Confessions of an industrial mathematican Chris Budd

What can industry learn from maths? What can maths learn from industry? Why does this matter, and is it worth the effort?

Two perspectives on applied maths Some ‘commonly held views’ Maths is useless and is best kept that way Applied maths is bad maths Industrial maths is even worse than applied maths and is only done for the money! All mathematicians are mad

My own view: Almost all maths can be applied almost everywhere almost surely … And this simple fact is truly amazing!!!! We can learn lots of new maths from almost all applications Calculus, Fourier analysis, Nonlinear Dynamics Applied maths is a two way process of learning new ideas and transferring them from one application to another.. And this is hard!!

Maths appears in rocks Swallow tail catastrophe

Is maths also present in human behaviour?

Good applications of maths can change the world Vectors, Maxwell, Radio, FFT, digital revolution, computers GoogleMatrices, eigenvalues

How does industry fit into all of this? Can maths be of any possible use in industry?

Traditional industrial users of maths are Telecommunications, aerospace, power generation, iron and steel, mining, oil, weather forecasting, security, finance But they equally well be … Retail, food, zoos, sport, entertainment, media, forensic service, hospitals, air-sea-rescue, education, transport, risk, health, biomedical, environmental agencies

All have problems which can potentially be formulated, and solved using mathematics Maths connects with all areas and knows no bounds! Too few people recognize that the high technology so celebrated today is essentially a mathematical technology Edward David, ex-president of Exxon R&D

Success story for CJB:Non-smooth dynamics But we can also learn new maths from industry! How do things rattle, bounce and slide?

For example: Aircraft undercarriage

Impact oscillators: the simplest non-smooth system obstacle

Novel bifurcations as parameters vary. Period doubling Grazing Theory Experiment

Period-adding route to chaos Transition to a periodic orbit Non-impacting orbit

Impacts and complexity in human behaviour

Scramble crossing

Escape from a lecture theatre!

Sona African sand patterns Used to tell stories (3,4) (2,4) Another example: Some early maths from the entertainment industry

(4,8) Almost identical to Celtic Knot designs

(2,2) 2 (3,2) 1 (5,3) 1 (4,4) 4 How many paths are needed? HCF … proved by a geometrical version of Euclids Algorithm

Chased Chicken Design What patterns can we see here?

But what are the problems of working with industry? What industry wants What do universities want Short term solutions Confidentiality Money Long term and deep research Open publication Training of young people Seem irreconcilable.. But there is a middle way!

Study groups: a tale of zoos and fish Study Group Model (in use all over the world) Bring academics and industrialists together Pose industrial problems on the first day Work on the problems for a week in teams Present a paper at the end Follow up with longer term projects

A wonderful way to Make new contacts Find really good research problems Train students and staff Get great examples for undergraduate teaching Make a fool of yourself in public! ESGI, ECMI, MITACS, PIMS, Australia …

= Example: Artis Zoo Amsterdam

So why bother? The challenges of industry make us think ‘out of the box’ and address new challenges Maths knows no bounds.. And … The maths needed to solve and drive industrial problems is boundless Pointless to differentiate between pure and applied maths!

We live in interesting times with the way that we apply mathematics in a process of great transition! 20 th century.. Great drivers of applied maths are physics, engineering and more recently biology Expertise in …. Fluids Solids Reaction-diffusion problems Dynamical systems Signal processing

Usually deterministic Continuum problems, modelled by Differential Equations Solutions methods Simple analytical methods eg. Separation of variables Approximate/asymptotic approaches Phase plane analysis Numerical methods eg. finite element methods PDE techniques eg. Calculus of variations Transforms: Fourier, Laplace, Radon Still pose MAJOR challenges eg. Exponential asymptotics

What are the drivers of the 21 st century applications? Information/Bio-informatics/Genetics? Commerce/retail sector? Complexity? What new techniques do we need to consider? Discrete maths Data and data assimilation Stochastic methods Very large scale computations Complex systems and networks Optimisation (discrete and continuous)

We cannot afford to differentiate between pure and applied maths either in research or in teaching if we are to meet these challenges in the future!

Example: From Farm to Fork and Beyond Maths and the food industry We use maths to help grow, store, freeze, defrost, transport, cook, eat and digest food Maths can do lots of what if? experiments in complete safety

Example : Finding land mines Land mines are hidden in foliage and triggered by trip wires Land mines are well hidden.. we can use maths to find them

Find the trip wires in this picture

Digital picture of foliage is taken by camera on a long pole Radon transfor m x y f(x,y) R(ρ,θ) Points of high intensity in R correspond to trip wires θ ρ Isolate points and transform back to find the wires Radon Transform x x x

Mathematics finds the land mines! Who says that maths isn’t relevant to real life?!?