**Title:** A Reynolds-robust preconditioner for the 3D stationary Navier–Stokes **Abstract:** When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed. Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased. Building on the work of Schöberl, Olshanskii and Benzi, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier–Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines a tailored finite element discretisation, augmented Lagrangian stabilisation, a custom prolongation operator involving local solves on coarse cells, and an additive patchwise relaxation on each level that captures the kernel of the divergence operator. We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds number 10 to 5000. **Title:** Computing disconnected bifurcation diagrams of partial differential equations **Abstract:** Computing the distinct solutions $u$ of an equation $f(u, \lambda) = 0$ as a parameter $\lambda \in \mathbb{R}$ is varied is a central task in applied mathematics and engineering. The solutions are captured in a bifurcation diagram, plotting (some functional of) $u$ as a function of $\lambda$. In this talk I will present a new algorithm, deflated continuation, for this task. Deflated continuation has three advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to an existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available. Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier’s problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem. We will also apply it to discover previously unknown solutions to equations arising in liquid crystals and quantum mechanics.

Speaker: **Roman Shvydkoy** (University of Chicago)

Title:** Lectures on alignment models of collective behavior**

Abstract: In this series of lectures we present basic principles of emergent dynamics in systems governed by laws of self-organization. Such systems arise in a variety of applications including biological (swarming behavior of animals), social (opinion dynamics, social networks) and technological contexts (cosmology, robotics, etc). A particular focus will be placed on the so-called Cucker-Smale models, which encode one of the simplest communication protocols that lead to two fundamental phenomena of collective action: alignment and flocking. We will start with elementary proofs of alignment in various scenarios involving local and global communications kernels, address the issues of collision between agents.

Mean-field limit and hydrodynamic models will be presented in the second part of the lectures, and we address the problems of global well-posedness, long time behavior, and stability of flocks on the macroscopic (large crowd) level. Finally, we discuss singular and topological kernels within the same Cucker-Smale context, which have been recently introduced to demonstrate how global collective phenomena emerge from purely local communication. These models present many new challenges to the regularity theory of fractional parabolic PDEs, which we will also discuss if time

permits.

Speaker: **Roman Shvydkoy** (University of Chicago)

Title: **Lectures on alignment models of collective behavior**

Abstract: In this series of lectures we present basic principles of emergent dynamics in systems governed by laws of self-organization. Such systems arise in a variety of applications including biological (swarming behavior of animals), social (opinion dynamics, social networks) and technological contexts (cosmology, robotics, etc). A particular focus will be placed on the so-called Cucker-Smale models, which encode one of the simplest communication protocols that lead to two fundamental phenomena of collective action: alignment and flocking. We will start with elementary proofs of alignment in various scenarios involving local and global communications kernels, address the issues of collision between agents.

Mean-field limit and hydrodynamic models will be presented in the second part of the lectures, and we address the problems of global well-posedness, long time behavior, and stability of flocks on the macroscopic (large crowd) level. Finally, we discuss singular and topological kernels within the same Cucker-Smale context, which have been recently introduced to demonstrate how global collective phenomena emerge from purely local communication. These models present many new challenges to the regularity theory of fractional parabolic PDEs, which we will also discuss if time permits.

Speaker: **Anna Marciniak-Czochra** (Applied Analysis and Modelling in Biosciences www.biostruct.uni-hd.de)

Title:** Post-Turing tissue pattern formation: Insights from mathematical modelling**

Abstract: Cells and tissue are objects of the physical world, and therefore they obey the laws of physics and chemistry, notwithstanding the molecular complexity of biological systems. What are the mathematical principles that are at play in generating such complex entities from simple laws? Understanding the role of mechanical and mechano-chemical interactions in cell processes, tissue development, regeneration and disease has become a rapidly expanding research field in the life sciences.To reveal the patterning potential of mechano-chemical interactions, we have developed two classes of mathematical models coupling dynamics of diffusing molecular signals with a model of tissue deformation. First we derived a model based on energy minimisation that leads to 4-th order partial differential equations of evolution of infinitely thin deforming tissue (pseudo-3D model) coupled with a surface reaction-diffusion equation. The second approach (full-3D model) consists of a continuous model of large tissue deformation coupled with a discrete description of spatial distribution of cells to account for active deformation of single cells. The models account for a range of mechano-chemical feedbacks, such as signalling-dependent strain, stress, or tissue compression. Numerical simulations show ability of the proposed mechanisms to generate development of various spatio-temporal structures. We compare the resulting patterns of tissue invagination and evagination to those encountered in developmental biology. We discuss analytical and numerical challenges of the proposed models and compare them to the classical Turing patterns as well as reaction-diffusion ODE models coupling diffusion-based cell-to-cell communication with intracellular signalling.

]]>Speaker: **K.R. Rajagopal**

Title: **How should the Aorta be modeled?**

Speaker: **Magnus Svärd**

Title: **On the physicality of the Navier-Stokes equations**

Abstract: In the first part of the talk, a number of physically problematic features of the compressible Navier-Stokes equations will be discussed. These problems are caused by the way viscous effects are introduced into the model via a Lagrangian frame.

In the second part, a new model for viscous and heat conducting flows is proposed, that does not suffer from the same drawbacks as the standard Navier-Stokes equations. It is obtained by modelling diffusion, rather than viscosity, in an Eulerian frame. It can also be derived from the Boltzmann equation, if the latter is defined in an Eulerian frame.

]]>Speaker:** Christoph Allolio**

Title: **The Interplay of Membrane Curvature Elasticity, Charge and Specific Molecular Interactions**

Abstract: Molecular species, such as ions and peptides, can have dramatic effects on shape and elasticity of lipid membranes. A case in point is calcium and its crucial role in vesicle fusion[1], another example is the entry mechanism of cell penetrating peptides.[2] Using molecular dynamics, calcium and cell penetrating peptide mediated vesicle fusion were simulated in atomistic detail.[2] With a new, local approach[3] it became possible to extract the effect of the adsorbates on curvature elastic properties from the simulations. The results indicate that calcium specifically induces negative spontaneous curvature in those anionic lipid membranes which are are known to be susceptible to calcium mediated fusion. The molecular basis of curvature induction by cationic adsorbates is shown to be the induction of local stress through clustering.

In the following, I give an outline of my research program. The aim of the program is to use my simulation-based approach to advance the continuum theory of membrane elasticity. Specifically, the way proteins influence membrane properties is currently not well integrated with the Helfrich[4] theory. The unknown coupling terms are to be determined by generalization from specific examples. I introduce example systems chosen to represent the most important modes of interaction, but also address questions of practical relevance.

[1] E. Neher, R. Schneggenburger, Nature 406,889-893 2000

[2] C. Allolio, A. Magarkar, P. Jurkiewicz, K. Baxova, M. Javanainen, R.Sachl, M. Hof, D. Horinek, V. Heinz, R. Rachel, C. Ziegler, M. Cebecauer, A. Schrofel, P. Jungwirth, PNAS, 115, 11923-11928 2018

[3] C. Allolio, A. Haluts, D. Harries, Chem. Phys 514, 31-43 2018

[4] W. Helfrich, Z. Naturforsch. 28, 693-703 1973

Speaker: **Petr Stehlík** (University of West Bohemia)

Title: **Various solution types for reaction-diffusion equations on lattices and graphs**

Abstract: We discuss reaction-diffusion equations on lattices and graphs. The motivation comes both from theory (partial discretization) as well as from applications (especially mathematical ecology and population dynamics). We study and focus on properties which distinguish the discrete-space problem from the standard problem on continuous domain. This concerns mainly two solution types. First, we study the rich world of spatially heterogeneous stationary solutions and their stability. In the second part, we introduce the concept of bi- and multichromatic travelling waves on lattices. We observe that both problems are closely related to an interesting algebraic question regarding indefinite perturbations of semidefinite matrices.

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