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Cliff Harvey
Works at Functor AB
Attended Worcester Polytechnic Institute
Lives in Stockholm, Sweden
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Cliff Harvey

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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, 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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+Cliff Harvey you are the second person today to tell me something like that, sigh. 😉 I was writing software for the COBE interferometer at the time. 
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This is may be one of the more interesting and important developments in theoretical quantum gravity research lately. Stephen Hawking and Andrew Strominger both judge this to be a major step in understanding the black hole information paradox, though not yet decisive. I'm inclined to agree, to a point, but major questions remain.

The paper: Soft Hair on Black Holes
http://arxiv.org/abs/1601.00921

Quick Recap: The black hole info paradox is the fact that naively it seems like black holes are characterized by only a few numbers (mass, charge, angular momentum) while other considerations make it pretty clear they actually need to carry a huge amount of information (corresponding to everything it's gobbled up). To loose the information preserving property of physical laws would be huge.

The new insight here has to do with extra information that might be stored in zero-energy – so-called soft – photons and gravitons. Andrew Strominger describes his and Hawking's perspective on this in good depth in this interview. Pretty informative reading.

Like so many other things in physics, symmetry is at the heart of this. The basis of the new work are some previously-neglected symmetries of the laws of nature; of quantum field theory. And not just any symmetry but an infinite-dimensional symmetry in which "the point at infinity" plays a key role.

It seems pretty solid that these new properties of quantum field theory are basically right at the mathematical level, and it also seems likely that the importance for the black hole information paradox might be approximately what the authors argue (incomplete as the argument is).

However not everyone is fully on board with the exact interpretation presented by the authors. Here are a couple of responses from the usual theorist bloggers who cover these things.

http://motls.blogspot.se/2016/01/strominger-in-sciam-on-paper-with.html?m=1
http://backreaction.blogspot.se/2016/01/more-information-emerges-about-new.html

Im particularly partial to Lubos's quips that no solution to the black hole information paradox should really privilege points on the black hole horizon as special, since there isn't anything fundamentally special about them. Especially since the information in question is associated with soft particles, and zero-energy particles are supposed to be maximally-delocalized in spacetime according to the uncertainty principle. Also there seems to be some schizophrenic quality to these particles since they're supposed to save information that would otherwise be lost, but at the same time they also must be "not physically realizable" in some sense.

Like everyone else, my understanding of this is quite incomplete, but those are some of the questions that seem natural to ask after reading this new paper and interview.

Its always great to see more progress on black hole information, and great to see still more evidence that even insanely well-tested standard theories like general relativity and electromagnetism can continue to give up significant insights after all these decades. Quantum field theory continues to be a beautiful and mysterious beast.

I hope everybody had a great start to 2016, and plan to get more blogging in this year. Hopefully there will be many more great developments to discuss...

Via +Jenny Winder, +Omar Loisel .
The Harvard physicist explains the collaboration's long-awaited research on the black-hole information paradox
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As a mathematician, I take products of spaces at all hours of day or night, so that's how I normally talk... but normally I try to restrain my use of jargon in polite company!
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Homotopy Type Theory  - 
 
Conor McBride has been doing some insightful blogging lately, with an entertaining flair to boot. Here he discusses the problem of getting a computationally useful notion of univalence, considering Observational Type Theory as a prototype.

Thorsten Altenkirch, Wouter Swierstra and I built an extensionality boat. We might imagine that one day there will be a fabulous univalence ship: the extensionality boat is just one of its lifeboats. But nobody’s built the ship yet, so don’t be too dismissive of our wee boat. You might learn something about building ships by thinking about that boat. I come from Belfast: we built the Titanic and then some prick sailed it into an iceberg because they made a valid deduction from a false hypothesis.

Along the same lines, his previous couple posts about OTT are also worth a look. For reference I'll link the OTT paper as well:

https://pigworker.wordpress.com/2015/01/06/observational-type-theory-the-motivation/
https://pigworker.wordpress.com/2015/01/08/observational-type-theory-delivery/
http://strictlypositive.org/ott.pdf
"Why do you hate homotopy type theory?" is question I am sometimes asked, but I never answer it, because the question has an inaccurate presupposition. I am not happy when people forget that functi...
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Mathematics  - 
 
 
Diving for Calabi-Yau manifolds. Initially, i had written the following paragraphs in reverse order, more like being washed up ashore, saved from the deep. But i figured you might prefer to peacefully approach K3 surfaces, the secret topic of this post. Assuming you know what polynomials are, we'll start nice and slow:

What are Rational functions? You get those by dividing one polynomial by another N(x)/D(x). Their degree is handily defined to be the maximum power appearing in any of the two polynomials. Since the denominator mustn't be zero, we'll get funny "singularities" wherever the D(x) polynomial has a zero. Just as polynomials themselves, rational functions are quite an easy subject, so don't be shy. 

Some facts: They're representative for meromorphic functions. Just like holomorphic functions but singularities are allowed to appear at a finite set of points. You can "Taylor" them to your required precision, too.

https://en.wikipedia.org/wiki/Rational_function


The Elliptic integral appeared when people set out to calculate arc lengths of ellipses. It's an integral like this:

 x
∫ R(t, √P(t)) dt
 c  

That's a path integral, and you have to pick a starting point c for your path.

The result of the integration cannot, in general, be stated using elementary functions. But when we add the three Legendre canonical forms, also known as the elliptic integrals of 1st, 2nd, and 3rd kind, we can write the solutions as integral over rationals combined with these forms. The reduction formula does this.

https://en.wikipedia.org/wiki/Elliptic_integral

I just hate those notations, but luckily, there's also the Carlson symmetric form:
https://en.wikipedia.org/wiki/Carlson_symmetric_form


The Algebraic variety is the object of a projective geometric theory, in contrast to the linear algebraic groups, giving the better known affine theory. That's to be expected, because there's a division happening in rational functions, and you get projective spaces by "dividing" lines out, so you can look at the remainder.

One can define algebraic varieties as "the set of solutions of a system of polynomial equations". They're just like a regular manifolds, but may have singular points.

https://en.wikipedia.org/wiki/Algebraic_variety


A special case is the Abelian variety, defined as: A projective algebraic variety that is also an algebraic group. Such a group is given by regular functions, which are just another name for polynomials in this context. It means we call a curve Abelian variety when one can find a set of polynomial functions that generate a group.

Historically, the abelian variety played the role of generalizing the elliptic integral from involving polynomials of 3rd and 4th degree to ones involving 5th or higher degree. Originally studied over the complex numbers, over which they turned out to be exactly the complex tori that can be embedded in a complex projective space. Solomon Lefschetz worked that out, and also was the first to use the term Abelian variety.

Example, an elliptic curve is an Abelian variety of dimension 1.

https://en.wikipedia.org/wiki/Abelian_variety


Abelian varieties appear naturally as Jacobian variety of algebraic curves. We define: J(C) of a nonsingular algebraic curve C of genus g is the moduli space of degree 0 line bundles. You fill your surface with nonoverlapping lines, forget that a line has many points, and look at how lines are related.

As a group, the Jacobian of a curve is isomorphic to [...] divisors of rational functions. Our lines correspond to multiples of rational functions, just like any fraction can be given as a multiple of itself:

a/b = (x·a)/(x·b)

Note that plotting f(x) = x·a gives a line going through the origin.

https://en.wikipedia.org/wiki/Jacobian_variety


Abelian varieties also appear as Albanese variety of a curve C, a very natural generalization of the Jacobian to higher dimensions. The construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms (our "lines") generalizes straight forward to give the Albanese variety. Btw, when we're working over complex numbers, holomorphic means polynomial, 1-forms are the linear ones.

https://en.wikipedia.org/wiki/Albanese_variety


Enriques-Kodaira Classification of compact complex surfaces into 10 classes. For 9 of them we have a fairly complete understanding, but the remaining "general" type has too much in it. (Okay, our understanding for type VII depends on the global spherical shell conjecture.)

If you look at the diagram now you'll notice that the 9 well-understood classes are at most 1-d portions, lines of dots. The "general" type is the area left, i'm not surprised it's full of stuff.

https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification


One class are the K3 surfaces, named after Ernst Kummer, Erich Kähler and Kunihiko Kodaira. Actually there are four classes of K3 surfaces, but Andre Weil ran out of people to name fourth one. He chose the mountain K2 instead, which, as you can see, didn't really make it into modern language....

They're the complete smooth surfaces with trivial canonical bundle. That's exactly those 2-dimensional Calabi-Yau manifolds that aren't complex tori. The latter are simple Abelian varieties and much less interesting simpler, so they already got their own class.

Another way to define Calabi-Yau surfaces is as complete smooth surfaces with trivial canonical bundle. The canonical bundle is one passing through zero, it's called trivial when it's made of "lines". Maybe a better way to understand them is to state that all of them have Kodaira dimension 0...

https://en.wikipedia.org/wiki/K3_surface


The Kodaira dimension  k is one less than the number of algebraically independent generators. It's the rate of growth k such that P(d)/d^k is bounded. All Abelian varieties have zero Kodaira dimension! While k > 0 corresponds to hyperbolic, k = 0 to flat, sometimes, k is undefined or negative. We then say k = -oo and those cases correspond to surfaces with positive curvature.

So our surfaces are the "flat" ones in the above sense.


Initially, i had written this post in reverse, trying to figure out what this thread by +David Roberts mentioning K3 surfaces, Chern forms, and the magic number 24 was all about:
https://plus.google.com/+DavidRoberts/posts/CoBq5Y5Haqp


More links

Since the Jacobian variety is the connected component of the identity in the Picard group of our curve, i should have told you about Picard varieties and their associated Picard groups as well. I guess i'll have to do that next time, and maybe then we can also talk about how to blow something up:
https://en.wikipedia.org/wiki/Blowing_up

See also the Abelian surface, which are just Abelian varieties in 2d.
https://en.wikipedia.org/wiki/Abelian_surface

Hodge diamonds give another invariant, just like the Kodaira dimension is:
https://en.wikipedia.org/wiki/Homological_mirror_symmetry#Hodge_Diamond

The diagram is by R.e.b.:
https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira_classification#/media/File:Geography_of_surfaces.jpg

The portraits i found here:
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Kodaira.html
https://upload.wikimedia.org/wikipedia/commons/5/52/Federigo_Enriques.jpg

#ScienceSunday - #algebraic #geometry , #elliptic integrals, #jacobian #variety
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There's a nice Scholarpedia article by Yau: http://www.scholarpedia.org/article/Calabi-Yau_manifold
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This is some fantastic reporting by +Quanta Magazine on recent progress on the Monstrous Moonshine conjecture,  definitely one of the most fantastically intriguing and yet bafflingly inexplicable connections in mathematics and physics.

It's a story that relates number theory, string theory, the Monster group, and optimal sphere-packing patterns, among other things. The grandest insight in the story was proved in 1992: that the modular j-function which knows so much about number theory exactly describes bosonic string theory on a particular spacetime – a spacetime which corresponds to a particularly optimal packing of spheres. But there is clearly much more of this relationship to uncover, in particular relating to other sporadic groups. This latest article correlates to new proofs published on these other branches of the Moonshine story.

This incredible understanding will undoubtedly be a bonanza for pure mathematics, but there also seems a decent chance that physics will be impacted as well, at least as it pertains to string theory.
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+John Baez well, I feel background-dependence tends to be underestimated so that "basic ideas" is less unequivocal than it sounds. I also find that Moonshine with Ramanujan's constant and the like, exhibit features making it a fascinating real world candidate analogue to the monolith of "2001 Space Odyssey", together with the "magnetic anomaly" that draws attention to it in the movie.

BTW, the natural integer that Ramanujan's constant approximates so closely, cries for being rewritten
640320^3+744 and 640320 in turns flaunts divisibility by 2001 while its being cubed evokes the idea of a parallelepiped. A good enough allusion to the 2001 monolith, as far as I am concerned:)

I've published my own share of the article while introducing it with a discussion of Ramanujan's constant here:

https://plus.google.com/u/0/+BorisBorcic/posts/KAnpb4ht6pN
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Cliff Harvey

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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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String Physics Discussion  - 
 
 
M&m and little strings

It is amazing how efficiently String theorists build on previous work of their colleagues constructing step by step in a consistent way this beautiful framework of ideas and tools we call String theory.

There is an interesting troika of papers [1], [2], [3] following the pioneered work of Vafa et al [4], [5], [6] on M-strings (intersections of open M2 and M5 branes, refer to figure 2) which relate the M-strings of 6d SCFTs, the monopole Strings of 5d Supersymmetric Yang-Mills theories (coined m-strings) and the 6d little strings.

The basic result is that there is a direct correspondence between these type of Strings in the sense that monopole and little strings are basically bound states of M-strings!

In [1] it is proposed that the BPS degeneracies of bound-state of M-strings provide the elliptic genus of the moduli space of m-strings in the Nekrasov-Shatashvili limit.

This is based on the fact that BPS states of the 5d gauge theory are related via S-duality to magnetic monopole m-strings of the same theory but also via the appropriate compactification scheme to the M-strings in 6d.

The partition function of the 5d Supersymmetric Yang-Mills theory corresponds to an index that counts the degeneracies of BPS bound-states of W-bosons with instanton particles. Then the S-dual m-string picture can be shown to be the elliptic genus of certain moduli spaces.
The authors of [1] manage to match the partition function of the M-strings with this elliptic genus and thus establishing the correspondence.

To calculate the M-strings partition function they rely on the original work of Vafa et al [4], [5], [6] and use the powerful topological vertex formalism to calculate the topological string partition function of certain elliptically fibered Calabi-Yau 3-folds.

The key point here is that if we manage to geometrically engineer the 5d theory by compactifying M-theory on this CY the topological string partition function will give the partition function of the gauge theory. The gauge theory partition function on the other hand appears in the world-volume of certain (p,q) 5 branes of IIB which are associated to the corresponding M-theory branes configurations where the M-strings appear. From the IIB we can uplift to M-theory in 11 dimensions and thus find the appropriate compactification on the corresponding CY (it is well known that IIB (p,q) brane web diagrams are associated to CY toric diagrams in the context of toric geometry-branes correspondence).

Thus via the 5d theory the topological String partition function on the CY captures the BPS degeneracies of M-strings. This was suggested in the quite famous troika of the "parent" papers [4], [5], [6] where the partition function of the 5d theory was derived via three different methods, the topological string partition function, localization techniques in the 5d field theory and the (2,0) elliptic genus of the M-Strings (refer to figure 1).

Similar arguments are made in [3].

Now how the little strings come into the picture in [2]?

This can be achieved by compactifying the dimension along which the M5 branes are separated. Then even in case the M5 branes coincide (and thus M-strings become tensionless) there can always be M2 branes suspended between the first and last M5 and thus tensile strings; these are the little strings (little string theories are six-dimensional non-local quantum theories with non-gravitational string excitations which at energies far below the string scale flow to the SCFTs counterparts).

Another way for getting LSTs is in the world-volume of NS5 branes in IIA and IIB. The two LSTs (IIa and IIb respectively) are related by T-duality much like the parent string theories. This T-duality is reflected in the M5-M2 branes configuration setup by exchanging suitable compactification circles and in the topological string partition function computation by a fiber-base duality of the corresponding elliptically fibered CY.

There is a illuminating analogy that the authors make with Dp branes and open strings. Here is the relevant excerpt:

They compare multiple M5-branes on a transverse circle with multiple Dp-branes on a transverse circle and they interpret the M-strings (i.e. open M2-branes) as noncritical counterparts of open fundamental strings, while a little string ground state is the noncritical counterpart of a closed fundamental string. In the same way as multiple open fundamental strings on the Dp-branes can form a closed string and move freely in ambient ten-dimensional bulk spacetime, multiple open M2-branes ending on M5-branes can form a closed M2-brane and move freely in eleven dimensional spacetime but what makes the little strings very different from fundamental strings is that, in the decoupling limit (i.e. g-->0) the little strings are confined inside the five-brane worldvolume, i.e. the six dimensional spacetime the little string theories live in.

Now, by checking the compact partition functions (corresponding to little strings) to their non-compact counterparts (corresponding to M-strings) studied in [1], it is found that the counting function of compact BPS configurations can fully be constructed as a linear superposition of the non-compact ones. This implies that little strings can be viewed as bound-states of M-strings.

The overall result is stunning. As the authors state the fact that the little strings can be viewed as bound-states of M-strings means that "for the purpose of BPS counting of IIb little strings at least, one only needs to know BPS excitations of the (2,0) superconformal field theory, which is just the low-energy limit of the IIb little string theory".

As far as I know this is the first time that such relation between little strings and M-strings is proposed. In my mind this will have far reaching implications for the study of the elusive 6d SCFT where M-strings become tensionless.

Much more and other beautiful insights inside.

Figures were taken from [6] plus a bonus, a rare picture of uncle Albert giving a lecture on M5/M2 branes, M-strings and 6d SCFT.

[1] M String, Monopole String and Modular Forms

http://arxiv.org/abs/1503.06983

[2] Instanton-Monopole Correspondence from M-Branes on S1 and Little String Theory

http://arxiv.org/abs/1511.02787

[3] From strings in 6d to strings in 5d

http://arxiv.org/abs/1502.06645v2

[4] M-strings

http://arxiv.org/abs/1305.6322

[5] On orbifolds of M-Strings

http://arxiv.org/abs/1310.1185

[6] M-strings, Elliptic Genera and N=4 String Amplitudes

http://arxiv.org/abs/1310.1325
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Any TENSOR matrix of 4d draw the space time you want. Each terms is a string with itself clock = dimension = pole = string = fold axis, so you has 16 terms or strings, this do the framework into the multiplex topologic math could draw the increible number of 4^4 shapes of differents space time with the same matrix, just change your relativity speed clocking system to measure the fold. But like you have 16 terms in 16 positions you could have 16^16 space time in balance to draw the intradimensional leap, so on now all must know that it must exist the inverse matrix of energy TENSOR field, guessing that appear the other 16^16 if we operate mixing every clocking will get the full 32^32 strings = 2^5 x 2^5 = 4^10, how ever must exist the first reference to begin the secuence it would be 2^20 +1, but like any fold is a infinite we must renormalizate it with impliciti NEPER base, which reach a ladder to up amid infinities.
The TENSOR Matrix of manifold or variety or energy field put in their main diagonal position the reference polar direction or the principal, building the rest with their vectorial product doing it antisimetric if stay up or down, you only has to get the trace with the terms you prefer to paint the string into the infinite variety to proyect it in RIEMMAN - FRIEDMAN continous geometric shapes to begin the construction of SPACE structure to allow the moving of TIME passing to get absolute more power theorie of 6d in 4d making up a 24 dimension.
But if you join the first 4x4 to itself inverse in NEPER imaginary form, knowing that each clocks defines a imaginary base to compute itself maths, you could open the matrix to a 8x8 clocking system then blow your mind up to 64^64 dimension or stings in relativity moving to solve it with a good logic is needing use the CHESS PLAY, with this play each figure is one full fold the table is the GENERAL RELATIVITY SPACE TIME so each one has itself logic physic law to solve their movement problems.
Space is mass and light is time so are inverse meanings that paint a logic structure in algebra.
Now but what is time? Time is the direction of space, because without time is impossible to space exist but time can be without time.
To change relativity time clocking only is need direction in space not speed in moving, so you could change yourself clock without moving, just with a magnetic polar shift should be easier convert from light or from time any mass or space, if you want to move throw time you must convert in light, exist some polar interval in every mass that will do the same effect. With polar phase is easy transform a mass gravity into inertial and back.
;;))) Thanks for share!!

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This is a fantastic development, one I've been hoping for and advocating for for many years. I have very little doubt that if this can get through without being derailed the world will be much safer as a result.

The truth is that there exists a tremendous technological capacity to monitor a nuclear program and obtain strong guarantees of its non-weaponization. Many basic aspects of this verification regime have already been in place with the NPT Safeguards agreement, but from the details that were outlined yesterday, all weak points of this existing regime will be comprehensively addressed, over the entire fuel cycle from the Uranium mines to the enrichment itself. And of course besides the intense verification aspect, the significant technological limitations that have been agreed to will ensure any hypothetical decision to pursue weaponization would take significantly longer than it otherwise would – about a year just to get the requisite enriched uranium.

If there is any legitimate gripe to be had against this agreement, the idea that it will "pave the path" to a nuclear weapon is absolutely not among them.

Parameters for a Joint Comprehensive Plan of Action
https://www.whitehouse.gov/sites/default/files/docs/parametersforajointcomprehenisveplanofaction.pdf
The more you look at the terms of the nuclear framework agreement, the more favorable they look for the United States. That's surprising.
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Mathematics  - 
 
 
"Towards the proof of the geometric Langlands conjecture"
by Hebrew University of Jerusalem
36 videos, 2390 views. Last updated on 7 Jun 2014
Workshop: "Towards the proof of the geometric Langlands conjecture"

https://www.youtube.com/playlist?list=PLT-roSWIpp1FPjDpEXizVfqrvvZ0U0DnM

Workshop Website - https://sites.google.com/site/geometriclanglands2014/
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Way beyond me. Maybe I will have the chance to learn new advanced stuff during the summer. 
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An anomaly (or statistical fluctuation?) worthy of keeping an eye on at LHC13.
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Quantum Field Theory  - 
 
Simple Recursion Relations for General Field Theories
http://arxiv.org/abs/1502.05057

This looks like an exciting development in the program of reformulating quantum field theory in full generality:

On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.

Clifford Cheung, Chia-Hsien Shen, Jaroslav Trnka
Abstract: On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to ...
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Cliff Harvey

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This saga about the BICEP measurement of the would-be primordial B-mode radiation continues to be highly fascinating. I noticed this write up by +Matt Strassler that seems to do a pretty good job of explaining in general terms what’s happened recently as the BICEP and Planck experiments collaborate to compare their data, which he analogizes to two people with differently-colored sunglasses needing to communicate to infer the overall state of the sky.

Even the disappointing development of learning this signal can’t be unambiguously attributed to cosmic inflation has had some definite positive aspects in retrospect. It’s clearly the case that seemingly earth-shattering measurements often turn out to be wrong, for a variety of reasons, and watching it unfold is clearly conducive to developing the right kind of skepticism. It really has demonstrated why science simply works, when the right kind of substantive and careful criticism is brought to bear, as it was in this case. Beside this aspect, the initial surge of excitement around the possible signal compelled me to try to understand inflation in much more detail, especially how it generates the gravitational waves and the B-modes, and to learn about the challenges that confront attempts to make fundamental sense of inflation. So from my purely selfish considerations, this was a definite plus.

Of course it’s not just the positive signals that move science forward anyway, but excluding parameter space also gives tremendously valuable information. The limits from various experiments on the two main parameters characterizing inflation are illustrated at the chart in this post:

http://resonaances.blogspot.se/2015/02/weekend-plot-inflation15.html

It’s mentioned briefly on that blog, and its also been pointed out to me by +Urs Schreiber, that one of the only inflationary models that sits right in the center of the preferred region is the Starobinsky model or R^2 model. This model is so named because instead of depending in the most straightforward way on R – the Ricci curvature scalar of general relativity – it adds a term proportional to R^2 to the fundamental equation (the Einstein-Hilbert action). There are a couple reasons this could be a very fascinating possibility from the standpoint of fundamental physics. For one thing, it can embed into supergravity and which reduces the required size of the initial homogeneous region by a couple orders of magnitude. For more on that I’ll refer you to his post and well-sourced wiki page:

https://plus.google.com/108081058828040288656/posts/T1PimSeZEUH
http://ncatlab.org/nlab/show/Starobinsky+model+of+cosmic+inflation

It’s interesting to note in passing that this possibility, more generally called f(R) gravity, seems to be one of the very few ways that general relativity itself can be modified without instantly running into serious problems of one sort or another. I recall Nima Arkani-Hamed’s slogan “Don’t modify gravity, understand it,” which is particularly intriguing when you learn that R^2 gravity turns out to be physically equivalent to standard general relativity coupled to a new scalar field(!). It is this field that acts as the inflaton.

https://en.wikipedia.org/wiki/F(R)_gravity

It seems that no matter what new experimental signals turn up there will be many fascinating insights to uncover, and even if most of the models we focus on turn out to be wrong, if we get to better understand the space of consistent possibilities along the way, progress will still have been made.

#sciencesunday
© Matt Strassler [February 6, 2015] Unfortunately, though to no one's surprise after seeing the data from the Planck satellite in the last few months, the BICEP2 experiment's claim of a discovery o...
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All experts I had talked to in person always just highlighted that Starobinsky is the best fit, but now there is debate about how much better it is than other fits. Table 6 in http://arxiv.org/abs/1502.02114 has the answer to this, but I am not sure if I know how to read it. I gather we are to look at the third and fourth column for the Bayesian relative likelihood (which is what is amplified in the text on the preceding pages). The notation "ln B_0X" seems not to be introduced in this article, though. Help me: what is the meaning of an entry "-k" in one of these columns? Is it: "the corresponding model is e^-k times as likely as the R^2 model?"
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Co-Founder of Functor AB. I love high energy physics, computer science and giant robots
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Above all, Im passionate about physics. More generally Im interested in furthering understanding of this world at all levels, from particle physics to economics to galactic superclusters and everything in between. I have an artistic side I once cultivated aggressively but has somewhat fallen by the wayside. My inner artist/musician is howling to get let out soon.

As a physics major at WPI, I did a couple major research projects on quantum information theory, specifically on proofs of Bell's theorem involving 4 and 5 qubits. Those projects heavily shaped the way I view physics in general, and its deep foundational issues in particular. Despite the difficulty of the subject, I am convinced there is much more confusion than is necessary, as sloppy reasoning and explanations persist.

At the moment Im mostly spending my time learning quantum field theory and string theory. I am consistently amazed by the unity of physical and mathematical logic, and as I've studied the structure of this logic I've inexorably gravitated to the stringy school of thought. Despite the challenges, it seems to me an essentially indispensable set of puzzle pieces that allow the whole structure to make sense. It appears so deeply enmeshed that extrication just does not seem very likely. But, as a scientist, I of course try to challenge any presumptions of mine aggressively.

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