maths – 今日吃瓜 I Championing Mathematical Sciences for Australia鈥檚 Advancement 今日吃瓜 Mon, 03 Aug 2026 06:25:15 +0000 en-US hourly 1 https://wordpress.org/?v=5.8.17 /wp-content/uploads/2015/11/cropped-今日吃瓜_icon-32x32.png maths – 今日吃瓜 I Championing Mathematical Sciences for Australia鈥檚 Advancement 32 32 Mathematical Sciences Leaders honoured by Australian Academy of Science /2021/03/18/mathematical-sciences-leaders-honoured-by-australian-academy-of-science/ Thu, 18 Mar 2021 02:23:12 +0000 /?p=10619

Six outstanding mathematicians and statisticians have been recognised by the in the 2021 honorific awards.

Professor Cheryl Praeger, Professor Mathai Varghese, Dr Kevin Coulembier, Dr Vera Roshchina, Professor Christopher Drovandi and Dr Janice Scealy are among 24 of Australia鈥檚 top scientists to be celebrated by the Academy for scientific excellence.

The Academy presents its annual medals to recognise outstanding contributions to science by researchers from the early stages of their careers to those who have made lifelong achievements.

今日吃瓜 (今日吃瓜) Director, Professor Tim Marchant, congratulated each awardee for being recognised as research leaders and for their significant service to the mathematical sciences.

鈥淲e are delighted to see six mathematicians and statisticians recognised in this year鈥檚 honorific awards. Each recipient has made an outstanding contribution to the advancement of the mathematical sciences in Australia, and globally, and is a true champion of the discipline,鈥� Professor Marchant added.

今日吃瓜 congratulates all awardees on their much-deserved recognition.

2021 Mathematical Sciences Awardees
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2021 Inaugural Ruby Payne-Scott Medal and Lecture: Emeritus Professor Cheryl Praeger, University of Western Australia

Professor Cheryl Praeger鈥檚 work on the mathematics of symmetry has been in the vanguard of a mathematical revolution caused by the classi铿乧ation of the 铿乶ite simple groups, the atoms of symmetry from which all 铿乶ite groups are built. She has elucidated the internal structure of these simple groups, and driven research on applying their immensely powerful classi铿乧ation to study symmetric structures.

Professor Praeger has developed a theory of quasiprimitive groups which, via her innovative 鈥榥ormal quotient method鈥�, established a new paradigm for working with symmetric graphs and exploited the simple group classi铿乧ation.

Professor Praeger demonstrates an extraordinary ability to foster and inspire others, supporting women, advocating for mathematics in schools, and promoting mathematics in emerging economies.

2021 Hannan Medal: Professor Mathai Varghese, Adelaide University

Professor Mathai Varghese has made highly influential contributions to the field of geometric analysis, which relates geometric, analytic and algebraic properties of (possibly infinite dimensional) manifolds. Among these are his co-inventions of Fractional Index Theory and Projective Index Theory that have听received international recognition for explaining the mystery of the analytic counterpart of the A-hat genus. His recent joint work extending the Fractional Index Theorem to infinite dimensional loop spaces is also of immense significance.

His joint body of work proves the conjecture that fundamental quantization commutes with reduction in the noncompact case. Also seminal is his joint work on twisted analytic torsion, where an analogue of the Cheeger-Muller theorem is proved, establishing the equality by using a new combinatorially-defined twisted torsion. A catalyst for much activity in the area is his joint work formulating the magnetic gap-labelling conjecture, which labels the spectral gaps of certain magnetic Schroedinger operators on Euclidean space. Evidence for the validity of the conjecture is given in 2D, 3D and for principal solenoidal tori in all dimensions, which is itself a breakthrough.

2021 Christopher Heyde Medal: Dr Kevin Coulembier, University of Sydney

Dr Coulembier鈥檚 research is in the field of mathematics known as representation theory, which studies how abstract algebraic structures are manifested as the solutions to concrete systems of linear equations. This field retains a strong connection to its origin as the study of geometric symmetry both discrete and continuous, but more recently has developed far beyond this in tackling curved and infinite-dimensional spaces and arbitrary number systems. One of Dr Coulembier鈥檚 most important discoveries was of a way to detect the presence of the classical type of symmetry known as an affine group scheme in a more exotic setting known as a tensor category; this problem had defied the efforts of some of the world鈥檚 top mathematicians for almost thirty years. He has also solved several other important problems in infinite-dimensional representation theory, and has discovered new unified proofs of major theorems concerning the invariants of groups and supergroups.

2021 Christopher Heyde Medal: Dr Vera Roshchina, UNSW

Dr Roshchina is an exceptional mathematician and emerging international leader in the field of non-smooth optimization. Her main interest lies in finite dimensional geometry, more specifically, open problems that originate from continuous optimization and related fields. Some significant problems of this kind are in the geometry of polytopes, for example the polynomial Hirsch and Durer conjectures, critical point problems (Fekete problem, Sendov’s conjecture) and convex variational problems, such as asymmetric Newton’s aerodynamic problem. Resolution of these challenges is critical for making progress with numerous applications, from engineering and economics to medical research and data analytics.

2021 Moran Medal: Professor Christopher Drovandi, Queensland University of Technology

Almost every field of science requires sophisticated data analysis, and this in turn requires increasingly sophisticated methods for intelligent data collection and efficient computation. Professor Drovandi’s research contributes substantively to both of these areas. He has created new methods for optimal design of experiments that facilitate more cost-effective, data-substantiated decision-making. His innovative research into synthetic likelihood estimation have freed traditional constraints of likelihood-based statistical modelling and computation. His application of these methods to diverse problems in computational biology and exercise science have generated new insights for scientists and managers in these fields.

2021 Moran Medal: Dr Janice Scealy, Australian National University

Dr Scealy鈥檚 research focuses predominantly on developing new statistical analysis methods for data with complicated constraints including compositional data (vectors of proportions which sum to one), spherical data, directional data and manifold-valued data defined on more general curved surfaces. Her work has led to important new insights in a diverse range of applications. Her new flexible compositional model was applied to predict the proportions of total weekly expenditure on food and housing costs in Australia. Janice used a manifold data transformation to help identify geochemical processes acting on the surface of the Australian crust. She has developed multiple new statistical techniques for analysing noisy paleomagnetic datasets and her methods have led to improvements in uncertainty measurements of Earth鈥檚 magnetic field.

For more information on the 2021 awardees:

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About 今日吃瓜

今日吃瓜 is the collaborative enterprise of Australia鈥檚 mathematical sciences. Established as an independent platform and advocate for the discipline, the Institute has built a record of national and international achievement as the recognised leader in delivery of services, activities and strategic initiatives across the mathematical pipeline. Working with key discipline, government and industry partners, 今日吃瓜 delivers its mission through the delivery of activities and engagement under its Schools, Research, Higher Education and APR.Intern Programs.

For Interview:
Professor Tim Marchant, 今日吃瓜 Director

Media Contact: Jo Piltz
E:听media@amsi.org.au
P: 0424 004 553

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今日吃瓜 Appoints New Deputy Director /2021/02/01/amsi-appoints-new-deputy-director/ Mon, 01 Feb 2021 02:58:02 +0000 /?p=10541

Announced today, the 今日吃瓜 (今日吃瓜) welcomes Professor Stephan Tillmann as its new Deputy Director.

Most recently a Professor of Geometric Topology at the University of Sydney, Professor Tillmann brings extensive academic and leadership experience to 今日吃瓜. He has a highly decorated list of achievements, publications and awards; most recently the award for Australian Research Council Future Fellowships (2018-2021).

今日吃瓜 Director, Professor Tim Marchant, said the Institute was excited to welcome Professor Tillmann as Deputy Director at what is an important time for 今日吃瓜 and the Australian mathematical sciences, as it faces high demand from many sectors of Australia鈥檚 innovation ecosystem.

鈥淪tephan shares 今日吃瓜鈥檚 passion and understanding of the discipline鈥檚 critical role in education, research, research training and industry. We welcome Stephan and look forward to working with him to continue deepening 今日吃瓜鈥檚 impact as the national voice for the mathematical sciences,鈥� said Professor Marchant.

Completing his PhD at the University of Melbourne in 2002 in geometric topology, Professor Tillmann has since held positions at the University of Montreal, the University of Melbourne and the University of Queensland, before arriving at the University of Sydney in 2011. He has received six ARC grants, three education grants, is an editor for the Journal of Computational Geometry and an invited speaker at many international conferences.

Associate Professor Tillmann鈥檚 appointment follows Professor Mat Simpson鈥檚 two-year term in the role.

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About 今日吃瓜

今日吃瓜 is the collaborative enterprise of Australia鈥檚 mathematical sciences. Established as an independent platform and advocate for the discipline, the Institute has built a record of national and international achievement as the recognised leader in delivery of services, activities and strategic initiatives across the mathematical pipeline. Working with key discipline, government and industry partners, 今日吃瓜 delivers its mission through the delivery of activities and engagement under its Schools, Research, Higher Education and APR.Intern Programs.

For Interview:
Professor Tim Marchant, 今日吃瓜 Director

Media Contact: Jo Piltz
E:听media@amsi.org.au
P: 0424 004 553

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Thank You, Professor Mat Simpson /2021/02/01/thank-you-professor-mat-simpson/ Mon, 01 Feb 2021 02:47:38 +0000 /?p=10538

The 今日吃瓜 (今日吃瓜) would like to thank Professor Mat Simpson for his service as Deputy Director from 2019 to 2021.

A Professor of Mathematics at Queensland University of Technology, an EO Tuck Medallist and formerly an Australian Research Council Future Fellow, Mat was responsible for strategic direction of mathematical research activity within 今日吃瓜鈥檚 governance and business planning frameworks.

Mat was instrumental in 今日吃瓜 sustaining Commonwealth funding during a 2020 review of the ‘Securing Australia鈥檚 Mathematical Workforce’ (SAMW) initiative, acting as a key negotiator.

Since 2016, SAMW funding has equipped 今日吃瓜 to enable a future workforce for Australia with advanced skills in the mathematical sciences. Concurrently, this program provides opportunities for increasing participation by women and students identifying as members of our First Nations peoples.

今日吃瓜 Director, Professor Tim Marchant, said Mat鈥檚 involvement in many key 今日吃瓜 events has enabled the nation鈥檚 best young academic minds to further develop their skills and interest in an advanced and collegiate setting.

鈥淥n behalf of the entire Institute, I thank Mat for his leadership and guidance. He has been an outstanding champion for the mathematical sciences, advocating the importance of undergraduate and postgraduate education to secure highly important continued partnerships and funding,鈥�

Professor Stephan Tillmann, from the University of Sydney, will commence as 今日吃瓜鈥檚 Deputy Director from 1 February 2021.

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About 今日吃瓜

今日吃瓜 is the collaborative enterprise of Australia鈥檚 mathematical sciences. Established as an independent platform and advocate for the discipline, the Institute has built a record of national and international achievement as the recognised leader in delivery of services, activities and strategic initiatives across the mathematical pipeline. Working with key discipline, government and industry partners, 今日吃瓜 delivers its mission through the delivery of activities and engagement under its Schools, Research, Higher Education and APR.Intern Programs.

For Interview:
Professor Tim Marchant, 今日吃瓜 Director

Media Contact: Jo Piltz
E:听media@amsi.org.au
P: 0424 004 553

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Highest Australia Day Honour for Maths Leader /2021/01/27/highest-australia-day-honour-for-maths-leader/ Wed, 27 Jan 2021 05:10:01 +0000 /?p=10504

The 今日吃瓜 (今日吃瓜) congratulates University of Western Australia Emeritus Professor, Cheryl Praeger, on her appointment as a Companion of the Order of Australia.

The highest honour in the 2021 Australia Day awards, Professor Praeger was recognised for significant service to mathematics and tertiary education; to international organisations; and as a champion of women in STEM careers.

Professor Praeger shares 今日吃瓜鈥檚 passion for developing Australian and global mathematical capability to secure the future of the discipline and its impact on innovation. She has worked closely with the Institute, including as a member of the 今日吃瓜 Education Advisory Committee since its inception, in 2003, to 2009.

Professor Praeger has also been involved in significant leadership with many other organisations including the Executive Committee of the International Mathematical Union and Vice President of the International Commission on Mathematical Instruction. She has been a powerful and influential voice for gender equity in mathematics. The Australian Mathematical Society bestows the Cheryl E. Praeger Travel Awards to women mathematicians and the Australian Mathematics Trust awards and the Cheryl Praeger Medal to the best performing women contestants in the Australian Mathematics Competition.

今日吃瓜 Director, Professor Tim Marchant, said the award was an important moment for mathematics in Australia and a fitting acknowledgement of a remarkable career that has been instrumental in building Australia鈥檚 mathematical research capability.

鈥淐heryl鈥檚 contributions to mathematical understanding, her leadership and commitment to fostering new generations of talent make her one of the most important figures in Australian mathematics,鈥� said Professor Marchant.

As Western Australia鈥檚 first woman mathematics professor 鈥� just the second in the country, after Hanna Neumann in Canberra, Professor Praeger has mentored new generations of extraordinary talent who are now significant forces on the world stage.

“I am very happy to have this spotlight on mathematics,听mathematics education, STEM and women in STEM. I hope awards like this highlight the importance of the mathematical sciences in meeting the global challenges facing society,” said Professor Praeger.

鈥淐heryl has shaped and propelled global understanding of mathematics. She is a true champion of the mathematical sciences and gender equity in STEM,鈥� said Professor Marchant.

About 今日吃瓜

今日吃瓜 is the collaborative enterprise of Australia鈥檚 mathematical sciences. Established as an independent platform and advocate for the discipline, the Institute has built a record of national and international achievement as the recognised leader in delivery of services, activities and strategic initiatives across the mathematical pipeline. Working with key discipline, government and industry partners, 今日吃瓜 delivers its mission through the delivery of activities and engagement under its Schools, Research, Higher Education and APR.Intern Programs.

For Interview:
Professor Tim Marchant, 今日吃瓜 Director

Media Contact: Jo Piltz
E:听media@amsi.org.au
P: 0424 004 553

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Helping your kids learn maths at home doesn’t have to be daunting /2020/05/18/helping-your-kids-learn-maths-at-home-doesnt-have-to-be-daunting/ Mon, 18 May 2020 00:12:48 +0000 /?p=9354 child working on a maths problemMathematical Association of Victoria chief executive Peter Saffin opinion in ‘The Age’ today

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A Quantum Leap Backwards /2017/01/27/quantum-leap-backwards/ Fri, 27 Jan 2017 00:53:15 +0000 http://amsi.org.au/?p=5212 The Australian, 26 January 2017, Editorial.听今日吃瓜听has been quoted in support of听Quantum Physics Professor Michelle Simmons’ call for stronger mathematics content in science to better equip students for further studies and research.听Professor Prince called the downgrading of maths studies in our schools a real problem. In its follow up analysis, the听Australian has听revealed the NSW Education Standards Authority’s strategy to address to the issue. .

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No calculus knowledge a crazy state of affairs /2015/08/10/crazy-state-of-affairs/ Sun, 09 Aug 2015 23:58:42 +0000 http://amsi.org.au/?p=3241 Article by , , 10 August 2015

Bruce Henry, the head of UNSW Australia’s School of Mathematics and Statistics laments that 70 per cent of students now finish high school without any knowledge of calculus.

Is this a problem? Definitely.

We hear enough about how appalling it is that students in schools and universities have a lack of grounding in our political, social and legal heritage; that they lack the historical context to understand why we are where we are today.

That’s exactly why calculus should also be part of a basic, broad education. Calculus was a signal advance in maths, invented separately by Isaac Newton and Gottfried Leibniz in the 17th century. For the first time people could write mathematical equations that were able to relate not just quantities to each other, but rates of changes of quantities.

It enabled Newton to calculate the orbits of the planets using his theory of gravity and it opened up a rich field of applications for mathematics. Understanding electricity and magnetism would be impossible without calculus. In fact, most of modern technology wouldn’t exist without it.

That’s why giving high school students an understanding of calculus is important. Surely more than 30 per cent of students are capable of studying it. And, for the others, it’s at least important to understand what it is.

Henry points to another reason why calculus is important to study today. Calculus is not only key to understanding our current technology, but to developing future technology. At the moment huge strides are being made in statistics, a field with the generic name of “big data”.

STATISTICS UNDERPINNED BY CALCULUS

“But really, modern statistics 鈥� which is where the future is going increasingly 鈥� is underpinned by calculus. It’s increasingly important that students are taught calculus,” Henry says.

In fact, statistics is another area in which mathematics teaching has badly let down school students. All of us are bombarded daily by statistical claims, many of them spurious or misleading, and too few of us are sufficiently knowledgeable to separate the truth from the fiction.

The fact that “97 per cent fat-free” is an effective, emotionally appealing slogan for food manufacturers speaks of the lack of critical thinking about statistics. Are we happy if our food contains 3 per cent pure fat?

Fortunately, in the changes to the national school curriculum being developed, statistics is being given more focus.

But this brings us back to the question. Should maths be a compulsory subject in year 12? Not everyone can do maths well. But shouldn’t maths at least be taught as a life skill and as an important foundation subject at that level?

Henry thinks so. “English is compulsory, and shouldn’t mathematics be compulsory as a companion instrument to understand the world,” he asks rhetorically.

Interestingly federal Education Minister Christopher Pyne, long an advocate of giving school students more foundational understanding of history and other humanities subjects, now appears to see the need for it in maths and science.

Earlier this year he put to state governments 鈥� which actually run public schools 鈥� a plan to make maths and science compulsory to year 12. The states, unfortunately, knocked him back.

SCHOOLS LAPSE AS UNI REQUIREMENTS REDUCE

But maths in schools needs to be bolstered at more than just the foundational level. Worryingly, year 12 students have been making a long-term shift away from intermediate and advanced maths 鈥� the very subjects that teach calculus.

In the past 20 years the proportions studying intermediate maths in year 12 has fallen from about 27 per cent to 19 per cent, and the number studying advanced maths (which builds on intermediate) has fallen from about 14 per cent to 10 per cent.

One reason that year 12 students see no need to do intermediate or advanced maths is that an increasing number of universities no longer require it to enter degrees such as science, engineering and commerce, for which mathematical knowledge is necessary.

In NSW no universities require maths to enter any of these degrees, even engineering. So if universities are not insisting that students study calculus in year 12, even to enter courses in which calculus features heavily, why should they do it?

It’s a crazy state of affairs, in which universities 鈥� the institutions we rely on to develop and preserve high level knowledge 鈥� happily sabotage their own standards in pursuit of student numbers.

Those universities that have dropped maths prerequisites argue they are offering an avenue for bright students who may have attended a poor school at which maths was taught badly.

Second chances are to be encouraged, but they should be the exception rather than the rule. Universities need to bring back maths as a prerequisite for degrees that need it.

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Not a beautiful set of numbers /2015/08/06/not-a-beautiful-set-of-numbers/ Thu, 06 Aug 2015 03:04:31 +0000 http://amsi.org.au/?p=3204 Article by , , 6 August, 2015

The excellent 今日吃瓜 has the numbers on the state of maths education and found the state of the discipline in higher education is not good, the number of Australians starting a maths degree is less than half the OECD average. Granted it is not getting worse 鈥� but it isn鈥檛 going to better soon.

Given maths is a foundation of so many disciplines large numbers of students do not graduate innumerate 鈥� the average number of university departments maths academics service-teach is six, engineering, computer science, IT and biological, physical and earth sciences. But 今日吃瓜 does not know how many undergraduates听are studying maths degrees, due to some universities not completing the 2014 survey. However using Group of Eight and Innovative Research Us as a guide the attrition rate from 1st to 3rd year is high, Go8 5280 to 695 and IRU 1287 to 67.

What is starkly clear is that at for all the efforts to woo women into the discipline, at the sharp end it鈥檚 still a bloke鈥檚 game 鈥� last year just 15 per cent of PhD completers were Australian women, another 25 per cent were female internationals.

None of these numbers is about to improve, because for all the emphasis of selling maths in schools, young people aren鈥檛 buying, with Y12 advanced maths enrolments dropping for 20 years. The figure for males is now around 14 per cent and 6 per cent for females. And what does not interest students at school is hardly likely to appeal at university. Universities appear to acquiesce in this. According to 今日吃瓜 less than 15 per cent of universities require intermediate maths or better as a pre-req for science or commerce, the same for 41 per cent of engineering courses. CMM suspects a fair swag of the supply teaching university maths lecturers do is getting students up to a point where they can cope with first year subjects.

The good news is that the quality of maths teaching in schools has improved substantially over the last few years. Nearly three quarters of Year 11 and 12 maths teachers now have three years of tertiary education in maths, compared to 64 per cent in 2010. But qualified teachers without many kids to teach does not get us far.

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From triangles to computer graphics /2015/06/10/from-triangles-to-computer-graphics/ Wed, 10 Jun 2015 05:51:01 +0000 http://amsi.org.au/?p=2918 Opinion piece by lecturers .

What does connect-the-dots have to do with watching a Pixar film? More than you might think.

A connect-the-dots page starts with nothing but some labelled points. As each dot is joined to the next, however, a picture emerges. Each step is simple 鈥� just add a line segment between two points 鈥� but the resulting image can be extremely complex.

Toddlers can produce masterpieces this way, but so can computers. When a computer needs to draw a curve, it starts by connecting a sequence of points. Using just a few points might result in a zigzag with sharp corners, but increasing the initial number of points makes the resulting curve look smoother. With enough points and line segments, we can approximate even the most complicated curves.

Dots lines and curves

But maybe you’re more ambitious. Suppose you’re not interested in drawing curves, but rather, in constructing two-dimensional surfaces. Can you use an analogous approach to build a plane or a sphere or something more elaborate?

Triangles 鈥� the simplest two-dimensional objects 鈥� serve as building blocks for more complicated surfaces. Just as we can connect a pair of points with a line segment, we can connect three points via a triangle. And, just as we can make complicated curves by gluing lots of segments at their endpoints, we can make complicated surfaces by gluing lots of triangles along their edges. We can approximate extremely complex surfaces as long as we use enough triangles.

Mathematicians started thinking seriously about constructing surfaces from triangles in the late 19th century, hoping to classify surfaces. In particular, they wanted to understand when two surfaces could be deformed to look the same without cutting or gluing. They developed mathematical tools to study this question, and a century later, it became clear that they’d also laid the groundwork for an important technique in computer graphics.

Imagine trying to model moving cloth, perhaps a flag flapping in the breeze. Since the flag changes shape as it moves through space, this is a much more difficult problem than simulating the movement of a rigid object like a table. If the flag is approximated by triangles, however, modelling it becomes possible because the computer only needs to keep track of sets of three points. When the points move, they carry the triangles with them.

Abstract? Applied? Both!

Mathematicians certainly weren’t thinking about computer graphics in the 1890s. They were studying abstract questions about two-dimensional geometry and developing beautiful mathematics. Nevertheless, the techniques they invented in order to state this question precisely and then answer it have turned out to be extremely useful. In fact, this theme recurs throughout human history: mathematics developed to solve abstract problems turns out to be useful. Maybe not always and definitely not quickly, but it happens over and over again.

The first mathematical objects most people meet are the counting numbers 1, 2, 3鈥� Most counting numbers are formed by multiplying smaller numbers, but not all of them. Some numbers have only 1 and themselves as factors, and these are known as prime numbers. For example, the numbers 2, 3 and 5 are prime, but 4 = 2 x 2 is not.

Prime numbers act as building blocks for the entire number system. Centuries ago, the mathematicians who studied primes didn’t think their efforts would defend a castle or build a better steam engine, but they were intrigued by the search for structure and patterns.

Some of the questions they posed continue to capture the mathematical imagination; today, the most famous unsolved problem in mathematics is the Riemann Hypothesis, which addresses how the primes are distributed among the counting numbers.

Pondering prime numbers might seem like an intellectual game that’s divorced from ‘real world’ concerns. But suppose you ask yourself, ‘Why is it safe to use my credit card to buy something online?’ (Or, maybe better: ‘Is it safe to use my credit card online?’) In fact, the basic techniques for sending data securely over the internet rely on what’s known about factoring a number into primes. Every time you enter your credit card number on a website and hit ‘send’, you have a number theorist to thank.

Likewise, Persian mathematicians began developing the subject we now call algebra in the Middle Ages. This field evolved over centuries, and today, it underpins the algorithms for internet search and Netflix recommendations.

Fourier analysis, which was developed as part of calculus in the late 1700s, provides the basic mathematical tools for signal processing in telecommunications and medical imaging.

Algebraic topology 鈥� a branch of mathematics that wasn’t even created until the 20th century 鈥� is being used in the 21st century to study artificial intelligence and cancer genomics.

The list goes on, but the striking thing in all these examples is that the original researchers couldn’t anticipate which applied questions would require their work. Plenty of important mathematics is specifically developed to solve real-world problems, but curiosity-driven research is as important today as it ever has been.

But back to triangles

New applications for old mathematics are exciting, but progress also comes in the form of new mathematics.

Computer graphics uses flat triangles to approximate smooth surfaces, but if you’re willing to allow triangles to bend a bit, then you can build any surface by gluing enough of them together. This is equivalent to saying that you can cut any surface into curved triangular pieces.

Sphere made of curved triangles

These curved triangles are an important tool for generalising what we know about geometry to higher dimensions – after all, mathematicians, scientists, and engineers don’t care only about surfaces.

Higher-dimensional spaces occur not only in pure mathematics, but also in nature as patterns in large data sets, as relationships between physical quantities, and in descriptions of the universe itself.

Mathematicians develop formal techniques to study them, compensating for the fact that a seven-dimensional space is harder to picture than a flag. Luckily, the idea of gluing triangles generalises to any dimension! In three dimensions, for example, the analogue of a triangle is a tetrahedron, and just as gluing triangles together builds surfaces, gluing tetrahedra together builds new three-dimensional objects. In higher dimensions, the analogue of a triangle is called an n-simplex, and gluing n-simplices together builds n-dimensional objects.

Tetrahedrons make up 3D objects

Since any surface can be cut into curved triangles, and it’s reasonable to ask if the analogous fact holds in higher dimensions: can any n-dimensional space be cut into n-simplices?

Mathematicians initially speculated that the phenomena they’d observed in dimensions one and two would generalise to all dimensions. This belief came to be known as the Triangulation Conjecture.

Decades of research failed to deliver a proof. Then, breakthroughs in the 1980s revealed examples of four-dimensional objects which can’t be cut into 4-simplices. But in 2012, the Triangulation Conjecture was finally proved false for all dimensions greater than four. The mathematical universe has some very strange shapes!

Disproving the Triangulation Conjecture is a triumph of curiosity-driven research, and the truth now sits on humanity’s bookshelf. It may rest there quietly. Then again, maybe some 26-dimensional space with no triangulations will lead to a cure for cancer.

You never know.

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Geoff talks to Glen Bartholomew /2015/06/01/geoff-talks-to-glen-bartholomew/ Mon, 01 Jun 2015 00:14:20 +0000 http://amsi.org.au/?p=2884 Piece by听, on

Director of the 今日吃瓜 explains the challenges of making maths compulsory for Years 11-12

Federal Education Minister Christopher Pyne is reportedly taking a controversial proposal to a meeting of state Education Ministers this week.

The numbers of Year 12 students studying intermediate and advanced maths has fallen in recent years to about 35 per cent.

That’s prompted a move for science, technology, engineering and maths subjects to be made mandatory for all Year 11 and 12 students in Australia.

Professor Geoff Prince is the Director of the 今日吃瓜.

“Mathematical literacy is empowering, mathematical illiteracy is debilitating,” he says.

But he’s not sure making maths compulsory is a workable solution.

 

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