Conference in honour of Ian Sloan on the occasion of his 80th birthday,
17-19 June 2018, UNSW, Sydney, Australia.

## Programme

• Fractional Swift-Hohenberg Equation and Its Application to Modelling of Crystals
Mark Ainsworth, Brown University, USA
• Recent years have seen a surge of interest in fractional partial differential equations driven, in part, by the desire to develop models that are able to more accurately model systems which exhibit behaviour including sub-and super-diffusion. In the current work, we shall discuss a fractional version of the Swift-Hohenberg equation. The Swift-Hohenberg equation is a non-linear parabolic PDE which is known to have solutions display pattern formation, and which has been used in the physics literature to develop macroscopic models for crystals known as "phase field crystals." We consider a fractional version of the Swift-Hohenberg equation (FSHE) which gives a non-linear fractional parabolic problem. In particular, we show that FSHE is well-posed and admits a unique solution; develop a Fourier-Galerkin scheme and obtain error estimates; analyse pattern formation properties and discuss the use of FSHE to develop fractional phase field crystal models. This is joint work with Zhiping Mao (Brown University).

• Lattice rules with a twist
Ronald Cools KU Leuven, Belgium
• Abstract coming soon

• From CFD to Monte Carlo methods -- the joys of a mid-career change of field
Mike Giles University of Oxford, UK
• One noteworthy aspect of Ian Sloan's career is his switch from theoretical physics to computational mathematics. In this talk I will discuss my own mid-career switch from CFD (computational fluid dynamics) to mathematical finance and Monte Carlo methods, with a visit to Ian in UNSW in early 2007 being a key part of my switch.

• Uncertainty quantification in partial differential and integral equation models
Ivan Graham University of Bath, UK
• In late 1975 I received a very enthusiastic aerogramme from Ian Sloan telling me that "he and his collaborators had just discovered some exciting new numerical methods for solving integral equations", and offering me a PhD scholarship to work with him on this at UNSW. Going to Sydney in 1976 to work with Ian was probably the most important decision of my career and I'm pleased to say that we have kept in close touch and have continued to work together down the years. In fact recently some of the things we were working on in 1976 became useful again in the analysis of uncertainty propagation in reactor modelling, an industrial project I've been doing with Wood plc in the UK. I'll talk about this and, more generally, I'll talk about uncertainty quantification for various PDE models and how recent work by Ian and a number of other collaborators on high dimensional integration has allowed us to make substantial progress in this area.

• Pattern-avoiding permutations
Tony Guttmann Melbourne, Australia
• There are n! permutations of the integers 1,2,...,n. We say that a permutation avoids a given sub-permutation (called a pattern) if there is no sub-sequence of the permutation with digits in the same relative order as the digits in the pattern. Thus 1,2,3,4,5 avoids the pattern 2,1 as there are no two integers in decreasing order in the permutation. However 1,3,4,2,5 does not avoid the pattern, as the pairs 3,2, and 4,2 are both in decreasing order. We will describe some open questions, and discuss some numerical methods for exploring them.

• The Advantages of Sampling with Integration Lattices
Fred Hickernell IIT Chicago, USA
• Ian Sloan has been a key developer and proponent of integration lattices for numerical computation. His 1994 monograph has over 700 citations, and his more recent 2013 Acta Numerica article has over 200 citations. This talk highlights how well-chosen integration lattices can speed up numerical multidimensional integration in contrast to tensor product rules and simple Monte Carlo. It also describes recent work sampling with integration lattices when performing Bayesian cubature, where the integrand is assumed to be an instance of a Gaussian random process.

• Minimal Riesz energy problems in Sobolev spaces
Wolfgang Wendland University of Stuttgart, Germany
• The minimal energy problem for nonnegative charges on a closed surface $\Gamma$ in $\mathbb{R}^3$ goes back to C.F.~Gauss in 1839. The corresponding Riesz kernel is then on $\Gamma$ weakly singular. More general, in $\mathbb{R}^n, n\geq 2$, the constructive and numerical solution of minimizing the energy relative to the Riesz kernel then defines on $\Gamma$ a weakly singular single layer boundary integral operator. The construction and numerical solution of minimizing the corresponding energy then provides the distribution of charges on $\Gamma$ whose single layer potential is the solution of the Dirichlet problem inside $\Gamma$. For more general Riesz kernels the corresponding minimizing problem leads to a charge distribution which provides the density of Sloan’s integration points on $\Gamma$. If the surface consists of two separate parts $\Gamma_1$ and $\Gamma_2$ with positive charges on $\Gamma_1$ and negative charges on $\Gamma_2$ then the minimizing charge distribution can be approximated by piecewise constant charges via a Galerkin-Bubnov method. Wavelet matrix compression is applied for solving the discrete system. This is joint work with Helmut Harbrecht (Basel), Günther Of (Graz) and Natalia Zorii (Kiev).