## Meet the speaker: Dr Daniel Mathews

In News

We chat with Summer School 2018 lecturer Dr Daniel Mathews, from our host Monash University.

Tell me about your research field: what drew you to this area and its impacts on discoveryits 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 topologyalthough, 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 ellipsoidthese 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 spacethis 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 resultsthey 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 foreseenthis 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 surfacea 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 topologythe shape of thingsin 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 mathematicianspractising mathematicians, and budding mathematiciansand 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 doand 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 mostmathematically, and otherwiseare 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 suchand I’ve studied other things as well. But I have done many other things tooI’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.

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