MPGP 2015

Warning: The descriptions may be totally off and contain tons of errors since there was at most one talk that I’d feel somewhat comfortable talking about myself.

  • Existence/uniqueness for a crystalline curvature flow (Antonin Chambolle): Proves existence with some level set methods that looked interesting; however, I have to admit I couldn’t really follow the talk when it became technical (which was pretty early on)
  • Numerical Approximation of Interacting Particles in Lipid Membranes (Ralf Kornhuber): There were different approaches on how to deal with particles (e.g. receptors/proteins in general) that are in a membrane and how their orientation, rotation, etc. affects the PDEs’ domains and thus their behavior. Apparently this is pretty hard even after simplifying assumptions like only looking at a graph-like section of the membrane and neglecting some physical forces.
  • An Adaptive Moving Mesh Method for Geometric Evolutions Laws and Bulk-Surface PDEs (John MacKenzie): This talk was pretty “FEM-heavy” and the one of the few things I understood was that one shouldn’t naively choose the number of grid points on a surface proportional to its curvature since some serious bunching occurs. To avoid that (I’m not sure if that was the speaker’s idea, but he talked about it quite a bit) one introduces some kind of floor function for the mesh node distributions which yields better numerical results.
  • Numerical tools for inter- and extrapolation in the space of shells (Benedikt Wirth): Based on a scheme to evaluate Bezier curves (pointwise) given the concept of Bezier curves (and hypersurfaces in general) has been generalized to Riemannian manifolds, using the log map as distance function. After playing around a little bit one can simplify a lot (under the assumption of the hypersurface being at least C1).
  • Geometric graph-based methods for high dimensions (Andrea Bertozzi): That one was particularly interesting since it combined common methods from PDEs and the spectral embedding framework from machine learning. I will try to read the related papers as soon as possible and write more about it then. As an intermediary comment: Some kind of mincut can be found by evolving a phase field model by the flow induced by the Ginzburg-Landau functional. The speaker tried a discretization and what is surprising is that after only very few iterations (~4 if I remember correctly) this method produces great results (~96% on the MNIST dataset, given only ~3%(???) as training data). [Warning: Numbers may be completely mixed up]
  • Finite Elements on polytopic meshes and beyond (Andrea Cangiani) & A Stable Finite Element Approximation for the Dynamics of Fluidic Biomembranes (John Barrett): I’ll be honest and admit I didn’t understand anything of those two talks (very FEM-heavy).
  • Shape control through active materials: some case studies with applications to manipulation and locomotion (Antonio DeSimone): Great and very engaging talk. Started off with wondering how snakes can move if the only forces they experience are friction, translated that into PDEs and showed how the spontaneous curvature results in directed movement. Then this has been used to analyze how certain cells are moving.
  • Diffuse interface methods for bulk-surface variational problems (Martin Burger): Various smaller topics, mainly error analysis which was quite technical. Particular attention has been paid to low frequency perturbations and apparently their model is quite robust against those.
  • Numerical methods for large bilayer bending problems (Sören Bartels): Given two surfaces on top of each other that react (i.e. expand) differently to some external input (heat, electricity) they looked for a model describing that kind of behaviour. If I remember correctly they looked for a model that didn’t produce “cat ears” when evolving a rectangular surface. Gamma convergence has been shown and their model has some Laplacian of curvature in it, so we get a 4th order, nonlinear system of PDEs which are generally not understood too well.
  • Anisotropic mean curvature flow as a formal singular limit of the nonlinear bidomain and multidomain models (Giovanni Bellettini): I wouldn’t call it “technical”, but it was by far the most “mathematical” talk, in the sense of following a typical definition>theorem>proof style, so it is pretty hard to summarize.
  • Minimising a relaxed Willmore functional for graphs subject to Dirichlet boundary conditions (Klaus Decklenick): Given some closed surface the Willmore functional (surface integral over the mean(?) curvature) can be shown to be something like 2π with some power depending on the dimension. In this scenario the speaker considered the case of Dirichlet boundary conditions (in particular not a closed surface anymore; closed in the sense of nonexisting boundary in the manifold’s induced topology) in the restriction to a graph-like setting and proved several estimates.
  • A phase field model for Willmore’s energy with topological constraint (Patrick Dondl): Also a very engaging talk that basically dealt with the question on how to formulate some “topological invariance” (no splitting into disjoint subsets) of phase field models into a penalty functional.
  • Mathematical models for tissue growth (John King): It was that day’s last talk and about 5-10 variables have been introduced in the first few slides so I didn’t really get anything.
  • A coupled surface-Cahn-Hilliard bulk-diffusion system modelling lipid raft formation in cell membranes (Harald Garcke): This model is a funny one because many terms have been chosen such that it everything converges against something non trivial and physical interpretations have been found afterwards (at least that was my impression).
  • On anisotropic Willmore Flow (Paola Pozzi): Similar to Bellettini’s talk this also made heavy use of dual maps in order to extend the definition of an anisotropic Willmore flow (for strictly convex unit balls in the metric induced by some Minkowski functional) in such a way that the dual of the unit ball’s surface behaves like an n-sphere, i.e. (for a prescribed volume) it minimizes the (generalized) Willmore functional. Doing FEMs with this generalized Willmore functional seems to be incredibly hard though because of many, many nonlinearities that occur.
  • Finite element methods for the stochastic mean curvature flow (Xiaobing H. Feng): Additive and multiplicative perturbations (white noise independent or dependent on the solution, repectively) have been dealt with and some error bounds were established (uniformly in time and H^3 in space if I remember correctly). Apparently the Ito calculus doesn’t seem to be the “right” choice, instead the Stratonovich integral has been used (chain rule of ordinary calculus still holds). The problem with multiplicative white noise seems to be that it will change the sign and which makes error estimates pretty hard. From what I’ve understood one thus adds some term delta, so that f(u)(eps W’) becomes f(u)(delta+eps W’), proves that for delta going to zero this gives the correct solution and then proves error estimates only for eps being smaller than sqrt(2delta) or something like that.
  • Coupled bulk-surface free boundary problems arising from a mathematical model receptor ligand dynamics (Charlie Elliott): This talk started off with a video of yesterday’s bonfire night showing some politicians and popes getting incinerated. Followed up by some technical difficulties it got quite mathematical after all and again I couldn’t really follow since it already built on some prior work one should have been familiar with. However, it looked like it’d be very interesting if you could follow it.