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

## Programme

 Sunday 17 June Monday 18 June Tuesday 19 June 8:30 - 9:00 (* by invitation only) Breakfast 9:00 - 9:30 Tony Guttmann 9:30 - 10:00 James Sethian 10:00 - 10:30 Honorary degree conferral* Nalini Joshi 10:30 - 11:00 Honorary degree conferral* Coffee 11:00 - 11:30 Honorary degree conferral* Ronald Cools 11:30 - 12:00 Morning tea* Douglas Arnold 12:00 - 12:30 Morning tea* Mark Ainsworth 12:30 - 1:00 Lunch 1:00 - 1:30 Registration and lunch Lunch 1:30 - 2:00 Registration and lunch Fred Hickernell 2:00 - 2:30 Opening - Emma Johnston Welcome - Bruce Henry Markus Hegland 2:30 - 3:00 Edward Redish Henryk Wozniakowski 3:00 - 3:30 Wolfgang Wendland Afternoon tea and farewell 3:30 - 4:00 Coffee 4:00 - 4:30 Mike Giles 4:30 - 5:00 Linda Petzold 5:00 - 5:30 Dinner Ivan Graham 5:30 - 9:00 Dinner

Opening from the Dean of Faculty of Science
Emma Johnston UNSW, Sydney.

Welcome from the Head of School of Mathematics and Statistics
Bruce Henry UNSW, Sydney.

Math, Applied Math, and Math in Physics
Edward Redish University of Maryland, USA
In 1971, I was a recently minted mathematical physicist hunting for a future. Along came a visitor from the other side of the world bearing a set of new tools and new ways of mathematical thinking that changed how I thought about complex nuclear reactions. Twenty years later, I transitioned to a new career in which I think about the role math plays in scientific thinking and how “mathematicians and physicists are two disciplines separated by a common language.” I’ll provide reminiscences and examples of what I’ve learned as a mathematical physicist and as a cognitive education researcher.

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).

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.

The Master Clock: Structure and Function
Linda Petzold, UCSB, USA
In the mammalian suprachiasmatic nucleus (SCN), noisy cellular oscillators communicate within a neuronal network to generate precise system-wide circadian rhythms. In past work we have inferred the functional network for synchronization of the SCN. In recent work we have inferred the directionality of the network connections. We discuss the network structure and its advantages for function.

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 University of 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 Mathematics of Moving Interfaces: From Industrial Printers and Semiconductors to Medical Imaging and Foamy Fluids
James Sethian University of California, Berkeley, USA
Moving interfaces appear in a large variety of physical phenomena, including mixing fluids, industrial printers, medical images, and foamy fluids. One way to frame moving interfaces is to recast them as solutions to fixed domain Eulerian partial differential equations, and this has led to a collection of PDE-based techniques, including level set methods, fast marching methods, and ordered upwind methods. These techniques easily accommodate merging boundaries and the delicate 3D physics of interface motion. We will give a brief overview of these methods, and then talk about a few selected new applications from fluids, materials science, and image analysis.

Symmetry through Geometry
Nalini Joshi University of Sydney, Australia
Symmetry is an essential part of our description of the world. The quality of being made up of exactly similar parts facing each other is all around us: one day reflects another and the days fill out the year in the same way that similar hexagonal compartments fill out a honeycomb. The mathematical description of symmetries is built from only two operations: reflections and translations. In two dimensions, these give rise to triangular, hexagonal and square tilings of the plane. But in higher dimensions, many more tiling patterns are available. One of the many questions that arise is how to go from higher dimensional tilings to two dimensional ones. I will show how to use these ideas to link two major theories that arise in mathematical physics.

Lattice rules with a twist
Ronald Cools KU Leuven, Belgium
In 1991 Ian invited me to Sydney to join forces, and in 1992 I went. At that time one of his research passions were lattice rules for periodic functions. Mine was cubature formulas that are exact for trigonometric functions. Together we investigated minimal cubature formulas, and we obtained rules for arbitrary polynomial degrees of precision with free parameters. These are all equal weight rules with points shifted in a particular way. Within this continuum there are lattice rules. So we were both happy with the result. Why it took 4 more years before the paper appeared in print is another story. In a series of papers that started at an Oberwolfach meeting in 1992, Ian and Harald Niederreiter modified lattice rules for non-periodic function. They modified the weights and kept the points. Dirk Nuyens and I picked up this thread for the occasion of Ian's 80th birthday.

Spiky eigenfunctions
Douglas Arnold, University of Minnesota, USA
From our first encounters with partial differential equations and Fourier analysis, we encounter eigenfunctions of elliptic operators which are oscillatory and global: sines, cosines, Bessel functions, and so forth. But when the coefficients of the PDE are not smooth but disordered, an entirely different sort of eigenfunction appears: spiky and highly localized. Such localization is highly relevant in physical applications, especially in quantum mechanics. A large body of mathematics has been developed to explain it, but many aspects of localization remain mysterious. In particular, most known results are probabilistic, providing statistical information for a class or operators with random coefficients, but not information specific to a particular realization of such an operator. We will describe a new and very different perspective on localization. The new viewpoint is deterministic, enabling us, for the first time, to deduce the localized spectrum of a disordered operator without actually calculating its eigenfunctions and eigenvalues. This approach holds out the promise of harnessing disorder to create devices and materials with new and desired properties.

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).

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.

The combination technique for high dimensional approximation
Markus Hegland ANU, Australia
Ian Sloan has made substantial contributions to high dimensional quadrature and approximation which include the establishment of quasi Monte Carlo Methods, and results about the ANOVA decomposition of functions. Combining ideas from algebra and analysis he and his collaborators have been able to control the curse of dimensionality to a large extent. He has actively nurtured Australian research in high-dimensional computing, and the development of sparse grid approximation. Sparse grids have been used for the solution of differential equations and recently in uncertainty quantification and fault-tolerant algorithms. In my presentation I will review some of the development of the sparse grid combination technique and its foundations in algebra, extrapolation and ANOVA decompositions.

Ian Sloan and Tractability
Henryk Wozniakowski Columbia University, USA and University of Warsaw, Poland
I will try to summarize over 25 years of our collaboration and friendship with Ian Sloan. Most of our papers deal with tractability of multivariate problems such as multivariate integration and approximation. I will cite the results of our first and last paper.