Mapping class groups and curves in surfaces

Firstly, thanks to Rachael for inviting me to write this post after meeting me at the ECSTATIC conference at Imperial College London, and to her and Anna for creating such a great blog!

My research is all about surfaces. One of the simplest examples of a surface is a sphere. We are all familiar with this – think of a globe or a beach ball. Really we should think of this beach ball as having no thickness at all, in other words it is 2-dimensional. We are allowed to stretch and squeeze it so that it doesn’t look round, but we can’t make every surface in this way. The next distinct surface we come to is the torus. Instead of a beach ball, this is like an inflatable ring (see this post by Rachael). We say that the genus of the torus is 1 because it has one “hole” in it. If we have g of these holes then the surface has genus g. The sphere doesn’t have any holes so has genus 0. We can also alter a surface by cutting out a disc. This creates an edge called a boundary component. If we were to try to pass the edge on the surface, we would fall off. Here are a few examples of surfaces.

surfaceexamples

As with the sphere, topology allows us to deform these surfaces in certain ways without them being considered to be different. The classification of surfaces tells us that if two surfaces have the same genus and the same number of boundary components then they are topologically the same, or homeomorphic.

Now that we have a surface, we can start to think about its properties. A recurring theme across mathematics is the idea of symmetries. In topology, the symmetries we have are called self-homeomorphisms. Strictly speaking, all of the self-homeomorphisms we will consider will be orientation-preserving.

Let’s think about some symmetries of the genus 3 surface.

Here is a rotation which has order 2, that is, if we apply it twice, we get back to what we started with.

hyperelliptic

Here is another order 2 rotation.

halfrotation

And here is a rotation of order 3. Remember that we are allowed to deform the surface so that it looks a bit different to the pictures above but still has genus 3.

thirdrotation

However, not all symmetries of a surface have finite order. Let’s look at a Dehn twist. The picture (for the genus 2 surface) shows the three stages – first we cut along a loop in the surface, then we rotate the part of the surface on just one side of this loop by one full turn, then we stick it back together.

twist

A Dehn twist has infinite order, that is, if we keep on applying it again and again, we never get back to what we started with.

If we compose two homeomorphisms (that is, apply one after the other) then we get another homeomorphism. The self-homeomorphisms also satisfy some other properties which mean that they form a group under composition. However, this group is very big and quite nasty to study, so we usually consider two homeomorphisms to be the same if they are isotopic. This is quite a natural relationship between two homeomorphisms and roughly means that there is a nice continuous way of deforming one into the other. Now we have the set of all isotopy classes of orientation-preserving self-homeomorphisms of the surface, which we call mapping classes. These still form a group under composition – the mapping class group. This group is much nicer. It still (usually) has infinitely many elements, but now we can find a finite list of elements which form a generating set for the group. This means that every element of the group can be made by composing elements from this list. Groups with finite generating sets are often easier to study than groups which don’t have one.

An example of a mapping class group appears in Rachael’s post below. The braid group on n strands is the mapping class group of the disc with n punctures (where all homeomorphisms fix the boundary pointwise). Punctures are places where a point is removed from the surface. In some ways punctures are similar to boundary components, where an open disc is removed, but a mapping class can exchange punctures with other punctures.

So how can we study what a mapping class does? Rachael described in her post how we can study the braid group by looking at arcs on the punctured disc. Similarly, in the pictures above of examples of self-homeomorphisms the effect of the homeomorphism is indicated by a few coloured curves. More precisely, these are simple closed curves, which means they are loops which join up without any self-intersections. Suppose we are given a mapping class for a surface but not told which one it is. If we are told that it takes a certain curve to a certain other curve then we can start to narrow it down. If we get information about other curves we can narrow it down even more until eventually we know exactly what the mapping class is.

Now I can tell you a little about what I mainly think about in my research: the curve graph. In topology, a graph consists of a set of points – the vertices – with some pairs of vertices joined by edges.

graphexample

Each vertex in the curve graph represents an isotopy class of curves. As in the case of homeomorphisms, isotopy is a natural relationship between two curves, which more or less corresponds to pushing and pulling a curve into another curve without cutting it open. For example, the two green curves in the picture are isotopic, as are the two blue curves, but green and blue are not isotopic to each other.

isotopic

Also, we don’t quite want to use every isotopy class of curves. Curves that can be squashed down to a point (inessential) or into a boundary component (peripheral) don’t tell us very much, so we will ignore them. Here are a few examples of inessential and peripheral curves.

inessential

We now have infinitely many vertices, one for every isotopy class of essential, non-peripheral curves, and it is time to add edges. We put an edge between two vertices if they have representative curves which do not intersect. So if two curves from these isotopy classes cross each other we can pull one off the other by an isotopy. Here’s an example of some edges in the curve graph of the genus 2 surface. In the picture, all of the curves are intersecting minimally, so if they intersect here they cannot be isotoped to be disjoint.

I should emphasise that this is only a small subgraph of the curve graph of the genus 2 surface. Not only does the curve graph have infinitely many vertices, but it is also locally infinite – at each vertex, there are infinitely many edges going out! This isn’t too hard to see – if we take any vertex, this represents some curve (up to isotopy). If we cut along this curve we get either one or two smaller surfaces. These contain infinitely many isotopy classes of curves, none of which intersects the original curve.

So why is this graph useful? Well, as we noted above, we can record the effect of a mapping class by what it does to curves. Importantly, the property of whether two curves are disjoint is preserved by a mapping class. So not only does a mapping class take vertices of the curve graph (curves) to vertices, but it preserves whether or not two vertices are connected by an edge. Thus a mapping class gives us a map from the curve graph back to itself, where the vertices may be moved around but, if we ignore the labels, the graph is left looking the same. We say that the mapping class group has an isometric action on the curve graph, so to every element of the group we associate an isometry of the graph, which is a map which preserves distances between elements. The distance between two points in the graph is just the smallest number of edges we need to pass along to get from one to the other. When we have an isometric action of a group on a space, this is really useful for studying the geometry of the group, but that would be another story.

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.

A correspondence between braids and arcs

I’ve been thinking a lot about the braid group recently, and different ways of studying it. You can read my original post on the braid group here. The arcs I will talk about in this post have really taken over my PhD in the past few weeks, so much so that I’ve even started drawing them on beer mats in the pub!

There is a rather nice correspondence between braids in the braid group and arcs on a punctured disc (think pancake with a few little holes in the middle). In this post I will try and explain this correspondence in the case of the three strand braid group. In this instance we have to imagine a disc with 3 punctures (little holes):
punctured-disc

We will start of with the arc that corresponds to the identity braid, which we draw from the centre of the bottom of the disc to the left puncture.

identity-2

Below we show the identity arc on the left and the identity braid on the right.

identity

So what happens when we have a braid with a twist, like the one below? How does the arc have to change to incorporate information about this braid?

braid

Consider the braid: it has three strands and we have three punctures! The left strand crosses over the middle strand and we can incorporate this into the picture by starting with the identity arc and letting our left puncture and middle puncture rotate around each other to swap places, with the left one taking the upper path (so a clockwise rotation). We let the arc follow the puncture as though it is attached, making sure it does not intersect itself or the other punctures. This process is pictured in the two images below.

arc

single-braid

For every braid in the braid group (not just the simple ones with one crossing) an arc can be drawn. When you smoosh (see the braid group post) some simpler braids together to make a longer braid, you can draw the corresponding arc one step at a time, following the braid in a downwards direction to find out the next step you should take.

Below is an image of the arc changing as the braid grows longer. At the first crossing the arc moves over the left puncture as before but then there is a second crossing where now the middle strand of the braid moves under the right strand and therefore the middle puncture will rotate with the right one. This time the middle one moves under since the middle strand moves under on the braid, giving us an anticlockwise rotation.

time-series

Below is an example of a more complicated arc. If you want a fun challenge you could try to draw out the braid that it is related to!

twisty-arc

You might wonder what use it is to draw these arcs when we already know how to work with the braids. One reason is that we can learn more about certain braids by drawing the corresponding arcs and recognising patterns or algorithms that weren’t obvious from studying the braids. This technique is used quite a lot in mathematical proof, where an answer may be a lot simpler to see when the question is formulated in a different way (for example with arcs instead of braids).

I’ll end with a couple of pictures from my ‘everyday life’. Here is a wee set up I put together to help me figure out the twists and turns for some arc examples I was working on (a hard mornings work…)

string

And finally here is a shot of my current blackboard, a mess of arc drawing in progress!

blackboard

 

Hall of mirrors: Coxeter Groups and the Davis Complex

I’ve spent a lot of time this summer thinking about the best way to present maths. When someone gives a talk or lecture they normally write on the board in chalk or present via Beamer slides. Occasionally though, someone comes along with some great hand drawn slides that make listening to a talk that wee bit more exciting. So the images in this blog are part of my tribute to this new idea.

I’ve talked about Coxeter groups before (here), but I’ll start afresh for this post. It is worth mentioning now that Coxeter groups arise across maths, in areas such as combinatorics, geometric group theory, Lie theory etc. as well as topology.

A Coxeter group is a group generated by reflections, and “braid type” relations. Informally, you can imagine yourself standing in the middle of a room with lots of mirrors around you, angled differently. Your reflection in a single mirror can be viewed as a generator of the group, and any other reflection through a series of mirrors can be viewed as a word in the group. Here is a silly picture to show this:

silly picture

Formally, a Coxeter group is defined by a Coxeter matrix on the generating set S. This is an S by S matrix, with one entry for each ordered pair in the generating set. This entry has to be 1 on the diagonal of the matrix or a whole number that is either bigger than 2 or infinity (\infty ) off the diagonal. It also has to be symmetric, having the same entry for (t,s) as (s,t) . See below:

coxeter groups1

Given this matrix you can then define the Coxeter group to be the group generated by S with relations given by the corresponding entry in the matrix.

coxeter groups2

In particular notice that the diagonal of the matrix being 1 gives that each generator squares to the identity, i.e. it is an involution. It can be hard to see what is going on in the matrix so there is a nicer way to display this information: a Coxeter diagram. This is a graph with a vertex for every generator, and edges which tell you the relations, as described below:

coxeter groups3

The relation (st)^{m_{st}} can also be rewritten as ststs...=tstst... where there are m_{st} elements on each side. This is reminiscent of the braid relations, hence why I called them “braid like” before. In the mirror analogy, this says the mirrors are angled towards each other in such a way that you get the same reflection by bouncing between them m times, independent of which of the two mirrors you turn towards to begin with.

There exist both finite and infinite Coxeter groups. Here is an example of a finite Coxeter group, with two generators s and t. If you view them as reflections on a hexagon (as drawn) then doing first s and then t gives a rotation of 120 degrees, and so doing st 3 times gives the identity, as required.

coxeter example

On the other hand, if you add another generator u with a braid-3 relation with both s and t , then the group is infinite. You can imagine tiling the infinite plane with triangles. If you take s , t and u to be reflections in the 3 sides of one of these triangles then they satisfy the relations they need to, and you can use these three reflections to transport the central triangle to any other one. If you think about this for a while, this shows the group is infinite. A (somewhat truncated) picture is shown below.

coxeter example 1

Examples of Coxeter groups don’t just live in 2-D Euclidean space. There is another finite group which acts as reflections on the permutahedron:

coxeter-complex

And other Coxeter groups which act as reflections on the hyperbolic plane.

The mathematical object I am working with at the moment is called the Davis Complex. You can build it out of finite subgroups of a Coxeter group (side note for the mathematicians: taking cosets of finite subgroups and forming a poset which you can then realise). Even infinite Coxeter groups have lots of finite subgroups. The great thing about the Davis complex being built out of finite things is that there is a classification of finite Coxeter groups! What this means is that when you have a finite Coxeter group its diagram either looks like one of the diagrams below, or a disjoint collection of them.

classification of finite coxeter

So because we only have a few diagrams to look at in the finite case, we can prove some things! Right now I am working on some formulas for staring at the Coxeter diagrams and working out the homology of the group. I’m using the Davis complex and all its nice properties to do this. I’ll leave you with a picture of the Davis complex for our first example.

davis example 2

davis example

Deleting edges to save cows – guest post by Dr Kitty Meeks

many-trades

I met Rachael at the LMS Women in Mathematics Day in Edinburgh in April, where I gave a talk about using graph theory to understand the spread of disease in livestock, and she invited me to write a blog on this topic.  Everything I’m going to say here is based on joint work with Dr Jess Enright from the University of Stirling (supported by Scottish Government as part of EPIC: Scotland’s Centre of Expertise on Animal Disease Outbreaks), who knows a lot more about cattle than I do!

It seems obvious that, when we want to understand how a disease spreads through a population, we should consider which members of the population have contact with each other – this simple observation has given rise to the field of Network Epidemiology.  If we’re trying to understand the spread of disease in humans, one of the key challenges is to guess the underlying contact network – who has contact with whom – and a lot of interesting work has been done on this.  Looking at the spread of disease in livestock, however, we don’t have to worry about this: all movements of farm animals have to be reported to the relevant government agencies, so we (well, people I collaborate with, who provide advice to the Scottish government on controlling animal disease outbreaks) know exactly which animals have contact with each other.

Depending on the specific disease we are interested in, there are several different kinds of contact we might need to consider.   If animals need to be in close contact to transmit the disease, then there are two main ways the disease could spread from one farm to another: it could spread between farms that are geographically adjacent (if cows on opposite sides of the fence decide to have a chat) or by trade when an animal is sold from an infected farm to another farm that is not yet infected.  Some other diseases are spread by biting insects, in which case any farm within a certain radius of an infected farm would be at risk.

Mathematically, we can represent these potentially infectious contacts by means of a graph: a vertex is used to represent each farm, and two farms are connected by an edge if there is the potential for the disease to spread from one to the other (in the case of infection due to trade, this would normally be one-way, so we could capture more information by using a directed edge in the direction of the trade).  My first picture illustrates what part of this graph might look like if we are just considering the spread of disease across shared fence lines.  The graph we get in this situation will normally be planar, that is it can be drawn in the plane without any edges crossing (although there are some circumstances where this is not necessarily true, for example if one farmer owns several fields that are not connected to each other).

farms-planar

Now we can add to this graph the contacts that come from trade.  It’s not so obvious what we would expect the graph of trade links to look like, but this is exactly what Jess and I have been trying to understand.

many-trades

First, though, I should explain why we want to know about the structural properties of these contact graphs.  We’re interested in restricting the spread of disease, and Jess and her colleagues are asked by policy makers to investigate how it might be possible, in theory, to modify the network to make it less vulnerable to an epidemic.  When we talk about modifying the network, we’re considering actions that correspond roughly to deleting edges and/or vertices in the graph: to “delete” a vertex, we might vaccinate all animals at the particular farm (so that it can no longer play any role in the transmission of disease), whereas “deleting” an edge might correspond to putting up a double fence-line between two farms (keeping animals on adjacent farms just a little bit further apart) or introducing extra monitoring on a specific trade route.  What do we want to achieve by modifying the network?  Well, there are lots of different parameters we could look at which relate to how vulnerable a particular graph is to an epidemic, but the simplest one we could think of to start with is the maximum component size (the largest number of vertices that can be reached from any single starting vertex in the graph by traveling along edges).  In the picture below, if the farm circled in red is infected, then all of the others are potentially at risk.

cow-network-1

However, if we delete one edge from this network, we can be sure that the disease will not spread to the three farms on the right hand side (now highlighted in green).

cow-network-2

Of course, all of these possible ways to modify the network are costly, so we want to do this in the most efficient way possible, i.e. make the fewest modifications required to achieve our goal.  This motivates the following question: given a graph, what is the minimum number of edges (or vertices) we need to delete so that the maximum components size of the resulting graph is at most h?  This is an example of a graph modification problem, a kind of problem which comes up a lot on theoretical computer science.  It turns out that, like many graph modification problems, this particular problem is NP-complete: unless most mathematicians and computer scientists are wrong, and P=NP, there is no efficient algorithm to answer this question for all possible inputs.  But we don’t have to give up there!  Just because a problem is intractable on arbitrary inputs doesn’t mean we can’t find a fast way to solve it on input graphs with specific structural properties – most of my research is based on exactly this idea – so in order to solve the problem efficiently on the real data we want to understand the structure of the livestock contact networks.

Jess and I haven’t yet considered the general problem as I described it above: instead, we’ve been looking only at the trade network, and only at edge deletion.  This special case is relevant in the management of a particular kind of persistent cattle trade link that exists in Scotland, so it seemed as good a place to start as any.  Thinking about this problem, we did what any graph theorist does when faced with a computationally hard problem: we tried to solve it on trees (graphs that don’t have any cycles, so there’s exactly one way to get between any two vertices).  And, after dealing with a few slightly awkward details, it worked in exactly the way any graph theorist would expect: we can solve the problem recursively, starting with the leaves (vertices with only one neighbour) and working our way up the tree – the fact there are no cycles means we never have to rethink the answers we’ve already calculated.

trees

But cattle trade networks aren’t trees, so is this any use in the real world?  Somewhat surprisingly, the answer seems to be yes!  Although the trade networks aren’t exactly trees, many of them seem to have small treewidth – this is a hard-to-define graph parameter that essentially captures how “tree-like” a particular graph is.  And the great thing about graphs of small treewidth is that algorithms that work on trees can usually be adapted (taking just a bit more time) to work on graphs of small treewidth in exactly the same way.  The next picture shows how the treewidth of the Scottish cattle trade network grows as we add in more trades, based on real data from 2009.  The trades that took place that year are added to the graph one day at a time, and the treewidth (in fact, for technical reasons, an upper bound on the treewidth) of the resulting graph is shown.

2009_data

We can see from this graph that, even if we look at trades taking place in the whole year (much longer than the timescale we would need to consider for many disease outbreaks) the treewidth of the trade network is still less than 20, for a graph that has nearly 3000 vertices and 8000 edges – this is much lower than we would expect for a random graph with the same number of vertices and edges.

Initial experiments suggest that our algorithm that exploits the treewidth performs significantly better on some real datasets than a generic approach based on constraint programming (which doesn’t take into account any structural properties of the data): there were examples where our algorithm found the optimal solution in under a minute, whereas the alternative couldn’t solve the problem in 6 hours.

We’ve really only scratched the surface of this problem, though.  The planar graphs that arise from geographical proximity definitely don’t have small treewidth, so we will need a different approach here – but they certainly have other distinctive structural properties that we hope to exploit.  What interests me most to investigate next, though, is why many cattle trade networks have surprisingly low treewidth: farmers aren’t looking at the horrible definition of this parameter when they decide where to sell their animals, so there’s something else going on here – and if we could find a good way to model this “something else” mathematically, would it let us devise even better algorithms?

SIAGA: Polyhedral Geometry

Seven pictures from Applied Algebra and Geometry: Picture #3

The Society for Industrial and Applied Mathematics, SIAM, has recently released a journal of Applied Algebra and Geometry called SIAGA. The poster for the journal features seven pictures. In this blog post I will talk about the third picture. See here for my blog posts on pictures one and two. And see here for more information on the new journal.

In the first section of this post “The Context”, I’ll set the mathematical scene and in the second section “The Picture” I’ll talk about this particular image, representing Polyhedral Geometry.

Fig3big

The Context

A polyhedron is a combinatorial object that crops up in many places. For example, it is the shape of the feasible region for a linear optimization problem. It is a convex shape in d-dimensional space \mathbb{R}^d described by an intersection of finitely many closed half-spaces P = \{ x \in \mathbb{R}^n : Ax \leq b \}
where A is a d \times n matrix describing the angles of the half spaces, and b \in \mathbb{R}^d encodes their translational information.

A compatible collection of polyhedra is called a polyhedral complex. It is possible to associate a polyhedral complex to an algebraic variety, allowing us to use combinatorial tools to understand the variety. A typical way to do this is via the methods of toric geometry. This approach has been applied in areas such as phylogenetics, integer programming, economics, biochemical reaction networks and computer vision (from where this particular picture arose).

Tropical geometry gives one method, called tropicalization, for getting a polyhedral complex from an algebraic variety. The new object gives us useful information. For example, the dimension of the original variety equals that of the new polyhedral complex and the latter is much easier to compute.

Polyhedral geometry also arises at the interface with the life sciences. For instance, Gheorghe Craciun recently announced a proof of the Global Attractor Conjecture for toric dynamical systems. This result is important for systems biology, and polyhedral fans play a key role in the proof.

The Picture

This picture describes one example of using polyhedral tools to understand an algebraic variety, this shedding light on the application from which the variety arose.

In the field of computer vision, and in the real world, ‘taking a photo’ is a map from the three-dimensional world to a two-dimensional world. As any good photographer knows, the resulting features of the photo depends heavily on the angle and location of the camera.

We photograph three-dimensional projective space \mathbb{P}^3. Each camera, A, is a 3 \times 4 matrix which determines a map \mathbf{x} \mapsto A\mathbf{x} to two-dimensional projective space \mathbb{P}^2. This map tells us where each point of the original world ends up in the photograph.

More information can be gained by considering multiple cameras ( A_1 , A_2, A_3) at different locations. Then we have a map from the real world to three photographs:
\phi : \mathbb{P}^3 \dashrightarrow (\mathbb{P}^2)^3
\mathbf{x} \longmapsto (A_1 \mathbf{x}, A_2 \mathbf{x}, A_3 \mathbf{x} ).

The closure of the image of this map is an irreducible variety. For example, if the A_i are the coordinate projections, the variety in (\mathbb{P}^2)^3 is cut-out by the Gröbner basis
\{ z_0 y_2 - x_0 z_2 , z_1 x_2 - x_1 z_2, z_0 y_1 - y_0 z_1, x_0 y_1 x_2 - y_0 x_1 y_2 \} \text{ .}
When we take the initial monomials of these generators, their zero-set decomposes into seven pieces: one copy of \mathbb{P}^1 \times \mathbb{P}^1 \times\mathbb{P}^1, and six copies of \mathbb{P}^1 \times\mathbb{P}^2.

Our picture shows the three-dimensional shape we get from this zero-set when we identify each projective space \mathbb{P}^i with the i-simplex. For example, \mathbb{P}^2 corresponds to the two-dimensional simplex \Delta_2 – the triangle – under the map:

\mathbb{P}^2 \ni (x_0:x_1:x_2) \longleftrightarrow \frac{1}{x_0 + x_1 + x_2} (x_0 , x_1, x_2) \in \Delta_2

and we identify \mathbb{P}^1 with the one-dimensional simplex \Delta_1 – the unit-length line.

Our zero-set is represented by a collection of polytopes which are faces of (\Delta_2)^3. The \mathbb{P}^1 \times\mathbb{P}^1 \times\mathbb{P}^1 corresponds to \Delta_1 \times \Delta_1 \times \Delta_1. This is the dark blue cube. Each of the six pieces \mathbb{P}^2 \times\mathbb{P}^1 correspond to \Delta_2 \times \Delta_1. This gives the six triangular prisms in the picture.

Each piece has been separated a little to make it easier to see, and the close-by parallel faces show how the different parts meet. Meeting at a triangle \Delta_2 means the projective spaces meet at a copy of \mathbb{P}^2. If the shared facet is a square \Delta_1 \times \Delta_1, the projective spaces meet at a copy of \mathbb{P}^1 \times\mathbb{P}^1. The original picture, and other nice ones, can be found in the paper “A Hilbert Scheme in Computer Vision” by Aholt, Sturmfels and Thomas (2013). It was made using Michael Joswig’s software Polymake.