## Autumn 2018

All seminars (unless otherwise stated) will take place on **Tuesdays at 3.00pm in ****Room B15 in the Darwin Building** - see the map for further details. There will be tea afterwards in **Maths Room 606** (6th Floor, 25 Gordon Street) - see map for further details. If you require any more information on the Applied seminars please contact Prof Jean-Marc Vanden-Broeck (e-mail: j.vanden-broeck AT ucl.ac.uk or tel: 020-7679-2835) or Prof Ilia Kamotski (e-mail: i.kamotski AT ucl.ac.uk or tel: 020-7679-3937).

## 09 October 2018

### Speaker: Miss Belgin Seymenoglu (UCL)

##### Title: Invariant manifolds of models from Population Genetics

**Abstract: **

In the past four years, I investigated two continuous-time models in classical Population Genetics: one is a fertility-mortality model proposed by Nagylaki and Crow, while the other is the Selection-Recombination model. After making many phase plots for both systems, I (almost) always found a stubborn curve or surface, which are so-called "invariant manifolds". I derived conditions for the manifold to exist in each model. You can also look forward to a gallery of colourful phase plots showing that the manifold in the Nagylaki-Crow model need not be unique, smooth, decreasing or convex.

If time permits, I will also discuss the special case of the Selection-Recombination model with the recombination rate set to zero, whose equations of motion are identical to replicator dynamics from Game Theory.

## 16 October 2018

##### See the Departmental Colloquium webpage

## 23 October 2018

### Speaker: Dr Charlotte Perrin (Université d'Aix-Marseille)

##### Title: Compression effects in heterogeneous media

**Abstract: **

This talk addresses the mathematical analysis of fluid models including a maximum packing constraint. These equations arise naturally for instance in the modeling of mixtures like suspensions or in the modelling of collective motion. I will present recent results on two classes of PDEs systems which correspond to two modeling approaches: the "soft" approach based on compressible equations with singular constitutive laws, pressure and/or viscosities, close to the maximal constraint, and the "hard" approach based on a free boundary problem between a congested domain with incompressible dynamics and a free domain with compressible dynamics.

## 30 October 2018

### Speaker: Professor Sheehan Olver (Imperial College London)

##### Title: Sparse spectral methods for PDEs on triangles with multivariate orthogonal polynomials

**Abstract: **

Univariate orthogonal polynomials have a long history in applied and computational mathematics, playing a fundamental role in quadrature, spectral theory and solving differential equations with spectral methods. Unfortunately, while numerous theoretical results concerning multivariate orthogonal polynomials exist, they have an unfair reputation of being unwieldy on non-tensor product domains, and their use in applications has been limited. In reality, many of the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate orthogonal polynomials, including the existence of Jacobi operators, fast evaluation of expansions using Clenshaw’s algorithm and the ability to construct sparse partial differential operators, a la the ultrapsherical spectral method [Olver & Townsend 2012]. We demonstrate these computational aspects using multivariate orthogonal polynomials on a triangle, including the fast solution of general partial differential equations.

## 06 November 2018

##### READING WEEK - NO SEMINAR

## 13 November 2018

### Speaker: Professor Herbert Huppert (Cambridge University)

##### Title: Fluid and elasto dynamics of flows through rocks: fracking and dyking.

**Abstract: **

After a short introduction to the formation of dykes in the Earth and the mechanisms and politics of fracking, the talk will concentrate on the fluid mechanics and elastodynamics of driving fluid into cracks and the quite different response when the pressure is released and the fluid flows back out. Development of the governing equations will be presented along with their numerical solution and asymptotic analysis in various useful limits.

A desktop experiment will be performed along with videos of more carefully controlled laboratory experiments and the results successfully compared with the theoretical predictions. Directions for further research will be outlined at the conclusion of the talk.

## 20 November 2018

### Speaker: Dr Mark Blyth (University of East Anglia)

##### Title: Critical free-surface flow over topography

**Abstract: **

Two-dimensional free-surface flow over a localised bottom topography at critical Froude number (F=1) is examined with an emphasis on calculating steady, forced solitary-wave solutions. In particular we focus on the case of a Gaussian topography. We study the flow in the weakly-nonlinear limit by way of the forced KdV equation, but also compute some fully nonlinear solutions using a conformal mapping method. Boundary-layer theory is used to construct asymptotic solutions in appropriate limits.

One point of interest here is an internal boundary layer which mediates a change from an initial exponential decay of the free-surface to algebraic decay in the far-field. (Algebraic far-field decay is a characteristic feature of steady critical flow solutions.) Many solution branches are identified including branches with multiple waves trapped over the main part of the topography, which cannot be described by boundary-layer theory. Solutions on the first few branches are also found for the fully nonlinear problem. The stability of the steady solutions is also considered.

## 27 November 2018

### Speaker: Dr Ricardo Bardos (Loughborough University)

##### Title: Large amplitude internal waves in multi-layered fluids

**Abstract: **

In this talk we adopt the strongly nonlinear theory to study large amplitude internal waves in two basic configurations: a two-layer fluid with a top free surface; a three-layer fluid confined between two rigid walls. In both cases, solitary-wave solutions are governed by a Hamiltonian system with two degrees of freedom. We will be considering ranges of speeds for which solitary waves correspond to homoclinic orbits at a saddle-center. Focus will be given to the three-layer configuration in certain regimes relevant to real oceanic applications and laboratory experiments, for which the richness of the dynamical systems becomes apparent. New classes of solutions, characterised by multi-humped profiles, are revealed in the case when the stratification is weak and the density transition layer is thin. In contrast with classical one-hump solutions described by the KdV theory, such solutions cannot exist for a continuum set of speeds. Our analytical predictions based on asymptotic theory are then corroborated by a numerical study of the full dynamical system.

## 04 December 2018

### Speaker: Dr Tao Gao (University of Bath)

##### Title: Hydroelastic Waves - From Mathematics to Sea Ice

**Abstract: **In the autumn of 2017, a workshop on sea-ice phenomenon was held at Isaac Newton Institute for mathematical sciences in Cambridge. The programme covered a range of topics in floating ice such as multi-scale modelling of ice characteristics and behaviour, ice-fluid interaction, ice-structure interaction, ice fracture and cracks and etc., cf. Smith et al. (Phil. Trans. R. Soc. A. 2018). In this talk, we focus on the problem of hydroelastic wave which is mainly concerned with the interactions between deformable ice sheets and water flows beneath. The recent advances in modelling the deformation of a thin ice sheet are presented. They are followed by an introduction to the linear and weakly nonlinear theory, and the fully nonlinear computation. Some analytic and numerical results are to be discussed.

## 11 December 2018

### Speaker: Professor Paul Milewski (University of Bath)

##### Title: Understanding the Complex Dynamics of Faraday Pilot Waves

**Abstract: **

Faraday pilot waves are a hydrodynamic structure that consists a bouncing droplet which creates, and is propelled by, a Faraday wave. These pilot waves can behave in extremely complex ways, are a macroscopic wave-particle entity, and result in dynamics which mimic behaviour usually thought to be unique to quantum mechanics. I will show some of this fascinating behaviour and will present a surface wave-droplet fluid model that captures many of the features observed in experiments, focussing on the statistical emergence of complex states.