Riddle. Mystery. Enigma.

I appear in a rather excellent and fun episode of the ABC Radio National program Radiotonic.

You can listen to the whole thing here. From the website:

For most of us, codes are riddles wrapped in enigmas wrapped in mysteries. Even the Saturday crossword or Sudoku can be a stretch. Yet we’re surrounded by codes everyday, whether it’s a puzzle in the paper, the GPS on our Smartphones, or every time we browse the internet.

Initially stumped by that most common—and frustrating—of codes, a cryptic crossword clue, writer Sunil Badami finds himself surfing an internet ‘Wikislide’: discovering the connections between code and poetry, evading shadowy government agencies, and ending up in the murky corners of the dark web, where he’ll try to unlock the secret to one of the Web’s most intriguing and eerie mysteries.

Riddle. Mystery. Enigma. takes you on a journey from the familiar to the unexplored, through the mysterious history of cryptology and coding, and how it affects our lives today. Part radio documentary; part radio code breaking exercise; definite radio goose chase. Plus you get the opportunity to solve our own intriguing secret code, encrypted into the episode.

Yes, this is secret coded radio. Keep your ears open, you don’t want to miss anything.

Introduction to the AJ conjecture, Melbourne, August 2015

On 14 August 2015, I gave a talk in the Melbourne University Knot Invariants seminar.

Title:

Introduction to the AJ conjecture

Abstract:

The AJ conjecture is an important outstanding conjecture in knot theory, relating two very different polynomial invariants: the A-polynomial and the coloured Jones polynomial. In this talk I’ll explain what the AJ conjecture is, and some of the things that are known about it. I’ll also outline some ideas that have been used to prove it in some cases, including skein modules and character varieties.

Geometric quantisation and A-polynomials, June 2015

On 12 June, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar.

Title: Geometric quantisation and calculation of A-polynomials

Abstract: Dimofte has developed a method for calculating A-polynomials via geometric quantisation and symplectic reduction. This method allows us not only to compute the classical A-polynomial but also a quantisation which conjecturally annihilates the 3-d index and appears in the AJ conjecture. I’ll discuss his method and present an example.

Every world in a grain of sand: John Nash’s astonishing geometry

(Published at The Conversation.)

As has been widely reported, John Forbes Nash Jr died tragically in a car accident on May 23 of this year. Many tributes have been paid to this great mathematician, who was made famous by Sylvia Nasar’s biography A Beautiful Mind and the subsequent movie based on that book.

Much has been said about Nash’s work on game theory. But less has been said about Nash’s other mathematical achievements. Many mathematicians who understand Nash’s work would agree, I think, that although his work in game theory had the most impact on other fields, Nash made other breakthroughs which were even more impressive.

Apart from game theory, Nash worked in fields as diverse as algebraic geometry, topology, partial differential equations and cryptography.

But perhaps Nash’s most spectacular results were in geometry. To honour Nash’s life, I would like to try to give a flavour of some of this work.

John Nash and pure mathematics

A great deal of Nash’s work was in the field of geometry. But this kind of geometry – differential geometry – is very different from the geometry learned at high school. It is not about trigonometry or Pythagoras, as found in secondary maths textbooks. Rather, it is about topics like surfaces, curvature and smoothness.

Like all pure mathematicians, Nash proved theorems: logical statements that are rigorous, precise and absolutely true, with no tolerance for vagueness. The world of pure mathematics is austere and often abstruse, but its claims to truth are eternal and absolute.

Well, that’s the theory at least. Breakthroughs in pure mathematics are often at the very limits of human understanding. It takes time, even for those in the field, to fully comprehend new developments.

Nash’s work was an extreme case. His papers could be chaotically presented, hard to follow and his approaches to problems were often unlike anything that had come before him, bamboozling students and experts alike. But he was almost otherworldly in his creativity.

While mathematical arguments are tightly constrained by the rigorous requirements of logic, Nash’s constructions and methods were wild. And nowhere was this more so than in his work on geometry.

Nash’s geometry

Take a flat sheet of paper. You can bend it, but without ripping it or creasing it, what shapes can you make? You can’t make a sphere, or even a section of a sphere, because a sphere is curved, while the paper is flat.

But you can make a cylinder. And even a cone, as you’ll know if you’ve ever seen a dunce’s hat. (This fact is also useful for making waffle cones, as shown below.)

Waffle cones start off as flat surfaces. Source: Gotham3 on Imgur.

As it turns out, even though a cylinder or a cone looks curved, it is intrinsically flat. In an undergraduate course on differential geometry (such as the one I teach at Monash), one studies this intrinsic curvature, and it turns out that there are lots of flat surfaces.

This surface might not look flat, but it is. Source: Richard Morris on Wikipedia.

These ideas were around for hundreds of years before Nash, but Nash took them much further.

The embedding problem

Nash took up the idea of “embedding” a surface: placing it into space without tearing, creasing or crossing itself. An embedding which does not distort the surface’s intrinsic geometry is “isometric”. In other words, the surfaces above are “isometric embeddings” of the plane into 3-dimensional space.

The isometric embedding question can be asked not just for the plane, but for any possible surface: spheres, donuts (which mathematicians call tori to try to sound respectable) and many others.

As it turns out, there are surfaces that are so strongly curved or tangled up that they cannot be embedded into 3-dimensional space at all. In fact, they can’t even be embedded into 4-dimensional space.

But Nash showed that any surface can be embedded into 17-dimensional space. Extra dimensions, far from making the problem even more difficult, actually make it easier – giving you more room to embed your surface! Later on, Nash’s work was improved by others, and we now know that any surface can be embedded into 5-dimensional space.

However, surfaces are only 2-dimensional. And Nash was interested in surfaces of any possible dimension. These higher dimensional analogues of surfaces are known as “manifolds”.

Nash proved that you can always embed a manifold into space of some dimension, without distorting its geometry. With this momentous result, he solved the isometric embedding problem.

Nash’s proof of the isometric embedding problem came as a complete surprise to much of the mathematical community. His methods were revolutionary. The great mathematician Mikhail Gromov said that Nash’s work on the embedding problem struck him to be “as convincing as lifting oneself by the hair”. But after great effort, Gromov finally understood Nash’s proof: at the end of Nash’s lengthy argument, Gromov said, Nash “miraculously, did lift you in the air by the hair”!

Isometric embedding in action

Gromov went on to develop his own ideas, inspired by Nash’s work. He wrote a book – similarly renowned among mathematicians for its incomprehensibility, just like Nash’s work – in which he developed a method called “convex integration”.

Gromov’s method had several advantages. One is that it is easier to draw pictures of an embedding made with his convex integration method. Prior to Gromov, we knew isometric embeddings existed, and had wonderful properties, but had a very tough time trying to visualise them, not least because they were often in higher dimensions.

In 2012, a team of French mathematicians produced computer graphics of isometric embeddings using Gromov’s convex integration methods. They are extremely intricate, almost fractal-like, yet smooth. Some are shown below.

The world in a grain of sand

Nash’s work on the isometric embedding problem has many facets and has led to huge amounts of subsequent research.

One particularly amazing aspect is how isometric embeddings are constructed. Nash’s work, combined with subsequent work by Nicolaas Kuiper, showed that if you wanted to isometrically embed a surface in 3-dimensional space, it’s enough to be able to shrink it.

If you have a “shrunken” embedding of your surface – that is, with all lengths decreased – then Nash and Kuiper show how you can obtain an isometric embedding of your surface just by adjusting your shrunken version a bit.

This sounds ridiculous. For instance, take a sphere – say the surface of a tennis ball – and imagine shrinking it down to have a nanometre radius. Nash and Kuiper show that by “ruffling” the surface sufficiently (but always smoothly; no creasing or folding or ripping or tearing allowed!) you can have an isometric copy of your original tennis ball, all contained within this nanometre radius. This type of “ruffling” of the surface was reproduced in the French team’s computer graphics.

The French team considered taking a flat square piece of paper. Glue the top side to the bottom side, to get a cylinder. Now glue the left side to the right side. If you think about it, you might be able to see that you get a donut. But you’ll find the paper is now creased or distorted.

Can you embed it into 3-dimensional space without distortion? Nash and Kuiper say “yes”. Gromov says “use convex integration”. And the French mathematicians say “this is what it looks like”!

Isometric embedding of the square flat torus in ambient space. Hevea Project, CC BY-SA.

More pictures are available at the Project’s website.

But the mathematical theorem doesn’t just apply to tennis balls or donuts: the theorem holds for any manifold of any dimension. Any world can be contained in a grain of sand.

How did he do it?

Nash had a rare combination of genius and hard work. In her biography of Nash, Sylvia Nasar details his formidable intensity and effort spent working on the problem.

As is well known from the movie, Nash came to believe in outlandish conspiracy theories involving aliens and supernatural beings, as a result of his schizophrenia. When later asked why he, an extremely intelligent scientist, could believe in such things, he said those ideas “came to me the same way that my mathematical ideas did. So I took them seriously”.

And frankly, if my head told me ideas as accurate and as insightful as those needed to prove the isometric embedding theorem, I’d likely trust it on aliens and the supernatural too.

The A-polynomial, symplectic geometry, and quantisation, May 2015

On 15 May, 2015 I gave a talk at the University of Melbourne, in the Moduli Spaces seminar.

Title: The A-polynomial, symplectic geometry, and quantisation

Abstract: The A-polynomial is a knot invariant based on the SL(2) representation theory of a knot group. It has long been known to have close connections to hyperbolic geometry, but it has returned with a vengeance in recent years, appearing in various guises from physics, and motivating mathematical statements such as the AJ conjecture. In this talk I’ll outline the A-polynomial and some work of Dimofte in which he considered its relation to Chern-Simons theory. In particular, I’ll talk about methods to calculate the classical A-polynomial, and its quantisation, via symplectic geometry.

Paranoid defence controls could criminalise teaching encryption

(A version of this article was also published at The Conversation.)

You wouldn’t think that academic computer science courses could be classified as an export of military technology.

But unfortunately, under recently passed laws, there is a real possibility that innocuous educational and research activities could fall foul of Australian defence export control laws.

Under these laws, despite recent amendments, such “supplies of technology” — and possibly a wide range of other benign activities — come under a censorship regime involving criminal penalties of up to 10 years imprisonment.

The Defence and Strategic Goods List

How could this be?

The story begins with the Australian government’s list of things it considers important to national defence and security. It’s called the Defence and Strategic Goods List (DSGL). Goods on this list are tightly controlled.

Regulation of military weapons is not a particularly controversial idea. But the DSGL covers much more than munitions. It includes many “dual use” goods – goods with both military and civilian uses – including for instance substantial sections on chemicals, electronics, and telecommunications.

Disturbingly, the DSGL veers wildly in the direction of over-classification, covering activities that are completely unrelated to military or intelligence applications. To illustrate, I will focus on the university sector, and one area of interest to mathematicians like myself — encryption — which raises these issues particularly acutely. But similar considerations apply to a wide range of subject material, and to commerce, industry and government.

Encryption: An essential tool for privacy

Encryption is the process of encoding a message, so that it can be sent privately; decryption is the process of decoding it, so that it can be read. Encryption and decryption are two aspects of cryptography, the study of secure communication.

As with many technologies subject to “dual use” regulation, the first question is whether encryption should be covered at all.

Once the preserve of spies and governments, encryption algorithms have now become an essential part of modern life. We use them almost every time we go online. Encryption is used routinely by consumers to guard against identity theft; by businesses to ensure the security of transactions; by hospitals to ensure the privacy of medical records; and much more. Given that email has about as much security as a postcard, encryption is the electronic equivalent of an envelope.

Encryption is perhaps “dual use” in the narrow sense that it is useful to both military/intelligence agencies as well as civilians; but so are other “dual use” technologies like cars.

Moreover, while States certainly spy on each other, essentially everyone with an internet connection is known to be spied on. Since the Snowden revelations — and much earlier for those who were paying attention — we know about mass surveillance by the NSA, along with its Five Eyes partners, which include Australia.

While States have no right to privacy — this is the whole point of Freedom of Information laws — an individual’s right to privacy is a fundamental human right. And in today’s world, encryption is essential for citizens to safeguard this human right. Strict control of encryption as dual-use technology, then, would not only be a misuse of State power, but the curtailment of a fundamental freedom.

How the DSGL covers encryption

Nonetheless, let’s assume for the purposes of argument that there is a justification for regarding at least some aspects of cryptography as “dual use”. (Let’s also put aside the efforts of government, stretching back over decades now, to weaken cryptographic standards and harass researchers.)

The DSGL contains detailed technical specifications covering encryption. Very roughly, it covers encryption above a certain “strength” level, as measured by technical parameters such as “key length” or “field size”.

The practical question is how high the bar is set: how powerful must encryption be, in order to be classified as “dual use”?

The bar is set low. For instance, software engineers debate whether they should use 2048 or 4096 bits for the RSA algorithm, but the DSGL classifies anything over 512 as “dual-use”. It’s probably more accurate to say that the only cryptography not covered by the DSGL is cryptography so weak that it would be foolish to use.

Moreover, the DSGL doesn’t just cover encryption software: it also covers systems, electronics and equipment used to implement, develop, produce or test it.

In short, the DSGL casts an extremely wide net, potentially catching open source privacy software, information security research and education, and the entire computer security industry, in its snare. This is typical of its approach.

Most ridiculous, however, are some badly flawed technicalities. As I have argued elsewhere, the specifications are so poorly written that they potentially include a little algorithm you learned at primary school called division. If so, then division has become a weapon, and your calculator (or smartphone, or computer, or any electronic device) is a delivery system for it.

These issues are not unique to Australia: the DSGL encryption provisions are copied almost verbatim from the Wassenaar Arrangement, an international arms control agreement. What is unique to Australia is the harshness of the law relating to the list.

Criminal offences for research and teaching?

The Australian Defence Trade Controls Act (DTCA) regulates the list, and enacts a censorship regime with severe criminal penalties.

The DTCA prohibits the “supply” of DSGL technology to anyone outside Australia without a permit. The “supply” need not involve money, and can consist of merely providing access to technology. It also prohibits the “publication” of DSGL technology, but after recent amendments, it only applies to half the DSGL: munitions only, not dual-use technologies.

What is a “supply”? The law does not define the word precisely, but the Department of Defence seems to think that merely explaining an algorithm would be an “intangible supply”. If so, then surely teaching DSGL material, or collaborating on research about it, would be covered.

University education is a thoroughly international and online affair — not to mention research — so any such supply, on any DSGL topic, is likely to end up overseas on a regular basis.

Outside of academia, what about programmers working on international projects like Tor, providing free software so citizens can enjoy their privacy rights online? Network security professionals working with overseas counterparts? Indeed, the entire computer security industry?

Examples of innocuous, or even admirable, activities potentially criminalised by this law are easily multiplied. Such activities must seek government approval or face criminal charges — an outrageous attack on academic freedom, chilling legitimate enquiry, to say the least.

To be sure, there are exceptions in the law, which have been expanded under recent amendments. But they are patchy, uncertain and dangerously limited.

For instance, public domain material and “basic scientific research” are not regarded as DSGL technology. However, researchers by definition create new material not in the public domain; and “basic scientific research” is a narrow term which excludes research with practical objectives. Lecturers, admirably, often include new research in teaching material. In such circumstances none of these provisions will be of assistance.

Another exemption covers supplies of dual-use technology made “preparatory to publication”, apparently to protect researchers. But this exemption will provide little comfort to researchers aiming for applications or commercialisation; and none at all to educators or industry. A further exemption is made for oral supplies of DSGL technology, so if computer science lecturers can teach without writing (giving a whole new meaning to “off the books”!) they might be safe.

Unlike the US, there is no exception for education; none for public interest material; and indeed, the Explanatory Memorandum makes clear that the government envisions universities seeking permits to teach students DSGL material – and, by implication, criminal charges if they do not.

On a rather different note, the DTCA specifically enables the Australian and US militaries to freely share technology.

Thus, an Australian professor emailing an international collaborator or international postgraduate student about a new applied cryptography idea, or explaining a new variant on a cryptographic algorithm on a blackboard in a recorded lecture viewed overseas — despite having nothing to do with military or intelligence applications — may expose herself to criminal liability. At the same time, munitions flow freely across the Pacific. Such is Australia’s military export control regime.

Now, there is nothing wrong in principle with government regulation of military technology. But when the net is cast as broadly as the DSGL — especially as with encryption — and the regulatory approach is censorship with criminal penalties — as with the DTCA’s permit regime — then the result is a vast overreach. Even if the Department of Defence did not exercise its censorship powers, the mere possibility is enough for a chilling effect stifling the free flow of ideas and progress.

The DTCA was passed in 2012, with the criminal offences schedule to come into effect in May 2015. Thankfully, emergency amendments in April 2015 have provided some reprieve.

Despite those amendments, the laws remain paranoid. The DSGL vastly over-classifies technologies as dual-use, including essentially all sensible uses of encryption. The DTCA potentially criminalises an enormous range of legitimate research and development activity as a supply of dual-use technology, dangerously attacking academic freedom — and freedom in general — in the process.

This story illustrates just one of many ways in which basic freedoms are being eroded in the name of national security.

Unless further changes are made, criminal penalties of up to 10 years prison will come into effect on 2 April 2016.

The day after April fool’s day. Jokes should be over by then.

The CIA 119

Years and years on, abuses continue.

The Bureau of investigative Journalism, together with the Rendition Project, is still trying to piece together the CIA’s kidnapping (“rendition”) and torture programme.

Only in December 2014 did the US Senate Intelligence Committee release its <summary of its report into the programme — a programme which, at least according to this report summary, effectively ended in 2006.

It took nearly ten years after the fact for an official report to arrive.

And this report, despite arriving so late on the scene, had only its summary published — the rest of the report is still classified to this day — and even the summary was the subject of bitter controversy among politicians. (Though what counts as controversial among US mainstream politicians is not a very good guide as to what matters are deserving of controversy: take global warming, for instance.)

Only with this report, well over a decade after most of the facts, only then did we learn the most basic facts about the program, like the number of people captured under it. The answer to that question, at least according to the report, is 119. They appear to have included people from dangerous terrorists, through to innocents sold to the CIA for profit.

The Bureau’s report begins to pull together the evidence to find out what happened to them. They were disappeared from their lives, disappeared into unaccountable captivity, disappeared into a legal black hole — and, in several cases, disappeared from history. The Bureau was unable to determine the fate of 39 of the abductees.

It is a story of no accountability, brutality and incompetence. To be sure, it apprehended some terrorists — though it appears that following a proper legal process, in every case, would have led to better results in terms of security and preventing terrorism, as well as, of course, following the law and abusing human rights. But other cases are ridiculous.

There is Laid Saidi, who was tortured by submersion in a bathtub of icy water and interrogated about a conversation in which he talked about aeroplanes (as if that were a crime) — except it turns out, thanks to faulty translation, he was talking about tyres. Saidi was later released — except he was released to the wrong country, so had to be taken back into custody and released again months later.

There is Khaled el Masri, who was detained by Macedonian authorities and held in a hotel in Skopje, then handed over to the CIA and taken to Afghanistan. There he was tortured by beatings, solitary confinement, and sodomy. His crime? Having a name similar to that of an alleged terrorist. He eventually won damages from Macedonia in the European Court of Human Rights, but his case is unusual in having won some recompense.

Of course, this is only one of many programs of the CIA as part of the “War on Terror” — a “war” which, for the most part, appears to have consisted of terror. And the CIA is only one of numerous US government agencies to have engaged in abuses. And, the United States is only one of many nations to have engaged in abuses — indeed, they all do, though the US still reigns supreme in its ability to project force around the globe. Australia has assisted many of these abuses.

Almost fourteen years after September 11 2001, more than ten years after most of the kidnappings, the struggle remains ongoing to find out what happened and why. These events offer not just a window into a particular time and circumstances, but the institutional circumstances in which unaccountable force is used and unpunished (or even “legal”) crimes are committed.

In Australia we have heard a lot recently about “lest we forget”. We should above all remember the abuses perpetrated by ourselves and our allies — lest we forget them, and in so doing enable them to happen again. The struggle of people against power has always been the struggle of memory against forgetting.

There is also the constructive question, in examining abusive organisations and programmes like this one, to identify what factors caused, or at least allowed, such horrors to happen. What better set of institutions can we build to ensure that similar abuses never happen again — and maintain peace and security for all?

The lower classes of things

Everything is free to move across borders, except… some lesser things.

It’s a long-standing principle of law, in the “developed” world at least, that “freedom” means the ability to move across borders without hindrance or restriction. This is commonly called “globalization”. Borders fade away and become irrelevant; non-discrimination becomes a defining, enlightened principle; and the world becomes one cosmopolitan village. Except, of course, that this otherwise laudable, advanced, cosmopolitan version of “freedom” applies only to inanimate material objects. To be fair, it does also apply to immaterial objects such as transfers of capital that exist only as abstract ideas, entries in spreadsheets or bits of information.

But one only needs to try to catch a boat from Indonesia to Australia to find out how much this well-established “freedom” and crowning glory of inanimate objects applies to living, breathing, feeling, thinking human beings.

Nonetheless, though it may be a great hypocrisy, this “freedom” of inanimate objects to move across borders is well-established. Such is the world we live in, where consumer goods such as cars and washing machines have advanced rights that humans do not have. This principle is enshrined in international treaties such as the General Agreement on Tariffs and Trade, and the various protocols adhered to be all member States of the World Trade Organisation.

However, this glorious liberty granted to inanimate objects, and even abstract objects, does not quite apply to all objects. Exceptions can be made, provided there is a special reason for it.

And, our world, divided into nation-states, is so organised that the highest decision-making authorities in the world pertain to geographic regions established largely by war, conquest and colonisation. So there is no more sanctified reason to limit freedoms than the military interests of States. In particular, weapons of war have much less freedom to flow across borders. The flow of weapons is tightly regulated — or at least, when it suits a State’s interest to do so.

Such is the idiosyncrasy and backwardness of human civilization in the early 21st century. Rights are given to inanimate objects — even abstract immaterial objects — but not sentient beings. Power lies with a tumultuous collection of clashing commonwealths, whose military interests are the highest good. Destructive weapons plague the world, but weapons are almost alone among inanimate objects in being subject to regulation.

Weapons are deprived of the rights accorded to other inanimate objects, and in this lie with other lower classes of things, such as hazardous waste, disease carriers, dangerous chemicals, plants, animals, and human beings.