On a tangential note, let me point out John Baez’ most recent “This Week’s Finds”, which has an accessible but fairly in-depth discussion of climate modelling.  There have been many years of very loud public discussion of this which, for reasons of politics, seems to involve putting the “Mathematical models are inherently elitist gibberish” and “Science knows everything so shut up, moron” positions on display and letting viewer decide.  This is known in the journalism trade as “balance”.  Obviously, within the research community working on them, there’s a mountain of literature on what the models model, how detailed they are, how they work, etc., but it mostly goes over my head, so John’s post strikes a nice balance for me.

Like most computer simulation models, they’re basically discrete approximations to big systems of differential equations – but exactly which systems, how they’re developed, how accurately they model the real thing, and the relative merits of simple vs. complex models is the main point.  The use of Monte Carlo methods and Bayesian analysis to tune the various free parameters is a key part of the matter of how accurate they should be.  Anyway – check it out.

Meanwhile, the TQFT club at IST recently started up its series of seminars.  The first few speakers were Rui Carpentier, Anne-Laure Thiel, and Marco Mackaay.  Rui is faculty here at IST, and a former student of Roger Picken (his thesis was on a topic closely related to what he was talking about).  Anne-Laure is a post-doc here at IST, mainly working with Marco, who, however, is actually at the University of the Algarve in Faro, Portugal, and had to come up to Lisbon specially for the seminar.  Anne-Laure and Marco were both speaking mainly about some of the Soergel bimodule stuff which came up at the Oporto meeting on categorification, which I posted about previously, so I’ll go over that in a bit more detail here.

First, though, Rui Carpentier’s talk:

## 3-colourings of Cubic Graphs and Operators

All these talks involve algebraic representations of categories that can be represented by some graphical calculus, but in this case, one starts with a category whose morphisms are precisely graphs with loose ends.  (The objects are non-negative integers, or, if you like, finite sets of dots which act as the vertices of the loose ends).  The graphs are trivalent (except at the input and output vertices, which are 1-valent), hence “cubic graphs”.  This category is therefore called $\mathbf{CG}$, and it has a small number of generators, which happen to be quite similar to those which generate the category of 2D-cobordisms (one of the connections to TQFT), though the relations are slightly different.

Roughly, and without drawing the pictures: the generators are cup and cap (the shapes $\cup$ and $\cap$), two different trivalent vertices (a $Y$, and the same upside-down), the swap (an $X$ where the strands cross without a vertex), and the identity (just a vertical line).  There are a number of relations, including Reidemeister moves, on these generating pictures, which ensure that they’re enough to identify graphs up to isotopy of the pictures.

Then the point is to describe graphs using operators – that is, construct a representation $F :\mathbf{CG} \rightarrow \mathbf{Vect}$.   Given any such representation, these generators provide all the structure maps of a bialgebra – chiefly, unit, counit, multiplication and co-multiplication – and the relations imposed by isotopy make this work (though unlike some other situations, it’s neither commutative nor cocommutative).  The representation $F$ he constructs is based on 3-colourings of the edges of the graphs.  At the object level, it assigns to a dot the 3-dimensional vector space $V= span(e_1,e_2,e_3)$.  Being monoidal, $F$ takes the object $n$ to $V^{\otimes n}$ – the tensor product of the spaces at each vertex.

The idea is that choosing a basis vector in this space amounts to picking a colouring of the incoming and outgoing edges.  For morphisms, we should note that the rule that says when a colouring is admissible is that all the edges incident to a given vertex must have different colours.  Then, given a morphism (graph) $G : m \rightarrow n$, we can describe the linear map $F(G)$ most easily by saying that the component in the matrix, given an incoming and outgoing basis vector, just counts the number of admissible graphs that agree with the chosen colourings on the in-edges and out-edges.

There’s another functor, $\hat{F}$, which counts these graphs with a sign, which marks whether the graph contains an odd or an even number of crossings of differently-coloured edges – negative for odd, positive for even.  This  is the “Penrose evaluation” of the graph.

So these maps give the “operators” of the title, and the rest of the point is to use them to study graphs and their colourings.  One can, in this setup, rewrite some graphs as linear combinations of others – so-called “Skein relations” hold, for example, so that, after applying $F$, the composite of multiplication and comultiplication (taking two points to two points, through one cut-edge) is the same as the identity minus the swap.  This sort of thing appears in formal knot theory all the time, and is a key tool for recoupling in spin networks, and so on…

Given this “recoupling” idea, there are some important facts: first, any graph can be rewritten as a linear combination of planar graphs, and any planar graph with cycles can be reduced to a sum of planar graphs without cycles.  (Rui gave the example of decomposing a pentagonal cycle as a linear combination of four other graphs, three of which are disconnected).  So in fact any graph decomposes as a linear combination of forests (cycle-free graphs, the connected components of which are called “trees”, hence the name).  Another essential fact is that, due to the Euler characteristic of the plane, any planar graph can be split into two parts with at most five edges between them (the basis of the solution to the three utilities puzzle).  Then it so happens that the space of graphs connecting zero in-edges to five out-edges is a 6-dimensional space, $\mathcal{V}^o_5$, generated by just six forests (including one lonesome tree).

So one theorem which Rui told us about, which can be shown using the so-called Penrose relations (provable using the representations $F$ and $\hat{F}$), is that there’s just one such graph (which he described in the particular basis above) that evaluates to zero when composed with some other graph.  The proof of this uses the Four Colour Theorem (3-colouring of graph edges being related to 4-colouring of planar regions); in fact, the two theorems are equivalent so if anyone can find an alternative proof of this one, the bonus is another proof of the FCT.

Finally, he gave a conjecture that, if true, would help recognize planar graphs just by the operators produced by the representation $\hat{F}$ (at least it proposes a necessary condition).  This conjecture says that if a planar graph with five output edges (the maximum, remember) is written in the basis mentioned above, then the sum of the coefficients of the five disconnected trees is nonnegative.  (Thus, the connected tree doesn’t contribute to this measure).  This is still just a conjecture – Rui said that to date neither proof nor counterexample has been found.

## Soergel Bimodules, Singular and Virtual Braids

As I mentioned up top, I previously posted a bit about work on Soergel bimodules when describing Catharina Stroppel’s talk at the meeting in Faro in July.  To recap: they are associated with categories of modules over rings – specifically, rings of certain classes of symmetric functions.  Even more specifically, given a partition $\lambda$ of an integer $n$, there is a subgroup of the symmetric group $S_{\lambda} \subset S_n$ which fixes the partition.  All such groups act on the ring of $n$-variable polynomial functions $R =\mathbb{Q}[x_1, \dots, x_n]$, and the ones fixed by $S_{\lambda}$ form the ring $R^{\lambda}$.

Now, these groups are all related to each other in a web of containments, hence so are the rings.  So the module categories $R^{\lambda}$ are connected by various functors.  Given a containment $R^{\lambda '} \subset R^{\lambda}$, modules over $R^{\lambda}$ can be restricted to ones over $R^{\lambda '}$, and modules over $R^{\lambda '}$ can be induced up to ones over $R^{\lambda}$.  The restriction and induction functors can be represented as “tensor with a bimodule” (this is much the same classification as that for 2-linear maps which I’ve said a bunch about here, except that those must be free).  Applying induction functors repeatedly gives abitrarily large bimodules, but they are built as direct sums of simple parts.  Those simple parts, and any direct sums of them, are Soergel bimodules.  The point is that such bimodules describe morphisms.

So in the TQFT club, Marco Mackaay gave the first of a series of survey talks on this topic, and Anne-Laure Thiel gave a talk about the “Categorification of Singular Braid Monoids and Virtual Braid Groups”.  Since Marco’s talk was the first in a series of surveys, and a lot of what it surveyed was work described in my post on the Faro meeting, I’ll just mention that it dealt with the original motivation of a lot of this work in categorifying representation theory of Lie algebras (c.f. the discussion of the Khovanov-Lauda categorification of quantum groups in the previous post), and also got a bit into some of the different diagrammatic calculi created for that purpose, along the lines of the talks by Ben Webster and Geordie Williamson at that meeting.  Maybe when Marco has given more of these talks, I’ll return to this one here as well.

Now, the starting point of Anne-Laure’s talk was that the setup above lets one define a category with a presentation like that of the Hecke algebra (a quotient of the group algebra of the braid group), where exact relations become isomorphisms.  That is, we go from a category where morphisms are braids (up to isotopy and Reidemeister moves and so forth as usual) to a 2-category where the morphisms are bimodules, which happen to satisfy the same relations.  (The 2-morphisms, bimodule maps, are what allow relations to hold weakly…)

Specifically, the generators of the braid group are $\sigma_i$, the braids taking the $i^{th}$ strand over the $(i+1)^{st}$.  The parallel thing is $B_i = R \otimes_{R^{\sigma_i}} R$, where here we’re talking about the subgroup generated by the transposition of $i$ and $i+1$.  In the language of partitions, this corresponds to a $\lambda$ with one part of size two, $(i,i+1)$, and the rest of size one.  Now, since this bimodule is actually built from polynomials in $R$, it naturally has a grading – this corresponds to the degree of $q$, since the Hecke algebra involves a quotient giving q-deformed relations – so there is a degree-shift operation that categorifies multiplication by $q$.  This much is due to Soergel.

Anne-Laure’s talk was about extending this to talk about a categorification, first of the braid group in terms of complexes of these bimodules (due actually to Rouquier), then virtual and singular braids.  These, again, are basically creatures of formal knot theory (see link above).  They can be described by a presentation similar to that for braids – just as the braid group has a generators-and-relations presentation in terms of over-crossings of adjacent strands, these incorporate other kinds of crossings.  Singular braids allow a sort of “through” crossing, where the $i^{th}$ strand goes neither over nor under the $(i+1)^{st}$.  Virtual braids (the braid variant on virtual knots) have a special type of marked crossing called the “virtual crossing”, drawn with a little circle around it.  These are included as new generators in describing the virtual braid group, and of course some new relations are added to show how they relate to the original generators – variations on the Reidemeister moves, for example.

To categorify this, Anne-Laure explained that these new generators can also be represented by bimodules, but these ones need to be twisted.  In particular, twisting the bimodule $R$ by the action of a permutation $\omega \in S_n$ gives $R_{\omega}$, which is the same as $R$ as a left $R$-module, but is acted on by an element $a \in R$ on the right through multiplication by $\omega(a)$, so that $b \cdot p \cdot a = bp(\omega(a))$.  Then the new generators, beyond the $B_i = R \otimes_{R^{\sigma_i}} R$, are of the form $R_{\omega} \otimes_{R^{\omega '}} R$.  These then satsify the right relations for this to categorify the virtual braid group.