The Annual Summer School is one of the Flagship events on the AMSI Research and Higher Education calendar and is one of the most anticipated events for Honours, Masters and PhD students studying mathematics nationally.

Hosted by Monash University in January 2018, the program attracted 169 students from around Australia. Taking place over a jam packed four weeks, students studied a range of specialised topics that they wouldn’t normally get to study at their home universities. In addition to this, they were given the opportunity to network with lecturers and academics from around Australia, meet with members of industry, get tips on becoming job ready, as well as meet other students from their cohort who are just as passionate about mathematics as they are. All this while fitting in mid-week barbecues, movie nights and sight-seeing expeditions in between.

This year welcomed Professor Nick Trefethen FRS all the way from Oxford University as the Summer School’s Public Lecturer. He presented on the topic *Discrete or Continuous* to an enthusiastic and captivated audience.

AMSI Summer School is jointly funded by the Department of Education and Training and the Australian Mathematical Sciences Institute. AMSI would like to thank partners and hosts, Monash University for their time and dedication to delivering such a successful event, as well as other event partners, Australian Mathematical Society, Australian and New Zealand Industrial and Applied Mathematics (ANZIAM), Statistical Society Australia, BHP Billiton Foundation, Department of Defence, Monash Centre for Quantitative Finance and Investment Strategies, and Commonwealth Bank of Australia.

**ITERATIVE METHODS FOR SPARSE MATRICES**

Associate Professor Timothy Moroney, Queensland University of Technology

**LOW-DIMENSIONAL TOPOLOGY**

(Sponsored by the Australian Mathematical Society)

Dr Daniel Mathews, Monash University

Associate Professor Jessica Purcell, Monash University

**MATHEMATICAL RELATIVITY AND LORENTZIAN GEOMETRY**

Dr Andy Hammerlindl, Monash University

Associate Professor Todd Oliynyk, Monash University

**MATHEMATICS OF EXTENSIONAL FLOWS
**(Sponsored by Australian and New Zealand Industrial and Applied Mathematics)

Associate Professor Yvonne Stokes, University of Adelaide

**PROBABILISTIC METHODS AND RANDOM GRAPHS**

Professor Nick Wormald, Monash University

Dr Jane Gao, Monash University

**PROBABILITY, COMPLEX ANALYSIS AND LATTICE MODELS**

Dr Laurie Field, Australian National University

Dr Gregory Markowsky, Monash University

**STATISTICAL MACHINE LEARNING
**(Sponsored by the Statistical Society of Australia)

Dr Ivan Guo, Monash University

Dr Tiangang Cui, Monash University

**TOPOLOGICAL DATA ANALYSIS**

Dr Vanessa Robins, Australian National University

Dr Katharine Turner, Australian National University

*Tell me about your research field, what drew you to this area and its impacts on discovery – its real-world applications?*

Computational topology and topological data analysis have developed rapidly over the past 20 years or so. The goal is to establish methods for quantifying the shape of data. It is quite simple for a person to look at a 2D pattern of points and “see” the shape those points trace out, but it requires a lot of mathematics and good algorithms to get a computer to do the same. And if the points live in a higher-dimensional space, it is difficult even for a person to identify the shapes.

I’ve always loved geometry and patterns and the way they manifest in nature. And I’ve always loved seeing how different areas of mathematics work together to help describe real-world phenomena. My first research project as an honours student investigated patterns that arise in models of plasma fusion experiments.

The shape of the magnetic fields controls how much of the space the plasma particles can move through. If the fields fill out a smooth surface the plasma remains contained, while if certain field lines break up and develop large holes, the particles will escape and fusion will fail. This was well quantified in two-dimensional models, but much more difficult to study in the more realistic higher-dimensional models. The question of how to detect holes in higher-dimensional point data then inspired my doctoral research on computational topology. For example, holes in the trajectories traced out by a particle signal the presence of non-linear periodicities, or the detection of small-scale local holes allows us to discriminate between different classes of random point processes. This helps scientists understand and test the validity of their models for processes that underlie the data generated. My thesis explored the mathematical foundations for extracting topological quantities from point data. Simultaneous studies by others around the same time led to efficient algorithms and a field of research now called topological data analysis.

*You are a researcher at ANU. What are you working on currently? Can you tell me about some recent achievements?*

I have two main areas of research at ANU. One is working with the ANU x-ray CT group who developed world-leading lab-based x-ray imaging techniques. We have adapted the primary tool of TDA (persistent homology of a filtration) to analyse 3D voxel data rather than point-cloud data. Our work established the first good definition of critical saddle point for a function on a digital grid and has also led to efficient algorithms for the simultaneous skeletonisation and partitioning of binary images. I’m currently using these tools to analyse x-ray CT images of porous sandstone rock cores to establish correlations between their pore-space geometry and connectivity and their physical properties. For example, the capacity for capillary trapping of CO2 gas for permanent carbon sequestration. Another example of work with this group is our recent *Nature Communications* paper “Pore configuration landscape of granular crystallization”, where we show how TDA gives clear insights into the local mechanisms behind the order-disorder transition in packings of spherical beads.

My other main area of research (with Stephen Hyde’s group) is on topological crystallography of one and two-dimensional structures embedded in 3D space. One of the biggest areas of research in solid-state physics and chemistry is on framework materials built via the self-assembly of small molecules. A fundamental question here is to describe the range of structures that are possible within some physically relevant classes. Some of the more complicated frameworks consist of multiple inter-threaded components. Ideally we would like to have robust invariants that tell us when two structures are topologically equivalent. Mathematically this translates into a need for computable invariants of linked and knotted graphs in the 3-torus.

*What are the biggest challenges in this area and more broadly facing the global mathematics community?*

A major challenge for TDA is to extend its methods to more complex situations. One of the main tools of TDA is the persistent homology of a filtration: this tracks the way topology changes as you add elements to the object. So it works only for growing sequences ordered by a single parameter. But many scientific problems have objects that grow and then shrink, or have two or more parameters associated with them. A number of groups around the world are working to develop mathematical theories and computable invariants for this situation.

Another challenge is to make the techniques of TDA easily accessible to scientists who might want to use them. This is something that applies to mathematics more generally. The mathematical community has been improving its expository skills and this is something to be encouraged and expanded. Mathematics is an important part of our culture and global knowledge base, and communicating the key insights and motivations to others is necessary to maintain its vitality. The results don’t need to be immediately applicable, but it is worth taking the time to explain why they are interesting.

*You are lecturing on topological data analysis at AMSI Summer School 2018. Can you give us the elevator pitch for your session?*

TDA is a rapidly developing field of applied mathematics that uses topology – previously thought of as a very “pure” area of maths – to probe the structure of data from real scientific applications. TDA provides a complete picture of the geometric and topological structure of the data set over a wide range of length scales. This helps scientists to identify and visualise the important structural elements present in their data and provides a means to compare real data with models. What could be more exciting than learning about elegant mathematical quantities and practical algorithms for computing them – tools that can lead to new scientific insights in many different fields.

*How important are opportunities such as AMSI Summer School as we seek to strengthen national and international engagement within the mathematical sciences and prepare emerging research talent to drive innovation?*

I’m really excited by the opportunity to pass on my knowledge of applied topology to as many students in Australia as possible. This is a very active area of research in North America and Europe, so the school will certainly help strengthen national and international engagement in this area. The AMSI summer school also provides a great opportunity for mathematics students from around Australia to form friendships and collegial networks that will last throughout their careers.

*What do you see as the biggest barriers to driving innovation? How important are initiatives to provide industry experience and knowledge to graduates and address issues such as participation of women and indigenous Australians?*

To me, innovation comes from having the freedom to daydream a little and explore questions that just grab your attention and mean something to you. It is fostered by open access to information and engagement with other like-minded people. Industry experience and knowledge is helpful for “keeping it real” and reminding academics what matters to the wider world. A natural barrier to innovation is the very human tendency to keep doing things the way they have been done in the past – it takes a lot of energy to change. One thing I’ve learnt is that you can combine interests in just about any choice of fields if you think hard enough. So non-Euclidean geometry can be used in fashion design, the physics of nuclear isotopes is vital to dating archaeological artefacts, machine learning is being used to data-mine 18^{th}-century French literature. Perhaps a wider recognition of these interdisciplinary possibilities would help under-represented groups find a way into their fields of interest.

*As part of Choose Maths, we are in the process of establishing a mentoring program particularly in relation to encouraging the participation of women. Who are your biggest maths influences or mentors, how have they impacted your maths journey and career?*

As a woman in mathematics and physics, I was generally one of only a few in my classes. It has been a struggle at times to overcome the self-doubt instigated by being in the minority and having very few role models. One thing that has kept me going is the encouragement from my academic advisors along the way, and to see other women around me succeed. I’m particularly happy to see the energy and enthusiasm of my younger female colleagues as they develop their careers. The current swell of support for women in disciplines they are under-represented in is also a welcome boost to morale.

*Did you grow up mathematical or did maths find you along the way? Was it always a career dream?*

I always had an aptitude for mathematics, particularly geometry, and loved learning about all areas of natural science. I started my undergraduate degree in physics, but always felt I needed to know more mathematics to understand the physical models better. So it’s not at all surprising that I’ve ended up doing what I do now.

]]>*Tell me about your research field, what drew you to this area and its impacts on discovery – its real-world applications?*

I work in topology, which can be described as the mathematics of shape and space. Topologists try to answer questions about shapes. For example, they may study knots. If someone hands you an extension cord with the ends fused together, should you bother trying to unknot it? How would you know whether or not it could be unknotted? If not, can you at least remove a lot of crossings? How many crossings can you remove? How long will it take you? How many knots are there with a fixed number of crossings, and how do you know? How can you tell them apart? Sometimes such questions can be answered by looking at more complicated objects, such as the space around the knot, or higher dimensional objects whose boundaries are related to the knot. Topologists try to analyse all these spaces and shapes and others.

*You are a researcher at Monash University. What are you working on currently? Can you tell me about some recent achievements?*

I am working on the interplay of hyperbolic geometry and 3-dimensional spaces called manifolds. Hyperbolic geometry is a negatively curved geometry that arises when spaces have saddles or flared ends. To go back to the knot example, the space around a knot is 3-dimensional. It often admits a unique hyperbolic structure, which gives a very natural way of measuring distances and volumes. One aspect of my research involves trying to determine hyperbolic metric properties of knots. A problem I have been investigating recently is to determine, out of all knots with a fixed crossing number, the knot with the largest hyperbolic volume. While we are still a long way from answering this question, we have some good candidates. Some recent work of mine investigates the geometric properties of these candidates, and shows that at least in a limiting sense, they have as much volume per crossing as possible. I have also been looking at other limiting behaviour of other knots, and showing that these limits can have surprising geometric properties.

*What are the biggest challenges in this area and more broadly facing the global mathematics community?*

I believe one of the biggest challenges in my area is relating and applying new knowledge to old problems, or new instances of similar problems. In the last 15 years, my specific area of mathematics has seen a phenomenal amount of progress. Many old questions and conjectures have been answered, beginning with the century old Poincare conjecture that was proved in 2002-3 and confirms a suspected property of 3-dimensional spheres. This progress and success has led to great tools to study 3-dimensional manifolds, and a better understanding of their structure. However, it is still very difficult to apply these tools to examples that arise in practice, both in low-dimensional topology and in other fields. To give an example, physicists have been working with quantum invariants associated to knots and 3-dimensional manifolds, and experimentally some of these invariants seem intricately related to hyperbolic geometry. But we cannot prove this! Moreover, we really don’t have a good understanding of why this happens in a lot of cases. Many remaining challenges in my subfield involve relating results on geometry to other invariants, and giving geometric reasons why the relations hold.

More broadly, I think the biggest challenges facing the global mathematics community involve communication. Mathematics thrives when researchers are able to work together and communicate. It requires global societies that are open and value the process of seeking answers. It also requires work on the part of the mathematicians. We need to be better at sharing our ideas, and trying to communicate the value of what we do. We need to have the courage to look at problems from different points of view, and to try to understand the work that is done in other fields and to contribute.

*You are lecturing on *Low Dimensional Topology* at AMSI Summer School 2018, can you give us the elevator pitch for your session?*

Low-dimensional topology is the mathematics of spaces in dimensions 2, 3, and 4. We will encounter spaces that can often be described by sketching pictures or diagrams encoding their forms. The mathematical tools we develop will help us to analyse the pictures and diagrams, and confirm or contradict our intuition, and help us understand spaces when diagrams are not available.

*We will start by considering surfaces. We’ll look at maps from a surface back to itself, and determine a set of generating maps. We will use these to build up 3-manifolds, and discuss related topics along the way such as knots and applications to 4-dimensional spaces.*

Questions from low-dimensional topology are becoming more common and more broadly interesting in our world. Triangulations and properties of surfaces and 3-manifolds are important in computer vision and visualisation. As mentioned above, 3-dimensional manifolds appear in quantum physics. The knots you find in your extension cords and shoe laces are also appearing in biochemistry, as knotted and folded proteins and DNA strands, and the knotting seems to affect function. There is much current research in these directions. However, our session will focus on the developing the theory and not much on the applications.

*How important are opportunities such as AMSI Summer School as we seek to strengthen national and international engagement within the mathematical sciences and prepare emerging research talent to drive innovation?*

The AMSI Summer School is a great way for students to learn about aspects of mathematics that are deeper or broader than what they see in a standard curriculum. The extra knowledge should help them prepare for a future in which knowledge is important. Another huge benefit of the AMSI Summer School that I see is the opportunity for mathematically inclined students to meet each other, and to interact with other students who do well in mathematics and who love it. My research career has been shaped by interactions with people I met as a student, especially fellow students. These are my mathematical friends, now residing all over the world. I go to them to bounce off ideas or share intriguing problems and projects. I hope the students take the opportunity to form their own mathematical friendships at the AMSI Summer School.

*What do you see as the biggest barriers to driving innovation? How important are initiatives to provide industry experience and knowledge to graduates and address issues such as participation of women and indigenous Australians?*

In mathematics a lot of innovation comes from the sharing of ideas. Putting restrictions on the people who are encouraged to participate in this sharing of ideas is a big barrier to innovation. Another barrier to innovation comes from the attitudes we sometimes communicate as a society. I have found that students sometimes believe that they will be happier if they avoid challenges. In fact the opposite is true. Working on challenging problems can lead to deep satisfaction. Providing experience with industry and programs like the AMSI Summer School can help graduates develop the skills they will need to tackle challenging problems, and to enjoy the challenge!

As part of Choose Maths, we are in the process of establishing a mentoring program particularly in relation to encouraging the participation of women. Who are your biggest maths influences or mentors, how have they impacted your maths journey and career?

As an undergraduate, I had a mentor in mathematics who helped me immensely. She was able to point out which classes would be most important to help me reach my goals, and suggest opportunities for further development and exploration, such as summer schools. I would have had no idea these options existed without her mentoring. Since then, I have found mentors among colleagues at different institutions at different stages of my career. They have all helped me to achieve my goals, and navigate changes.

*Did you grow up mathematical or did maths find you along the way? Was it always a career dream?*

I liked mathematics in school, but I didn’t understand that mathematics could be a career. I remember telling someone early in my undergraduate days that I had already learned nearly everything there was to learn about mathematics — I couldn’t imagine maths beyond multivariable calculus. With the support of strong mentors and the encouragement of my professors, I continued taking more and more mathematics courses, and discovering that in fact there was more and more to learn.

A few years later, I did have a hard time deciding whether or not to get a PhD in mathematics. While I was doing well in my maths classes, I imagined that a mathematician had to be clever and quick – the kind of person who did really well on maths contests and competitions. I was not that kind of person. Again with some encouragement, I built up enough confidence to try a PhD. And again I found that I liked it, and continued to enjoy mathematics most of the time, and I realised that the day to day job of a mathematician was not much like a maths contest. In any case, I am happy to be a mathematician now.

]]>*Tell me about your research field: what drew you to this area and its impacts on discovery***—***its real-world applications? (Think how you’d explain what you do at a family BBQ)*

My mathematical research is in the broad field of geometry and topology**—**although, depending on the day, it may also involve lots of algebra or physics or any number of other things. To a guy on the train the other day it looked like some alien hieroglyphics burning a hole in his brain!

Topology is the study of the shape of things. It’s a type of geometry where you do not care about lengths or angles. A cube, a sphere, an ellipsoid**—**these are all the same to a topologist. The classic description of a topologist is someone who can’t tell the difference between a coffee cup and a donut!

Topology is a huge and deep field. It concerns itself with things like the possible shapes of spaces. For instance, what are the possible shapes of the universe? It’s apparently a 3-dimensional space, but what are the weird and wonderful ways in which a 3-dimensional space can connect up with itself? Topology also concerns itself with things like the ways a loop of string can be tied up in space**—**this is the subject of knot theory, for instance.

But this is just scratching the surface. Advances in topology in recent years demonstrate how it is deeply connected to a lot of ideas from all over mathematics and indeed from all over science. It has real-world applications to everything from string theory and quantum field theory, to chemistry and biology, where molecules and DNA strands may be topologically knotted.

Of course, this is all extremely vague, and for a precise version of the above you should take our summer school course on low-dimensional topology!

*You are a researcher at Monash University. What are you working on currently? Can you tell me about some recent achievements? (E.g. new papers, examples of innovation or direct impact of your research)*

The mathematical research questions I’ve worked on recently are quite abstract, dealing with the properties of curves, surfaces, knots, and different types of topology and geometry. I like to work with types of geometry such as hyperbolic, symplectic and contact geometry. Hyperbolic geometry is a type of negatively-curved geometry which is amazingly related to the topology of 3-dimensional spaces. Symplectic and contact geometry are types of geometry that don’t care about lengths or angles, but do care about certain types of areas in a sense that is closely related to physics.

These are deep results**—**they take a long time to figure out, and a long time to prove and write down. Because it’s so abstract, the applications cannot be foreseen**—**this is a common feature of fundamental, or basic research, such as a lot of pure mathematics.

A good thing about pure mathematics research is that you can make up your own question. If you can ask an interesting mathematical question, and give a new answer to it, then you have advanced mathematics. In formulating those questions you are limited only by your imagination.

For one fairly recent example, together with a former student and a Monash colleague, we asked a simple question about the number of ways that curves can be arranged on a surface. That’s a pure, abstract question, which we managed to answer. But in finding the answer, we uncovered a wonderful and deep structure. To answer the question, we used ideas from quantum physics and complex analysis and a very interesting type of recursion. Yet all this arises simply from looking at the way that curves are arranged on a surface**—**a down-to-earth situation that happens all the time. So, you just never know what you will find: the universe in a grain of sand, so to speak.

*What are the biggest challenges in this area and more broadly facing the global mathematics community?*

For the fields of geometry and topology, as in any pure mathematical field, there are always challenges in the form of open problems! In knot theory, just to pick two I like, there are volume and AJ conjectures, which propose deep and tantalising connections between topology, algebra, geometry and physics. Do these connections exist, and if so why?

Internally to the field, sometimes there are difficult challenges for new researchers and PhD students, because the questions are so abstract and sometimes proofs are so lengthy and intricate that their status is in doubt. This has been a problem for symplectic geometry, where much recent research builds on enormous works of analysis over which some researchers have raised question marks. But thankfully the researchers involved are talking to each other and I think eventually the scientific process will arrive at the truth.

Externally to the field, with all pure mathematics there is the problem of public communication: it’s a difficult subject and it’s not always the easiest thing to explain. In physics or chemistry, for instance, when Nobel Prizes are announced, it’s usually possible to explain to a general audience at least a rough idea of what the prizes are for. But with mathematics and Fields medals, it’s much more difficult, and we usually settle, in our public communications, for descriptions that are woefully vague, if not downright wrong. Sometimes, indeed, it may be an impossible task to explain without a full course in pure mathematics; but sometimes it is not. I think we need to try harder.

For mathematics in Australia, there is the problem of research funding: research grants have an extremely low funding rate, not because of the low quality of the research, but because of the low amount of funding available.

There is also the problem of education. The number of students taking advanced mathematics is declining, and so students are arriving at university with weaker backgrounds. We at the universities then need to bring them up to speed! With mathematics, and other STEM fields, now so essential to our economy and society, we need to turn this around. I tend to think this is a cultural problem more than anything else: we need to be a society, and a culture, that respects and values scientific and mathematical thinking. But the relationship between science and mathematics, and the general public, goes both ways; the scientific and mathematical communities also need to be a culture that respects and values the broader community. It needs to listen, educate when necessary, avoid arrogance, and take a stand when necessary.

Finally, on a related note, there is the very general problem of a crisis of confidence in science, and in facts more generally, with the rise of fake news and so on, as new technologies, especially through manipulation of social media, are used to bypass our critical faculties and stimulate the worst in us. We should not think mathematics stands apart from this. Mathematics, learned well, is a course in intellectual self-defence and critical thinking.

*You are lecturing on Low Dimensional Topology at AMSI Summer School 2018, can you give us the elevator pitch for your session?*

I’m really looking forward to this course. We’re going to look at topology**—**the shape of things**—**in low dimensions, 2, 3, and maybe 4. Two-dimensional spaces are also known as surfaces, and there are beautiful mathematical theories about them. Three-dimensional spaces, or 3-manifolds as they’re sometimes known, are a fundamentally important topic, not least because our own world is 3-dimensional!

We’re going to cover some of the foundational results in this subject, and some beautiful theorems, about maps of surfaces, about decompositions of 3-dimensional spaces. We’ll also talk about knots and we may get a little into 4-dimensions, which is an area full of open questions. For topologists, 4 is still considered a “low” number of dimensions!

How can you tell different knots apart? What are the possible symmetries of a surface? We’ll look at these questions and many more.

*How important are opportunities such as AMSI Summer School as we seek to strengthen national and international engagement within the mathematical sciences and prepare emerging research talent to drive innovation?*

I think the AMSI summer school is a fantastic innovation. There are always great courses on a wide range of topics and it’s a place where interested students from around the country can come and learn mathematics and solve problems together. It builds a community of mathematicians**—**practising mathematicians, and budding mathematicians**—**and equips them with new knowledge, new skills and new connections.

*What do you see as the biggest barriers to driving innovation? How important are initiatives to provide industry experience and knowledge to graduates and address issues such as participation of women and indigenous Australians?*

Quite frankly I’m a bit sceptical of all the rhetoric we see these days about driving innovation. If we want to think about what’s most important for our economy right now, it’s much more important that we avoid climate change and become carbon neutral and get off fossil fuels as soon as possible, than whether we have the most support for startups building the latest app.

The future of the planet is at stake, and the present is a crucial time. The innovation required to get Australia, and the world, living renewably, is considerable. The biggest barriers to that, however, reside in governments that don’t even accept the science of climate change, and in well-funded climate denier networks. We need to innovate these dinosaurs out of existence.

It’s true that women and indigenous Australians are woefully underrepresented in mathematics. We need to lift our game. A few recent developments are promising, such as the Athena SWAN program, and AMSI’s “We are more than numbers” initiative. It’s a process of cultural change: we need to be a society where all people think of maths, and science more generally, as a living, breathing, exciting thing that they can do**—**and by this I mean people of all colours and genders. Not as something that’s done by freaks and geniuses only; not as something that’s done by men only; not as something that’s too hard or dry or repetitive, but something that is intriguing and challenging, imaginative, curious, and free.

I think we mathematicians ourselves need to lift our game too. When we can, we should be going out in public, in our schools, and telling people about ourselves. That’s not something many of us are comfortable with, but we are in a pretty privileged position and we ought to use our privilege in an inclusive way.

*As part of Choose Maths, we are in the process of establishing a mentoring program particularly in relation to encouraging the participation of women. Who are your biggest maths influences or mentors, how have they impacted your maths journey and career?*

I got interested in mathematics through my involvement in the Olympiad programme. That’s a really valuable programme for talented students and indeed several of my Australian colleagues at Monash also got into mathematics that way.

As for mentors and influences, the people whose views have impacted me the most**—**mathematically, and otherwise**—**are giants of humanity, as well as mathematics, like Bertrand Russell, Noam Chomsky and Albert Einstein.

*Did you grow up mathematical or did maths find you along the way? Was it always a career dream?*

The mathematics Olympiad found me, I suppose! I was fortunate enough to have some very good teachers at school, like Dr Michael Evans, who got me into it, and supported and encouraged involvement in these activities. But it was never a career dream as such**—**and I’ve studied other things as well. But I have done many other things too**—**I’m also a fully qualified lawyer, for instance, though I’ve never practised law.

Mathematics is something that I enjoy doing, that is creative and useful work, and which gives me the freedom to pursue goals I value.

]]>The AMSI Summer School is one of the key events on the calendar of the Australian mathematical sciences community, due to its focus on mentoring and engaging with the new cohort of emerging mathematical talent in Australia, and its role in public engagement and outreach.

The AMSI Summer School provides the opportunity for students to build both professional networks and peer friendships, which will aid them in their future careers. Students gain valuable knowledge from the specialised subjects offered, and they are exposes to some of the best young lecturing talent in Australia, while the Careers Afternoon remains an important activity for students seeking input into future career paths.

The Public Lecture was highly successful, with over 200 members of the public attending Margaret Wertheim’s talk on “Corals, Carbon and the Cosmos: The Story of Hyperbolic Space”.

AMSI Summer School 2016 was funded jointly by the Department of Education and Training, and the Australian Mathematical Sciences Institute, and supported by RMIT University, AustMS, ANZIAM, Australian Government Department of Defence, BHP Billiton Foundation (part of the Choose Maths initiative), CSIRO, City West Water, and Optiver.

**CALCULUS OF VARIATIONS**

Julie Clutterbuck, Monash University

Anja Slim, Monash University

**COMPLEX NETWORKS**

Stephen Davis, RMIT University

**CONIC PROGRAMMING**

Vera Roshchina, RMIT University

**DESIGN AND ANALYSIS OF EXPERIMENTS**

Stelios Georgiou, RMIT University

**LINEAR CONTROL THEORY**

Yoni Nazarathy, The University of Queensland

**MODERN NUMERICAL METHODS**

Jerome Droniou, Monash University

**PROJECTIVE GEOMETRY**

John Bamberg, The University of Western Australia

**STOCHASTIC MODELLING**

Giang Nguyen, The University of Adelaide

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.