# Optics and functors

At the moment, I don’t think we’ve got a very good physical intuition for functors — just something which is completely and utterly abstracted away from the physical universe.

Well, we’ve got full image, essential image, and from optics a whole slew of machinery for dealing with images of things. Virtual images, focal depth, whatnot.

* Do functors have focal depth?
* Are there things which are not quite categories in which a functor could be diffracted?
* Think about imaging technologies: MRI, cat scans — anything which we use to construct visual representations of things might have a functorial analogue. Functorial holography? (and no, I don’t mean this: “holographic principle of higher category theory”
* Just how many of these w: optical phenomena have functorial analogues?

# notes for complex analysts

1. we need a realtime renderer. this sequence of “ah, I have a transcendental meromorphic function I want to visualize, I shall type it into my favorite software package and make a picture, and wait to see the result”‘s days is numbered. By the time that rolls around, spending laboriously long time proving extremely persnickety results is going to be dated.

A. it would be good if it could Q or SL(2,Z) highlight images.

B. It would be nice to have a mathematician orientated special functions/fractal platform running over cuda, one that could make things as easy as Adobe Illustrator in some ways.

2. You’ve got a copy of Wegert’s Visual Complex Functions and NIST Handbook of Mathematical Functions, right? Also worth looking at is the Bateman project, Abramowitz and Stegun, Gasper’s book about hypergeometric series, and the Tata lectures on theta functions. It would be nice to start working through that material.

3. You’ve read Jim Belk and Bradley Forrest’s A Thompson Group for the Basilica… it would be nice if we could make videos of these transformations for arbitrary things.

4. Do not trust one renderer. That having been said, we need to agree on a hue model (which is mostly settled at the present?) Why? So we can encode sheaf-theoretic information in it! (Nevanlinna theoretic information as well)

5. If you write a paper about some transcendental meromorphic function which varies on a parameter, make a movie and post it to youtube along with the paper, or on your site.

6. This ‘dip it in formaldehyde, move like insects’ is dangerously slow.

7. Moar pictures. And don’t list points. Make a picture.

8. It would be nice if we could figure out a way for the nonprofessional mathematicians (Burner/Fractaltribe types) to contribute in a meaningful way. But at the moment everything is so Balkanized that there are a lot of disparate communities making very very slow progress on one thing or another.

(I’m sort of at the saddle point between Burners and complex analysts, which makes talking to either somewhat aggravating)

# lattice of (topological) pants

(just imagine this extends to infinity…)
probably has some meaning in Teichmuller theory, but I’m a complex analyst, and want to string lacunaries across those circles, and insist that we can only link the pants when the lacunaries match.

# an integral related to the zeta function

The integral in question is:

$\int_{1/2}^{\infty} \left[\zeta\left(t+i\mathrm{Im}(\rho_{n})\right)-1\right] dt$

Since the limit starting from a nontrivial root going parallel to the real axis of the zeta function is 1, we may subtract that off so that it converges.

144 evaluations, text file, done with mpmath

What’s amazing, is that when the values of this integral corresponding to nontrivial roots of the zeta function are plotted, some interesting structure revealed:

# air on swinging back and forth

The core of the iteration here is a process, seeded at s=1, such that for each cycle of the iteration we have:

S = f(z/s)
s = f(S/z)

It doesn’t actually have to be division, just some inverse process, and because I am fond of giving bright and strange names to things, I propose that we call these functions hā.ʻukē.ʻukē

hā.ʻukē.ʻukē: vi. To swing back and forth with a bang or clatter; to click as the teeth; to bounce back and forth, as breasts.

# iterated special functions on complex parameters

Frequently a special function will have many complex parameters: the polylogarithm, for instance: $\mathrm{Li}_{s}(z)$. Usually we’ll iterate on $z$, but we don’t have to. I’m currently rendering $\mathrm{Li}_{\mathrm{Li}_{\mathrm{Li}_{z}(z)}(z)}(z)$

Suppose we have a $latex${}_{5}F_{4}\$ hypergeometric function:

${}_{5}F_{4}\left({t,x,c,v,b\atop a,s,d,f};z\right)$

And we started looking at monstrosities like this:

${}_{5}F_{4}\left[{{}_{5}F_{4}\left({z,x,c,v,b\atop a,s,d,f};t\right),{}_{5}F_{4}\left({t,z,c,v,b\atop a,s,d,f};x\right),{}_{5}F_{4}\left({t,x,z,v,b\atop a,s,d,f};c\right),{}_{5}F_{4}\left({t,x,c,z,b\atop a,s,d,f};v\right),{}_{5}F_{4}\left({t,x,c,v,z\atop a,s,d,f};b\right)\atop {}_{5}F_{4}\left({t,x,c,v,b\atop z,s,d,f};a\right),{}_{5}F_{4}\left({t,x,c,v,b\atop a,z,d,f};s\right),{}_{5}F_{4}\left({t,x,c,v,b\atop a,s,z,f};d\right),{}_{5}F_{4}\left({t,x,c,v,b\atop a,s,d,z};f\right)};z\right]$

# Fukinsei and large scale structure in mathematics

There’s a “let’s classify and taxonomize all teh things” historical thread in mathematics. The “now that we know it exists, let’s dip it in formaldehyde”. Unfortunately the conversations between those in online fractal forums and mathematicians at universities doing formal complex analysis/dynamics (in the benighted language of category theory) are scarce.

Nature is zebra stripes: westerners will adjoin ‘irregular’ alongside ‘pathological’ or ‘malign’, and seek to impose fences: structures which are exactly regular and satisfy a group law of a sort. I have some contentions about the zebra stripes of nature:

* I suspect they have interesting long-range correlations which are kind of absent in fences.

* I think that fences are local. Even on the grand cosmic scale — the little fear that the universe might repeat exactly that lurks in the consciousness and science and religion occasionally do little to ameliorate — if you’re arguing that reality is mathematics (a la Frenkel and Tegmark), then the recipe-ish nature, a la put in the same things and get the same results back, whether applied to physical phenomena (where it works) or people (where it is foul and doesn’t). profoundly disregard the foamy and frothy and irrepetetive nature of the (whatever local context-spanning word you want to use).

* ‘fukinsei‘ is probably the best word for what I’m going after here, because all English words appear to carry connotations of problematical evil and horror, in a way.

And, just watch the first few seconds of this video:

I know that these can be accurately modeled by reaction diffusion systems, but the fingering here is strongly reminiscent of the fingering (I prefer ‘phalanges’)

Sometimes there are features which merely quantitative descriptions (here is a list of the roots) miss.

# special functions identities for 5 year olds

So you’re looking at the contents of the <a href=”http://dlmf.nist.gov/”>NIST Digital LIbrary of Mathematical Functions</a>, and there are all these equations, and (well I,) you say to yourself, ‘this is an enormous amount of wallpaper, how am I ever going to get this into my kinesthetic memory? I’ve started copying down bits from the section about q-hypergeometric series, but I think there’s a better way to do this:

So far I’ve made videos demonstrating the Borwein cubic theta functions identity:

And two showing how the j-function is invariant under SL(2,Z):

And you look at papers about the Borwein Cubic theta functions, or libraries of modular forms, and it’s all oily, stringy, two dimensional symbol manipulation. That creates an enormous barrier of entry to the material. Here is what I’m proposing:

We translate the content of the DLMF into realtime interactives http://www.visual.wegert.com/ phase portraits</a> on the complex plane that a five year old could interact with on an ipad or other tablet with suitable fast graphics processor.

Looking at something like this is worthwhile:

https://www-m10.ma.tum.de/bin/view/Lehrstuhl/StefanKranichPickIt#pickit

(insert a I am not a reformer or revolutionary tag here.I’m just your friendly transcendental kuroko 🙂

There is the absurdly barbaric suggestion that mathematics education should start with drilling addition, subtraction, multiplication, and division, or that all mathematics necessarily *starts* there. Imagine that being a Ramanujan could be common if only we got around to making a realtime user interface for all this special functions material.

(the longer term goal is to construct a new user interface for mathematics that doesn’t involve computer simulacra of two dimensional symbol manipulation with a cursor or mouse doing the input. I’d like to use a didgeridoo to drive real time special functions fractal generation and have that be the arena in which I do mathematics in rather than dealing with messy collections of symbol-spaghetti which are prone to error, but hey, I get this nice cultural syncretism between mathematics and a 40,000 year old instrument, and lots of rainbows, so potential win)

additionally, I know people usually kvetch about colored equations, but so many q-hypergeometric identities seem to look like wallpaper without colors:

# uru mai: indistinguishable points of different q-deformations

Since the negative of the derivative of the Gamma function evaluated at 1 is the Euler-Mascheroni constant $-\Gamma'(1) = \gamma$, and the,er, most obvious way of q-deforming this is the q-digamma function, here is a phase portrait of (one possible q deformed Euler Mascheroni Constant:

$-\psi_{q}(1)$

But the Euler-Mascheorni constant $\gamma$ can also be expressed as an infinite series involving the Riemann zeta function:
$\gamma=\sum_{m=2}^{\infty} (-1)^m\frac{\zeta(m)}{m}$
and poking around google seems to indicate there are a few different ways of q-deforming the Riemann zeta function. Now, this suggests the question… “where do these different q-deformations agree”. I’m using the Maori expression uru mai to denote points where two different q-deformations are indistinguishable. For instance, there are three Jackson q-Bessel functions, and I can certainly make phase portraits of their differences and look for zeros — points uru mai with respect to a number of q-defromations.

(‘uru mai’ is Maori for consistent, just because the word ‘consistent’ has an overuse problem)

# relief

Suppose I want to get a really good idea about the behaviour of gamma(z) near z=1. What about doing the following:

$\Gamma(\theta_{3}(z,q))\prod_{n=1}^{\infty}\frac{1}{\Gamma\left[(1+2q^{2n-1}\cos(2z)+q^{4n-2}\right]}$

Say this is a relief of the Gamma function in the third Jacobian theta function. Gamma could be replaced with any function whose limit as z approaches 1 is 1. Places where the relief is not one are places where the numerator differs from the denominator, thus taking a very fine microscope to the way the gamma function behaves.

Similarly, we can do the same for infinite sums. All you need is a function which goes to zero as its argument goes to zero. Sometimes you can even do self-reliefs:

$\sin(z) = \sum_{n=0}^{\infty} \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}$

So the self-relief of sine in sine would be:

$\sum_{n=0}^{\infty}\sin\left[ \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}\right] - \frac{(-1)^{n}z^{2n+1}}{(2n+1)!}$