In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the whole blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of poroelastic equations is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the rest of the circulatory system. We discuss well-posedness analysis for this multiscale model, as well as solution methods focused on a detailed comparison between functional iterations and an energy-based operator splitting method and how they handle the interface conditions.

We address an error analysis of an Eulerian finite element method employed to solve a linearized Navier-Stokes problem posed in a time-dependent domain. In this study, we assume the known evolution of the domain, independent of the solution to the problem at hand. The numerical method under consideration combines a standard BDF-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, we employ Nitsche's method to enforce the boundary conditions.
The main focus of this presentation is a convergence analysis of the method for various inf-sup stable velocity--pressure elements.
The analysis provides an optimal order convergence estimate in the energy norm for the velocity component and a scaled $L^2(H^1)$-type norm for the pressure component.

Shape-memory alloys are materials which display a striking thermodynamic behaviour. They undergo a solid-solid phase transition in which symmetry is reduced upon the passage from the high to the low temperature phase. This gives rise to a 'memory' for these materials and to fascinating microstructures determined by a rich energy landscape. In this talk we will approach these structures through the calculus of variations and scaling laws. This allows us to thus rigorously capture some of the complexity of the underlying microstructures. A particular emphasis will be on the analysis of a striking dichotomy between extreme rigidity and flexibility in the mathematical description of these materials.

Spin glasses are models of statistical mechanics in which a large number of elementary units, called spins, interact in a disordered manner. They can serve as simplified models for a broad class of problems encountered in theoretical computer science, combinatorics, or high-dimensional statistics. In the simplest case, called the SK model, there are direct interactions between all pairs of spins. The first part of the talk will present the main ideas underpinning the analysis of this model. However, some models that are very similar remain less well-understood. For instance, one can consider a bipartite version of the SK model, in which the spins are divided into two groups, and direct interactions are only present between spins of different groups. I will highlight some of the challenges posed by models of this type, and discuss some partial progress.