Tutte meets Homfly

Graphs are pretty important objects in mathematics, and in the world — what with every network of any kind being represented by one, from social connections, to road and rail systems, to chemical molecules, to abstract symmetries. They are a fundamental concept in understanding a lot about the world, from particle physics to sociology.

Knots are also pretty important. How can a loop of string get knotted in space? That’s a fairly basic question and you would think we might know the answer to it. But it turns out to be quite hard, and there is a whole field of mathematics devoted to understanding it. Lord Kelvin thought that atoms were knots in the ether. As it turns out, they are not, and there is no ether. Nonetheless the idea was an interesting one and it inspired mathematicians to investigate knots. Knot theory is now a major field of mathematics, part of the subject of topology. It turns out to be deeply connected to many other areas of science, because many things can be knotted: umbilical cords, polymers, quantum observables, DNA… and earphone cords. (Oh yes, earphone cords.) Indeed, knots arise crucially in some of the deepest questions about the nature of space and time, whether as Wilson loops in topological field theories, or as crucial examples in the theory of 3-dimensional spaces, or 3-manifolds, and as basic objects of quantum topology.

Being able to tell apart graphs and knots is, therefore, a pretty basic question. If you’re given two graphs, can you tell if they’re the same? Or if you’re given two knots, can you tell if they’re the same? This question is harder than it looks. For instance, the two graphs below look quite different, but they are the “same” (or, to use a technical term, isomorphic): each vertex in the first graph corresponds to a vertex in the second graph, in such a way that the connectedness of vertices is preserved.


(Source: Chris-martin on wikipedia)

Telling whether two graphs are the same, or isomorphic, is known as the graph isomorphism problem. It’s a hard problem; when you have very large graphs, it may take a long time to tell whether they are the same or not. As to precisely how hard it is, we don’t yet quite know.

Similarly, two knots, given as diagrams drawn on a page, it is difficult to tell if they are the “same” or not. Two knots are the “same”, or equivalent (or ambient isotopic), if there is a way to move one around in space to arrive at the other. For instance, all three knots shown below are equivalent: they are all, like the knot on the left, unknotted. This is pretty clear for the middle knot; for the knot on the right, have fun trying to see why!


(Source: Stannered, C_S, Prboks13 on wikipedia)

Telling whether two knots are equivalent is also very hard. Indeed, it’s hard enough to tell if a given knot is knotted or knot — which is known as the unknot recognition or unknotting problem. We also don’t quite know precisely how hard it is.

The fact that we don’t know the answers to some of the most basic questions about graphs and knots is part of the reason why graph theory and knot theory are very active areas of current research!

However, there are some extremely clever methods that can be used to tell graphs and knots apart. Many such methods exist. Some are easier to understand than others; some are easier to implement than others; some tell more knots apart than others. I’m going to tell you about two particular methods, one for graphs, and one for knots. Both methods involve polynomials. Both methods are able to tell a lot of graphs/knots apart, but not all of them.

The idea is that, given a graph, you can apply a certain procedure to write down a polynomial. Even if the same graph is presented to you in a different way, you will still obtain the same polynomial. So if you have two graphs, and they give you different polynomials, then they must be different graphs!

Similarly, given a knot, you can apply another procedure to write down a polynomial. Even if the knot is drawn in a very different way (like the very different unknots above), you still obtain the same polynomial. So if you have two knots, and they give you different polynomials, then they must be different knots!

Bill Tutte and his polynomial

Bill Tutte was an interesting character: a second world war British cryptanalyst and mathematician, who helped crack the Lorenz cipher used by the Nazi high command, he also made major contributions to graph theory, and developed the field of matroid theory.

He also introduced a way to obtain a polynomial from a graph, which now bears his name: the Tutte polynomial.

Each graph \(G \) has a Tutte polynomial \(T_G \). It’s a polynomial in two variables, which we will call \(x\) and \(y\). So we will write the Tutte polynomial of \(G\) as \(T_G (x,y)\).

For instance, the graph \(G\) below, which forms a triangle, has Tutte polynomial given by \(T_G (x,y) = x^2 + x + y\).

So how do you calculate the Tutte polynomial? There are a few different ways. Probably the easiest is to use a technique where we simplify the graph step by step in the process. We successively collapse or remove edges in various ways, and as we do so, we make some algebra.

There are two operations we perform to simplify the graph. Each of these two operations “removes” an edge, but does so in a different way. They are called deletion and contraction. You can choose any edge of a graph, and delete it, or contract it, and you’ll end up with a simpler graph.

First, deletion. To delete an edge, you simply rub it out. Everything else stays as it was. The vertices are unchanged: there are still just as many vertices. There is just one fewer edge. So, for instance, the result of deleting an edge of the triangle graph above is shown below.

The graph obtained by deleting edge \(e\) from graph \(G\) is denoted \(G-e\).

Note that in the triangle case above, the triangle graph is connected, and after deleting an edge, the result is still connected. This isn’t always the case: it is possible that you have a connected graph, but after moving an edge \(e\), it becomes disconnected. In this case the edge \(e\) is called a bridge. (You can then think of it as a “bridge” between two separate “islands” of the graph; removing the edge, there is no bridge between the islands, so they become disconnected.)

Second, contraction. To contract an edge, you imagine shrinking it down, so that it’s shorter and shorter, until it has no length at all. The edge has vertices at its endpoints, and so these two vertices come together, and combine into a single vertex. So if edge \(e\) has vertices \(v_1, v_2\) at its endpoints, then after contracting \(e\), the vertices \(v_1, v_2\) are combined into a single vertex \(v\). Thus, if we contract an edge of the triangle graph, we obtain a result something like the graph shown below.

The graph obtained by contracting edge \(e\) from graph \(G\) is denoted \(G.e\).

Contracting an edge always produces a graph with 1 fewer edges. Each edge which previous ended at \(v_1\) or \(v_2\) now ends at \(v\). And contracting an edge usually produces a graph with 1 fewer vertices: the vertices \(v_1, v_2\) are collapsed into \(v\).

However, this is not always the case. If the edge \(e\) had both its endpoints at the same vertex, then the number of vertices does not decrease at all! The endpoints \(v_1\) and \(v_2\) of \(e\) are then the same point, i.e. \(v_1 = v_2\), and so they are already collapsed into the same vertex! In this case, the edge \(e\) is called a loop. Contracting a loop is the same as just deleting a loop.

So, that’s deletion and contraction. We can use deletion and contraction to calculate the Tutte polynomial using the following rules:

  1. Let’s start with some really simple graphs.
    • If a graph \(G\) has no edges, then it’s just a collection of disconnected vertices. In this case the Tutte polynomial is given by \(T_G (x,y) = 1\).
    • If a graph has precisely one edge, then that it consists of a bunch of vertices, with precisely one edge \(e\) joining them. If \(e\) connects two distinct vertices, then it is a bridge, and \(T_G (x,y) = x\).
    • On the other hand, if \(G\) has precisely one edge \(e\) which connects a vertex to itself, then it is a loop, and \(T_G (x,y) = y\).
  2. When you take a graph \(G\) and consider deleting an edge \(e\) to obtain \(G – e\), or contracting it to obtain \(G.e\), these three graphs \(G, G-e, G.e\) have Tutte polynomials which are closely related:

    \(\displaystyle T_G (x,y) = T_{G-e} (x,y) + T_{G.e} (x,y)\).

    So in fact the Tutte polynomial you are looking for is just the sum of the Tutte polynomials of the two simpler graphs \(G-e\) and \(G.e\).
    However! This rule only works if \(e\) is not a bridge or a loop. If \(e\) is a bridge or a loop, then we have two more rules to cover that situation.

  3. When edge \(e\) is a bridge, then \(T_G (x,y) = x T_{G.e} (x,y)\).
  4. When edge \(e\) is a loop, then \(T_G (x,y) = y T_{G-e} (x,y)\).

These may just look like a random set of rules. And indeed they are: I haven’t tried to explain them, where they come from, or given any motivation for them. And I’m afraid I’m not going to. Tutte was very clever to come up with these rules!

Nonetheless, using the above rules, we can calculate the Tutte polynomial of a graph.

Let’s do some examples. We’ll work out a few, starting with simpler graph and we’ll work our way up to calculating the Tutte polynomial of the triangle, which we’ll denote \(G\).

First, consider the graph \(A\) consisting of a single edge between two vertices.


This graph contains precisely one edge, which is a bridge, so \(T_A (x,y) = x\).

Second, consider the graph \(B\) consisting of a single loop on a single vertex. This graph also contains precisely one edge, but it’s a loop, so \(T_B (x,y) = y\).

Third, consider the graph \(C\) which is just like the graph \(A\), consisting of a single edge between two vertices, but with another disconnected vertex.

This graph also contains precisely one edge, which is also a bridge, so it actually has the same Tutte polynomial as \(A\)! So we have \(T_C (x,y) = x\).

Fourth, consider the graph \(D\) which consists of a loop and another edge as shown. A graph like this is sometimes called a lollipop.

Now let \(e\) be the loop. As it’s a loop, rule 4 applies. If we remove \(e\), then we just obtain the single edge graph \(A\) from before. That is, \(D-e=A\). Applying rule 4, then, we obtain \(T_D (x,y) = y T_{D-e} (x,y) = y T_A (x,y) = xy\).

Fifth, consider the graph \(E\) which consists of two edges joining three vertices. We saw this before when we deleted an edge from the triangle.

Pick one of the edges and call it \(e\). (It doesn’t matter which one — can you see why?) If we remove \(e\), the graph becomes disconnected, so \(e\) is a bridge. Consequently rule 3, for bridges, applies. Now contracting the edge \(e\) we obtain the lollipop graph \(E\). That is, \(E-e=C\). So, applying rule 3, we obtain \(T_E (x,y) = x T_{E-e} (x,y) = x T_C (x,y) = x^2 \).

Sixth, let’s consider the graph \(F\) consisting of two “parallel” edges between two vertices. We saw this graph before when we contracted an edge of the triangle.

Pick one of the edges and call it \(e\). (Again, it doesn’t matter which one.) This edge is neither a bridge nor a loop, so rule 2 applies. Removing \(e\) just gives the graph \(A\) with one vertex, which has Tutte polynomial \(x\). Contracting \(e\) gives a graph with a single vertex and a loop. Applying rule 4, this graph has Tutte polynomial \(y\). So, by rule 2, the Tutte polynomial of this graph \(F\) is given by \(\displaystyle T_F (x,y) = x + y \).

Finally, consider the triangle graph \(G\). Take an edge \(e\); it’s neither a bridge nor a loop, so rule 2 applies. Removing \(e\) results in the graph \(E\) from above, which has Tutte polynomial \(x^2\). Contracting \(e\) results in the graph \(F\) from above with two parallel edges; and we’ve seen it has Tutte polynomial \(x+y\). So, putting it all together, we obtain the Tutte polynomial of the triangle as

\(\displaystyle T_G (x,y) = T_{G-e} (x,y) + T_{G.e} (x,y) = T_E (x,y) + T_F (x,y) = x^2 + x + y.\)

Having seen these examples, hopefully the process starts to make some sense.

However, as we mentioned before, we’ve given no motivation for why this works. And, it’s not even clear that it works at all! If you take a graph, you can delete and contract different edges in different orders and get all sorts of different polynomials along the way. It’s not at all clear that you’ll obtain the same result regardless of how you remove the edges.

Nonetheless, it is true, and was proved by Tutte, that no matter how you simplify the graph at each stage, you’ll obtain the same result. In other word, the Tutte polynomial of a graph is actually well defined.

H, O, M, F, L, Y, P and T

Tutte invented his polynomial in the 1940s — it was part of his PhD thesis. So the Tutte polynomial has been around for a long time. The knot polynomial that we’re going to consider, however, is considerably younger.

In the 1980s, there was a revolution in knot theory. The excellent mathematician Vaughan Jones in 1984 discovered a polynomial which can be associated to a knot. It has become known as the Jones polynomial. It was not the first polynomial that anyone had defined from a knot, but it sparked a great deal of interest in knots, and led to the resolution of many previously unknown questions in knot theory.

Once certain ideas are in the air, other ideas follow. Several mathematicians started trying to find improved versions of the Jones polynomial, and at least 8 mathematicians came up with similar ways to improve the Jones polynomial. In 1985, Jim Hoste, Adrian Ocneanu, Kenneth Millett, Peter J. Freyd, W. B. R. Lickorish, and David N. Yetter published a paper defining a new polynomial invariant. Making an acronym of their initials, it’s often called the HOMFLY polynomial. Two more mathematicians, Józef H. Przytycki and Pawe? Traczyk, did independent work on the subject, and so it’s often called the HOMFLY-PT polynomial.

Like the Tutte polynomial, the HOMFLY polynomial is a polynomial in two variables. (The Jones polynomial, however, is just in one variable.) It can also be written as a homogeneous polynomial in three variables. We’ll take the 3-variable homogeneous version.

Strictly speaking, to get a HOMFLY polynomial, your knot must be oriented: it must have a direction. This is usually represented by an arrow along the knot.

HOMFLY polynomials also exist for links — a link is just a knot with many loops invovled. So even if there are several loops knotted up, they still have a HOMFLY polynomial. (Each loop needs to be oriented though.)

So, if you’re given an oriented knot or link \(K\), it has a HOMFLY polynomial. We’ll denote it by \(P_K (x,y,z)\). So how do you compute it? By following some rules which successively simplify the knot.

  1. If the knot \(K\) is the unknot, then \(P_K (x,y,z) = 1\).
  2. If you take one of the crossings in the diagram and alter it in the various ways shown below — but leave the rest of the knot unchanged — then you obtain three links \(L^+, L^-, L^0\). Their HOMFLY polynomials are related by

    \(\displaystyle x P_{L^+} (x,y,z) + y P_{L^-} (x,y,z) + z P_{L^0} (x,y,z) = 0\).

    Source: C_S, wikimedia

    A relationship like this, between three knots or links which differ only at a single crossing, is called a skein relation.

  3. If you can move the link \(L\) around in 3-dimensional space to another link \(L’\), then this doesn’t change the HOMFLY polynomial: $latex P_L (x,y,z) = P_{L’} (x,y,z).
  4. If the oriented link \(L\) is split, i.e. separates into two disjoint (untangled) sub-links \(L_1, L_2\), then you can take the HOMFLY polynomials of the two pieces separately, and multiply them, with an extra factor:

    \(\displaystyle P_L (x,y,z) = \frac{-(x+y)}{z} P_{L_1} (x,y,z) P_{L_2} (x,y,z)\).

  5. If \(L\) is a connect sum of two links \(L_1, L_2\), then you can take the HOMFLY polynomials of the two pieces, and multiply them:

    \(\displaystyle P_L (x,y,z) = P_{L_1} (x,y,z) P_{L_2} (x,y,z)\).

    What is a connect sum? It’s when two knots or links are joined together by a pair of strands, as shown below.

    Source: Maksim, wikimedia

And there you go.

Again, it’s not at all clear where these rules come from, or that they will always give the same result. There might be many ways to change the crossings and simplify the knot, but H and O and M and F and L and Y and P and T showed that in fact you do always obtain the same result for the polynomial.

Let’s see how to do this in a couple of examples.

First of all, for the unknot \(U\), by rule 1, its HOMFLY polynomial is \(P_U (x,y,z) = 1\).

Second, let’s consider two linked unknots as shown below. This is known as the Hopf link. Let’s call it \(H\).

Source: Jim.belk, wikimedia.

Let’s orient both the loops so that they are anticlockwise. Pick one of the crossings and consider the three possibilities obtained by replacing it according to the skein relation described above, \(H^+, H^-, H^0\). You should find that \(H^+\) corresponds to the crossing as it is shown, so \(H^+ = H\). Changing the crossing results in two unlinked rings, that is, \(H^- =\) two split unknots. By rule 4 above then, \(P_{H^-} (x,y,z) = \frac{-(x+y)}{z} P_U (x,y,z) P_U (x,y,z)\); and as each unknot has HOMFLY polynomial \(1\), we obtain \(P_{H^-} (x,y,z) = \frac{-(x+y)}{z}\). On the other hand, smoothing the crossing into \(H^0\) gives an unknot, so \(P_{H^0} (x,y,z) = P_{U} (x,y,z) = 1\).

Putting this together with the skein relation (rule 2), we obtain the equation

\(\displaystyle x P_{H^+} (x,y,z) + y P_{H^-} (x,y,z) + z P_{H^0} (x,y,z) = 0\),

which gives

\(\displaystyle x P_H (x,y,z) + y \frac{-(x+y)}{z} + z = 0\)

and hence the HOMFLY of the Hopf link is found to be

\(\displaystyle P_H (x,y,z) = \frac{ y(x+y)}{xz} – \frac{z^2}{xz} = \frac{xy + y^2 – z^2}{xz}\).

 

When the Tutte is the HOMFLY

In 1988, Francois Jaeger showed that the Tutte and HOMFLY polynomials are closely related.

Given a graph \(G\) drawn in the plane, it has a Tutte polynomial \(T_G (x,y)\), as we’ve seen.

But from such a \(G\), Jaeger considered a way to build an oriented link \(D(G)\). And moreover, he showed that the HOMFLY polynomial of \(D(G)\) is closely related to the Tutte polynomial of \(G\). In other words, \(T_G (x,y)\) and \(P_{D(G)} (x,y,z)\) are closely related.

But first, let’s see how to build an link from a graph. It’s called the median construction. Here’s what you do. Starting from your graph \(G\), which is drawn in the plane, you do the following.

  • Thicken \(G\) up. You can then think of it as a disc around each vertex, together with a band along each edge.
  • Along each edge of \(G\), there is now a band. Take each band, and put a full right-handed twist in it. You’ve now got a surface which is twisted up in 3-dimensional space.
  • Take the boundary of this surface. It’s a link. And this link is precisely \(D(G)\). (As it turns out, there’s also a natural way to put a direction on \(D(G)\), i.e. make it an oriented link.)

It’s easier to understand with a picture. Below we have a graph \(G\), and the link \(D(G)\) obtained from it.

A graph (source: wikimedia) and the link (source: Jaeger) obtained via the median construction.

Jaeger was able to show that in general, the Tutte polynomial \(T_G (x,y)\) and the HOMFLY polynomial \(P_{D(G)} (x,y,z)\) are related by the equation

\(\displaystyle P_{D(G)} (x,y,z) = \left( \frac{y}{z} \right)^{V(G)-1} \left( – \frac{z}{x} \right)^{E(G)} T_G \left( -\frac{x}{y}, \frac{-(xy+y^2)}{z^2} \right),\)

where \(V(G)\) denotes the number of vertices of \(G\), and \(E(G)\) denotes the number of edges of \(G\). Essentially, Jaeger showed that the process you can use to simplify the link \(D(G)\) to calculate the HOMFLY polynomial, corresponds in a precise way to the process you can use to simplify the graph \(G\) to calculate the Tutte polynomial.

In addition to this excellent correspondence — Tutte meeting HOMFLY — Jaeger was able to deduce some further consequences.

He showed that the four colour theorem is equivalent to a fact about HOMFLY polynomials: for every loopless connected plane graph \(G\), \(P_{D(G)} (3,1,2) \neq 0\).

Moreover, since colouring problems for plane graphs are known to be very hard, in terms of computational complexity — NP-hard — it follows that the computation of the HOMFLY polynomial is also NP hard.

Said another way: if you could find a way to compute the HOMFLY polynomial of a link in polynomial time, you would prove that \(P = NP\) and claim yourself a Millennium prize!

Polytopes, dualities, and Floer homology

(41 pages)

on the arXiv – published in the Proceedings of the Conference on Gromov-Witten Theory, Gauge Theory and Dualities, ANU/Kioloa, January 2016.

Abstract: This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.

Local pdf available here (703 kb).

polytopes_dualities_floer_homology

Adani: icon of Australian climate infamy

Here we are, in the year 2017. With now 25 years of climate-change international agreements behind us, here we are still trying to build oil pipelines and coal mines.

It is sad. Sad for humanity.

It is no longer a question of reducing the speed at which we are approaching the cliff. It is now a question of counting the metres to the cliff, as it approaches so fast.  It is no longer a question of reducing climate emissions to a reasonable level. It is now a question of counting the remaining tons which may be emitted, budgeting them carefully and switching off them with an emergency.

There are various different ways the budget can be calculated. The MCC Carbon Clock calculates that to limit increase in global average temperature to 2 degrees Celsius, the CO2 budget remaining is 734 gigatonnes, on a moderate (neither optimistic nor pessimistic) set of assumptions. At the current rate of emissions, that budget will be exhausted by 2035. That is time at which, continuing as we are now, scientific laws predict failure.

But there is a strong argument that a 2 degrees Celsius increase is too much. It means a vast range of climate impacts – for instance, at 2 degrees tropical coral reefs do not stand a chance. A limit of 1.5 degrees increase is an altogether better goal. Indeed, the Paris agreement aims to hold increase in global average temperature to 2 degrees Celsius, but “to pursue efforts” to limit the increase to 1.5 degrees Celsius. And the MCC Carbon Clock, under the same set of assumptions, calculates the remaining CO2 budget, to limit the increase to 1.5 degrees Celsius, as 41.3 gigatonnes. At the current rate, this budget will be exhausted in September 2018 — in just over a year’s time.

One year. Just one year.

Either way, it is a clear and present danger, urgent, all-absorbing, putting all other tumults to silence. All efforts must be to get off fossil fuels immediately, now, yesterday.

Yet where are we in Australia? We are about to build our biggest coal mine ever.

Adani, corrupt, lawbreaking Adani. They still want to build their Carmichael mine in the Galilee Basin in Queensland.

And Australian governments still fall over themselves to assist them. Approvals and re-approvals flow from the federal Liberal government. The Queensland Labor premier’s interventions have convinced Adani to go ahead. The total emissions of the Carmichael project — producing and burning the coal – will be 4.7 gigatonnes of CO2. That’s over 10% of the remaining budget for 1.5 degree increase. Just this one mine.

The project is itself barely financially viable. Adani says a special loan from the Northern Australian Infrastructure Facility (NAIF) is critical to their financing. Nonetheless contracts were announced in July. They say it will employ 10,000 people: the reality, as given by Adani’s own expert under oath, is closer to 1,500.

Like everything else in Australia, the mine would be built on Aboriginal land. The traditional owners, the Wangan and Jagalingou people, released a statement: Stop Adani destroying our land and culture.

If the Carmichael mine were to proceed it would tear the heart out of the land. The scale of this mine means it would have devastating impacts on our native title, ancestral lands and waters, our totemic plants and animals, and our environmental and cultural heritage. It would pollute and drain billions of litres of groundwater, and obliterate important springs systems. It would potentially wipe out threatened and endangered species. It would literally leave a huge black hole, monumental in proportions, where there were once our homelands. These effects are irreversible. Our land will be “disappeared”.

Native Title claims being too much of an uncertain quantity for Adani – and courts showing an increasing level of respect for indigenous desires to control their land — legislation was dutifully passed by the federal parliament in June to smooth Adani’s way.
Just like the Keystone XL and Dakota Access pipelines in the US, which have seen such inspiring resistance, if the Adani Carmichael mine is built it will be game over for the climate.

It cannot go ahead. It must not.

Adani continues to acquire property along its proposed rail corridor, even as new accusations of fraud emerge against them. But largely they are currently playing a waiting game. The minister responsible for NAIF, Matt Canavan, stepped down over the citizenship farce; and his replacement is Barnaby Joyce. Until the High Court rules, and possibly byelections are held, it may wait.

That gives a crucial opportunity to press the opposition to the mine and decisively stop it. Protests continue.  A few days ago, religious leaders promised civil disobedience.

As these leaders argued, it is a simple moral choice.

It is a simple scientific choice too.

Stop Adani’s mine, and switch this sunburnt country to renewable energy now.

At least mathematics is commendable

Today the Australian government announced a proposal to force tech companies to provide government agencies with the contents of encrypted communications.

I don’t think any draft of proposed legislation exists yet — my understanding is that a bill will be introduced later in the year — but the most recent announcement today and the press conferences by Turnbull and Brandis essentially follow on from the G20 statement last week, which has a paragraph including such ideas.

Since there are no specifics, it’s hard to comment beyond generalities. But in general the whole proposal seems to me to be, to the extent it is not technically impossible or entirely misconceived, a threat to the privacy and safety of everyone.

The best thing to come out of the Turnbull’s press conference was that he said

The laws of mathematics are very commendable but the only laws that apply in Australia is the law of Australia.

I am very glad to see that Turnbull thinks mathematics is commendable. In this case, he should, for instance, take seriously the results of applying the laws of mathematics in climate models, which show just how dire the planetary climate situation is. He would be better advised to spend his precious days as Prime Minister bringing the laws of Australia into line with the laws of mathematics as applied to climate, than to try to fight the mathematics behind encryption by legislation.

I am afraid that, however commendable Turnbull thinks they are, the laws of mathematics simply cannot be avoided, whatever he thinks of them, and they cannot be legislated away. That’s the way the universe works.

You cannot legislate that messages sent by properly implemented end-to-end encryption be decrypted any more than you can legislate that pi is 3. Central results in cryptography show that properly implemented encryption schemes make decryption practically impossible. (This is putting aside potential futuristic technologies like quantum computers.)

So, in practice what this means is that the government wants to force tech companies to not implement end-to-end encryption properly, but to make some modification, whether by using a weakened implementation or malware or a backdoor of some sort, so that the government can access it. Such proposals by law enforcement and intelligence have a long and ignominious history going back to at least the 1990s and the Clipper Chip. Technical dificulties aside, the important point which has come out of all that history is that there is no way to make encryption subject to government-mandated decryption without making it vulnerable to other attacks as well. If encryption is weak enough that a conversation can be decrypted by someone other than the parties to the conversation, then it is weak enough to be decrypted by many others, hackers, other governments, and so on. If it is implemented through government-mandated malware, then anyone who gains access to that malware has similar power — and we have seen precisely this happen, for instance, with NSA malware and WannaCrypt attacks.

The government’s approach with the present proposal appears to be to transfer responsibility to tech companies. Rather than legislate government backdoors, they seem to want to legislate that the tech companies must do what they can to assist. They want to use the legal language of rendering “proportionate” and “reasonable” assistance. But breaking end-to-end encryption, or implementing backdoors, is not at all proportionate or reasonable. If a company makes such a change, then they no longer implement end-to-end encryption and the promises of privacy provided to their users are null and void. There is no proportionate way to break an algorithm which mathematically provides secure encryption. It is either secure, or it is not.

In recent years there has been a mass takeup of encrypted messaging by people around the world. End-to-end encryption has been implemented by many major technology companies. This is largely sparked by revelations of mass warrantless surveillance by the NSA, not only of individuals, but also of those very tech companies. People are right to be wary of their privacy.

The Australian authorities, I’m afraid, do not inspire a great deal of confidence. They have already been given draconian powers. Quite aside from other draconian laws which, for instance, criminalise government leaks and whistleblowing from within refugee detention centres, metadata laws have come into effect. These metadata laws allow many government agencies, without any court warrant, to access the metadata of almost any Australian’s online activity. These agencies have been invested with great power, and yet even the mild protections for journalists have been violated, as we found out in April, when the Australian Federal Police admitted that a journalist’s data had been accessed. No charges were laid and no action was taken, so far as I’m aware, beyond the Federal Police holding a press conference. Given the approach the AFP takes to journalists — a class of people with special legal protections — one wonders what approach they take to ordinary citizens. How will they then treat whistleblowers, activists, and government critics?

Police and intelligence already have enormous powers of surveillance and monitoring. Terrorism, child pornography and sex trafficking are important issues, but these proposals are not the way to deal with them.

Holy h-principle, Batman!

(With apologies and tribute to the late Adam West)

Here’s a situation known to any beginning skier.

You are at the top of a mountain slope. You want to go down to the bottom of the slope. But you are on skis, and you are not very good at skiing.

Despite your lack of skill, you have been deposited by a ski lift at the top of the slope. Slightly terrified, you know that the only honourable way out of your predicament is down — down the slope, on your skis.
Pointing your skis down the slope, with a rush of exhilaration you find yourself accelerating downwards. Unfortunately, you know from bitter experience that the hardest thing about learning to ski is learning to control your speed.

If you are bold or stupid, your incompetent attempt to conquer the mountain likely ends in a spectacular crash. If you are cowardly and have mastered the snowplough, your plodding descent ends with a whimper and worn out thighs from holding the skis in an awkward triangle all the way down.

But you know that the more your skis point down the mountain, the faster you go. And the more your skis point across the slope, the slower you go.

When your skis point down the slope, they are pointing steeply downwards; they are pointing in quite a steep direction. But when your skis point across the slope, they are not pointing very downwards at all; they are pointing in quite a flat direction.

As an incompetent skier, the best way to get down the slope without injury and without embarrassment is to go take a path which criss-crosses the slope as much as possible. You want your skis to point in a flat direction as much as possible, and in a steep direction as little as possible.

The problem is that each time you change direction, you temporarily point downwards, and risk runaway acceleration.

Is it possible to get down the mountain while always pointing in a flat direction?

Of course the answer is no. But there is an area of mathematics which says that the answer is yes, almost, more or less.

This, very roughly, is the beginning of the idea of Gromov’s homotopy principle — often abbreviated to h-principle.

* * *

The ideas of the h-principle were developed by Mikhail Gromov in his work in the 1970s, including work with my PhD advisor Yakov Eliahsberg. The term “h-principle” first appeared and was systematically elaborated by Gromov in his (notoriously difficult) book Partial differential relations. Gromov, who grew up in Soviet Russia, was not a friend of the authorities and in 1970, despite being invited to speak at the International Congress of Mathematicians in France, was prohibited from leaving the USSR. Later he finally made his “gromomorphism” to the US, and now he works at IHES just near Paris.

The ideas of the h-principle are about a “soft” kind of geometry. If you are prepared to treat your geometry like playdough, and morph various things around, within certain constraints, then the $h$-principle tells you how to morph your playdough-geometry to get the nice kind of geometry you want. Technically, morphing playdough has a fancy name: it’s called homotopy.

An example of the h-principle (or, more precisely, of “holonomic approximation”) in the skiing context would be as follows.

Your ideal skiing path down the slope would be to go straight down, but always have your skis pointing flat. That is, your skis should always point horizontally, but you should go straight down the mountain. That is the ideal.

That, of course, is a ridiculous ideal path. But it’s an ideal path nonetheless. You want to go straight down the mountain, and the ideal path does this; and you want to have your skis pointing safely horizontally, and the ideal path does this too. This is the ideal of the incompetent skier: go down the mountain as directly as possible, with your skis always being completely flat.

Now the ideal path cannot be achieved in practice. In practice, if your skis are pointing horizontally, you go horizontally. (We ignore skidding for the purposes of our mathematical idealisation.) In practice, you go in the direction your skis are pointing.

The “ideal path” is a path down the mountain, which also tells you which way your skis should point at each stage (i.e. horizontally), but which doesn’t satisfy the practical principle that you should go in the direction you’re pointing. As such, it’s a generalisation of the real sort of path you could actually take down the mountain. (The technical name for this type of path is a “section of a jet bundle”.)

A realistic path down the mountain — one where you go in the direction your skis are pointing — is also known as a holonomic path.

One of the first results towards the h-principle, says that if you are prepared to make a few tiny tiny adjustments to your path, then you can take an actual, holonomic path down the mountain, where you are very very close to the ideal path — both in terms of where you go, and in the direction your skis point. You stay very very close — in fact, arbitrarily close — to the path straight down the mountain. And your skis stay very very close — again, arbitrarily close — to horizontal at every instant on the way down.

How is this possible? Well, you have to make some adjustments.

First, you make some adjustments to the path. You might have to make a wiggly path, rather than going in a straight line. Actually, it will have to be very wiggly — arbitrarily wiggly.

And, second, you’ll have to make some adjustments to the mountain too. You’ll have to adjust the shape of the mountain slope — but only by a very very small, arbitrarily small amount.

Well, perhaps these types of alternations are rather drastic. But without moving the mountain, you won’t be able to go down the mountain and stay very close to horizontal. You must alter the ski slope, and you must alter your path. But these movements are very very small, and you can make them as small as you like.

How do you alter the mountain? Roughly, you can make tiny ripples in the slope — and roughly, you turn it into a terraced slope. Just like rice farming in Vietnam, or for growing crops in the Andes.

Terraced farmland in Peru. By Alexson Scheppa Peisino(AlexSP).

As you go along a terrace, you remain horizontal! We don’t want our terraces to be completely horizontal though — we want them to have a very gentle downwards slope, so that we can stay very close to horizontal, and yet eventually get to the bottom of the mountain.

And also, we’ll need to be able to go smoothly from one terrace down to the next, so each terraces should turn into the next. So perhaps it’s more like Lombard Street, the famously windy street in San Francisco. (Which is not, however, the most crooked street in the US — that’s Wall Street, of course. Got you there.)

Lombard Street. By Gaurav1146

Perhaps a more accurate depiction of what we want is the figure below, from Eliashberg and Mishachev’s book Introduction to the h-principle. We want very fine, very gently sloping terraces, and we want them extremely small, so that the mountain is altered by a tiny tiny amount. And to go down the slope we need to take a very windy path — with many many wiggles in it. To go down the slope is almost like a maze — although it’s a very simple, repetitive maze.

A modified, lightly terraced, very windy, ski slope.

Thus, with a astronomical number of microscopic terraces of the mountain, each nano-scopically sloped downwards, and an astronomical number of microscopic wiggles in your path down the terraces, you can go down the mountain, staying very close to your idealised path. You go very very close to straight down the mountain, and your skis stay very very close to horizontal all the way down.

And then, you’ve done it.

This is the flavour of the h-principle. More precisely this result is called holonomic approximation. Holonomic approximation says that even an incompetent skier can get down a mountain with an arbitrarily small amount of trouble — provided that they can do an arbitrarily large amount of work in advance to terrace the mountain and prepare themselves an arbitrarily flat arbitrarily wiggly path.

* * *

The h-principle has applications beyond idealised incompetent skiing down microscopically terraced mountains. Two of the most spectacular applications are sphere eversion, and isometric embedding. In fact they both preceded the h-principle — and Gromov’s attempt to understand and formalise them directly inspired the development of the h-principle.

Sphere eversion  is a statement about spheres in three-dimensional space. Take a standard unit sphere in, but again regard it as made of playdough, and we will consider morphing (erm, homotoping) it in space. We allow the sphere to pass through itself, but never to crease, bend or rip. All the sphere can intersect itself, each point of the sphere must remain smooth enough to have a tangent plane. (The technical name for this is an immersion of the sphere into 3-dimensional space.)

Smale’s sphere eversion says that it’s possible to turn the sphere inside out by these rules — that is, by a homotopy through immersions. This amazing theorem is all the more amazing because Smale’s original 1957 proof was an existence proof: he proved that there existed a way to turn the sphere inside out, but did not say how to do it! Many explicit descriptions have now been given for sphere eversions, and there are many excellent videos about it, including Outside In made in the 1990s. My colleague Burkard Polster, aka the Mathologer, also has an excellent video about it.

Smale has an interesting and admirable biography. He was actively involved in the anti-Vietnam War movement, even to the extent of being subpoenaed by the House Un-American Activities Committee. His understanding of the relationship between creativity, leisure and mathematical research was epitomised in his statement that his best work was done “on the beaches of Rio”. (He discovered his horseshoe map on the beach at Leme.)

Isometric embedding is more abstract; see my article in the conversation for another description, but it is even more amazing. It is a theorem about abstract spaces. For instance, you could take a surface — but then, put on it an abstractly-defined metric, unrelated to how it sits in space. Isometric embedding attempts to map the surface back into 3-dimensional space in a way that preserves distances, so that the abstract metric on a surface corresponds to the familiar notion of distance we know in three dimensions.

Isometric embedding is largely associated with John Nash, who passed away a couple of years ago and is more well known for his work on game theory, and from the book and movie A Beautiful Mind. The proof is incredible. Gromov describes how he came to terms with this proof in some recollections. He originally found Nash’s proof “as convincing as lifting oneself by the hair”, but after eventually finding understanding, he found Nash’s proof “miraculously, did lift you in the air by the hair”!

The Nash-Kuiper theorem says that if you can map your abstract surface into 3-dimensional space in such a way that it decreases distances, then you can morph it — homotope it — to make it preserve distances. (Actually, it need not be a surface but a space of any dimension; and it need not be 3-dimensional space, but space of any dimension.) And, just like on the ski slope, this alteration of the surface in 3-dimensional space can be made very very small — arbitrarily small.

The h-principle is another mathematical superpower, and it comes up in many places where geometry is “soft”, and you can slightly “morph” or “adjust” your geometrical situation to find the situation we want.

Morse structures on partial open books with extendable monodromy

With Joan Licata (19 pages) – on the arXiv – published in the 2016 MATRIX annals.

Abstract: The first author in recent work with D. Gay developed the notion of a Morse structure on an open book as a tool for studying closed contact 3-manifolds. We extend the notion of Morse structure to extendable partial open books in order to study contact 3-manifolds with convex boundary.

Local pdf available here (1.2 MB).

morse_structures_partial_open_books

Eighty years ago, Spanish people responded to the far right with social revolution

Eighty years ago to the day, the far right was in its ascendancy, and still rising. Hitler was in complete control of Germany, Mussolini had been in charge of a police state in Italy for a decade. The world failed to stop them, and war was to break out within a few years, consuming the world in the deadliest conflict in human history.

But a little to the southwest, in Spain, war had already broken out.

In July 1936, Franco and his co-conspirators made their coup attempt against the elected Republican government, dividing the country and driving it into war. By the end of the year they controlled several major cities and had laid siege to Madrid.

The Spanish civil war had erupted, and while the supposedly democratic powers refused to come to the assistance of the besieged Republic — indeed Britain made some moves in the opposite direction — Mussolini and Hitler had no such qualms supporting the forces of reaction. Their support expressed itself in, among other military attacks, the bombing of Guernica. This attack, the world’s first targeting of civilians by aerial bombardment, inaugurated a new type of crime, and a new type of terror, from which humanity has been suffering ever since. The terror and destruction – though mild by the bombings that were to come, of London, Dresden, Tokyo, Hiroshima, Nagasaki, and elsewhere – has never really left us. Conscientious, idealistic and passionate volunteers from abroad fought for the Republic, Orwell’s account being perhaps the most well known.

But the various forces of Spanish reaction — monarchists, Carlists, phalangists, fascists and more, along with mercenaries, German Nazis and Italian fascists — were not merely met with resistance on the battlefield. Many groups within Spanish society took that very moment to make a social, political and economic revolution.

As Chomsky wrote in his Objectivity and Liberal Scholarship,

During the months following the Franco insurrection in July 1936, a social revolution of unprecedented scope took place throughout much of Spain. It had no “revolutionary vanguard” and appears to have been largely spontaneous, involving masses of urban and rural laborers in a radical transformation of social and economic conditions that persisted, with remarkable success, until it was crushed by force. This predominantly anarchist revolution and the massive social transformation to which it gave rise are treated, in recent historical studies, as a kind of aberration, a nuisance that stood in the way of successful prosecution of the war to save the bourgeois regime from the Franco rebellion. Many historians would probably agree with Eric Hobsbawm that the failure of social revolution in Spain “was due to the anarchists,” that anarchism was “a disaster,” a kind of “moral gymnastics” with no “concrete results,” at best “a profoundly moving spectacle for the student of popular religion.” … In fact, this astonishing social upheaval seems to have largely passed from memory.

In other words, at the same time, and in the same place as this horrific war, also eighty years ago to the day, an extraordinary experiment was underway — which has arguably seen no parallel before or since. It was not undertaken as a sideshow, or as a footnote to the ongoing war, but because of it. It was associated with those most optimistic of political philosophies — anarchism, or libertarian socialism — which hold that human beings can organise their lives without the illegitimate authority of bosses or States, without property in capital, and with equality, freedom, democracy, and free association.

How did this happen? All at once. In Catalonia, the question of how far to push for libertarian revolution under conditions of war against fascism presented itself immediately — indeed, within a day of the initial coup. While the Catalan president Luis Companys had refused to issue arms, the anarchists that had stormed the barracks at Atarazanas, defeating the local military coup plotters, seizing weapons and obtaining de facto power. The result was that the anarchists were forced to respond to the question of taking political power in the most dramatic way: they were literally offered State power. Companys, a Catalan nationalist, but unusually sympathetic to anarchists, presented the anarchist leaders Juan Garcia Oliver, Buenaventura Durruti and Diego Abad de Santillán with a proposition, a mixture of generous support, alliance, realpolitik (for the anarchists were more heavily armed than nearby official Republican forces), self-preservation, and acknowledgment of the justice of their cause.

As Anthony Beevor describes it in his history of the war:

On the evening of July 20, Juan Garcia Oliver, Buenaventura Durruti and Diego Abad de Santillán met with President Companys in the palace of the Generalidad. They still carried the weapons with which they had stormed the Atarazanas barracks that morning. In the afternoon they had attended a hastily called meeting of more than 2,000 representatives of local CNT [anarchist] federations. A fundamental disagreement arose between those who wanted to establish a libertarian society immediately and those who believed that it had to wait until after the generals were crushed. …
At the meeting… Companys greeted anarchist delegates warmly:

… Today you are the masters of the city and of Catalonia because you alone have conquered the fascist military… and I hope you will not forget that you did not lack the help of loyal members of my party…  But you have won and all is in your power. If you do not need me as president of Catalonia, tell me now, and I will become just another soldier in the fight against fascism. If, on the other hand… you believe that I, my party, my name, my prestige, can be of use, then you can depend on me and my loyalty as a man who is convinced that a whole past of shame is dead.

To my knowledge this is the only offer of its kind ever made by any State to an anarchist organisation. It was an incredible dilemma for the anarchists:

Garcia Oliver described the alternatives as ‘anarchist dictatorship, or democracy which signifies collaboration’. Imposing their social and economic self-management on the rest of the population appeared to violate libertarian ideals more than collaborating with political parties. Abad de Santillán said that ‘we did not believe in dictatorship when it was being exercised against us and we did not want it when we could exercise it only at the expense of others’. At their Saragrossa conference only seven weeks before, the anarchists had affirmed that each political philosophy should be allowed to develop ‘the form of social co-existence which best suited it’. This meant working alongside other political bodies with mutual respect for each other’s differences. Though genuine, this was a simplistic view, since the very idea of worker-control and self-management was anathema both to businessmen and the communists.

Even if the anarchist leaders sitting in Companys’ ornate office, having just been offered the keys of the kingdom, could have foreseen the future, it is doubtful whether their choice would have been made any easier. They had the strength to turn Catalonia and Aragon into an independent non-state almost overnight. But Madrid had the gold, and unofficial sanctions by foreign companies and governments could have brought them down in a relatively short space of time. However, what influenced their decision the most was concern for their comrades in other parts of Spain. The demands of solidarity overrode other considerations. They could not abandon them in a minority which might be crushed by the Marxists.

Accordingly, a Central Committee of Anti-Fascist Militias was organised, allowing pluralism among the various Republican factions in the government of Catalonia; the anarchists decided to share, rather than take, power.

Being in the majority, the least we can do is to recognize the right of minorities to organize their own lives as they want and to offer them our cordial solidarity.

The result in Barcelona, as described by the journalist John Langdon-Davies, was

the strangest city in the world today, the city of anarcho-syndicalism supporting democracy, of anarchists keeping order, and anti-political philosophers wielding political power.

In December 1936, while Madrid was being attacked, George Orwell arrived in Barcelona and described what he saw.

I had come to Spain with some notion of writing newspaper articles, but I had joined the militia almost immediately, because at that time and in that atmosphere it seemed the only conceivable thing to do. The Anarchists were still in virtual control of Catalonia and the revolution was still in full swing. To anyone who had been there since the beginning it probably seemed even in December or January that the revolutionary period was ending; but when one came straight from England the aspect of Barcelona was something startling and overwhelming. It was the first time that I had ever been in a town where the working class was in the saddle. Practically every building of any size had been seized by the workers and was draped with red flags or with the red and black flag of the Anarchists; every wall was scrawled with the hammer and sickle and with the initials of the revolutionary parties; almost every church had been gutted and its images burnt. Churches here and there were being systematically demolished by gangs of workmen. Every shop and cafe had an inscription saying that it had been collectivised; even the bootblacks had been collectivized and their boxes painted red and black. Waiters and shop-walkers looked you in the face and treated you as an equal. Servile and even ceremonial forms of speech had temporarily disappeared. Nobody said ‘Señor’ or ‘Don’ or even ‘Usted’; everyone called everyone else ‘Comrade’ or ‘Thou’, and said ‘Salud!’ instead of ‘Buenos dias’. Tipping had been forbidden by law since the time of Primo de Rivera; almost my first experience was receiving a lecture from a hotel manager for trying to tip a lift-boy. There were no private motor-cars, they had all been commandeered, and the trams and taxis and much of the other transport were painted red and black. The revolutionary posters were everywhere, flaming from the walls in clean reds and blues that made the few remaining advertisements look like daubs of mud. Down the Ramblas, the wide central artery of the town where crowds of people streamed constantly to and fro, the loud-speakers were bellowing revolutionary songs all day and far into the night. And it was the aspect of the crowds that was the queerest thing of all. In outward appearance it was a town in which the wealthy classes had practically ceased to exist. Except for a small number of women and foreigners there were no ‘well-dressed’ people at all. Practically everyone wore rough working-class clothes, or blue overalls or some variant of militia uniform. All this was queer and moving. There was much in this that I did not understand, in some ways I did not not even like it, but I recognized it immediately as a state of affairs worth fighting for. Also, I believed that things were as they appeared, that this was really a workers’ State and that the entire bourgeoisie had either fled, been killed or voluntarily come over to the workers’ side; I did not realise that great numbers of well-to-do bourgeois were simply lying low and disguising themselves as proletarians for the time being.

Together with all this there was something of the evil atmosphere of war. The town had a gaunt untidy look, roads and buildings were in poor repair, the streets at night were dimly lit for fear of air-raids, the shops were mostly shabby and half-empty. Meat was scarce and milk practically unobtainable, there was a shortage of coal, sugar and petrol, and a really serious shortage of bread. Even at this period the bread-queues were often hundreds of yards long. Yet so far as one could judge the people were contented and hopeful. There was no unemployment, and the price of living was still extremely low; you saw very few conspicuously destitute people, and no beggars except the gypsies. Above all, there was a belief in the revolution and the future, a feeling of having suddenly emerged into an era of equality and freedom. Human beings were trying to behave as human beings and not as cogs in the capitalist machine. In the barbers’ shops were Anarchist notices (the barbers were mostly Anarchists) solemnly explaining that barbers were no longer slaves. In the streets were coloured posters appealing to prostitutes to stop being prostitutes. To anyone from the hard-boiled, sneering civilization of the English-speaking races there was something rather pathetic in the literalness with which these idealistic Spaniards took the hackneyed phrase of revolution. At that time revolutionary ballads of the naivest kind, all about the proletarian brotherhood and the wickedness of Mussolini, were being sold on the streets for a few centimes each. I have often seen an illiterate militiaman buy one of these ballads, laboriously spell out the words, and then, when he had got the hang of it, begin singing it to an appropriate tune.

More generally across the Republican zone, anarchists and socialists found themselves in positions of power. As Chomsky writes:

Workers armed themselves in Madrid and Barcelona, robbing government armories and even ships in the harbor, and put down the insurrection while the government vacillated, torn between the twin dangers of submitting to Franco and arming the working classes. In large areas of Spain, effective authority passed into the hands of the anarchist and socialist workers who had played a substantial, generally dominant role in putting down the insurrection.

Turning more specifically to the economic revolution carried out in the midst of the civil war, Beevor explains the collectives as follows.

The collective in Republican Spain were not the state collectives of Russia. They were based on the joint ownership and management of the land or factory. Alongside them were ‘socialized’ industries, restructured and run by the CNT and UGT as well as private companies under the joint worker-owner control. Co-operatives marketing the produce of individual smallholders and artisans also existed, although these were not new. They had a long tradition in many parts of the country, especially in fishing communities. There were estimated to have been around 100,000 people involved in co-operative enterprises in Catalonia alone before the civil war. The [anarchist] CNT was, of course, the prime mover in this development, but [socialist union] UGT members also contributed to it. The UGT or UGT—CNT organized about 15 per cent of the collectives in New Castile and La Mancha, the majority in Estremadura, very few in Andalucia, about 20 per cent in Aragon, and about 12 per cent in Catalonia.

The regions most affected were Catalonia and Aragon, where about 70 per cent of the workforce was involved. The total for the whole of Republican territory was nearly 800,000 on the land and a little over a million in industry. In Barcelona workers’ committees took over all the services, the oil monopoly, the shipping companies, heavy engineering firms such as Vulcano, the Ford motor company, chemical companies, the textile industry and a host of smaller enterprises.

Any assumption by foreigners that the phenomenon simply represented a romantic return to the village communes of the Middle Ages was inaccurate. Modernization was no longer feared because the workers controlled its effects. Both on the land and in the factories technical improvements and rationalization could be carried out in ways that would previously have led to bitter strikes. The CNT wood-workers union shut down hundreds of inefficient workshops so as to concentrate production in large plants. The whole industry was reorganized on a vertical basis… Similar structural changes were carried out in other industries as diverse as leather goods, light engineering, textiles and baking. … One of the most impressive feats of those early days was the resurrection of the public transport system at a time when the streets were still littered and barricaded.  …

At the same time as the management of industry was being transformed, there was a mushroom growth of agricultural collectives in the southern part of Republican territory. They were organized by CNT members, either on their own or in conjunction with the UGT. The UGT became involved because it recognized that collectivization was the most practical method of farming the less fertile latifundia.

There are lessons here for the present day: the economic consequences of technological innovation — whether “modernization” by mechanisation or digitalisation, whether “automation” by production line or software — need no longer be feared when production is under the control of workers. We should, however, be clear that while collectivization was often voluntary — indeed spontaneous — it was sometimes coerced.

To many people’s surprise the anarchists made attempts to win the trust of the middle classes. If a shopkeeper complained to the CNT that his goods were being taken by workers’ patrols, a sign would be put up stating that the premises belonged to the supply committee. Small firms employing fewer than 50 people were left untouched if the management had a good record. …

In Aragon some collectives were installed forcibly by anarchist militia columns, especially Durruti’s. Their impatience to get the harvest in the feed the cities, as well as the fervour of their beliefs, sometimes led to violence. Aragonese peasants resented being told what to do by over-enthusiastic Catalan industrial workers, and many of them had fears of Russian-style collectives. …

There were few villages which were completely collectivized.  The ‘individualists’, consisting chiefly of smallholders who were afraid of losing what little they had, were allowed to keep as much land as a family could farm without hired labour. In regions where there had always been a tradition of smallholding, little tended to change. The desire to work the land collectively was much stronger among the landless peasants, especially in less fertile areas where the small plots were hardly visible.

However, persuasion was often recognised as not just the most principled but also the most effective tactic. The Austrian Marxist writer Franz Borkenau, visiting Spain, wrote:

The anarchist nucleus achieved a considerable improvement for the peasants and yet was wise enough not to try to force the conversion of the reluctant part of the village, but to wait till the example of the others should take effect.

Indeed, as Beevor continues,

the anarchists tried to persuade the middle classes that they were in fact oppressed by an obsession with property and respectability. ‘A grovelling existence,’ they called it. ‘Free yourselves socially and morally from the prejudices that have dominated you until today.’ …

The anarchists continually tried to persuade the peasants that the ownership of land gave a false sense of security. The only real security lay within a community which cared for its own members by providing medical facilities and welfare for the sick and retired.

Collectivization of agriculture was, in economic terms, a qualified success:

whatever the ideology, the self-managed co-operative was almost certainly the best solution to the food-supply problem. Not only was non-collectivized production lower, but the ‘individualists’ were to show the worst possible traits of the introverted and suspicious smallholder. When food was in short supply they hoarded it and created a thriving black market, which, apart from disrupting supplies, did much to undermine morale in the Republican zone. The communist civil governor of Cuenca admitted later that the smallholders who predominated in his province held onto their grain when the cities were starving. …

In terms of production and improved standards for the peasants, the self-managed collectives appear to have been successful. They also seem to have encouraged harmonious community relations. There were, however, breakdowns of communication and disputes between collectives. The anarchists were dismayed that collective selfishness should seem to have taken the place of individual selfishness, and inveighed against this ‘neo-capitalism’.

It should be made clear just how much opposition the anarchists faced — not just from the Nationalists, but also from other factions in the Republican camp, liberals and communists.

The most outspoken champions of property were not the liberal republicans, as might have been expected, but the Communist Party and its Catalan subsidiary, the PSUC. La Pasionaria and other members of their central committee emphatically denied that any form of revolution was happening in Spain, and vigorously defended businessmen and small landowners (at a time when kulaks were dying in Gulag camps). This anti-revolutionary stance, prescribed by Moscow, brought the middle classes into the communist ranks in great numbers. Even the traditional newspapers of the Catalan business community Vanguardia and Noticiero, praised ‘the Soviet model of discipline’.

…

There were [for collectivized industries] serious problems in obtaining new machinery to convert companies which were irrelevant, like luxury goods, or under-used because of raw-material shortages, like the textile industry. They were caused principally by the Madrid [Republican!] government’s attempt to reassert its control by refusing foreign exchange to collectivized enterprises.

…

[Moreover, for Catalonian industry a] sizeable part of the home market had been lost in the rising. The peseta had fallen sharply in value on the outbreak of the war, so imported raw materials cost nearly 50 per cent more in under five months. This was accompanied by an unofficial trade embargo which the pro-Nationalist governors of the Bank of Spain had requested among the international business community. Meanwhile, the central government tried to exert control through withholding creidts and foreign exchange.  [Republican Prime Minister] Largo Caballero, the arch-rival of the anarchists, was even to offer the government contract for uniforms to foreign companies, rather than give it to CNT textile factories. (The loss of markets and shortage of raw materials led to a 40 per cent decline in textile output, but engineering production increased by 60 per cent over the next nine months.)

The communists’ Popular Front strategy of defending commercial interests so as to win over the middle class was perfectly compatible with their fundamental opposition to self-management. As a result their Catalonian affiliate, the PSUC, started to persuade [socialist union] UGT bank employees to use all possible means to interfere with the collectives’ financial transactions.

…

[Prime Minister] Giral’s government in Madrid did not share the anarchists’ enthusiasm for self-managed collectives. Nor did it welcome the fragmentation of central power with the establishment of local committees. Its liberal ministers believed in centralized government and a conventional property-owning democracy…. They were appalled at having no control over the industrial base of Catalonia. But… [the Republican government’s] continued control of supply and credit held out the prospect that concessions might gradually be wrung from the revolutionary organizations”

Facing such obstacles — and of course, eventually, destruction under the victorious military forces of fascism — it is perhaps surprising that the collectives achieved even a fraction of the success they did.

* * *

Apparently there is an old Spanish proverb, “History is a common meadow in which everyone can make hay”. No doubt I am making hay of it in my own way. Let us be clear that all sides in the Spanish Civil War were responsible for atrocities, and many collectivizations were forced. Perhaps the libertarianism of the anarchists is too optimistic for human nature; perhaps, left to run its own course, with increasing complexity of industry, self-management might have floundered. But we do know that the revolution did not fail for those reasons; nor did it fail for inefficiency, or bureaucracy, or authoritarianism — on the contrary. It failed because it was crushed, opposed by every other faction both among both enemies and allies. The situation left them no chance. Between the military attacks of the fascists, the opposition by erstwhile Republican allies, ranging from political interference to outright military attack — supported by all the greatest monsters of European history, Hitler, Mussolini and Stalin — and abandonment by the liberal democratic powers, it is impossible to say how they might have developed, leaving fodder for cynic and dreamer alike, and everyone in between.

Let us limit ourselves to a few obvious remarks. People fight harder for something worth defending. When an existing political or economic system is at an ebb, giving rise to the worst forces of reaction, xenophobia, nationalism and authoritarianism, is precisely when changes are most possible. The time for the most realistic utopian thinking is in the time of catastrophe. The history of all great reforms is a span which begins with a demand for the impossible and ends with the acceptance of the inevitable. Even an anarchist can be offered the keys to the kingdom. And even those offered power can decline — and try to build something better instead.

But let us leave the history to speak for itself — a reminder of what can be achieved even under the greatest adversity.
Eighty years ago, Spanish people responded to the rise of the far right with social revolution. What will you do?