## August 22, 2007

### Justificatory Narratives

#### Posted by David Corfield

Eighteen months ago, back at my old blog, I discussed a paper by Robert W. Batterman, On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician). It has now appeared in the latest edition of the British journal of the Philosophy of Science, so I’ll copy the post to the Café. As you will see, what bothers me is a readiness to delineate sharply justificatory narratives of pieces of mathematics into two classes: ‘mathematics’ and ‘physics’. Also, I find it unfair that we get to hear much more up-to-data accounts from the physics side than from the mathematics side.

I’ve just read an interesting paper by Robert W. Batterman - On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician). Here’s an extract:

…it seems reasonable to maintain that many special functions are special for more than simple pragmatic reasons. They are not special simply because they appear in the physicist’s, applied mathematician’s, and engineer’s toolboxes. Furthermore, special functions are not special simply because they share some deep mathematical properties. Recall this is the point of view of Truesdell and of Talman/Wigner. On their proposals, what makes some functions special is that despite “surface” differences, they are each solutions to the “F-equation” (for Truesdell), or they possess similar group representations (for Talman and Wigner). While these classificatory schemes suffice to bring some order to the effusions of the Divine Mathematician, they do not fully capture the special nature of the special functions.

From the point of view presented here, the shared mathematical features that serve to unify the special functions–the universal form of their asymptotic expansions–depend upon certain features of the world. What Truesdell, Talman and many others miss is how the world informs and determines the relevant mathematical properties that unify the diverse special functions.

As I noted, in many investigations of physical phenomena we find dominant physical features–those features that constrain or shape the phenomena. These are things like shocks and the highly intense light appearing in the neighborhood of ray theoretic caustics. They are features that are most effectively modeled by taking limits.

Limiting idealizations are most effective for examining what goes on at places where the “laws” break down–that is, at places of singularities in the governing equations of the phenomena. These “physical” singularities and their “effects”–how they dominate the observed phenomena–are themselves best investigated through asymptotic representations of the solutions to the relevant governing equations. The example of the Airy integral is a case in point. By using Stokes’ asymptotic representation we get superb representations of the nature of the diffraction relatively far (large z) from the dominating “physical” singularity–the caustic.

There’s a curious tendency when it comes to discussing the applicability of mathematics to polarise one’s response to the question of which of mathematics and science owes the other the most. It seems each side is only too happy to exaggerate the role of their favoured discipline, finding the benefits mathematics bestows as miraculous, or deflating mathematics to a bunch of tautologies which physics is gracious enough to give an interpretation to. In the philosophical literature of the past century, with one or two notable exceptions, the latter attitude has been prevalent. Of the two, it is the role of mathematical understanding which tends to get passed over or taken for granted.

Batterman is more sensitive to the narratives various mathematicians and physicists tell about special functions, but I wonder if there might not be a richer mathematical story which would make the still sharp dichotomy he maintains between physical and mathematical considerations less sharp. Is it not possible that the richer mathematical understanding of special functions gained since Talman (1968) could intersect with the physical considerations treated by Batterman? For instance, might the asympotic results he discusses have something to do with current ideas from the group representation understanding of special functions? And how do q-deformed special functions fit in? It’s surely common to find mathematics and physics narratives creatively intertwined.

Batterman concludes:

I hope that the discussion here leads us to question the anthropocentric role of the mathematician’s appreciation for beauty (or formal analogy) as an important criterion for what arguably should be paradigm examples of mathematics’ applicability to the world; namely, the special functions.

I’d like to hear some contemporary mathematicians’ aesthetic narratives about special functions.

Posted at August 22, 2007 9:59 AM UTC

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### Re: Justificatory Narratives

It’s because academia is a Darwinian struggle for resources. Only one department can win, and by God it will be us!

Posted by: Walt on August 22, 2007 4:35 PM | Permalink | Reply to this

### Re: Justificatory Narratives

I feel I should say something, but I’m not very excited about special functions these days. I’m not sure why. I guess it’s just that none of the questions I’m most fascinated by now involves special functions. This hasn’t always been the case — it’s just the way it is right now.

Well, okay, I must admit that recently I was excited when I learned that the ‘hypergeometric differential equation’ is the answer to one of the simplest Riemann–Hilbert problems. I never really liked hypergeometric functions until I realized this and realized that Riemann-Hilbert problems are really interesting!

So, let me talk about this a little. In a Riemann–Hilbert problem you seek a linear differential equation on some Riemann surface, with only regular singular points, and with given monodromies around these points.

If it helps — and for me it really does! — you can think of this differential equation as specifying a flat connection on some holomorphic vector bundle over the Riemann surface with some points removed. The goal is then to find a connection of this sort that gives specified holonomies around these ‘punctures’.

Note: ‘monodromy’ and ‘holonomy’ mean approximately the same thing, though people in differential equations like to say ‘monodromy’, while people in gauge theory like to say ‘holonomy’. I guess people also tend to limit ‘monodromy’ to the case of connections that are flat except at certain singular points: ‘branch points’, or ‘punctures’.

As far as I’m concerned, the Riemann–Hilbert problem is fun because it’s a question about gauge theory. If our punctured surface is $S$, then the holonomies (or monodromies) of our differential equation give some homomorphism

$\rho :{\pi }_{1}\left(S\right)\to \mathrm{GL}\left(n,ℂ\right)$

if the equation has $n$ linearly independent solutions. Can you get any homomorphism you want from some linear differential equation? That’s the Riemann–Hilbert problem.

To make things really simple, suppose you take your surface to be the Riemann sphere $ℂ{P}^{1}$, and remove 3 points. Then it doesn’t really matter where these punctures are, because there’s a conformal transformation mapping any 3 distinct points to any 3 others. For 4 points this ain’t true, so things get more complicated! But for 3, we might as well take our punctures to lie at $0$, $1$, and $\infty$ — the three coolest numbers ever invented. Then, it turns out we can solve the Riemann-Hilbert problem: we can get any holonomy you want:

$\rho :{\pi }_{1}\left(ℂ{P}^{1}-\left\{0,1,\infty \right\}\right)\to \mathrm{GL}\left(2,ℂ\right)$

from a suitable 2nd-order linear differential equation. And, this is the hypergeometric equation!

As far as I can tell, the fun really gets started when we think about this stuff using $D$-modules. In fact it was in my (so far rather preliminary) attempts to learn about $D$-modules that I learned all this stuff.

So, as far as ‘justificatory narratives’ go, I justify my (extremely limited) interest in hypergeometric functions through their relation to ideas I’m currently interested in, like gauge theory, flat bundles, and $D$-modules. But of course, other people have wholly different reasons for being interested in hypergeometric functions. I feel my understanding of them is very shallow.

Posted by: John Baez on August 25, 2007 9:35 PM | Permalink | Reply to this

### Re: Justificatory Narratives

$\rho :{\pi }_{1}\left(S\right)\to \mathrm{GL}\left(n,ℂ\right)$

So that’s about representations of the fundamental group, right? Will your categorified gauge theory allow you up the ladder, like to 2-representations of the fundamental 2-groupoid of some space? Then you could pose the Baez-Riemann-Hilbert problem to ask if any 2-rep comes from solutions of some, hmm what is a categorified linear differential equation?

Posted by: David Corfield on August 26, 2007 8:26 AM | Permalink | Reply to this

### Re: Justificatory Narratives

Ah, the store of knowledge that is this blog. We already have some exposition about how

a local system on a manifold M is a representation of ${\pi }_{1}\left(M\right)$ (or perhaps on this post that should be the fundamental 1-category).

and its generalisation as constructible sheaves on stratified spaces.

a representation of the fundamental n-category of a stratified space.

Posted by: David Corfield on August 26, 2007 12:11 PM | Permalink | Reply to this

### Re: Justificatory Narratives

David writes:

$\rho :{\pi }_{1}\left(S,*\right)\to \mathrm{GL}\left(n,ℂ\right)$

So that’s about representations of the fundamental group, right?

Right! I think we can rephrase a certain version of the Riemann–Hilbert problem as asking whether, for any representation of the fundamental group of a punctured Riemann surface, we can find a linear differential equation

${a}_{n}\left(z\right)\frac{{d}^{n}f}{d{z}^{n}}\phantom{\rule{thickmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}\cdots \phantom{\rule{thickmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}{a}_{1}\left(z\right)\frac{df}{dz}+{a}_{0}\left(z\right)f=0$

with holomorphic coefficients ${a}_{i}\left(z\right)$, such that the monodromies of the solutions realize this representation.

And the answer, by the way, is yes.

There are also versions in higher dimensions.

Will your categorified gauge theory allow you up the ladder, like to 2-representations of the fundamental 2-groupoid of some space?

Maybe so! Certainly Urs and I know what a ‘flat 2-connection’, and the holonomies of such a thing define a homomorphism

$\rho :{\Pi }_{2}\left(M,*\right)\to G$

from the fundamental 2-group of the base manifold $M$ to the gauge 2-group $G$.

We’re a bit less advanced when it comes to understanding ‘2-vector 2-bundles’, so we could replace $G$ by $\mathrm{GL}\left(V\right)$ for some ‘2-vector space’ $V$. Actually this isn’t so hard formally — we can always use Baez–Crans 2-vector spaces to define 2-vector 2-bundles, and we know what it means to put 2-connections on these. But, we’d need to work with some examples to get a feel for what these things are actually like.

And, I think we reach some really unexplored territory when we ask the question analogous to the Riemann–Hilbert problem.

Then you could pose the Baez–Riemann–Hilbert problem to ask if any 2-rep comes from solutions of some, hmm what is a categorified linear differential equation?

Good question. Solve it, and we’ll be talking about the ‘Corfield–Riemann–Hilbert problem’.

Posted by: John Baez on August 27, 2007 10:57 AM | Permalink | Reply to this

### Re: Justificatory Narratives

So in higher dimensions the problem’s called the Riemann-Hilbert correspondence, where you move to the monodromies of the solutions of partial differential equations on a complex manifold. I wonder how much of the homotopy of the manifold is at stake.

Posted by: David Corfield on August 29, 2007 9:00 AM | Permalink | Reply to this

### Re: Justificatory Narratives

I’m not very excited about special functions these days

Who can put the ‘special’ back into ‘special functions’ for John?

Will Michael Berry succeed or Walter Van Assche?

Gelfand has shown that many special functions such as Bessel and Whittaker functions, Jacobi and Legendre polynomials appear as matrix coefficients of irreducible representations. This interpretation of special functions immediately explains the functional and differential equations for these functions. It is clear now that [almost] all special functions studied in the 19-20 centuries can be interpreted as matrix coefficients or traces of representations of groups or their quantum analogs (e.g., works of Tsuchiya-Kanie, I. Frenkel-Reshetikhin on the representation theoretic interpretation of hypergeometric (respectively q-hypergeometric) functions, works of Koornwinder, Koelink, Noumi, Rosengren, Stokman, Sugitani and others on representation theoretic interpretation of Askey-Wilson, Macdonald, and Koornwinder polynomials). (Works of I. Gelfand on the theory of representations, Kazhdan)

Posted by: David Corfield on August 26, 2007 8:52 AM | Permalink | Reply to this

### Re: Justificatory Narratives

Thanks for the link which describes Gelfand’s work on representation theory.

Some parts which really interested me were:

Gelfand believed that the space $\stackrel{^}{G}$ of irreducible representations of (a locally compact Lie group) $G$ is a reasonable “classical space”….

… it was natural to guess that the total space $\stackrel{^}{G}$ is also an algebraic variety.

Could someone could explain the basic results here to me, in layman’s terms? I’m going to expose my shocking ignorance here and admit I only know the basics about representations of compact Lie groups, and my algebraic geometry is pretty shoddy.

The reason I’d like to know is that when one works with 2-representations of Lie groups, it becomes useful to think of the space of irreducible representations as a geometric space. At least, that’s the case for finite groups… and it’s probably also the case for locally compact Lie groups; after all, this seems to be in the spirit of Yetter’s Measurable Categories (though I haven’t grasped this paper yet).

So I’m interested in this theme of thinking of $\stackrel{^}{G}$ as a geometric space. I know this is really a huge theme and I have no hope of understanding the serious stuff… but just a glimpse? Does $\stackrel{^}{G}$ form a nice algebraic variety? What’s a nice example?

Posted by: Bruce Bartlett on August 26, 2007 7:15 PM | Permalink | Reply to this

### Re: Justificatory Narratives

Bruce wrote:

Could someone could explain the basic results here to me, in layman’s terms? I’m going to expose my shocking ignorance here…

I’m shockingly ignorant too. The buzzword you want is “Plancherel formula”. A very readable introduction is here.

Briefly, if the locally compact group $G$ has a Haar measure that’s invariant under both left and right translations, we say it’s unimodular.

It then makes sense to look at the space $\stackrel{^}{G}$ of all irreducible unitary representations of $G$, and ask if there’s a measure on $\stackrel{^}{G}$, the Plancherel measure $\nu$, which lets us write down this nonabelian generalization of a formula familiar from Fourier theory:

${\int }_{G}\mid f\left(g\right){\mid }^{2}dg={\int }_{\stackrel{^}{G}}\parallel \stackrel{^}{f}\left(\pi \right){\parallel }^{2}d\nu \left(\pi \right)$

(For details of what this formula actually means, read the reference I just gave.)

Murray and von Neumann discovered that such a measure doesn’t always exist! However, if if $G$ is ‘type I’, it does. And in this case, $\stackrel{^}{G}$ is metrizable and locally compact.

I don’t want to say what ‘type I’ means. But, you’ll be happy to know that real or complex simple Lie groups qualify, at least if they’re ‘algebraic’ — that is, defined by algebraic equations. For example, $\mathrm{SL}\left(2,ℝ\right)$ is algebraic, but its universal cover is not.

In fact we don’t need our group to be simple: any reductive algebraic group $G$ has a Plancherel measure on $\stackrel{^}{G}$. So, groups like $\mathrm{GL}\left(n,ℝ\right)$ and $\mathrm{GL}\left(n,ℂ\right)$ have a Plancherel measure.

So, take your favorite group of this sort, and try to actually get your hands on $\stackrel{^}{G}$ and its Plancherel measure. Unless your group is compact, this is usually really hard! This is why people like Harish–Chandra and Vogan are famous: they made real progress on these questions.

I certainly haven’t seen people saying that $\stackrel{^}{G}$ is an algebraic variety. Is this true?

I think we could look up stuff about the irreducible unitary representations of $G=\mathrm{SL}\left(2,ℝ\right)$ and get some sense of what’s going on. You’ve got your discrete series, your principal series, your complementary series, …

… but basically, in this case I think $\stackrel{^}{G}$ consists of a countable discrete set of points, two lines, and a closed unit interval, perhaps glued together in some way.

Posted by: John Baez on August 27, 2007 10:26 AM | Permalink | Reply to this

### Re: Justificatory Narratives

Thanks John, this is a big help.

Posted by: Bruce Bartlett on August 28, 2007 1:54 AM | Permalink | Reply to this

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