L3

It is well known that the Navier-Stokes equations of viscous fluid flow do not give good predictions of when a viscous flow is likely to become unstable.  When classical linearized theory is used to explore the stability of a viscous flow, the Navier-Stokes equations predict that instability will occur at fluid speeds (Reynolds numbers) far in excess of those actually measured in experiments.  In response to this discrepancy, theories have arisen that suggest the eigenvalues computed in classical stability analysis do not give a full account of the behaviour, while others have suggested that fluid instability is a fundamentally non-linear process which is not accessible to linearized stability analyses.

In this talk, an alternative account of fluid instability and turbulence will be explored.  It is suggested that the Navier-Stokes equations themselves might not be entirely appropriate to describe the transition to turbulent flow.  A slightly more general model allows the possibility that the classical viscous fluid flows predicted by Navier-Stokes theory may become unstable at Reynolds numbers much closer to those seen in experiments, and so might perhaps give an account of the physics underlying turbulent behaviour.

Persistent homology is a tool for identifying topological features of (often high-dimensional) data. Typically, the data is indexed by a one-dimensional parameter space, and persistent homology is easily visualized via a persistence diagram or "barcode." Multi-dimensional persistent homology identifies topological features for data that is indexed by a multi-dimensional index space, and visualization is challenging for both practical and algebraic reasons. In this talk, I will give an introduction to persistent homology in both the single- and multi-dimensional settings. I will then describe an approach to visualizing multi-dimensional persistence, and the algebraic and computational challenges involved. Lastly, I will demonstrate an interactive visualization tool, the result of recent work to efficiently compute and visualize multi-dimensional persistent homology. This work is in collaboration with Michael Lesnick of the Institute for Mathematics and its Applications.

LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. During discharge, LiFePO4 material can undergo phase separation, between a highly and lowly lithiated form. Discharge of LiFePO4 crystals has traditionally been modelled by one-phase Stefan problems, which assume that phase separation occurs.

Recent work has been using phase-field models based on the Cahn-Hilliard equation, which only phase-separates when thermodynamically favourable. In the past year or two, this work has been having considerable impact in both theoretical and experimental electrochemistry.

Unfortunately, these models are very difficult to solve numerically and involve large, coupled systems of nonlinear PDEs across several different size scales that include a range of different physics and cannot be homogenised effectively.

This talk will give an overview of recent developments in modelling LiFePO4 and the sort of strategies used to solve these systems numerically.

Acto-myosin network growth and remodeling in vivo is based on a large number of chemical and mechanical processes, which are mutually coupled and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed detailed physico-chemical, stochastic models of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work sheds light on the interplay between the chemical and mechanical processes, and also will highlights the importance of diffusional and active transport phenomena. For example, we showed that molecular transport plays an important role in determining the shapes of the commonly observed force-velocity curves. We also investigated the nonlinear mechano-chemical couplings between an acto-myosin network and an external deformable substrate.

Fix positive integers p and q with 2 \leq q \leq {p \choose 2}. An edge-colouring of the complete graph K_n is said to be a (p, q)-colouring if every K_p receives at least q different colours. The function f(n, p, q) is the minimum number of colours that are needed for K_n to have a (p,q)-colouring. This function was introduced by Erdos and Shelah about 40 years ago, but Erdos and Gyarfas were the first to study the function in a systematic way. They proved that f(n, p, p) is polynomial in n and asked to determine the maximum q, depending on p, for which f(n,p,q) is subpolynomial in n. We prove that the answer is p-1.

We also discuss some related questions.

Joint work with Jacob Fox, Choongbum Lee and Benny Sudakov.

Recent experimental work has determined the atomic structure of a quasicrystalline Cd-Yb alloy. It highlights the elegant role of polyhedra with icosahedral symmetry. Other work suggests that while chunks of periodic crystals and disordered glass predominate in the solid state, there are many hints of icosahedral clusters. This talk is based on a recent Mathematical Intelligencer article on quasicrystals with Marjorie Senechal.

The seminar will be followed by a drinks reception and forms part of a longer PDE and CoV related Workshop.

To register for the seminar and drinks reception go to http://doodle.com/acw6bbsp9dt5bcwb

In elasticity theory, one naturally requires that the Jacobian determinant of the deformation is positive or even a-priori prescribed (for example incompressibility). However, such strongly non-linear and non-convex constraints are difficult to deal with in mathematical models. In this talk, which is based on joint work with K. Koumatos (Oxford) and E. Wiedemann (UBC/PIMS), I will present various recent results on how this constraint can be manipulated in subcritical Sobolev spaces, where the integrability exponent is less than the dimension.

In particular, I will give a characterization theorem for Young measures under this side constraint, which are widely used in the Calculus of Variations to model limits of nonlinear functions of weakly converging "generating" sequences. This is in the spirit of the celebrated Kinderlehrer--Pedregal Theorem and based on convex integration and "geometry" in matrix space.

Finally, applications to the minimization of integral functionals, the theory of semiconvex hulls, incompressible extensions, and approximation of weakly orientation-preserving maps by strictly orientation-preserving ones in Sobolev spaces are given.

Attempting to extend the methods of critical point theory (e.g., those of Morse theory and Lusternik-Schnirelman theory) to the study of strong local minimizers of integral functionals of the calculus of variations I will describe how the obstruction method of algebraic topology can be successfully used to tackle the enumeration problem for various homotopy classes of maps in Sobolev spaces and that how this will result in precise lower bounds on the number of such local minimizers in terms of convenient topological invariants of the underlying spaces. I will then move on to dicussing variants as well as applications of the result to some classes of geometric nonlinear PDEs in particular problems in nonlinear elasticity.