A duality of pictures

Duality relates objects, which seem different at first but turn out to be similar. The concept of duality occurs almost everywhere in maths. If two objects seem different but are actually the same, we can view each object in a “usual” way, and in a “dual” way – the new vantage point is helpful for new understanding of the object.  In this blog post we’ll see a pictorial example of a mathematical duality.

How are these two graphs related?


bg2In the first graph, we have five vertices, the five black dots, and six green edges which connect them. For example, the five vertices could represent cities (San Francisco, Oakland, Sausalito etc. ) and the edges could be bridges between them.

In the second graph, the role of the cities and the bridges has swapped. Now the bridges are the vertices, and the edges (or hyperedges) are the cities. For example, we can imagine that the cities are large metropolises and the green vertices are the bridge tolls between one city and the next.

Apart from swapping the role of the vertices and the edges, the information in the two graphs is the same. If we shrink each city down to a dot in the second graph, and grow each bridge toll into a full bridge, we get the first graph. We will see that the graphs are dual to each other.

We represent each graph by a labeled matrix: we label the rows by the vertices and the columns by the edges, and we put a 1 in the matrix whenever the vertex is in the edge. For example, the entry for vertex 1 and edge a is 1, because edge a contains vertex 1. The matrix on the left is for the first graph, and the one on the right is for the second graph.


We can see that the information in the two graphs is the same from looking at the two matrices – they are the same matrix, transposed (or flipped). The matrix of a hypergraph is the transpose of the matrix of the dual hypergraph.

Mathematicians are always on the look-out for hidden dualities between seemingly different objects, and we are happy when we find them. For example, in a recent project we studied the connection between graphical models, from statistics, and tensor networks, from physics. We showed that the two constructions are the duals of each other, using the hypergraph duality we saw in this example.


Combing braids

I’m going to a conference next week, and it’s all about braids! So I thought I would write a wee post on combing, a technique which dates back to Artin in the 1940s. In fact the paper where he introduces the concept of combing finishes with the following amusing warning:

“Although it has been proved that every braid can be deformed into a similar normal form the writer is convinced that any attempt to carry this out on a living person would only lead to violent protests and discrimination against mathematics. He would therefore discourage such an experiment.” – Artin 1946

but I really don’t see it as so bad!

Combing is a technique for starting with any braid (see my introductory post on braids here) and ending up with a braid in which first the leftmost strand moves and the others stay put, then the next strand moves while the rest stay put etc etc. It’s much nicer to show this in pictures.

We want to start with any old braid, say this one:



and transform it into a braid where the strands move one at a time, like the following one. I’ve coloured the strands here so you can see that, reading the braid from top to bottom, first the red strand moves (i.e. all crossing involve the red strand, until it is finished), and then the green, and then the blue.

final coloured


For convenience I’ll only look at braids called pure braids, where each strand starts and ends at the same position. You can easily comb non-pure braids, you just need to add an appropriate twist right at the end to make them finish in the correct positions.

So how do we do this? Consider the first strand, I’ve coloured it red to make it clear. We want all the crossings between red and black strands to happen before (higher up than) a crossing of two black strands. So in this case the crossing circled in yellow are okay, because they happen lower down than any crossing involving the red strand. The crossings circled in blue and green need to be changed.

strand 1 highlight


We can slide some crossings of black strands down past the red and black crossings, as they don’t interfere. Here we can do it with the crossing circled in blue, as shown:

strand 1 step 1


We can start to do it with the crossing circled in green, but we encounter a problem as it wont simply slide past the red strand crossing below it. Moving this crossing down requires using some of the braid relations (see braid post) to replace a few crossings with an equivalent section in which the red strand moves first, as follows:

strand 1 step 2

Even though this braid looks different than the previous one they are in fact the same (you can always test this with string!). Now we have a braid in which the first strand moves before any others. Since all the first stand action is now at the top of the braid, we can now ignore the first strand all together, and consider the rest of the braid, as show below:

strand 1 forgetting


we only need to consider the following section now, and again we can put this into a form where only the first strand moves.

strand 2 beginning

In this case using braid relations gives us the following:

strand 2 final

And we can now ignore the green strand!

strand 2 forgetting

Colouring the first strand in this final section gives us no crossing that don’t involve the first strand:

strand 3 start

and we colour the last strand yellow for fun!

strand 3

Remembering all the pieces we have ignored gives us the full combed braid, where we focus on the leftmost strand until it ‘runs out of moves’ before looking to the next one.

final coloured

And this is exactly the same as the original braid, which looks a lot messier when coloured:

original coloured

Why might we want to do this? In some cases it makes mathematical proofs a lot easier. For me, recently I have been focusing only on what the first strand is doing, and so I want a technique to push the other strands down and away!

Defining topology through interviews. Interview seven with Jeremy Mann.

The final interview (*cry*) in the Defining topology through interviews series is with Jeremy Mann, who is a PhD student in mathematics at the University of  Notre Dame, studying geometry and topology.

1. What would your own personal description of  “topology” be?

Topology studies features we call “qualitative”: ones that don’t change if the system is gently* disturbed. In some sense, we created topology in order to give precise answers to qualitative questions. In my day to day life, I reason qualitatively. I rarely wonder “Will the temperature outside be greater than 23 degrees?” I ask: “Is it warm outside?” I would call the first question quantitative, and my second one topological. In other words, topology is created to give precise answers to the types of questions we, as humans, are naturally interested in.

* What one means by “gently” depends enormously on the context, and one has a lot of freedom in choosing what that means. For these reasons, despite being wonderfully vivid, topology is at times unavoidably abstract.

2. What do you say when trying to explain your work to non-mathematicians?

I fudge the details and I lie. If a careful mathematician were listening, they might interject with a few “well, actually—”s. But the details can obscure content, and people enjoy fiction, so I try not to lose sleep over it. That being said, I might tell a story like this:
By the age of three, I could pick two peaches out of a bag without knowing the first thing about the symbol “2.” A number was something like a bunch of stuff contained within a box. A number could bounce around and bruise. I could hold it in my hands.
If I had a sack of plums and a sack of peaches, I could add them together by pouring them both into a bigger sack.
But these terms didn’t help me add the grains of sand in a bucket, or the stars spread before my eyes. So I dropped this way of adding, in favor of an algebra with lots of symbols like “2,” and “376,” and eventually “x.”
Since then, I’ve made another shift. These days, my conceptualization of arithmetic is a lot closer to a child’s. This approach has many names, but my favorite is Factorization Algebras. I see a number as a collection of objects contained within a region of space.
But now, my numbers can interact. Symbols are no longer rich enough to capture their structure. Sometimes my numbers feel like exotic creatures. They can circle each other suspiciously.


Symbols see this as “2=2=2=2,” but this picture shows us there’s a lot more going on.


“1 + 1 = 2”.
Two numbers can be enemies. When I add them together, they remove each other from existence. “1 + (-1) = 0”. Sometimes, I play this in reverse, watching two enemies spontaneously born from empty tranquility.
I guess I’m interested in more than just writing down the final answer. I want to see their costumes. I want to know how they come together. I want to feel the content in their choreography. My work helps me do this.



3. How does your work relate, if at all, to the Nobel prize work?

The Nobel Prize was awarded for insights into the behavior of certain forms of matter at very low energy, where their behavior becomes “topological.” Strictly speaking, the structures I consider are not “topological” — despite “being a topologist,” my work does in fact know the difference between a coffee cup and a donut. It’s much higher-energy.*
Many physicists are interested in a material’s low energy behavior because these conditions contain a huge amount of information about a material’s possible phases. This even includes more “exotic” phases of matter, some with potential applications to quantum computers.
I’d like to point out the following: often, “exotic” means “outside of one’s comfort zone.” So, when physicists say “exotic topological phases of matter,” I suspect they are expressing how the low energy behaviors of certain materials are outside the comfort of zone of many members of the physics (and mathematics) community. This “exotic” behavior defies common intuition. However, when a material behaves in this manner, to a topologist, it enters very familiar territory. The topological is not exotic to a topologist.

** hotter, but certainly not sexier.

Defining topology through interviews. Interview six with Cécile Repellin.

The penultimate interview in this Defining topology through interviews series is with Cécile Repellin, who is a postdoc at the Max Planck Institute for the physics of complex systems, Dresden and works on condensed matter theory.

1. What do you say when trying to explain your work to non-mathematicians?

Since I’m not a mathematician myself, let’s rather pretend that I’m trying to explain my work to a non-physicist.
Understanding the different phases of matter is one of the most important goals of my field, condensed matter physics. Sometimes we can intuitively grasp the difference between various phases: you can think of water, which can appear in liquid form, but also as solid ice, or as steam, which is in gas form. Sometimes, it is more subtle, like the difference between a material that can carry an electrical current — a metal — and an insulator. Quantum mechanics leads to phases of matter even less intuitive than this, with electrical properties that are neither those of an electrical conductor like copper, nor those of a simple insulator, or even semiconductor like silicium. In these materials, everything happens as if the electrons carrying the electrical current were split into three or more parts. This phenomenon arises from the collective behavior and interactions (electric and magnetic) between the electrons.
My work consists in finding new phases of (quantum) matter, and more specifically new ways that electrons could split up. Among the many forces competing at the microscopic level, I try to figure out which ones are essential, to help predict in which materials these phases might appear.

2. What would your own personal description of  “topology” be?

One often gives the example of a mug and a donut to explain the concept of topology. Imagine that you have a mug made of an extremely elastic material. By stretching it, you can transform your mug into an object with the shape of a donut. Had you started from a bun, it would have been necessary to pierce a hole to achieve the same result. The number of holes is a global property of an object, or topological invariant: if you stand too close, you can’t tell how many holes there are. You need to take a step back and look at the whole object. On the other hand, the details of the mug do not matter to determine the number of holes. In the context of physics, you would not be talking about the number of holes, but about something that you can measure in an experiment, like the conductivity or resistivity. In a quantum Hall experiment, a thin layer of semiconductor is sandwiched between two thicker layers, and subjected to a large magnetic field. If you apply a voltage on either side, you will observe the apparition of a voltage in the opposite direction. Another way of saying this is that the transverse resistance (or Hall resistance) is finite. If the temperature is low enough (around -273C), this resistance evolves step by step by forming plateaus as you tune the magnetic field. There is something very special about the value of the Hall resistance on the plateaus: it does not depend on the sample that you are looking at, nor does it depend on the material. It is in fact related by a simple proportionality rule to physical constants: the Planck constant h and the charge of the electron e, or rather the ratio e^2/h. This property is very unique and is so robust that it is used in metrology to define the ratio e^2/h. The robustness is a consequence of the Hall resistance being a topological invariant, much like the number of holes in an object.


3. How does your work relate, if at all, to the Nobel prize work?

A lot of my research relates to the work of David Thouless and Duncan Haldane, two of this year’s Nobel prize winners in physics. In 1988, Haldane proposed a lattice model where the quantum Hall effect could be realized in the absence of a magnetic field. The first projects I worked on as a PhD student consisted in understanding the physics of this model (and other similar ones) when the electrons hopping on the lattice strongly repel one another. One way or another, my research interests are in large part related to topology in condensed matter physics. I was attending a conference on topological phases of matter when I heard about the Nobel prize, and it was very nice to share this moment with colleagues and see the community react and celebrate the great news.

Defining topology through interviews. Interview five with Renee Hoekzema.

First of all I would like to thank the AMS blog on math blogs for their recent mention of this series. It is always nice for Anna and I to know that people are reading and enjoying this blog!

The next of my  Defining topology through interviews  series is with Renee Hoekzema. Renee is a PhD student in mathematics at the University of Oxford and her research is in manifold theory.

1. What do you say when trying to explain your work to non-mathematicians?

Consider the five Platonic solids: the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron (Fig. 1).

Figure 1

It turns out that if you take the number of vertices of any one of them, subtract the number of edges and add the number of faces, the result is 2 for each shape. This number is called the Euler characteristic χ, and the reason it is the same for each of the Platonic solids is the fact that χ is a topological invariant. Each Platonic solid is topologically a sphere, with different decompositions. In fact, you don’t even need to take such a nice decomposition of the sphere to get the same number. You can also think of a sphere as a disc that is glued along its boundary to a single point (Fig. 2). χ  = 1 point – 0 edges +1 face = 2 again.

Figure 2

However, if we decompose a shape that is topologically different, such as a torus (i.e. doughnut, Fig. 3), we get a different number for χ, in this case 0. A double torus (i.e. a surface with two holes) gives χ = – 2, and in general for a surface with g holes we have χ = 2 – 2 g.

Figure 3

We call a shape orientable if it has an inside and an outside, and all orientable surfaces are one of these spheres with g holes. You might notice that they all have an even Euler characteristic (thus in particular, they can only be cut up into an even number of bits). For surfaces that are not orientable, this is not necessarily true. Fig. 4 shows the real projective plane, a non-orientable surface with χ = 1.

Figure 4

In higher dimensions, surfaces generalise to (closed) manifolds: shapes that are smooth everywhere and bounded in their size (essentially just nice shapes). Both orientability and the Euler characteristic can be generalised to higher dimensional manifolds. Being orientable means that the manifold has an inside and an outside. To calculate χ, we again cut up our manifold into bits and take the number of points, subtract the number of edges, add the number of discs, subtract the number of three-dimensional balls, add the number of four-dimensional balls, et cetera.

One of the things I did in my research was to generalise the statement that orientable surfaces have an even Euler characteristic, for some notion of “k-orientable“, where 0-orientable means “not necessarily orientable” and 1-orientable just means orientable:

Theorem: A k-orientable manifold (i.e. nice shape) has an even Euler characteristic unless the dimension is a multiple of 2k+1.

2. What would your own personal description of ‘topology’ be?

Topology is the study of shapes and spaces, where we consider two shapes to be the same if they can be deformed into each other without punching any holes. There are many different areas of research in topology. An example of a research direction in topology that I’m interested in is trying to make a list of all manifolds of a given dimension, in the way that I said above that all orientable surfaces are topologically a sphere with g holes for some integer g. One way to approach this question is by asking how manifolds can be cut up along manifolds of one dimension less and, oppositely, what fundamental building blocks are needed to build all manifolds. For example, any orientable surface can be built from gluing the building blocks shown in Fig. 5 along circles. A torus for example can be glued as: cup – co-pants – pants – cap (Fig. 6).

Figure 5
Figure 6


Alternatively, we can consider the shapes in Fig. 5 from the point of view of the circles: the pair of pants (yes that’s what we call it!) is the surface swept out by two circles merging over time, as described in Thomas’s interview. We call this a cobordism: manifolds evolving over time. Any cobordism between circles can also be built from the building blocks in Fig. 5.

3. How does your work relate, if at all, to the Nobel Prize work?

My work relates to the Nobel Prize because it is related to Topological Quantum Field Theories (TQFT’s). These theories are the mathematical framework behind the physics that was awarded the Prize, but they are also very interesting tools in mathematics. TQFT’s link cobordisms of manifolds on the one hand to algebraic structures on the other hand. The pair of pants, for example, takes two circles and merges them to one. This can be related to multiplying two numbers: you start with two and you merge them into one!

It turns out that the entire structure of gluing cobordisms of circles can be encoded as an algebraic structure called a Frobenius algebra (more specifically, the circle is the algebra and the cobordisms are operations such as multiplying two elements). A TQFT (in two dimensions) is an assignment of a specific algebra to the circle.

I personally think mostly about the cobordism side of TQFT’s. How can we understand the ways in which manifolds can be constructed from the cobordism pieces? Which manifolds are cobordant, i.e. related to each other by a cobordism? (Two manifolds are cobordant precisely if we can build one from the other with surgeries, see Carmen’s interview.)

Linking back to the Euler characteristic, it turns out that the parity of χ (whether it is odd or even), is preserved by cobordisms. That means that whenever I can evolve one manifold into another over time, sweeping out a cobordism, the Euler characteristic is either odd or even for both. So in order for both odd and even Euler characteristics to appear for a given dimension of manifolds of a certain type (e.g. “k-orientable”), there need to be at least two manifolds of that type that have no cobordism between them.

Defining topology through interviews. Interview four with Tom Hockenhull.

The next of my  Defining topology through interviews  series is with Tom Hockenhull. Tom is a PhD student in mathematics at Imperial College London and his research is in knot theory. You should also check out the cool topology themed comics he does for chalkdust magazine!

1. What do you say when trying to explain your work to non-mathematicians?

I’m in the odd position that my field gets more intelligible for non-mathematicians the more specific I am (well, to a point). It’s actually quite hard to capture the idea of ‘topology’ in enough generality, I find – but it’s rather easy to explain the subfield of topology I work in, which is knot theory. I usually talk about unknot recognition. If I get a piece of string, tie a knot in it, and then fuse the ends together, I get a knot which is trapped in the string — I can’t untie it without cutting the string again.

Tie a knot and fuse the ends

The question is, if you give me a big mess of string, can I tell easily whether I can just straighten it out into a big unknotted loop without cutting the string at all? People can understand quite easily that this is a hard question in general.

Can you make this into a simple loop of unknotted string without cutting it?

So how might I go about doing it? Well, one way to recognise if two things are different in general is to look at some easy to discern property of them and see if they’re different. For instance, if I’m trying to work out whether two insects are of the same species, I might look at how many legs they have.

Are these insects from the same species?

If I have a big tangle of string, I might try and work out some property of it that is different from that of a plain loop of string – then I can tell that it’s a different knot altogether. What’s the analogous notion for knots? Well, this is sort of what my work is in: there are a whole bunch of different properties we might try and use to compare knots. One which is easy to understand is the property of being able to colour a picture of the knot using exactly three colours (no less!), so that at each crossing I have three different colours, or all the colours are the same.

Colouring a crossing using exactly three colours

You can see that I can’t colour a simple loop of string with three colours (only one) – but I can colour the picture of the knot below.

However you draw an unknotted loop you can’t colour it using three colours
You can colour this one with exactly three colours!

It follows that I can’t turn one into the other without cutting the string!

2. What would your own personal description of ‘topology’ be?

There’s the standard ‘rubber sheet geometry’ or ‘the study of spaces up to deformation’, which are probably the most generally accurate, although they don’t really tell you much about what doing topology looks like or feels like. I suppose, though, that the problem is that ‘topology’ now encompasses a whole bunch of different areas that are rooted in the same place but are vastly different in their techniques, language and flavour.

3. How does your work relate, if at all, to the Nobel Prize work?

I’m not aware of any direct relevance – although that could be down to my ignorance! The word ‘quantum’ tends to pop up in descriptions of the Nobel work and in relation to a number of things to do with my work, but in my experience the use of the word quantum in my area seems to carry little relevance to its meaning in the world of physics.

Defining topology through interviews. Interview three with Ruben Verresen.

The third interview in my Defining topology through interviews series is with Ruben Verresen, who was a masters student alongside Anna and I. Ruben is now a PhD student at the Max Planck Institute for the Physics of Complex Systems and is one of two physicists I am thrilled to have participating in this series.

1. What do you say when trying to explain your work to non-mathematicians and non-physicists?

Traditionally theoretical physics follows the doctrine of ‘separate but equal’, where theories for different aspects of the universe are quite disjoint, with the smallest scales being described by particle physics and the largest by cosmology. (And perhaps not so ‘equal’ since particle physics is considered the building block for all the rest.) This has changed in recent years. The dusty 18th century divisive thinking is making way for a more thematic structure, as it is slowly becoming clear that various different fields give rise to similar concepts. Instead of defining a topic in terms of some mechanical parameters (e.g. ‘the size of an atom’), one can categorize physics in terms of relevant concepts (such as ‘topology’, more about this later). If you allow me one tacky metaphor, to me concepts are the jewels of physics, forged in the heat of discussion, mined out of theories and eventually polished until they reach their purest form. A successful theoretical framework you can forget after a year, but a beautiful and insightful concept, once germinated in the mind, will never leave you. So when people ask me what I do, I usually just end up talking about one of these concepts (which usually helps my understanding as well).

Three such concepts that I am actively exploring are emergence, quantum entanglement and topology. These three ideas turn out to run quite deeply and seem to be intimately connected—with the Nobel prize recognizing this. Emergence is the idea that the best way to understand something is not by thinking in terms of its constituents but rather by some new effective way of thinking—for example if we want to understand a hurricane, there is not much point in thinking about air molecules.

Waves on the Elbe are an example of emergent phenomena: to understand them we shouldn’t think about water molecules separately, but about collective motions where the molecules lose their individuality.

Entanglement is a less intuitive property about the universe: it says that any two objects—say you and the red eye of Saturn—can share an instantaneous connection, such that one can affect the other faster than the speed of light. Since its discovery 80 years ago, there have been strong disagreements about what this property actually means, but in the last few decades physicists have been discovering that no matter what it ‘is’, it is key to many new physical features, some of which may very well change our future (for example through quantum computing, teleportation, et cetera). And finally there is topology…

2. What would your own personal description of  “topology” be?

I really like this question. There are some standard go-to answers for the general public, but I don’t like them. The first common one is that topology has to do with holes, the typical example being that a coffee mug and a donut are topologically the same because they have the same amount of holes. A second—and closely related—common answer is that topology is about the properties of an object that don’t change if you stretch, squeeze and deform it—but you’re not allowed to tear it! Again, you can imagine a plasticine donut being reshaped into a coffee mug. Maybe I just don’t like these answers because I have heard them too many times before, but more importantly: although they seem easy to understand, I would say they are not. The issue is that they seem very arbitrary: what’s so special about holes? Or about tearing versus stretching? If that’s all you know about topology, there is no reason to think it would be interesting. But it is! Very much so. Aside from leading to beautiful mathematics, in recent years it has been rising to ever increasing prominence in almost all fields of physics. There has even been a feedback effect: studying the topology of quantum fields—a physical question—has given new fundamental insights into mathematics itself. There is no hint of this in the above. Indeed, my biggest peeve with those answers is that there is no fire in them: if I explain topology by comparing a donut to a coffee mug, I can just see my listener slowly turning off. This is an immense pity, since topology is currently one of the most exciting concepts around.

So, what then is topology? Imagine having an object, like a mountain lion, or the universe, or yourself. Now think of all the properties it might have, like height or color or holes. Some of these properties you can assign to arbitrarily small regions: if you know the color of the mountain lion at every small piece of its hide, you can say you know all about the colors of the cat as a whole. We say these properties are local. Topology is simply the study of the properties which are not local! If I look at my door frame, I can assign sizes and lengths to every small piece, and together they add up to for example the height of my door, so height is a local property. But the fact that my door (luckily) has a hole in it is not local: none of the four sides can be said to have a hole, it is only when I take them all at once that it is there. In that sense my (open) door is topologically different from my wall.

The topology of my door: you can’t say any of the four sides of my doorframe has a hole, it is only when they are all put together that it’s there. We call such a feature ‘non-local’/topological.

Given this characterization, it is then perhaps not surprising that topology enters in those fields of inquiry where we are learning that ‘the (w)hole is more than the sum of its parts’. It can be a fun game to go outside and try to identify ‘local’ versus ‘non-local’ properties. But maybe we don’t even need to look too far: perhaps the human mind cannot be understood by piecing together local properties of the brain, and somehow only emerges when everything is considered at once.

3. How does your work relate, if at all, to the Nobel prize work?

To better understand what this year’s Nobel Prize in physics is about, we just need to appreciate how and why these three concepts of emergence, entanglement and topology are so closely related. Luckily we are now in a position to do exactly that! To see this, let me just reword and recap:

  • Emergence is the idea that a system consisting of many particles can have properties which cannot be assigned to any of its individual particles, but only emerge when the system is taken as a whole. (H20 molecules do not have ‘wetness’, but a glass of water does.)
  • Quantum entanglement tells us that we cannot think of a ‘thing’ as merely being the sum of its parts. In a classical world we know that if we study each particle in the universe separately, then we know the state and motion of the complete universe. In quantum theory we would know almost nothing, since most of the information is stored in these instantaneous connections between different particles. This is sometimes called quantum non-locality—and it is worth pointing out that this has been experimentally verified!
  • Finally, as explained above, topology is about those properties of a ‘thing’ that cannot be attributed to any arbitrarily small piece of it.

So we see that these three concepts are all about some form of ‘non-locality’. More concretely, the Nobel Prize went to physicists who vastly aided our understanding by theoretically investigating physical systems where (some or all of) these phenomena play a crucial role. Personally I am most excited about this on a conceptual level, as it feels that we are learning something new and deep about how the universe works. But it is also of great importance for the more practical-minded: learning to harness the power of ‘non-locality’ means we have exponentially more possibilities at our disposal. In particular, ‘non-locality’ is often associated to strong stability (because if information is not stored locally it is much harder to accidentally destroy it) and physicists hope it could lead to the development of quantum computing, which is a whole exciting topic in and of itself. But anyway, despite there being deep links between emergence, entanglement and topology, it is still a basic question as to how deep the similarities really run. I’m certainly trying to find out!