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**The geometry of music revealed!**The

*red*lines connect notes that are a

**major third**apart. The

*green*lines connect notes that are a

**minor third**apart. The

*blue*lines connect notes that are a

**perfect fifth**apart.

Each triangle is a chord with three notes, called a

**triad**. These are the most basic chords in Western music. There are two kinds:

A

**major triad**sounds happy. The major triads are the triangles whose edges go

*red-green-blue*as you go around clockwise.

A

**minor triad**sounds sad. The minor triads are the triangles whose edges go

*green-red-blue*as you go around clockwise.

This pattern is called a

**tone net**, and this one was created by David W. Bulger. There's a lot more to say about it, and you can read more in this Wikipedia article:

http://en.wikipedia.org/wiki/Neo-Riemannian_theory

and this great post by Richard Green:

https://plus.google.com/u/0/101584889282878921052/posts/bgpNTT8WqHx

The symmetry group of this tone net is important in music theory, and if you read these you'll know why!

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A Capella Science crushes it once again, not only musically, but in every other way.

Via +Jenny Winder, +Jennifer Ouellette

Via +Jenny Winder, +Jennifer Ouellette

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A pretty good collection of various lectures on physics including strings, QFT, and several other things.

Via +Giotis Mth.

(I guess I can't just share to communities directly anymore? What gives?)

Via +Giotis Mth.

(I guess I can't just share to communities directly anymore? What gives?)

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Learn about Category Theory from a master, French mathematician Pierre Schapira, writing for the new Inference magazine: "Here I will consider uniqueness, or, rather, the concept of identity, and how it functions. I will therefore address the status of equality in mathematics and its variants, namely isomorphism, equivalence, and so on. This issue has until recently been totally ignored, but is of such importance that it may potentially lead us to question set theory itself." Fun and illuminating piece. The French version is available as well:

http://inference-review.com/article/categories-de-zero-a-linfini

http://inference-review.com/article/categories-de-zero-a-linfini

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**The trouble with QED**

If you're trying to understand charged particles and radiation in a way that takes

*special relativity*and

*quantum mechanics*into account, you need

**QED**.

That stands for

**quantum electrodynamics**. Feynman, Schwinger and Tomonaga invented this theory - with lots of help - around 1948. In QED we often compute answers to physics problems as power series in the fine structure constant

α ≈ 1/137.036

This number says how strong the electric force is. For example, if you have an electron orbiting a proton, on average it's moving about 1/137.036 times the speed of light.

We can compute lots of things using QED. A great example is the magnetic field produced by an electron. The electron is a charged spinning particle, so it has a

*magnetic*field in addition to its

*electric*field. How strong is this magnetic field?

With a truly heroic computation, physicists have used QED to compute this quantity up to order α⁵. This required computing and adding up over 13,000 integrals. If we also take other Standard Model effects into account we get agreement with experiment to roughly

*one part in a trillion!*

This is often called the most accurate prediction of science. However, if we continue adding up terms in this power series, there is no guarantee that the answer converges. Indeed, in 1952 Freeman Dyson gave a heuristic argument that makes physicists expect that the series diverges, along with most other power series in QED!

I explain that argument in this blog article. I'm especially happy because I think I've made it a bit more precise. But it's not a proof: just an argument that

*something very strange must happen*if the answer converges.

Currently, the consensus among physicists is that ultimately QED is inconsistent. I explain why. But again, there's no proof. We need some mathematicians to help settle these questions!

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**Einstein's stance on Gravitational Waves**(For +Urs Schreiber, extracted from Kennefick's

*Traveling at the Speed of Thought*.)

Citation marks refer to the book above.

1912 Max Abraham worked on his own version of relativistic theory of gravity, coming to the conclusion that gravitational radiation cannot be a component since there does not exist

*dipole radiation*.

1915 Publication of Einstein's paper on GR

1915-1916 Discover of Schwarzschild solution

February 19, 1916 In communication to Schwarzschild, Einstein indicated that he did not believe in gravitational radiation, for a reason which essentially boils down to the impossibility of dipole gravitational radiation, which he deduced from the non-existence of negative mass. [K p39]

Mid 1916 Publication of "Approximate Integration for the Field Equations", in which Einstein discussed linearization of his namesake equations, and showed that the linearized equations can be solved with retarded potentials, thereby showing existence of gravitational waves.

1918 Publication of new paper which corrected an error of the 1916 paper, after a letter from Nordstrom. (The formula in 1916 paper seems to indicate the existence of

*monopole radiation*. The 1918 paper correctly found that the lowest order terms are

*quadrupole*.) In particular he recognized that certain apparent monopole and dipole radiation that does not appear to carry energy can be gauged away with coordinate transforms. [K p65]

mid 1936 In a letter to Max Born, Einstein claims that gravitational waves in fact do not exist in the full theory. He claims that the full nonlinear theory puts additional constraints such one cannot upgrade the solutions to the linearized system to a solution of the full system. He arrived at this conclusion after studying plane-wave solutions with Nathan Rosen, and found that the solutions must be singular. [K p79]

(Their paper was sent to John Tate for

*Physical Review*. Tate sent it to HP Robertson to referee, and some comments were returned. Einstein threw a hissy fit and withdrew the submission on July 30, 1936.)

Early 1937 Publication of the aforementioned paper, now however with the conclusion altered to

*support*the existence of gravitational waves, in a different journal. This is the Einstein-Rosen wave paper.

It turns out that HP Robertson spotted a mistake in the original submission, that the singularity they observed is really just a coordinate singularity and can be removed if one assumes one is working on a cylindrical, instead of rectangular, coordinate system. [K p81 - 88]

---------------

As far as I am aware that is the summary of Einstein's involvement in the theory. There are papers published in the twenties concerning exact gravitational wave solutions (PP wave solutions seem to have been first written down by Brinkmann in '25), but Einstein was apparently oblivious to those papers during his investigation with Rosen.

For obvious reasons Einstein was not involved in the "controversies" about gravitational waves between the 50s and the later 70s.

After 1937, Einstein's focus on GR (besides the grand unified theory) seems to be on studying stationary solutions (where among other things, in the linearized regime he proved the rigidity part of the positive mass theorem) and on studying particle motions (the series of papers by Einstein, Infeld, and Hoffman, in some combination).

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**Signal Processing with GW150914 Open Data**

Besides securing a prominent position in scientific history, LIGO looks to be taking a lead role in the movement for open science. They've put out a tutorial in the form of an iPython notebook where they show you how to process the raw data from the gravitational wave signal, compare it to a numerical relativity prediction, convert it to audio, and so on:

https://losc.ligo.org/s/events/GW150914/GW150914_tutorial.html

This is a phenomenal thing to do. Communicating your science to the public shouldn't be limited to just explaining it in absolute basic language (though thats surely important too), but I also think there should be some effort put towards helping people with intermediate levels of knowledge and a desire to learn be able to do so. And of course, it is also extremely desirable to be able to see and verify the steps taken to analyze the data, so that it's easier to reproduce, understand, and ultimately be made that much more accountable.

I'm not sure to what extent this notebook exactly matches what they did to prepare their paper – it would certainly be especially great if that were the case. But either way this looks like a great contribution by LIGO for learning both about gravitational wave science and about the concepts involved in their data analysis.

It would be very cool to see something like this from +ATLAS Experiment or +CMS Experiment!

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The signal has probably reached you that the long search for gravitational waves has finally paid off!

LIGO observed a dramatic signal on September 14, consistent with black holes of 29 and 36 solar masses merging, with 3 solar masses of the combined energy converted into gravitational waves. They traveled for something like 1.3 billion years before being detected on Earth.

One of the facts that always amazes me about these processes and the waves they produce is that they actually lie in the audible frequency range. So playing this signal as sound is an especially meaningful way to experience it, as this video they released shows. It first plays the signal in its true form a couple times, then plays it shifted up in tone to be more audible. You can also really clearly see the signals as detected at the two LIGO sites (along with their Fourier transforms).

All the details of the discovery, including uncertainties in the numbers I mentioned can be found in the paper:

I don't have anything novel to say about this, other than spacetime and general relativity are amazing, science is awesome, and congrats to everybody who made this happen!

LIGO observed a dramatic signal on September 14, consistent with black holes of 29 and 36 solar masses merging, with 3 solar masses of the combined energy converted into gravitational waves. They traveled for something like 1.3 billion years before being detected on Earth.

One of the facts that always amazes me about these processes and the waves they produce is that they actually lie in the audible frequency range. So playing this signal as sound is an especially meaningful way to experience it, as this video they released shows. It first plays the signal in its true form a couple times, then plays it shifted up in tone to be more audible. You can also really clearly see the signals as detected at the two LIGO sites (along with their Fourier transforms).

All the details of the discovery, including uncertainties in the numbers I mentioned can be found in the paper:

**Main:**http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102**Backup:**https://drive.google.com/file/d/0B9ftavPIb7rGa1Y2LWJvblhqdW8/view?usp=sharingI don't have anything novel to say about this, other than spacetime and general relativity are amazing, science is awesome, and congrats to everybody who made this happen!

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I hope this holds up. It would be awesome to know that humanity has succeeded in directly measuring vibrations in spacetime, and, longer term, to learn what's within earshot at LIGO's new level of sensitivity (since September).

This is as big as the detection of the Higgs,

I got into physics so I could do general relativity, which lead me to this: http://math.ucr.edu/home/baez/gr/oz1.html and thence to being the category theorist I am today :-) But gravitational waves are incredibly cool, and if I'm not wrong, the direct detection of which is just about the last major un-verified prediction of GR.

**if**it turns out that is what they are seeing.I got into physics so I could do general relativity, which lead me to this: http://math.ucr.edu/home/baez/gr/oz1.html and thence to being the category theorist I am today :-) But gravitational waves are incredibly cool, and if I'm not wrong, the direct detection of which is just about the last major un-verified prediction of GR.

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**The relation between the icosahedral group and E8**in classical discussions remains a bit mysterious. N=2 super Yang-Mills theory clarifies it.

Various seemingly unrelated structures in mathematics fall into an "ADE classification" [1]. Notably finite subgroups of SU(2) and compact simple Lie groups do. The way this works usually is that one tries to classify these structures

*somehow*, and ends up finding that the classification is goverened by the combinatorics of Dynkin diagrams.

While that does explain a bit, it seems the statement that both the icosahedral group and the Lie group E8 are related to the same Dynkin diagram

*somehow*is still more a question than an answer. Why is that so?

The first key insight is due to Kronheimer in 1989 [3]. He showed that the (resolutions of) the orbifold quotients C^2/Gamma for finite subgroups Gamma of SU(2) are precisely the generic form of the gauge orbits of the direct product of U(ni)-s acting in the evident way on the direct sum of Hom(C^ni, C^nj)-s, where i and j range over the vertices of the Dynkin diagram, and (i,j) over its edges.

This becomes more illuminating when interpreted in terms of gauge theory: in a "quiver gauge theory" the gauge group is a product of U(ni) factors associated with vertices of a quiver, and the particles which are charged under this gauge group arrange, as a linear representation, into a direct sum of Hom(C^ni, C^nj)-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge orbit, and Kronheimer says that if the quiver is a Dynkin diagram, then this gauge orbit looks like C^2/Gamma.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a "simple" Lie group, in the technical sense [4], such as SU(N) or E8. The mechanism that relates the two classes of examples is spontaneous symmetry breaking ("Higgsing"): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point "spontaneously" singles out the corresponding stabilizer subgroup [5].

Now here is the final ingredient: it is N=2 super Yang-Mills gauge theories [6] ("Seiberg-Witten theory") which have a potential that is such that its vacua break a simple gauge group such as SU(N) down to a Dynkin diagram quiver gauge theory. One place where this is reviewed, physics style, is section 2.3.4 in the thesis [7].

More precisely, these theories have two different kinds of vacua, those on the "Coulomb branch" and those on the "Higgs branch" depending on whether the scalars of the "vector multiplets" (the gauge field sector) or of the "hypermultiplet" (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave "dually".

So that then finally is the relation, in the ADE classification, between the simple Lie groups and the finite subgroups of SU(2): start with an N=2 super Yang Mills theory with gauge group a simpe Lie group. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of C^2 by a discrete subgroup of SU(2).

[1] http://ncatlab.org/nlab/show/ADE+classification

[2] http://ncatlab.org/nlab/show/ADE+--+table

[3] http://ncatlab.org/nlab/show/ALE+space#Kronheimer89

[4] http://ncatlab.org/nlab/show/simple+Lie+group

[5] http://ncatlab.org/nlab/show/stabilizer+group

[6] http://ncatlab.org/nlab/show/N=D2+D=4+super+Yang-Mills+theory

[7] http://ncatlab.org/nlab/show/N=2+D=4+super+Yang-Mills+theory#Albertsson03

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