The *South Wales Analysis and Probability Seminar (SWAP)* is a regional seminar bringing together mathematicians based in South Wales, with interests in analysis, probability, PDEs, SDEs, dynamical systems, and other related fields.

The next meeting is scheduled for **14 December 2018** at **Swansea University**. The schedule is:

10:00-10:30 | Coffee & Welcome | |

10:30-11:05 | Carlo Mercuri (Swansea) |
Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations |

11:10-11:45 | Baptiste Morisse (Cardiff) |
The onset of instabilities for non hyperbolic systems |

11:45-13:15 | Lunch | |

13:15-13:50 | Andrew Neate (Swansea) |
Semiclassical diffusions |

13:55-14:30 | Junyong Zhang (Cardiff) |
Strichartz estimate and Schrödinger operator in a conic singular space |

14:30-15:00 | Coffee | |

Biomath Colloquium: |
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15:00-16:00 | Chandrasekhar Venkataraman (Sussex) |
Multiscale modelling of biological problems |

Carlo **Mercuri** (Swansea) - Quantitative symmetry breaking of groundstates for a class of weighted Emden-Fowler equations

Joint work with Ederson Moreira Dos Santos.

We prove that symmetry breaking occurs in dimensions N\geq 3 for the groundstate solutions to a class of Emden-Fowler equations on the unit ball, with Dirichlet boundary condition. We highlight this phenomenon considering large values of a certain exponent for a radial weight function appearing in the equation, which is reminiscent of the classical Hénon equation. The particular weight function we consider possesses a spherical shell of zeroes centred at the origin and of radius R. A quantitative condition on R for this symmetry breaking phenomenon to occur is given by means of universal constants, such as the best constant for the subcritical Sobolev embedding. Moreover, combining energy estimates and Liouville type theorems we study some qualitative and quantitative properties of the groundstate solutions.

Baptiste **Morisse** (Cardiff) - The onset of instabilities for non hyperbolic systems

We consider systems of first-order, quasilinear PDEs. In the case of (strongly) hyperbolic systems, it is well-know that the system is well-posed in Sobolev spaces. We study here the cases where hyperbolicity fails – that is, the spectrum of the symbol of the operator is non real. We disprove the Holder-type estimate of the propagator, even in Gevrey regularity, by constructing families of analytical solutions which present the typical growth in time and frequency linked to the particular failure of hyperbolicity we deal with. The talk is aimed to be self-contained and concise in order for everyoneto follow.

Andrew **Neate** (Swansea) - Semiclassical diffusions

We consider some examples of diffusion process associated with the semiclassical limits of coherent states for the harmonic oscillator and Coulomb potentials. The drifts in these diffusions can be viewed as arising from constrained Hamiltonian systems with complex valued Hamiltonians. The corresponding classical systems have periodic solutions, and by approximating our processes with appropriate asymptotic series, we can consider them as examples of random periodic processes.

Junyong **Zhang** (Cardiff) - Strichartz estimate and Schrödinger operator in a conic singular space

In this talk, we will discuss a Schrodinger operator in a conic singular space and the Strichartz estimate for the dispersive equation associated with this operator. The main techniques are the microlocal analysis method and spectral analysis argument. This is a joint work with Jiqiang Zheng (Institute of Applied Physics and Computational Mathematics, China).

Chandrasekhar **Venkataraman** (Sussex) - Multiscale modelling of biological problems

Problems involving multiple scales are ubiquitous in biology and their modelling presents a number of mathematical and computational challenges. In this talk we discuss two such problems.

In the first part of the talk, we consider problems from theoretical ecology for the dynamics of two competing species that reside in a heterogeneous environment. In particular, we wish to understand the role played by heterogeneous motility on invasion behaviour in mathematical models for competition between motile species. We study the effect of rapidly oscillating periodic motilities while performing simultaneous homogenization and strong competition limits. The limit problem is shown to be a free boundary problem of Stefan type with effective coefficients. We will also discuss the implications our analysis has on invasiveness of a species. The results will be supported by numerical simulations.

In the second part of the talk, we consider the derivation, analysis and simulation of mathematical models for cellular signalling processes in biological tissues. A coupled system of nonlinear bulk-surface partial differential equations is used to model the dynamics of signalling molecules in the inter- and intra-cellular spaces as well as for the cell membrane resident receptors. Using multiscale analysis techniques, we derive a macroscopic two-scale model for signalling processes defined on the tissue level. A two-scale numerical method is developed and implemented for simulations of the macroscopic bulk-surface problem and numerical results will be presented illustrating the role cell scale heterogeneities play in the dynamics of macroscopic concentrations.

Jonathan Ben-Artzi (Cardiff)

Dmitri Finkelshtein (Swansea)