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.

pic-a

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

pic-b

“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.

pic-c

 

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.

fqheexperiment

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).

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.

fig-2-1
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.

fig3
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.

fig-4
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).

fig-5
Figure 5
fig-6
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.

creating-a-knot
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.

knot-tangle
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.

bugs
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.

coloured-crossing
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.

colouring
However you draw an unknotted loop you can’t colour it using three colours
colouring-trefoil
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 two with Carmen Rovi.

The second of my interviews for the Defining topology through interviews series in with Carmen Rovi. Carmen was a PhD student in mathematics at Edinburgh whilst I did my undergrad, and is now a Postdoctoral Fellow at Indiana University in Bloomington. She works in the exciting world of algebraic surgery theory!

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

Topology studies the geometric properties of spaces that are preserved by continuous deformations. For a topologist any stretching or bending will not change the topological properties of an object, since we concentrate on much more essential aspects. This is why many times we hear the comment that a topologist can’t distinguish her coffee cup from her doughnut! Manifolds are the main examples of topological spaces, with the local properties of Euclidean space in an arbitrary dimension n. They are the higher dimensional analogs of curves and surfaces. For example, a circle is a one-dimensional manifold. Balloons and doughnuts are examples of two dimensional manifolds. A balloon cannot be deformed continuously into a doughnut, so we see that there are essential topological differences between them. An invariant of a topological space is a number or an algebraic structure such that topologically equivalent spaces have the same invariant. For example the essential topological difference between the balloon and the doughnut is calculated by the Euler characteristic, which is 2 for a balloon and 0 for a doughnut.

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

When I talk to non-mathematicians and I am asked this question I usually say “I do topology, which is an area of mathematics”. Unfortunately the conversation usually ends there when the other person says “mathematics… that’s difficult. I had a hard time in school with that”. Sometimes with a bit of luck they will want to know more and then I can explain that my work lies in a field called “surgery theory”. Like real surgeons, a mathematician doing surgery also does a lot of cutting and sewing. Of course like in real surgery, there are very precise rules for doing this! In the following figure, we perform a surgery on a 2-dimensional sphere. The first step is to cut out two patches (discs) from the sphere. This leaves two holes in the shape of two circles. We can then sew on a handle along those two circles. If we are careful on how we sew things together, we obtain a torus:

Surgery-sphere.jpg

This is an example of how surgery can be used to describe the classification of surfaces: sphere, torus, torus with two holes … Of course, surgery can also be used in much more sophisticated situations and it is a very powerful tool for investigating classification questions in topology. The aim of classifying complicated topological objects gives rise to the description of invariants which give you certain information about the objects you are trying to classify. As we have seen with the work of the physics laureates this year, such invariants can then be applied in “real life” contexts and produce stunning results.

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

One of the questions that the physics laureates from this year were trying to understand was the electrical conductance in a layer of condensed matter. Experiments done in extremely low temperatures and under very powerful magnetic fields showed that the electrical conductance of the material assumed very precise values, which is extremely rare in this kind of experiments. They also noticed that the conductance changed in steps, when the change in the magnetic field was significant. The strange behaviour of the conductance changing stepwise made them think of topological invariants. Of course, there is a wide gap between having such an intuition and being able to formalise exactly which topological invariant is relevant in this situation. The key invariant for them is called “Chern numbers”, which for me is very surprising as this is an invariant which features frequently in my work.

Defining topology through interviews. First up, Thomas Wasserman.

The first of my Defining topology through interviews  series is with Thomas Wasserman, a PhD student at Oxford University studying mathematics. In particular his work is in the realm of topological quantum field theory.

1. What do you say when trying to explain your work to non-mathematicians?
Suppose you have a bunch of circles. Now imagine some of those circles merging together to one, and some of them splitting themselves in two. This merging and splitting over time sweeps out a surface, an example is shown below.

diagram-1
A surface starting with 3 circles and ending with 2.

We view this surface as a kind of machine that takes your bunch of circles and spits out another bunch of circles. Observe that given two such machines, we can apply one after the other by sticking the first one on top of the second, as long as we have the same number of circles coming out of the first one as going into the second one.

As you’ll see from the other posts, us topologists have a lot of trouble telling circles apart. This means that if we want to say something about machines like this we need to do something clever. Let’s give each circle an extra bit of information and think of it (by flawed analogy) as a colour, chosen from some range of possibilities. On top of merging and splitting circles our machine can now also change the information associated to each circle, for example mix the colours if two circles merge.

If this new feature of our machine sounds completely arbitrary to you, you’re on the right track. In order to use this feature to study the original surface, it needs to be compatible with applying several machines in a row: the big machine formed from smaller machines should do exactly the same to the information as just applying the small machines in succession. I study how one can construct these extra features subject to these requirements.

2. What would your own personal description of ”topology” be?
Topology is the study of shapes. To me, that means studying shapes that you can imagine encountering in nature. As mathematicians, we have developed a way of describing these shapes that allows us to check our intuition about them and study them in a lot of detail. This is what makes topology such an interesting subject; it very explicitly creates a bridge between the intuition we have for the world around us and very abstract mathematical ideas.

3. How does your work relate, if at all, to the Nobel prize work?
My work fits very well into the way of thinking that the Nobel prize work has inspired in the last decades. In condensed matter physics, the area of physics that the Nobel prize work was done in, people spend a lot of time thinking about a type of (composite) particles called anyons. These anyons live on very thin slices of material and have the strange property that the way they behave is described only by their relative movement. That is, the only thing that matters is whether or not they merge, split or braid around each other. For example the image below shows a braiding followed by a merging.

diagram-3
A braiding and merging of 2 anyons (a blue one and a red one).

 

Comparing this kind of behaviour to what we discussed in question one, we can relate them as follows. Suppose that instead of a colour, we assigned a type of particle to each circle. The circles merging then corresponds to the particles merging, and the circles splitting to the particle splitting. The particles winding around each other is encoded by the surface winding around itself, as in the figure below, and this relates to my work on surfaces such as these!

diagram-2
The surface corresponding to the anyon winding and merging in the previous picture.