Interviews – ��ճԹ� I Championing Mathematical Sciences for Australia’s Advancement

Interview with Dr Daniel Mathews

Thu, 24 Aug 2017 05:00:39 +0000

��ճԹ� Summer School 2018—Lecturer Interview

Dr Daniel Mathews (Monash University)

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 ��ճԹ� 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 ��ճԹ� 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 ��ճԹ� 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 ��ճԹ�’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 Tao of Maths

Mon, 11 Jan 2016 22:46:39 +0000

Fields Medallist and member of ��ճԹ�’s Scientific Advisory Committee, Professor Terry Tao takes time out of his busy schedule at UCLA to give us an insight into what’s exciting him in mathematics, His recent collaborations and the ways in which he approaches complex problems.

What’s exciting you in mathematics at the moment?

It changes a lot from year to year – there are so many things going on in different parts of mathematics, it seems! I can name two recent breakthroughs in the last year or two which have generated a bit of excitement. The first is the recent proof of the Kadison-Singer conjecture by Marcus, Spielman, and Srivastava, which used radically new methods (in particular, interlacing polynomials) to solve a notoriously difficult problem in operator algebras and matrix analysis. It looks like there are other applications of this method (for instance, to theoretical computer science). The other is the breakthrough result of Matomaki and Radziwill earlier this year in understanding short sums of multiplicative functions in number theory. This has made several open problems in number theory (e.g. the Chowla conjecture, a cousin of the twin prime conjecture) look much more within reach. Recently I was able to use the Matomaki-Radziwill theorems to prove some partial results towards the Chowla conjecture, which could in turn be used to settle a long standing conjecture of Erdos on the discrepancy of sequences. I’m confident that we’ll be seeing other striking applications of Matomaki and Radziwill’s results in the near future.

Why do you think women are underrepresented in mathematics?

That’s a good question. Up to about the high school level, we seem to have fairly good parity these days; if anything, female maths students may even be slightly more numerous and a bit stronger. But then there is a lot of attrition at the undergraduate level and beyond. It seems there are a lot of reasons for this. One is that nearby disciplines (e.g. the life sciences) have much better gender balance and this can be more attractive than a discipline where one is in the minority. Another is the relative lack of high-profile female role models in mathematics, though there are excellent top female mathematicians who do their heroic best to counteract this. Then there is the fact that the graduate and postdoctoral portions of one’s career in mathematics can be rough on people who are also trying to start or raise a family. There are some little positive steps in these directions (for instance, child care availability is now taken as a serious issue in mathematics departments, institutes, and conferences, and more efforts are being made to overcome conscious or unconscious biases against minority candidates in hiring and in giving presentations), but there is still a long way to go here.

From a personal perspective what are your top three open problems in mathematics? What are their prospects for resolution?

Well, this is very subjective, and depends a lot on what you mean by “top”. There are statements which would have enormous implications if they could be definitively proved (e.g. the six remaining Millennium prize problems), but the likelihood of actually doing so is so remote, I don’t think these are the problems that we should be devoting the bulk of our mathematical manpower to attacking. (Though I do like to keep tinkering with an approach I have to disproving global regularity for the Navier-Stokes equations…) My philosophy is to focus on those open problems that are only a little bit out of reach of current techniques and methods – problems that require “only” one new breakthrough to solve, rather than a half-dozen. In number theory, I think the twin prime conjecture is getting close to this level of feasibility; in analysis, the Kakeya conjecture has already had much headway made against it from the previous four or five breakthroughs in the area, and one can hope that just one more is needed to finish it off. More ambitiously, I think the soliton resolution conjecture in PDE would be a fantastic result to settle, though this is currently well out of reach except in very special cases (e.g. completely integrable equations, perturbative data, or other very symmetric and special equations).

I read somewhere recently that as a child you thought that research was driven by a committee posing problems. Do you think the free ranging, creative side of mathematics comes off second best to problem solving for kids and adolescents with an interest in maths?

Well, I think even problem solving comes off as second best to the computation-intensive mathematics one sees in schoolwork. Certainly when I was a child, the only glimpses I saw of true mathematical research were in some more advanced level books I got from the library, or the informal discussions I had with some active and retired mathematicians in Adelaide. One big plus in today’s world though is that, with the internet, one can now listen to public lectures or other talks by some very good mathematical speakers, or see good examples of accessible mathematical writing online. Even just the mathematics section on Wikipedia is a wonderful resource which I would have very much enjoyed as a child. So it seems the hard part is to locate the kids with a potential interest in mathematics and inspire them to go explore for themselves.

When it comes to collaboration are you a workshop person? Or do you have a different MO?

I love collaboration; most of my papers are joint, and most of the mathematics I have learnt, I have learnt from my various co-authors. But the style is different for each co-author. One of them, for instance, likes to stick to the famous Hardy-Littlewood rules of collaboration (which include such counterintuitive rules that there is no obligation to respond to any research communication from the other author). I work with some authors almost exclusively by email, others by trying to secure a week at some conducive location where we can brainstorm at a blackboard. More recently, I’ve been involved with massively collaborative “polymath” projects where dozens of mathematics communicate through wikis and blogs to attack a single problem. Not every collaboration style is suited for every problem, but they are all fun!

You have a very different approach to Andrew Wiles say, did you ever make a conscious decision to have broad mathematical interests?

Actually I think it was my co-authors that helped me broaden the most. When I was a postgraduate student I was initially rather narrowly focused on harmonic analysis. But my co-author Allen Knutson got me interested in algebraic combinatorics and representation theory. My co-author Mark Keel got me into PDE, my co-author Ben Green got me into analytic number theory and additive combinatorics, my co-author Emmanuel Candes got me into signal processing, and so forth. I have a great respect for those mathematicians who drill deeply into a single field and extract some very profound results as a consequence, but I have always been more comfortable with entering a new field (usually with the assistance of a collaborator in that area) and seeing if any ideas or results from a previous one can be profitably applied to this new one.

What advice would you give to a philanthropist with deep pockets who wanted to invest in mathematics?

Well, that is certainly very admirable! I think as far as greatest need is concerned, prizes, scholarships and grants for junior mathematicians, e.g. to be able to attend conferences and have the opportunity to work with leaders in the field, are the most important. But unfortunately these don’t get nearly as much publicity and notice as the larger prizes that go to more established people for more visible accomplishments. It seems that a good compromise is to combine the two – to couple a larger prize with some smaller prizes aimed at junior mathematicians.

Which parts of maths do you think pay the greatest social dividend? Should we divert talent from say, the finance sector, into these areas and if so how?

Well, progress in mathematics isn’t just a matter of throwing money and resources into a given area; sometimes a field is just not yet ripe for dramatic progress, needing a little bit of serendipity to have someone find the key insight. Even very pure areas of mathematics can unexpectedly have tangible real world impact; I and several others had done some purely theoretical work on random matrices, for instance, that ended up being useful for compressed sensing, which is now used for instance to speed up MRI scans. The other thing is that while we certainly do need good people in mathematical research, not every person who is talented in, say, mathematical finance, would also be suitable for this; there are some qualities (e.g. the need to “play”, almost to the point of obsession, with mathematical concepts and problems) that are useful in research but perhaps not in other areas. So I don’t think we should actively try to divert people from a career that they already enjoy and are successful at, but we can certainly raise awareness that there are many areas of both pure and applied mathematics which need good people and which can be rewarding in many ways.

Terry is a long term member of ��ճԹ�’s Scientific Advisory Committee and ��ճԹ�’s first director, Garth Gaudry, was Terry’s mentor as a student at Flinders University. Terry continues to be a strong advocate for the Australian mathematical sciences community.

Heidelberg Laureate Forum Travel Fund Recipient: Melissa Lee

��ճԹ� — Wed, 12 Aug 2015 07:13:01 +0000

The Heidelberg Laureate Forum is an annual meeting bringing together winners of the most prestigious scientific awards in Mathematics (Abel Prize, Fields Medal and Nevanlinna Prize) and Computer Science (ACM Turing Award) with a select group of highly talented young researchers. Roughly 200 young scientists from all over the world get the unique opportunity to interact with their scholarly role models during lectures, panels and discussions. At the same time, the up and coming scientists can engage in inspiring and motivating conversations with the laureates during various social events. The Heidelberg Laureate Forum provides a platform for scientific dialogue across generations.

Each year ��ճԹ� and AustMS provide funding for young Australian researchers to attend.

Melissa Lee
University of Western Australia

Where are you in your career?��I’m currently studying my Masters in Pure Mathematics at the University of Western Australia, under the supervision of John Bamberg and Michael Giudici.

Why do you want to attend the HLF?��I am looking forward to participating in the HLF because it will give me the opportunity to meet many talented young researchers in computer science and mathematics from around the world and engage with outstanding mathematicians and computer scientists who have reached the top of their respective fields. I am excited to hear about their experiences and to gain inspiration from them as I start to look ahead to my own career in mathematics.

Tell us about your research and favourite applications of your work. My research concerns coverings of lines of flock generalised quadrangles, and the groups associated with them. The association schemes that arise from these line covers are reasonably rare, and so are of great interest to people working in other fields of mathematics.��

If you could meet any Fields Medalist or Abel Prize winner which would it be and why?��If I could meet any Fields Medallist or Abel Prize winner, it would be Maryam Mirzakhani because of the outstanding research she has done, and I would like to hear about her experiences as the first woman to win the Fields Medal.

Welcome Inge Koch: Choose Maths Executive Director

��ճԹ� — Mon, 10 Aug 2015 03:27:49 +0000

Associate Professor Inge Koch��completed an MSc at the University of Oxford, and a research MPhil at the University of London, and then worked for industry and the University of Aberdeen in the UK, and for the CSIRO in Canberra. She completed a PhD in statistics at the ANU in 1991 on theoretical problems in image analysis. Inge has joined ��ճԹ� as Executive Director and will head the Choose Maths initiative.

We caught up with Inge to find out about her maths, her stats,��her life and her goals for the secondment with us.

1) Why did you become a mathematician?

Mathematics was fun and a challenge all through my school years. Partly because of the widespread folklore that girls can’t do maths it was not until late in high school that I considered mathematics as a serious career option. By then I enjoyed mathematics more and took it more seriously than other subjects, and I was eager to learn more mathematics.

2) What are some areas of mathematics/statistics that you find particularly interesting?

I enjoy the interplay between theory, data analysis and solving real problems. I feel passionate about developing new statistical methods and theory in statistical learning or machine learning, dimension reduction and selection, and I am keen to apply and adapt these new methods to complex high-dimensional data in proteomics, other biotechnology applications and, more generally, in areas that deal with data with very many and typically too many variables.

3) Do you have any advice that may make more students choose maths as a future career path?

There are many branches of mathematics ranging from the purest pure, to applications in biology, medicine, sport, marketing, climate and the environment, television and finance to name just a few. Each part is important, and you need to work out which part of mathematics speaks to you, and what you like about it.

Go for the part of mathematics that you enjoy most.

Knowledge of mathematics does not have to be an end in itself, but can open doors to new areas and fascinating careers. If you are adaptable, the rigour and insight you learn in mathematics are��transferable to other areas that require analytical and thinking skills.

When looking for a job or career, don’t just search under ‘mathematician’, there is a big world out there that needs your skills and enthusiasm; convince them that you will be a good asset to them.

4) Biggest maths/stats regret?

I wish I had realised earlier how wonderful and exciting statistics can be. It is so much more than what you learn in school or in your early university education as ‘statistics’. It integrates areas of pure mathematics, statistical ways of thinking, computing and having to find efficient and workable solutions for real and diverse data and problems.

5) Biggest maths/stats success?

Classical multivariate statistical theory does not meet the needs of big data and problems arising in machine learning or data science. In the last few decades the often-ignored multivariate Gaussian theory has become increasing relevant again – driven by the demands of complex high-dimensional data and data experts who require answers to their problems. The combined research effort of statisticians such as Peter Hall, Iain Johnstone and Steve Marron and many others has led to new frameworks for principal component analysis and discriminant analysis which are suitable for the analysis of modern data with many more variables than observations. My own research in dimension-reduction methods and my research and graduate text Analysis of Multivariate and High-Dimensional Data are both contributions to integrating the classical with the recent theory and bridging the gap between theory, data analysis and computing for our modern big-data era.

6) What do you hope to achieve as the Executive Director of the Choose Maths program?

Promoting and enhancing an environment in which girls and young women can keenly embrace mathematics in their education and career choices is what makes the Choose Maths program exciting and worthwhile for me. I hope to make progress towards this goal by breaking down barriers, and by actively promoting and working towards change at educational and government levels and in the workplace. This will help to encourage girls and young women to pursue mathematical areas and applications with the passion that I feel for mathematics.

Heidelberg Laureate Forum Travel Fund Recipient: John Tsartsaflis

Wed, 29 Jul 2015 05:10:24 +0000

Each year��ճԹ� and AustMS provide funding for young��Australian researchers to attend.

John Tsartsaflis��
La Trobe University

Where are you in your career?��I’m a PhD student.

Why do you want to attend the HLF?��To meet all these great minds and have the opportunity to interact with��them.

Tell us about your research.��My research is on cohomology of nilpotent Lie algebras over a field of��characteristic 2. In more details, I study filiform Lie algebras and so��far our results are indicating that these algebras are special; they��don’t obey the same rules as they counterparts over a field ofcharacteristic zero.

What are your favourite applications of your work?��It’s pure mathematics! No applications here, thanks.

If you could meet any Fields Medalist or Abel Prize winner which would it be and why?

I would love to meet Maryam Mirzakhani, she seems to be very dedicated��and she has done incredible things in Geometry! Not that I really��understand her work, but I know that it has tremendous impact.

Heidelberg Laureate Forum Travel Fund Recipient: Matthew Tam

Fri, 24 Jul 2015 23:08:15 +0000

Each year��ճԹ� and AustMS provide funding for young��Australian researchers to attend.

Matthew Tam
Computed-Assisted Research Mathematics and its Applications (CARMA), University of Newcastle

Where are you in your career?��I am PhD student at��the��Centre for��Computed-Assisted Research Mathematics and its Applications (CARMA) at the��University of Newcastle.

Why do you want to attend the HLF?��The Heidelberg Laureate Forum, provides a unique opportunity to meet and to learn about out the working habit of��those best in our field. I also look forward to networking with other forum attendees.

Tell us about your research.��My research interests lie in variational and convex analysis, and their application to optimisation. The focus of my PhD is developing, understanding and applying iterative algorithms based on nearest point projections. These methods have a sound theoretical foundation in the presence of convexity but, nevertheless, still perform well when applied to non-convex problems, particularly to those having combinatorial or sparse structure.

What are your favourite applications of your work?��Together with Francisco Aragón Artacho and my supervisor Jonathan Borwein, we applied the Douglas-Rachford algorithm to a problem of reconstructing low-dimensional distance information which arises, for example, in protein conformation determination. In addition to being a mathematically interesting problem, it also generated some fun��.

If you could meet any Fields Medalist or Abel Prize winner which would it be and why?��William Thurston. His essay “On on proof and progress in mathematics” is an interesting and thought provoking discussion of what mathematics is, and what it means to be a mathematician.

Heidelberg Laureate Forum Travel Fund Recipient: Anna Tomskova

Thu, 25 Jun 2015 05:31:59 +0000

Each year��ճԹ� and AustMS provide funding for young��Australian researchers to attend.

Anna Tomskova
School of Mathematics and Statistics,��University of New South Wales

Where are you in your career?��I am a second year PhD student in Department of Pure Mathematics, School of Mathematics and Statistics, Faculty of Science, University of New South Wales, Australia.

Why do you want to attend the HLF?��The HLF is a great opportunity to meet scientists who are at the top of their research areas. It is essential for any young researcher to share their results with other scientists, to know their opinion about the results and to get possible directions from them. This is especially important when these scientists are the best experts in the area. For me, HLF is also a great chance to meet people who work in the same subject that I do for future collaborations. Of course, it is also great to meet people from different cultures with different or similar interests and make new friends.

Tell us about your research and favourite applications of your work. For the last two years I am working in noncommutative analysis. In particular, I study the Theory of Schur multipliers on operator ideals and more generally the Double (and Multiple) Operator Integration Theory.

This subject is very powerful in many questions from perturbation and scattering theory, which is of great importance in solid-state physics. I also like to work in Operator Integration Theory because it may give solutions to��the problems from different areas in mathematics. For example, the problem of Frechet differentiability of the norm��of noncommutative [latexpage]$L_p$ spaces (which is an important result in the geometry of Banach spaces) has been solved using the Multiple Operator Integration Theory.

If you could meet any Fields Medalist or Abel Prize winner which would it be and why?��I would love to meet Cedric Villani, who understands well the applications of mathematics in physics. I am sure that he could help me find more applications for my work in physics. I would also be delighted to meet top class experts in harmonic analysis.

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Heidelberg Laureate Forum Travel Fund Recipient: Philipp Bader

Thu, 25 Jun 2015 05:15:58 +0000

Each year��ճԹ� and AustMS provide funding for young��Australian researchers to attend.

Philipp Bader
Department of Mathematics and Statistics,��La Trobe University

Where are you in your career?��Postdoc

Why do you want to attend the HLF?��The Heidelberg Laureate Forum is a great opportunity to learn from and interact with great mathematical minds. It is very unique in the sense that it appears to generate a much more friendly and familiar environment in comparison with large scale conferences with similar high-impact participants, where actual contacts are usually confined to small parallel sessions and key speakers are only seen at plenary talks. I hope to create lasting contacts and meet international potential collaborators. Another key factor that caught my interest was the multidisciplinarity of the Forum. Even though many of the talks and interests of the participants will most likely be somewhat far from my area of expertise, I find that something can always be learned and could eventually prove to be useful by creating somewhat unexpected links when trying to solve a problem.

Tell us about your research.��I work on the design and analysis of Geometric Integrators for differential equations, in particular for the Schrodinger equation. Over the last decades, it was discovered that numerical solvers that preserve qualitative features of the exact solution such as energy, symplecticity, angular momentum, etc. are superior to standard black-box methods in accuracy and computational cost. The study of geometric integrators thus also involves the study of the properties of the exact solution which will then lead to the development of algorithms that can mimic its geometric properties.

What are your favourite applications of your work?��Lately, my interest has been directed towards the integration of optimal control problems and as a first test, we have applied geometric integrators to control the flight of a drone.

If you could meet any Fields Medalist or Abel Prize winner which would it be and why?��I would like to meet the fields medallist Edward Witten since he is the only physicist that was ever awarded a fields medal and I would like to talk to him about inspiration from physics to solve mathematical problems.

Interview with Nina Snaith

Thu, 21 May 2015 05:00:24 +0000

Nina Snaith is a Reader at the University of Bristol. The title Reader isn’t used often in Australia, it denotes a senior academic with a distinguished international reputation in research or scholarship. After meeting Nina I can see exactly why she has this title; she is humble and gracious, intelligent and passionate. It was a delight to get to know this deep, thoughtful thinker a little better.

Nina’s world has been full of numbers since she was born; her father, an academic mathematician, her mother and grandfather, mathematics teachers and her brother has a PhD in mathematics. Nina doesn’t believe this is simply a coincidence nor does she think it is purely genetic; she did point out that her sister, Anna, is a Professor in Twentieth-Century Literature at King’s College.

They aren’t all mathematicians, but she did marry one!

Despite being surrounded by mathematicians, Nina never thought she would become one. At school she found mathematics class a little dull and repetitive, however she still enjoyed the subject. She joined mathematics clubs, took part in��mathematics competitions and has fond memories of her father and grandfather introducing her to fun puzzles and ideas. So, while Nina has always been aware of the fun to be had with mathematics it wasn’t until university that the inkling to become a mathematician hit her.

Born in London and bred in Canada, Nina took an undergraduate degree in theoretical physics at McMaster University; her interest in Random Matrix Theory (RMT) saw her migrate back to the United Kingdom in the late 90s to begin her PhD research with Jon Keating. Nina completed her thesis, Random matrix theory and zeta functions, in 2000 at the University of Bristol.

Nina’s work is a good example of the two-way relationship between mathematics and physics. Nina explores the rich mathematical structure of RMT and continues to be fascinated by the connections it has with pure mathematics, even though the more��intuitive links are with fundamental physics and engineering.

As my day was beginning Nina’s was ending, but before hers was completely over she spoke to me about her past, her present and her future.��

��ճԹ� runs a Vacation Research Scholarship program over summer; I believe you participated in something similar?

I did. During these summer internships I realised how much I love learning and investigating new things. This experience is so different from learning material for assessment, such as an exam. I did three such projects throughout my time at McMaster; the first was in a molecular biology lab, the second was concerned with solid-state physics, but it was the final project that I did in theoretical physics that introduced me to random matrix theory. These projects showed me what it was like to study independently. If you want to see if research suits you, there is no better way than by doing a project like this, either in a summer or as part of an undergraduate program.

“Mathematics in Schools” is a course you designed at Bristol, what was the idea behind it and who benefits?

Our undergraduates have a placement in a local school in a maths class. They act as a role model for the students, provide support for the teachers and gain valuable experience for themselves about teaching maths and working in a school environment. So, it benefits everyone involved.

Because our students have chosen to study mathematics at university they are enthusiastic about it; this can help spark the interests of the school students.

You studied at McMaster University in Canada, what took you to Bristol?

I did my final year undergraduate project at McMaster on quantum chaos, which links with random matrix theory, the field I now work in. I knew I wanted to continue in the same sort of area, so I asked my project supervisor to suggest some names of people I could do a PhD with. I wrote to several mathematicians and physicists and the advisor I chose, Jon Keating, wrote me the nicest email. He used the phrase “don’t fret” when I was worried about having a background in physics instead of maths. I figured anyone who used that phrase had to be lovely!

A solid mathematics background is needed in many jobs; have your peers chosen other professions?

Software development and finance seem to have been popular career choices for my postgraduate peers. Also, many have continued in research or teaching mathematics in various environments and at various levels. But basically it prepares you for anything that doesn’t need honing of a specific skill, but rather the ability to think, learn, plan, be creative and investigate.

Theoretical physicist or applied mathematician?

What you are called depends on what country you are in. When I came to the UK the structure was very different to that in Canada. You would be called a physicist in Canada, but over here [UK] you’re called an applied mathematician.

I believe there is a continuous scale from maths to physics, and you can sit anywhere on that scale – from pure maths, to experimental physics. So, there doesn’t need to be any difference between applied mathematics and theoretical physics.

You study the connections between statistical properties of L-functions and RMT – layman’s terms?

The Riemann hypothesis — for the Riemann zeta function and other L-functions — says that these functions take value zero at points that lie on a straight line in the complex plane; we call it the critical line. These zeros are located at positions along this line – like beads on a wire. The statistical description of how these positions are located is the same for the zeros and for the eigenvalues of suitably defined random matrices (matrices with entries that are random, apart from appropriate symmetry constraints). It turns out that the similarity between the zeros and the eigenvalues is far reaching and means that properties of random matrices can be used to predict properties of the L-functions – even properties that don’t seem immediately connected to the positions of the zeros.

So, for example, properties of the values of the Riemann zeta function on the critical line, where it hops along between the zero values, can also be predicted by random matrix theory. Some problems of this sort are extremely hard to solve in number theory, so doing an easier random matrix calculation and having it provide a conjecture for what’s happening in number theory is invaluable.

What surprises you about your research?

Given that we are using this rather tenuous, unproven connection between random matrix theory and number theory, what surprises me the most is how our predictions can be so incredibly detailed and accurate!

Why do you believe women are underrepresented in maths?��

I’m sure there are many reasons. I think academia demands quite a lot of one. Of course it has its rewards too, but I think it is still the case that to do that top-notch research you have to put a lot of extra time into your job. I’ve heard female colleagues say that an academic career isn’t so important to them that they would sacrifice other parts of their life. There is also the issue of few role models and still some unconscious bias amongst current academics.

I started a “Women in Maths Group” at Bristol about 12 years ago to support women studying maths. We get together once a week and have lunch, talk about things that are going on I our research and non-research lives. We also hold events for undergraduate students to be role models for them, so that they can see there are women in senior academic positions that also have lives outside of academia.

How important are female role models to girls interested in maths?��

Very important. I remember vividly when one of my physics lecturers happened to mention that she had to run because she was picking up her son from football. My friend and I looked at each other with excitement because it was evidence that you could live a regular life, with kids, etc. and be a physicist. So many of our other lecturers were older men, and it was harder to imagine ourselves like them. I’m sure the same is true in maths.

Did anyone else in your family study mathematics?

My dad is an academic mathematician, my grandfather and mother were maths teachers, and my brother has a PhD in maths. I guess that is no coincidence.

Your brother is also a famous musician; do you see a similarity between studying mathematics and music?

The honest answer is that I do not know if there is a link between maths and music. Especially by the time you are performing music professionally. But, I think there is a similarity in the way we learn both of these things.

I observe my son doing his maths homework and I watch him at his piano lessons and I see exactly the same interest in the two activities. With the piano, he’s not interested in the tune or making a nice noise (he’s 7!) – the bit that captures his interest is the structure.��He asks about the length of notes, bars, rests.��Clearly it’s pattern and structure that interests him, and it’s the same with his mathematics.

Perhaps there is necessarily more of that in music than in other art forms, which is why music is often linked with maths?

As a young student you found mathematics a little dull, but you enjoyed the maths puzzles your father gave you to solve; what do you think the difference was?

We got to investigate, think about and solve the puzzles ourselves, rather than being “taught”, so I’m sure that was part of it.��Also, there’s a lot of repetition involved with gaining facility with maths as it is taught in school, which sometimes detracts from the excitement of learning something new.

Now when I want to learn something new I do lots of reading and I am learning about a method in order to solve a specific problem. It isn’t a matter of repeating calculations a million times like we have to in school, but rather learning about the area deeply enough to be able to tackle that particular, complicated problem. Learning new mathematics as a confident mathematician can be a lot more interesting then the way maths is often taught in school. We have to be creative with the maths��and figure out how to apply it.

Can your research be applied to practical applications?

Random matrix theory can be applied to many fields, from communication networks to aerospace engineering, but I was drawn away from the more practical applications to “applying” random matrix theory in pure maths. It’s a bit backwards – usually pure maths eventually finds applications in something more concrete but here it’s the other way around.��A technique that originated in nuclear physics is predicting results in one of��the most fundamental areas of pure mathematics: the theory of numbers.

I do my research because there are mathematical questions out there. And these questions are worth answering. I believe that 50 to 100 years from now, the maths we are developing now people will find uses for. But frankly, it’s unlikely to be me!

Your work deals quite closely with the Riemann hypothesis, regarded as the most important unproven proposition in mathematics, do you have hope that it will be proven in your lifetime?

It’s so hard to say. I don’t see signs that it’s about to crack any time soon, but with so many fantastic young mathematicians…who can say!

What has your research revealed about the Riemann hypothesis or the Riemann zeta function?��

Lots about the Riemann zeta function, such as how fast the function grows on average between its zeros as you move along the critical line, but not much about the Riemann hypothesis!

There has been considerable excitement about the connections between the Riemann Hypothesis and quantum mechanics. Do you know why RMT methods work in calculating the moments of the Riemann zeta function?

It’s actually a mystery! It’s part of the fun of this area of research. Of course you can observe, and partially prove, the similarity between the two situations and you can use this to shed light on questions in number theory, but fundamentally why this similarity exists…I don’t know!

Have these connections been used to suggest any answers to other long-standing and important problems relating to the zeta function?

The question of moments of the Riemann zeta function — that is, average values of the height of the function on the critical line — has been attacked by number theorists for 100 years, and no general answer has been forthcoming. This is clearly an immensely hard problem. Random matrix theory can predict the general answer.

You are coming to Australia to give the Hanna Neumann lecture at the AustMS and ANZIAM conference. What will you be presenting?��

As I am giving a plenary lecture, much of it will be devoted to the history of how random matrix theory and number theory came together, the people involved and the success this partnership has had. There are some pretty fun and interesting stories concerning those who have made contributions to the field. The talk will lead into the most exciting question at the moment: can random matrix theory shed light on the distribution of ranks of elliptic curves.

And have you had any success with the connection to elliptic curves?

Two of my past PhD students have been working on this project and we are in the middle of it, but we have done enough to have hope that something exciting might happen!

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