient solution methods for the coupled system of partial differential equations. Robust discretizations with respect to all the physical parameters are needed for this type of problems to obtain reliable numerical solutions. This is a very important task, and some efforts are being carried out in this address by the scientific community. Another important aspect in the numerical simulation of poromechanics problems deals with the efficient solution of the large systems of algebraic equations obtained after discretization. This is the most consuming part when real simulations are performed, and for this reason, a lot of effort has been made in the last years to design efficient solution methods for these problems. In this talk, we address the key points and the recent developments in numerical methods for poromechanics.