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Dan Piponi
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Dan Piponi

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I recently finished this book on the history of string theory. Although there is a lot of technical material, it's not essential to understand all of it in order to get something out of this book.

One of the most fascinating things about string theory is that it was originally discovered backwards, at least compared to any sensible development of the subject. When I was taught the subject (my lecturer was Peter Goddard, one of the discoverers of the no-ghost theorem) you started with generalising Hamilton's principle of least action from particles to strings. You derived some equations of motion. Did the standard quantum mechanics stuff that you'd do with any kind of extended oscillator. And then computed scattering amplitudes so you could predict what might happen in an experiment. (I left out the fun bit which is where it all goes wrong except in the critical 26 dimensions, because of the no-ghost theorem.) But historically what happened was that through the pure mathematical magic of complex analysis it was possible to guess what a possible scattering amplitude looks like globally from a few experimental clues. From that is was possible to reverse engineer what kind of physical system could have given that answer. It was really interesting to read about the details of how this process took place and see the continuity with other ideas going around at the time.

Other interesting things this book made me appreciate:

I think of quarks as the traditional way to do things and string theory as much newer. But in fact quarks date back to 1964 and strings to 1969. Gell-Mann was involved in the original adoption of both ideas. String theory is so old that Heisenberg probably attended at least one lecture on string theory.

Again backwards from how things are taught: I always thought of superstrings as supersymmetry applied to string theory. But supersymmetry developed out of string theory in around 1971. So, historically at least, it's more accurate to think of supersymmetry as superstrings minus strings. (Though there were some supersymmetrical ideas as far back as 1966.)

Back when I was a student there were a few papers on Liouville field theory which gave a way to make string theory work in non-critical dimensions.  (http://en.wikipedia.org/wiki/Liouville_field_theory) This gets a brief mention in the book but I wonder what happened to that idea.

This book only really gives a history of the subject up to around 1994 although it does have a chapter on more recent work. It'll probably take a decade or two more before a good history of that can be written.
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+Dan Piponi thanks for the feedback. I have been trying to advertize this and other neglected facts in my "string theory FAQ" [1]-

[1] http://ncatlab.org/nlab/show/string+theory+FAQ#DoesSTPredictSupersymmetry
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I think one of my earliest posts on Twitter linked to this Division by Three paper by Doyle and Conway: http://arxiv.org/abs/math/0605779

This is now a followup, Pan Galactic Divisionhttp://arxiv.org/abs/1504.02179

It shows how for any finite set N, and any injection from the set A×N to B×N, we can construct an injection from A to B without making use of the axiom of choice, in effect allowing us to cancel by N on both sides of the map. To make things easier it views the set B as a set of possible numbers or pictures on cards, and N as the set of possible suits, and constructs the injection by induction on N using a card game. Hence the attached artwork.

This problem has a long history going back to a "lost" proof from 1926.
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+John Baez I want to write something about this, and perhaps look at addressing Doyle and Qiu's remak "Still weaker systems would suffice: It would be great to know just how strong a theory is needed."
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Video gamers have casual games for when they don't have long blocks of time to devote to working their way through each level. I'm always on the lookout for casual mathematics. I found a nice example recently, Combinatorial Nullstellensatz. There's lots of deep and complex stuff you can do with it, but the statement is easy to understand, the proof is elementary, and once you know it you can start proving theorems in a couple of lines that until recently were considered tricky.

Let's start with this well known fact: If you have a polynomial of degree n in one variable, and a set with n+1 distinct numbers in it, you know that the polynomial must be non-zero for some element of the set.

By way of illustration, let me use this to prove something really trivial in a really stupid way. I have a set S of 4 distinct integers and I have the three properties {x≠1, x≠2, x≠3}. I want to show that some element of my set satisfies all of the properties. Yes, it's obvious, but here's an indirect way of doing it: each property can be represented as a (linear) polynomial giving the set {x-1, x-2, x-3}. Each "polynomial" is non-zero precisely when the corresponding property is true. We can "and" together all of the propositions by multiplying them to get p(x)=(x-1)(x-2)(x-3). This is cubic and so it can't be true that p gives zero for every element of S of size 4.

Combinatorial Nullstellensatz (CN) is a newish (1999) generalisation of the fact that a non-trivial degree n polynomial must be non-zero somewhere on a set of size n+1. It applies to multivariate polynomials. Its usefulness comes from the fact that using schemes like I did above, you can encode propositions about variables as (possibly linear) polynomials. Unlike my silly example, for multivariate polynomials you can get non-trivial results.

For example, it's great for proving things about all kinds of graph colourings. You can encode, as polynomials, the properties that you want neigbouring vertices (or coincident edges, or whatever) to have, multiply them all together, and apply CN to show that your colour scheme can be satisfied.

There are many problems that can be solved this way that aren't ostensibly about polynomials. So I find it surprising that this kind of algebra has anything to say about them.

CN is very useful for solving Mathematics Olympiad problems. (But I don't look at those things as they make me feel really stupid.)

An introductory document on the subject:
http://people.math.sfu.ca/~kya17/teaching/math800/Math800project.pdf

Another, this time with exercises:
http://mathcircle.berkeley.edu/archivedocs/2013_2014/lecture/1314lecturespdf/BMC-Advanced%20Sept.%2017th,%202013.pdf
(At least one of those exercises was a conjecture for 30 years but it just melts when you apply CN to it.)
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ha ha. indeed, when you put it that way. 
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Last and First Men is one of the most amazing works of science fiction ever written and it ought to be better known. It was written in the 1930s and describes the future of humanity over the next two billion years. If that isn't enough for you, the author also wrote a sequel Star Maker.

It's not what you'd conventionally call a good read. The first few chapters are a near future history but with hindsight from 2015 it's mostly annoying, though it does have a few insights. One edition recommends skipping the first few chapters though that's probably to spare Americans from insult rather anything else. But apart from that, much of the book is told with very broad brushstrokes. By "broad" I mean passages where Stapledon tells you that he'll skip over the next few hundred thousand years because not much happens, just a bunch of empires and tyrannies rising and falling across the Earth (or Mars, or Venus) or plagues almost wiping everyone out. Nothing important. There aren't many actual events as such in the book. It's more low resolution descriptions of the state of humanity over time. So don't expect anything like a plot or character development.

There are some neat things that I expect Stapledon got right. There isn't one humanity. There are Homo Sapiens versions 1 to 18 with a few side branches. In many cases version n+1 is the product of genetic engineering by version n. Version 4, for example, are basically giant brains constructed by version 3. I also enjoyed the way the Martians completely misunderstood the nature of intelligent life on Earth.

In many ways the book seems hopelessly wrong. Stapleton didn't anticipate modern information technology. At a certain level of abstraction life in 2×10⁹ A.D. doesn't seem all that different to life today despite the hive mind and the cannibalism. I think there's surprisingly little technological development over two billion years and I expect that much (but not all) of the technology Stapledon describes will actually exist within a few millennia. There's very little space travel with humans only relocating within the solar system. But there is terraforming. (Remember, this was written in the 30s.)

I don't know of any other work of science fiction that has a scale anywhere near as ambitious as this. But if you know of one, I'd love to hear about it.

Anyway, I currently have the audiobook, narrated by Travis from Blake's Seven, on my iPod for listening to in the night. Works perfectly as a cure for insomnia. (Don't take that as a disrecommendation.)

https://en.wikipedia.org/wiki/Last_and_First_Men
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City at the End of Time, by Greg Bear. Also strange, in a different way, of course, is In the Mountains of Madness, by H. P. Lovecraft. Not a word of dialog in a hundred pages of novella.
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With his movie scores John Carpenter has played a major role in the history of electronic music. Many will instantly recognise the Halloween theme music (https://www.youtube.com/watch?v=nQWqRGVE1Zk) and many artists claim Carpenter as an influence. But he's never released a standalone album that's not intended as a movie soundtrack. Until last month.

What can I say? It sounds exactly how you you expect music by John Carpenter to sound. Pretty excellent if you don't mind a bit of the old 80s analogue synth sound, except actually it's composed with Logic Pro. Sensible man.

You can watch a montage of his movies set to the first track here: http://johncarpenter.sacredbonesrecords.com
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If |x|<1, -log(1-x) =x+x²/2+x³/3+x⁴/4+…

So it seems reasonable to consider the function defined by:

Li(x) = x+x²/4+x³/9+x⁴/16+…

That converges for |x|<1 but you can analytically continue to the entire complex plane if you make a branch cut from 1 to +∞ or treat it as a multivalued function.

But until I read the first chapter of Zagier's introduction to the subject (http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-30308-4_1/fulltext.pdf) I had no idea how many astonishing properties this function, known as the dilogarithm, has.

Simple closed form expressions for the dilogarithm are only known at 8 points in the complex plane, four of them being -φ, -1/φ, 1/φ and 1/φ², where φ is the golden ratio.

Li also satisfies the really bizarre property that for any polynomial f of degree n without a constant term

Li(z) = C(f) + ∑ Li(x/a)

where the sum is over all n roots x of f(x)=z and all n roots a of f(a)=1, i.e. it's a sum of n² terms. C(f) is some complicated thing that depends only on the polynomial.

The dilogarithm has amazing connections with projective geometry, hyperbolic geometry, quantum field theory and even K-theory. (I can't imagine how that last one works.)

Li is just one of an infinite family of polylgarithms, which themselves generalise to multiple polylogarithms. I wonder if they have ladder operators for some group representation like the way I recently learnt Bessel functions do.

Zagier says in the introduction: "Almost all of [the dilogarithm's] appearances in mathematics, and almost all the formulas relating to it, have something of the fantastical in them, as if this function alone among all others possessed a sense of humor."
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The fact that you get from Li_{-n} to Li_{-n-1} via applying z(d/dz) looks like it really ought to have some sort of Weyl algebra action, or an action of something derived from the Weyl algebra.
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Dan Piponi

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Some days, finite puzzles won't do.
 
Be the first to solve my new puzzle!
  Can you solve my challenge puzzle? Cheryl   Welcome, Albert and Bernard, to my birthday party, and I thank you for your gifts. To return the favor, as you entered my party, I privately made ...
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Our paper has been accepted for the 2015 Mathematics of Program Construction Conference:

Polynomial Functors Constrained by Regular Expressions
http://dept.cs.williams.edu/~byorgey/pub/type-matrices.pdf

It's based on some blog posts from a couple of years ago and shows how to define a tree type in a language like Haskell where the leaves of the tree are constrained to match a regular expression. A basic example that many are familiar with is the type of lists where the elements alternate between two types. The picture corresponds to an example of a binary tree with regular expression b*1a* where 1 is the type with just one inhabitant so it functions as a "hole". These types have the property that it's completely impossible to build a tree that fails to satisfy the constraint.

As a side effect it also shows how you can think of Conor McBride's Jokers and Clowns paper as really being about divided differences (ie. (f(x)-f(y))/(x-y)) applied to types, even though you can't literally divide types: https://personal.cis.strath.ac.uk/conor.mcbride/Dissect.pdf

Sadly it's hard to find a programming language that can truly automate this process, especially as it involves matrices whose entries are types. Brent and I both tried independently with Agda, but even with dependent types we got stuck. But that doesn't stop you using the paper to hand craft the appropriate type. (I guess Template Haskell would work fine but that's cheating.)
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> Brent and I both tried independently with Agda, but even with dependent types we got stuck.

I'm not sure what it is that you were trying to do but after reading (and quite enjoying) your paper, I had a bit of fun using regular expressions to constrain the values or the types of the values stored in a tree structure.

Here are a few examples:
https://github.com/gallais/potpourri/blob/master/agda/poc/regexpfun/Data/Functor/Examples.agda

Here is the meat of the implementation:
https://github.com/gallais/potpourri/blob/master/agda/poc/regexpfun/Data/Functor.agda

It relies on the regexp library defined here:
https://github.com/gallais/aGdaREP

NB: my repo "potpourri" is a bit messy so it's probably hard to get the right set of files to typecheck Examples.agda so I'd say "don't try this at home". If there's interest, I guess I could upload a self-contained version somewhere.
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I just discovered the computational linguistics olympiad. These problems are a lot of fun. I tried some easy ones and they were easy. So I jumped to a hard one and now I'm stuck. Have fun!

(Don't post any spoilers, I haven't solved it myself yet. Brag about it if you have solved it though!)

http://www.ioling.org
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It's really not that hard. Try replacing words with symbols. Also, write down each catch.
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I recently finished Kazuo Ishiguro's latest book The Buried Giant.

I've been much entertained by the "controversy" that this book seems to have stirred up (http://www.theguardian.com/books/2015/mar/08/kazuo-ishiguro-rebuffs-genre-snobbery). Despite having written science fiction before, Ishiguro is apparently considered a "serious" writer and his fans are freaking out that the book doesn't come with a plain brown cover to hide the fact that they are reading a fantasy book starring ogres, pixies, magic as well as a dragon. And if snobbery from one side wasn't enough, Ursula Le Guin got all uppity about a "serious" writer having the temerity to invade her personal territory. Shocking stuff!

I could almost have believed that this book was written by Gene Wolfe. It has many of his signatures such as narrators with unreliable memories who speak with a highly affected style and enter into long expositions at the most inappropriate moments. In fact, it reminded me a lot of Wolfe's Wizard Knight duology, to the point of even sharing a character: Sir Gawain.

I recommend it.
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Sounds good.
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One of Project Loon’s earliest Eureka moments was the idea that we could provide continuous Internet connection not by keeping balloons stationary over a given location (which would require lots and lots of energy to work against the wind) but by coordinating a fleet of balloons to work with the wind, such that when one balloon leaves a location another moves into its place to continue providing connectivity. In theory, this means that any individual balloon would provide connection in one place and then, days later, provide connection at another location at the opposite end of the world. In our latest long distance LTE test this is exactly what we achieved!

Launched from New Zealand, our globe-connecting balloon made the first leg its journey travelling 9000 km over the Pacific Ocean. Approaching our test location in Chile at a speed of 80 km/h, a command was sent for the balloon to rise into a wind pattern that slowed it down to a quarter of its speed, allowing it to drift overhead members of the Loon operations team who were able to connect to the balloon via smartphones on our test-partner mobile network. 

Hanging around for half an hour to complete the connection testing, the balloon was then sent off on the winds over the South Atlantic ocean towards its next test location, over 10,000 km away in Australia! Our balloon completed this second leg of the journey in just 8 days, travelling over 1000 km per day and reaching a top speed of 140 km/h while whizzing over the ocean south of Africa. Once at the east coast of Australia the Loon Mission Control team implemented a series of altitude maneuvers to catch different winds and reverse the balloon path, lining it up to directly overfly our test location. Having travelled over 20,000 km around the world the balloon flew overhead at a ground distance of less than 500 meters away from our target (well within the 40,000 meter radius required for connection) to provide over 2 hours of Internet connection. That level of precision is like hitting a hole-in-one in golf from over 4 km away!

Tests like this give us real insight into how Project Loon can work at scale. With more balloons in the stratosphere and more Telco partners around the world capable of supporting Loon internet traffic, our ability to provide continuous connection in rural and remote areas will only increase. 
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And I assume you're not using helium, since that would be very irresponsible. 
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Here's a quote from the paper at: http://arxiv.org/abs/1411.6009

"[We predict] that the [supernova] will appear in the central image of the spiral host galaxy, at an approximate position of α = 11h49m36.01s, δ = +22◦23′48.13′′ (J2000.0) at a future time, within a year to a decade from now (2015 to 2025)."

So here's a puzzle. If the first time you know about a supernova is when you've already seen it happen, how can you possibly make such a prediction?

Scroll down for the answer.












As a result of gravitational lensing, this supernova, in galactic cluster MACS J1149.6+2223,  is visible in multiple locations. But the path length for each image is different so the image arrivals are staggered in time. If the astronomers have built their model correctly they can predict future image arrivals too.
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Thank you so much
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