The discovery of hyperbolic geometry in the nineteenth century helped to usher in a mathematical revolution, giving rise to new ways of mapping and analyzing curved surfaces. Such “non-Euclidean geometry,” now underlies the general theory of relativity and thus our understanding of the universe as a whole.

If the cosmos may be a hyperbolic manifold, at the molecular level carbon atoms can assemble into hyperbolic lattices, giving rise to exotic new materials. Meanwhile, on the Great Barrier Reef, the corals making hyperbolic structures are being threatened by global warming and the human deluge of carbon into our oceans.

In this multifacted talk bridging the domains of mathematics and culture, science writer and exhibition curator Margaret Wertheim discusses the story of hyperbolic space. How do hyperbolic forms arise in nature, in technology, and in art? And what might we learn about alternative possibilities for being from a mathematical discovery that redefined our concept of parallel lines.

**About the speaker
**Margaret Wertheim is an internationally noted science writer and exhibition curator whose work focuses on relations between science and the wider cultural landscape. The author of six books, including The Pearly Gates of Cyberspace, a groundbreaking exploration of the history of Western concepts of space from Dante to the Internet, she has written for the New York Times, Los Angeles Times, The Guardian, and many other publications. She is a contributing editor at Cabinet, the international arts and culture journal, where she often writes about mathematics.

Wertheim is the founder and director of the Institute For Figuring, a Los Angeles-based organization devoted to the aesthetic and poetic dimensions of science and mathematics. (www.theiff.org) Through the IFF, she has designed exhibitions for galleries and museums in a dozen countries, including the Hayward Gallery in London and the Smithsonian’s National Museum of Natural History in Washington D.C. At the core of the IFF’s work is the concept of material play, and a belief that abstract ideas can often be embodied in physical practices such as paper folding and crochet. By inviting audiences to play with ideas, the IFF offers a radical approach to maths and science engagement that is at once intellectually rigorous and aesthetically aware.

The IFF’s “Crochet Coral Reef” project, spearheaded by Margaret and her twin sister Christine – is now the largest participatory science-and-art endeavor in the world, and has been shown at the Andy Warhol Museum (Pittsburgh), Science Gallery (Dublin), New York University Abu Dhabi (UAE), and elsewhere. Through an unlikely conjunction of handicraft and geometry, the Crochet Coral Reef offers a window into the foundations of mathematics while simultaneously addressing the issue of reef degradation due to global warming. Wertheim’s TED talk on the topic has been viewed more than a million times, and translated into 20 languages, including Arabic.

In 2012 Wertheim served as the University of Southern California’s inaugural Discovery Fellow, designing participatory programming that engaged students across the campus from science, engineering and arts faculties. A highlight of the project was building a giant model of a fractal out of 50,000 business cards. Wertheim is currently Vice Chancellor Fellow for Science Communication at the University of Melbourne.

]]>1) What are the most interesting “big questions” in your field? And what kind of problems are you interested in broadly in the field? (maths as a whole)

My current research areas are under the umbrella of Applied Probability. Some of the big questions in this field include (a) developing numerical methods for stochastic differential equations and other continuous-time, continuous-space stochastic models, (b) rare-event simulations, (c) simulation optimisation, and (d) computing stationary performance measures of stochastic systems. Topics (a), (b) and (d) interest me the most.

What are some other areas of maths that are particularly interesting to you?

Besides Probability, Discrete Mathematics is my other great love, in particular Combinatorics and Graph Theory. I spent my Honours and PhD years studying the Hamiltonian Cycle Problem using tools from both Discrete Mathematics and Probability. Elegant and centuries-old, the HCP is connected to the Traveling Salesman Problem and, more broadly, to the unsolved P vs NP problem.

Why did you become a mathematician?

Growing up I got a lot of exposure to mathematics, as my father was a theoretical statistician. When it was time to decide my university major, I was fortunate enough to meet Professor Jerzy Filar, who encouraged and guided me through my undergraduate and postgraduate studies. Without my father and Professor Filar, I wouldn’t have become a mathematician.

Do you have any advice for future mathematicians?

When asked if he had any advice for up-and-coming chess players, the late Australian mathematician Greg Hjorth said, “Floss *before* brush.” I think that advice is good for future mathematicians as well.

Also, make sure you keep learning new tools while consolidating your expertise, because great results usually come from making unexpected combinations of insights from different areas.

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What are the most interesting “big questions” in your field?

I can identify two of them:

i) The design of numerical methods that respect physical properties of the model. A number of physical processes are modelled through partial differential equations (PDEs). This is for example the case of fluid flows. These equations have a structure that naturally enforce the expected physical properties of the solutions.

For example, the equation describing the evolution of a the concentration of a component in a mixture ensures that this concentration remains non-negative (a negative concentration does not make any sense!). However, when “discretising” these PDEs, i.e. trying to find numerical algorithms to approximate them, this magical structure which enforces physical bound is often lost. It is not clear how to preserve this structure, while taking into account engineering constraints (such as very complex grids encountered in some applications).

ii) The analysis of numerical methods under real-world assumptions. After designing a numerical method, the second job is to assess their accuracy, that is demonstrate that the approximate solution is close to the solution of the model. This can be done by testing the method in a number of situations, but this method of assessing the accuracy is highly dependent on the situations we consider. A more satisfactory way, for the mathematician, is to establish general results that are independent of any particular case/situation. This is often done by assuming some very strong assumptions on the solution or the model; in situations encountered in applications, these assumptions are not satisfied, and the analysis done in the ideal mathematical world cannot be applied. There are however ways to conduct a rigorous mathematical analysis under the exact conditions encountered in applications, at least for models that are not too complex (but include nonetheless some very meaningful physics). This is a recent topic in my field, and I believe one of the most important ones, as this participates in bringing rigorous mathematical analysis closer to real-world applications.

What kind of problems are you interested in broadly in the field?

I study partial differential equations, both at the theoretical and numerical levels. These equations appear as models in physics, mechanics, biology, etc. Their theoretical study consists primarily in establishing the existence and uniqueness of a solution to these equations. Since the models are usually quite complex, it is extremely rare to find formulas for the solutions.

Numerical analysis, which pertains to the design and analysis of algorithms to approximate these solutions, is often the only way to obtain qualitative information on their behaviour.

What are your favourite applications of your work?

When some of the mathematical tools I helped develop are found useful for the analysis (theoretical or numerical) of real-world models, such as equations modelling oil recovery.

What are some other areas of maths that are particularly interesting to you?

I like differential and Riemannian geometries, although I don’t practice often enough to be familiar with all the concepts in this field.

Why did you become a mathematician?

Because maths is like a game to me, and thus it’s fun. Moreover, I particularly enjoy when a theory comes together nicely and every piece fit into the other one and, in the end, this is what maths is about.

Do you have any advice for future mathematicians?

I think the most important quality of a mathematician is rigour. Depending on the kind of mathematics you do this can take different forms, but in any case each argument in a mathematical reasoning should be clearly justified and understood. If you’re not convinced yourself of your reasoning, then it need to be re-worked. Apart from that, choose your mathematical field based in your own interest and motivation, not on some trend or perceived career opportunities. Maths is a difficult field, but also a very rewarding one if you do what you like.

Biggest mathematical/statistical regret?

Not having the time to write all the paper I have ideas for.

Biggest mathematical/statistical success?

I can’t really identify one. I’d say that every year or so (sometimes even more frequently) I obtain a mathematical result of which I’m very proud. I consider it my biggest success, until another one supplants it the year after…

**What are the most interesting “big questions” in your field?**

In design of experiments the big question is to find the optimal design matrix that will minimize the cost and will lead to easy experimentation and good data collection.

**What kind of problems are you interested in broadly in the field? (maths as a whole)**

Solving discrete optimization problems for matrices (design matrices), Statistical analysis of different experiments and implementations

**What are your favourite applications of your work?**

My favourite application of my work is when people are taking the theoretical optimal designs and apply them in real situations to make their experiments and get their data.

**What are some other areas of maths that are particularly interesting to you?**

I am also particularly interested in combinatorics and discrete mathematics which have a tight connection to statistical experimental designs.

**Why did you become a mathematician?**

I become a Mathematician (statistician) because I believe that behind all sciences are some mathematics and statistics is hidden there. In more areas of research, statistics is widely used to justify, explain or argue on particular points.

**Do you have any advice for future mathematicians?**

Mathematics and Statistics build on previous knowledge. So, try to clear everything before taking the next step in mathematics.

**What kind of problems are you interested in broadly in the field? (maths as a whole)**

I’m very much at the applied end of the spectrum in the mathematical sciences; I am about using mathematical epidemiology and statistics to improve the scientific basis for managing wildlife disease.

**What are your favourite applications of your work?**

I worked on predicting plague outbreaks in rodents in Kazakhstan to more efficiently control and monitor plague in this country following its separation from the USSR. Visiting Kazakhstan, working with former Soviet Union scientists, and doing field work in Central Asian deserts was amazing.

I also work on mathematical models for tick-borne disease in the United States. I have a collaborator at Columbia University in New York and will be visiting her later this year. The ecology of these diseases is particularly complex and fascinating, and so keeping the mathematics relatively simple while increasing biological detail has given me plenty of room to be creative and clever.

**What are some other areas of maths that are particularly interesting to you?**

I’m also interested in complex networks, pattern recognition and applications of graph matching. We have a current project with the Australian Federal Police which is about automatic recognition of the quality of fingerprints accidentally left at the scene of a crime, disaster or terrorist act; it is surprising how mathematical such an applied project can be.

**Why did you become a mathematician?**

I’m as surprised as anyone really. I didn’t mean to. I went to University to do Chemistry because my experience at high school was that mathematics was repetitive and dull. That opinion changed quickly though and I was soon `in love’ with the ideas and creativity in mathematics as taught at University.

**Do you have any advice for future mathematicians?**

If you are able to talk with scientists from other disciplines then you will be in demand for the rest of your career. There are so many areas of research that can benefit from more abstract mathematical thinking, so get out there!

**Biggest mathematical/statistical regret?**

Not taking any courses in statistics and subsequently having to teach myself!

**Biggest mathematical/statistical success?**

First-author publication in *Nature* on percolation theory and disease dynamics.

**What kind of problems are you interested in broadly in the field? (maths as a whole) and what are your favourite applications of your work?**

I’m a fluid dynamicist particularly interested in slow and sticky flows occurring in geological and industrial applications. At the moment I’m most interested in modelling relevant to storing carbon dioxide underground and modelling relevant to the formation of nickel-copper-platinum group element ore deposits.

One of the biggest challenges in geological fluid dynamics is how to capture all relevant processes on scales from microns to kilometres in a tractable, yet rigorous model.

**What do you most like about your work?**

I love being at the frontier between maths and earth sciences; bringing the rigour of maths to help understand the earth. The Navier-Stokes equations allow us to model (almost) any geological flow, with the boundary conditions responsible for the huge variety in phenomena. The Navier-Stokes equations themselves are rarely tractable except numerically, but features of the flow such as slenderness can often be exploited to rigorously reduce the governing equations to the essentials. Here more can be said mathematically and deep insight can be gained.

**What are the most interesting “big questions” in your field?**

In finite projective geometry, the general consensus is that the following problems are the “biggest” open problems in our field:

– Is the order of a finite projective plane always a prime power?

– Is a finite projective plane of prime order always Desarguesian?

– The classification of ovoids of finite projective 3-spaces.

– The classification of ovals of Desarguesian projective planes.

– The Buekenhout-Metz Conjecture.

– The “Small-blocking set” Conjecture.

– Does there exist a 430-cap of PG(6,4) with two intersection numbers with respect to hyperplanes?

– Do there exist non-classical generalised hexagons or generalised octagons?

**What kind of problems are you interested in broadly in the field? (maths as a whole)**

I’m interested in problems that show the interplay between different areas of mathematics. In my own field, this can be between geometry and group theory, or between algebraic number theory and combinatorics.

**What are your favourite applications of your work?**

Some of my work in finite group theory has been used to classify combinatorial objects. I did not know back then that it would be used in this way down the track!

**What are some other areas of maths that are particularly interesting to you?**

Group theory, representation theory, algebraic combinatorics, number theory, the list goes on!

**Why did you become a mathematician?**

I really enjoyed mathematics at university, more so than at school. So I followed what I enjoyed, and I still enjoy mathematics to this day.

**Do you have any advice for future mathematicians?**

Read as much as you can. It’s never a waste, you will somehow use what you read later, sometimes indirectly, sometimes you will see surprising connections.

**Biggest mathematical/statistical regret?**

I never took that differential geometry course in 3rd year!

**Biggest mathematical/statistical success?**

I’ve been lucky to have encountered four mathematicians early in my career that were very supportive and an excellent influence. They know who they are!

]]>**What are the most interesting “big questions” in your field?**

There are several open problems related to conic optimisation, for example, what is left of the Hirsch Conjecture, the 9th problem of Stephen Smale and the generalised Lax conjecture.

The first problem I mentioned, the Hirsch conjecture, stated that in an n-facet polytope in d-dimensional Euclidean space every two vertices are connected by an edge path with length no more than n – d. This conjecture was proved to be wrong by Francisco Santos in 2010, but an important question remains regarding the upper bound on the length of the shortest path, in particular whether a polynomial bound exists.

The ninth problem of Stephen Smale is to find a strongly polynomial algorithm to decide the consistency of a system of linear inequalities. That is, given a real matrix A of size m×n and an m dimensional real vector b, we are looking for an algorithm that decides whether there exists an x such that Ax ≥ b, and does so in a polynomial number of arithmetic operations (as a function of n and m).

The generalised Lax conjecture asks whether a rather general kind of convex cones called hyperbolicity cones can be represented as an intersection of a linear subspace with the cone of positive semidefinite matrices.

**What kind of problems are you interested in broadly in the field? (maths as a whole)**

I like to think about open problems geometrically, so I tend to focus on problems that are finite dimensional and are essentially continuous. At the same time I find discrete mathematics truly fascinating, mostly because a lot of problems are so very easy to understand, and some proofs can be explained even to a relatively uninitiated person, whilst solving such problems is prohibitively difficult. So I enjoy learning about combinatorics and thinking about some open questions, but in terms of contribution I do not hold my hopes very high!

**What are your favourite applications of your work?**

Even though I work in optimisation, I am more interested in studying the geometry of a general class of problems rather than focussing on practical applications. What I find truly amazing though is the recent contribution of conic optimisation to solve theoretical problems. In particular, recent work by Frank Vallentin and his team on bounds for packing problems is fascinating, and of course the resolution of Kepler’s conjecture by Thomas Hales which heavily utilised numerical computation by means of linear programming is worth mentioning.

**What are some other areas of maths that are particularly interesting to you?**

Some of the most exciting research directions that I successfully explored came from reading random articles on the web or attending talks in seemingly unrelated areas of maths. This happened with some of the research I did on invisibility in dynamical billiards and that is also how I got interested in studying facial structure of convex cones, which is my main research interest at the moment. My preferences are defined by my limitations rather than anything else: my background in nonsmooth optimisation defines the range of ideas that I can truly appreciate.

**Why did you become a mathematician?**

Doing a PhD in maths was an easy choice: who wouldn’t want to move to a new country in their early twenties, being financed for three years of having fun? Unfortunately the job market is too competitive, and getting a stable position nowadays is more like winning a lottery, so I can say that maybe I was a lucky winner, and I would like to also encourage young people to keep their options open.

**Do you have any advice for future mathematicians?**

No, not really. It may be much more fun to figure things out by yourself!

**Biggest mathematical/statistical regret?**

I wish I had more time to study mathematics deeply when I was an undergraduate student.

**Biggest mathematical/statistical success?**

The end product of any academic research is communication of new ideas, and you also need someone to listen to what you have to say and to respond. So when this kind of communication happens—maybe you solve an open problem or introduce a new idea, then someone else takes notice and develops this further—I think that’s very rewarding.

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