We ask Associate Professor Regina Burachik why programs like AMSI Summer School are so important and what she hopes students will get out of her upcoming course in January.

**Can you tell us about your work? What drives your interest in this field?**

I do research on optimization, which consists of finding the best decision among a set of admissible choices. What I find fascinating in this area is the way in which this fundamental problem can be formulated as a particular case of more general ones. These formulations, in turn, provide a rich and unifying theory, which can be used for solving the original optimisation problem. The theoretical tools for solving optimisation problems are convex analysis, numerical analysis, set-valued analysis, variational inequalities, and maximal monotone operators. You never get bored when you work on optimisation: with so many tools available you will always find a new way for tackling your problem. It is also easy to find new problems to work on.

**What are the most interesting “big questions” or challenges facing researchers in your area?**

From my perspective, the “big question” is to find efficient ways for solving certain families of non-convex optimization problems. Examples that are very important are nonconvex polynomial optimization problems, or new duality methods for certain structured nonconvex and non-differentiable problems.

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

As examples, I have used my work for finding best dose calculation for cancer treatment, or for solving network communication problems using UAVs.

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

A topic I enjoy is maximal monotone operators. These are point-to-set maps that generalize derivatives of convex functions. There is a fascinating interplay between these operators and convex functions. Indeed, many surprising connections can be established between these two mathematical objects. I enjoy finding new examples of such connections in my work!

**Why did you become a mathematician/statistician?**

I did three years of chemistry before migrating to maths. My change happened because I realised that I had a lot of fun doing math, much more than with chemistry! This is a decision I will never regret, maths has given me a lot of happiness. It gives you a way to lift yourself from everyday life. It provides you a special world where you can have fun on your own, or with your colleagues. At the same time, it helps people in solving concrete real-life problems.

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

Choose your own problems, those that you enjoy. Maths is a source of enjoyment, and you should work on a problem only if it is fun and you feel passionate about it, otherwise it is not worth. Follow your own path, and this will make you unique as a mathematician.

**Deepening field knowledge and providing a networking platform, why are opportunities such as AMSI Summer School so valuable? What do you hope attendees take from your lectures?**

I do hope my students will have fun, as much as I will have! Optimization is a way of approaching fundamental problems, and you can see it as a language that provides you the best answer to a given question. Once you learn that language, you can formulate your problem and solve it in your unique way. You will share your views with other students, and learn from them as well. You will understand whether optimization is the topic for you (I hope so), and if so, you will discover new aspects of yourself as a mathematician, including your abilities and your preferences. This may be important for determining your relationship with this topic in the future.

**What do you consider your biggest achievement to date?**

I am a bit shy to speak about my own achievements, I prefer to leave this topic to others! I am just very happy with my choice of maths as a profession. This has allowed me to travel everywhere, meet amazing people all around the world, and be able to work together with these people on an equal level. Once you start working on maths, you become a member of the mathematical world, and you are identified with (and appreciated for) the maths that you produce. Being a (very tiny, infinitesimal), part of that beautiful constellation, is an achievement that makes me very happy!

Associate Professor Brurachik is delivering the** Optimisation **topic at the 2019 Summer School.

We catch up with Associate Professor Peter Kim to give us an insight into the big questions facing researchers in mathematical modelling.

**What are the most interesting “big questions” or challenges facing researchers in your area?**

I think that mathematically modelling social interactions will be a big new frontier in mathematical modelling. In addition, I think that combining ideas from financial mathematics with evolutionary modelling has potential to advance evolutionary modelling and bring it to a new level of sophistication.

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

Lately, I’ve enjoyed the application of mathematical modelling to questions in human evolution. Mathematics forces us to be specific about previously verbal arguments and hypotheses. Also, mathematically modelling social interactions and social cohorts has added a new dimension to modelling for me, which has required creative and innovative thinking.

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

I find trying to understand immune dynamics fascinating. The immune system is in many ways like a huge, diverse social system. After all, it is a mostly decentralised system consisting of myriad cells and diverse immune organs, such as lymph nodes. It is a mystery to understand more about how it works and how it responds to the environment.

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

I think that perseverance and the braveness to try random ideas helps a lot in research, at least in mathematical biology. After one gets into a research program and obtains a scholarship, I think that these traits even take precedence over one’s marks. So don’t worry too much about having what you think are “weak marks” or a “weak” technical background.

Associate Professor Kim is delivering the** PDE Models and Methods in Mathematical Biology **topic with Dr Justin Tzou at the 2019 Summer School.

We speak to Dr Zdravko Botev who explains why mathematics is the language of the cosmos and the challenges researchers face in the machine learning environment

**Why did you become a mathematician?**

Mathematics is the only knowledge which can claim absolute truths with the certainty craved by religion. Ever since ancient times our understanding of the world has kept changing – we had to change our views about the origin of the universe; we had to change our beliefs about what makes us ill and how we can cure diseases. Yet in all this whirlpool of constant change, only mathematics gives us absolute security and certainty. What the ancient Greeks proved more than 2000 years ago is still as valid as when it was discovered, and will be valid for all eternity. It is often said that if one day an alien intelligence visits our Solar System, then mathematics will be our only means of communication. Given all this, how can one not desire to study the only universal language of the cosmos?

**What drives your interest in this field?**

My focus is on using computer simulated random numbers to mimic processes that exhibit randomness. Also, I design algorithms for the accurate and efficient computation of the probability of occurrence of rare and high-impact events. The area is called Monte Carlo simulation. Originally, the method was proposed to help carry out difficult computations necessary for the construction of the atomic bomb during WWII. These days Monte Carlo method have become widespread with many applications in data mining.

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

Pick the topic that you find interesting, not necessarily the topic that is hot at the moment.

**What are the most interesting “big questions” or challenges facing researchers in your area?**

The biggest challenge is to create computer-driven systems that act and learn intelligently and autonomously, in a way similar to humans. Nobody knows how to create such computers. At the moment “artificial intelligence” and “deep-learning” are simply a buzzwords for old-fashioned “curve-fitting”.

Dr Botev is delivering the** Mathematical Methods for Machine Learning **topic at the 2019 Summer School.

Dr Michael Coons takes us through the things that pique his mathematical interest and the best piece of advice he can give to researchers of the future

**Can you tell us about your work? What drives your interest in this field?**

My research interests lie in pure mathematics as broadly defined. At heart I am a problem solver, and due to this, my research has connections to number theory, combinatorics, theoretical computer science, analysis, algebra, and dynamical systems. I have special interest and expertise in Diophantine approximation results related to structures in theoretical computer science and dynamics such as finite automata, regular sequences and block substitutions. My interest is piqued when a problem has many facets that are of interest to several areas within mathematics.

I often work on several problems simultaneously, both as a sole-researcher and in collaboration. My current research is mostly focussed on growth aspects of regular sequences—generalisations of automatic sequences. I am interested in how the maximal values of regular sequences in certain intervals can shed light on the finiteness conjecture for finite sets of integer matrices, and if the theory of Mahler functions can be used to give answers in this area.

**What are the most interesting “big questions” or challenges facing researchers in your area?**

Pure mathematics is full of questions—as is number theory. For my part, the problems I find most important are questions at the interface of theoretical computer science and number theory. The biggest of these is known as Borel’s conjecture. It states that the base expansion of a real irrational algebraic number is normal, meaning that it is essentially random. The point here is to examine how people think about things (where the easy things are algebraic using the standard algebraic operations like ‘plus’ and ‘times’) versus the native environment of computers, base expansions. Borel’s question is essentially asking if there is a fundamental difference in ‘human’ mathematics and the mathematics of ‘computers.’

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

I do mathematics for the sake of mathematics. I like solving problems and feel rewarded when I do. At times I have been influenced by Hardy. Like him, I subscribe to the idea that “Mathematics may, like poetry or music, ‘promote and sustain a lofty habit of mind’, and so increase the happiness of mathematicians and even of other people.” I cannot think of a better application.

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

I’m interested in any good problem. But usually it has to have some arithmetic flavour. As such, I am really interested in problems in classical analysis. For example, a classical result from about one hundred years ago says that a power series with coefficients from a finite set that is bounded in a sector of the unit circle is a rational function. This means that an irrational power series with coefficients from a finite set has the unit circle as a natural boundary. Well, okay. But then this means something much stronger than transcendence. At least transcendence over meromorphic functions (this I showed with Yohei Tachiya), but what more can be said about such a function? Can it satisfy an algebraic differential equation? I would love to know more about this.

**Why did you become a mathematician?**

I actually first went to university as a chemistry major to eventually lead to medical school. I kept taking mathematics courses because they were an easy good mark for me. Along the way I realised I did not want to help people in the way a medical doctor could, and I realised that I was pretty good at mathematics. So I just kept doing it. Actually, until just a few years ago my parents were still asking me if I would give up this maths thing and go to medical school. I like to remind them that I am a doctor… just not a medical one!

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

“Wear Sunscreen. If I could offer you only one tip for the future, sunscreen would be it. The long term benefits of sunscreen have been proved by scientists, whereas the rest of my advice has no basis more reliable than my own meandering experience… I will dispense this advice now…” But really, all I have for advice are quotes from people I’ve heard over the years. Probably the best of these was from George Willis, my colleague at Newcastle, and even this was something that someone had told him: “Never give a problem too much respect.” It could be hard, but its a problem. Try. You could fail. I have on several occasions—many, but countably many so far. Also, get used to failure. It happens a lot more than success. Be it problems, job applications, grant applications, prizes, promotions, problems… you get the idea. But each success, no matter how small, is sweet. And worth it. As the saying goes, “Mathematicians… we’ve got problems.” And that’s part of the fun.

**Deepening field knowledge and providing a networking platform, why are opportunities such as AMSI Summer School so valuable? What do you hope attendees take from your lectures?**

Knowledge can be gotten anywhere, the thing that the AMSI Summer School adds is exposure to different people and other places. The first time I went somewhere new, for some reason I had this weird thought that whatever mathematics I knew would not be the way things were at that new place. I know this is stupid—really, it is in retrospect—but I had to go somewhere new to figure this out. Now I’ve been many places… and math is the only thing that seems to stay the same!

**What do you consider your biggest achievement to date?**

I’m a pure mathematician and for some reason I don’t quite understand, I get to solve problems and do mathematics everyday and somebody pays me for it. Seriously, I don’t really understand why, but they do. One of my PhD supervisors (Peter Borwein) used to tell me that, “An applied mathematician gets a job at a university and they complain and say that they could get a better paying job in industry and that the university should be happy they are here. A pure mathematician gets a job at a university and thinks, ‘You suckers! I’d do this anyway… and you’re paying me!’ “

Dr Coons is delivering the** Analytic Number Theory **topic at the 2019 Summer School.

Find out Assoc. Prof. Lisa Alexander’s thoughts on what inspires her work in climate change research, the big questions facing researchers, and why she thinks AMSI Summer School is such a valuable experience.

**Can you tell us about your work? What drives your interest in this field?**

I work on understanding global and regional changes in climate extremes (think heatwaves, storms etc) in observations and models. This involves trying to work out how much of past changes are due to natural variations in the climate and how much is due to other factors such as human influence. I also look at projections of these events in the future based on a range of possible scenarios. The short story is that the amount of greenhouse gases we emit, will be crucial for determining how ‘extreme’ future changes will be. Our choices matter. I love this field because extremes have real societal impacts so it means that the results of my work can be very policy-relevant.

**What are the most interesting “big questions” or challenges facing researchers in your area?**

Questions include how changes in circulation and moisture content (background state) control extremes on a regional scale, and what processes underlie the spatial and temporal organisation of convective systems that lead to extreme rainfall and trying to understand the relative contributions from dynamics and thermodynamics to changes. Some other ‘big questions’ are relatively simple but equally difficult to answer e.g. are our observations actually good enough to answer the questions posed and how can we improve the simulation of extremes in climate models?

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

The American Society of Actuaries are using a dataset that my colleagues and I developed to create a risk index related to changes in extreme weather (http://actuariesclimateindex.org/about). This was completely unintended as the dataset was developed primarily for other climate researchers. I also helped to develop software that is used to derive climate indices that can be used to help sectors such as health and agriculture. The software is primarily used in developing countries.

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

I’m a data geek! Climate data are on the petabyte scale so it’s quite thrilling to be able to analyse that much information while trying to condense results into something meaningful that would benefit the field and other researchers.

**Why did you become a mathematician?**

It was my favourite subject at school so it was a natural undergraduate choice for me. While I also liked languages, I felt Maths would offer me much more choice once I graduated (and I think I was probably right).

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

Work hard, find your niche, collaborate, enjoy!

**Deepening field knowledge and providing a networking platform, why are opportunities such as AMSI Summer School so valuable? What do you hope attendees take from your lectures?**

Maths has so many exciting applications which you don’t always realise when you’ve got your head down and trying to pass exams. I personally have benefited so much from networking opportunities in the past to the extent that I wouldn’t be where I am today without those opportunities.

I would like lecture attendees to see that the subject they enjoy so much has real-world and important applications and how over the years many incredible mathematical tools, both simple and complex, have been developed that can address cutting-edge problems.

**What do you consider your biggest achievement to date?**

I was lucky enough to be a lead author on one of the Intergovernmental Panel on Climate Change (IPCC) Assessments. There have been five of these reports since 1990 and they are written as policy-relevant documents based on an assessment of climate science literature for the world’s governments to develop their climate policies. I got to see governments in action and how policy decisions are made. It was an eye opening experience, extremely hard work at times but an amazing sense of accomplishment when the final report was released.

Associate Professor Alexander is delivering the** Mathematics of Planet Earth **topic with Dr Shane Keating at the 2019 Summer School.

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