I have just noticed user Parallaxicality on youtube. He does nice half-hour long features on astro topics, the one below is part 10 of his series on water, and covers history (including the funny bits involving Herschel's discovery of Uranus), and all the facts you want to know about these curious worlds!

https://m.youtube.com/watch?v=WYPpTnGdfyE

#Neptune #Uranus #iceGiants

Amongst other cool info about Saturn, Michele Dougherty presents a very cool view on why Enceladus is very weird interesting. A must see!

https://youtube.com/watch?v=HiyHRp9O3-U

#saturn #cassini #enceladus #tigerStripes

Processing Juno data done right – Gerald Eichstädt, who has been gaining fame for his marvelous processing abilities ever since he's been showing off on Juno data: the latest and best from Jupiter, he talks about his tricks!

I spent a little time looking into these images myself, and while you can achieve astonishing improvements merely playing with image processing software, Gerald's results are so much better, they literally made me drop my jaws.

"what's an unsharp mask?"
https://plus.google.com/+RefurioAnachro/posts/GwhPk9M9xiv

"sharpened clouds"
https://plus.google.com/+RefurioAnachro/posts/1iqLCRmvmLs

He must have been using raw data, I thought by myself, somehow stretching the images to get rid of the subtile shifts in perspective between the red / green / blue filter bands.

Well, that's a grossly simplistic description of what he really does. Using the FITS header data he reconstructs the movement of Juno, its rotation, and that of the planet, effectively putting 3d coordinates on each sample!

That, and other, even fancier stuff! Look at Gerald's slides from the "Juno Ground-Based Support from Amateurs: Science and Public Impact" workshop held May 12-13, 2016:

http://www.ajax.ehu.es/Juno_amateur_workshop/talks/06_03_Junocam_processing_Eichstadt.pdf

...a few months before the spacecraft's orbit insertion on Jul 5th, so the example images show Earth as Juno saw it on its flyby in 2013...

For some reason I missed all this until I watched Tony Darnell's latest "Deep Astronomy" hangout: "Forget Backyard Imaging! Use NASA Telescopes Yourself" (Feb 21)

https://www.youtube.com/watch?v=Fu7JUd2Zx_0

...where Gerald gives a long interview! Last but not least, he likes to hang out on unmannedspaceflight,com, here's their Juno forum:

http://www.unmannedspaceflight.com/index.php?showtopic=8313

See you there!

#imageProcessing of #junoCam images taken by #juno while orbiting #jupiter

Locating the Amplituhedron, decompositions and projections – So, what is the positive region of a Grassmannian, say Gr(2,4)? Remember, any element of this Grassmannian can be written as a matrix:

x1 x2 x3 x4
y1 y2 y3 y4

Each row is a direction in the containing R^4. Let's fix that plane: what can we do to these vectors x, y such that the subspace span(x,y) stays the same? We can rotate one vector towards the other, and when they point into the same direction the space collapses by one dimension. It's even better to think about more than two vectors, the way we were looking at it works equally well with three fingers: our space also degenerates whenever we pass a finger between two others. Try four or more, it's not hard at all! We start in convex position, there's a plane spanned by every pair of fingers.

In high enough dimension we'd be looking at (k-1)-hyperplanes, but let's better call these planes edges: After all, we are dealing with one vector at a time, so the projective principle should hold and we can intersect our lines with a sphere to make all vectors have unit length and appear as vertices. Well, not really lines with a sphere, but rays with that cross-capped half sphere I mentioned last time. However, by restricting to the positive quadrant/octant/2^n-tant we can safely ignore this global topological pecularity.

So, our fingertips are cyclically ordered along a planar convex k-gon. Between them we have (k-1)-hyperplanes represented as lines, giving a complete graph. If k is large enough we also get (k-2)-hyperplanes selecting three vertices, and so on for every codimension space. Indeed, the positive region has facets in each codimension k-j but let's delay that thought. To exchange two adjacent fingers, one vertex has to cross the lines emanating from the other. We might have to flip some sign whenever that happens...

What are we doing? Hm, we have defined the inside of a particular point P in our Grassmannian using a positive representation. To get a full basis for R^n we would need another n-k vectors for which we have the same kind of freedom. All this keeps the subspace, our point P in the Grassmannian, fixed!

So... we've just worked out a quotient construction to describe Grassmannians!

Gr(k,n) = O(n)/O(k)×O(n-k)

O(n) is the orthogonal group, O(k) corresponds to all ways to represent a fixed R^k and O(n-k) is the just same for the space outside. But let's stick to the positive region of O(k)! Here is the positivity condition from last post:

<ViVj...Vk> > 0 for i<j<...<k

Starting with few fingers we've actually looked at a flag made of simplices: a vertex, an edge, a triangle, a tetrahedron, and so on until we reach dimension k. Doing that is called decomposition into Schubert cells and it includes one region for every dimension.

We could instead look at all possible ways to combine the independent vectors from our complete graph picture and end up with some factorial number of cells called matroid stratification, which is surely fine grained enough to answer questions like wether a point P is inside the positive region. It is actually so much information that fully understanding this implies as much wisdom to be gained about algebraic varieties, essentially killing the subject.

In other words: that's hopeless! But what We have been discussing so far suggests another way to triangulate (simplexify?) a polytope: we pick an initial segment, a Schubert cell like before, but we also allow cyclic shifts of our window! Postnikov calls this positroid stratification, if you're clever picking your favorite positive point you end up with matrices like these

1 a b c
0 0 1 d
0 0 0 1

I heard that Young tableaux are a nice thing to think of here...

Oh well, I wished I had more time. As the graph picture suggests you can get any positive Grassmannian from projecting a higher dimensional simplices to a lower-dimensional space. But to get an Amplituhedron you would project the positive Grassmannian yet again to another hyperplane...

The plan for today would have involved developing "plabic graphs" as notation for describing permutations of the vertices-on-a-circle as planar networks, which I could then relate to the logarithmic singularities of the differential operators in the dual cell complex living in physical space... all that and more in the next post. Stay tuned!


Links:

Part 1 of this series on the Amplituhedron:
https://plus.google.com/+RefurioAnachro/posts/gi1yjuSVK6N

As a consolation for the sudden ending of today's post, some stuff that should make you curious about how to really connect this to the physics:

The amplituhedron is a recurring topic in Nima Arkani-Hamed's recent work, and you can find more relevant papers just by searching for him on the arxiv. Or his recent talks on youtube, in which he eloquently entertains with history of physics and deep motivational considerations, while sketching lookouts into the formal side. My investigations were spurred by his 2017 talk "Physics and Mathematics for the End of Spacetime":

https://youtube.com/watch?v=z1-QDXReDTU

Edit: I just found a much shorter talk which also gives a nice and quick overview:

https://www.youtube.com/watch?v=82NatoryBBk

It's especially interesting because Nima mentions that the early Amplituhedron is fancy and primitive at the same time, that the general idea works in many places and that you are more likely to encounter Associahedra!

Arkani-Hamed, and Jaroslav Trnka's 2017 paper: "Unwinding the Amplituhedron in binary", which carried me like a lifeboat until I made it all the way through.

https://arxiv.org/abs/math/1704.05069

I have a video with Trnka in which he goes beyond the scope of my series on the Amplituhedron, but maybe you'd like to see the man in action:

Jaroslav Trnka, "MHV Non-planar Leading Singularities II"
https://www.youtube.com/watch?v=okROYP9nerA

He says his talk should be seen as part two of Freddy Cachazo's talk here, which should be closer to our discussion:

Freddy Cachazo, Planar On-shell Diagrams in N=4
https://www.youtube.com/watch?v=Gzte9sdUP9s

+John Baez' reshare of part 1 lead to a nice discussion here:
https://plus.google.com/+johncbaez999/posts/UEcmzAkwjwq

Additional thanks to +Allen Knutson for his helpful thoughts. He has written about "positroid stratification" here:
http://arxiv.org/pdf/0903.3694.pdf

#amplituhedron #grassmannian #geometry #sciencesunday

Maybe you know Tadashi Tokieda from +Numberphile, here's an hour long public talk he gave at Gresham College, published just a few days ago:

https://www.youtube.com/watch?v=j9SYMA8dhzY

"We want to see how mathematics breaks"

"Where incompatible regimes meet, nature has a tough decision to make"

#public #popMath #video #lecture

Preorder, Poset, and totally ordered – Definitions, examples, and their degeneration.


+John Baez is currently giving a free online course on category theory! It has already started, and attracted over 100 people on the serious end on a spectrum from officially participating as student of the Leiden applied categories seminar fest, ranging to the passer by spectator and time traveling lurker. Be one of them! Here is what looks like a nice and easy blog post, but really constitutes "Lecture 3: […] Posets":

https://forum.azimuthproject.org/discussion/comment/16267/#Comment_16267

I'm not sure if you can still officially apply, and where exactly John will draw a line, but if you do find following the course material and discussions interesting, tell the world about it! I suppose John Baez might appreciate getting mentioned. I'm trying to be cautious because the technology behind this course may not scale to a very large number of participants, and I haven't asked John wether it would be a good idea to invite yet more people... If in doubt, send curses and insults my way.

Here's the remainder of my orderly comment about ordered sets:


We all know about totally ordered sets. They are sets which can be put on a line, like a chain or a necklace, a string of pearls. These can be captured by four axioms like John said:

1. reflexivity: x≤x

2. transitivity: x≤y and y≤z imply x≤z

3. antisymmetry: if x≤y and y≤x then x=y

4. trichotomy: for all x,y we either have x≤y or y≤x.

And the canonical example are the integers with the ordering relation ≤ as we know it. Now, a poset is a structure where 4 doesn't hold, and a preorder is one where neither 3 nor 4 hold. Then what do these axioms mean, what do they do?

Axiom 4 makes sure that every pair of elements is related. It excludes parallel structures and this axiom is what causes the set to collapse to a string. When Axiom 4 doesn't hold we may get a directed acyclic graph (DAG) instead. Well, sort of, as a directed acyclic graph does not necessarily imply transitivity, a poset does. More on transitivity below.

Given a total order (like the integers with ≤) we can degenerate it by adding 'parallel structure' (e.g. an element ε which is just like 3 but not identical to it).

Axiom 3 suppresses cycles. It says that the only cycles are indentity arrows.

Given a total order (like the integers with ≤) we can create a cycle by adding a backward relation like (5≤2). Note that if axiom 3 / antisymmetry does not hold, axiom 4 cannot hold either, as the 'and' is incompatible with the 'either-or'.

What does the transitivity *axiom 2* do? That's a funny one. It implies associativity, and says that if you find two arrows that can be combined because their ends match, you're allowed to talk about the combined arrow. It actually says that the combined arrow has to be among our collected arrows.

You can break this structure by taking out a combined arrow, e.g. insisting that 2 and 4 cannot be compared, even though 2≤3 and 3≤4. Strike out the corresponding arrow, mark it red, you don't have enough gas to drive two routes...

As I said earlier, a poset is usually drawn as a DAG without outlining each and every matching combination of arrows. That's because people (and algorithms) have no trouble to understand that once we find a connected path between two points that those points are also connected. So people will often draw a DAG when they actually want to talk about posets.

#category #theory #course

Hunting for Amplituhedra in positive Grassmannian spaces – Let's delve into a beautiful generalization of projective geometry: into the world of Grassmannian geometry. To get an idea of the territory we'll encounter a few handful of delightful elementary concepts which can be elegantly fused to inspect polytopes inside the positive section of Grassmannians, a region which is much easier to understand than the general beast. Come along, we're starting right now:

Definition: Gr(k,n) is the space of all k-dimensional subspaces in R^n. Although other fields or manifolds might also work.


For k=1 we obtain the real projective space RP^(n-1) or just P^(n-1). Gr(1,n) is the space of all lines in n-dimensional space which pass through the origin. We need the origin because otherwise we don't get a subspace. Gr(1,3) is the space of all such lines in 3-space, and that one looks like a half-sphere with a twisted crosscap.

I've written about RP² here:
https://plus.google.com/+RefurioAnachro/posts/YbgFU8Xzbac

Remember? Any direction in R³ can be given by a 3-vector, and any line consists of all the points we can reach by stretching that vector while maintaing its direction. Since the length of the vector doesn't really matter we get a family of which any member describes a line: an element of, or a point in Gr(1,3).

p = t·v
p = t·(ax + by + cz)
p = [a:b:c] = [q·a:q·b:q·c]

This way to denote points on RP^n is called homogeneous coordinates. Note that we can also stretch [a:b:c] by a negative factor q and still end up describing the same point. Note also that there is no point deserving the name [0:0:0] so at least one coordinate has to be nonzero.


For k=2 it turns out we need two equations or vectors to describe a plane. This pattern holds and we can describe points in Gr(k,n) as a k×n matrix! For example:

(x1 y1 z2)
(x2 y2 z2)

Then Gr(2,3) should be the space of all planes in 3-space (containing the origin). But since every such plane can also be given by a 3-vector (its surface normal), we obtain the same space as Gr(1,3) = P². Which is a two-dimensional surface for which two coordinates seem to be enough, what's going on? Simple scaling won't rid us of the extra parameters...

Meet Plücker coordinates, sometimes called Grassmann coordinates because it was actually Hermann Grassmann who generalized Julius Plücker's notation for arbitrary k. It seems Graßman helped Plücker out a little because it was Plücker who found Gr(2,4): the first Grassmannian that isn't just a projective space.

https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates

Looking at our k×n matrix we can construct a square minor (matrix) of maximal dimension k by striking out the i'th column. Lets call the determinants of our minors delta Δ1 Δ2 Δ3. Remember, such a determinant gives the signed volume of a parallelpiped spanned by the vectors that ended up in our maximally sized minor.

Together with the scaling relation we recover the homogeneous coordinates for Gr(2,3). What about that smallest non-projective Gr(2,4)? It turns out that the scaling relations aren't enough to completely collapse families of coordinates pointing at the same element, we need more Plücker relations!

https://en.wikipedia.org/wiki/Pl%C3%BCcker_embedding

Because of the scaling property we find that Δij = -Δji so we don't need both, and that means that if there were any Δjj they'd have to be zero. Playing some more you might find another relation:

Δ13Δ24 = Δ12Δ34 + Δ14Δ23

You can write this one graphically as a sum of pairs of chords in a circle:

(X) = (||) + (=)

The general form for this Grassmann-Plücker relation looks like this:

Δ_i1,…ik·Δ_j1,…jk = sum_s=1^k Δ_js,i2,…ik·Δ_j1,…j(s-1),is,j(s+1)…jk

And here is the sign-flip for any k:

Δ_i1,…ik = (-1)^sign(w) Δ_iw(1),…iw(k)

For more on relations like these take a look at cluster algebras:
https://en.wikipedia.org/wiki/Cluster_algebra


In the end it is not quite trivial to understand Grassmannians in full generality, but we can make things much simpler by restricting our interest to the positive Grassmannian Gr+(k,n) which is a region or section of the full space that has the shape of a convex polytope!

The positivity condition we want is that all Plücker symbols must be positive. Well, as I said earlier, switching the indices like Δij = -Δji also flips the sign, so we need to pick a canonical order of the indices and demand their symbols to be positive. With the pyhsicist's short-hand for a determinant we might state this as:

<ViVj...Vk> > 0 for i<j<...<k

Or say that all cyclic maximal minors should be positive. As is true for any determinant, if you cyclically shift or "roll" the vectors, and k is odd, it changes sign. When k is even nothing happens, such determinants are cyclically invariant.

The advantage of the positive region is that we can stop worrying about crosscaps and other weird topological features. Our first example, the real projective plane P² is completely symmetric: any of its points is equivalent to all others, and if you cut it into pieces the crosscap is gone! So the positive part of Gr(1,3) is exactly an octant of a sphere!

But there is more. We will be interested in high-dimensional Grassmannians, and that means that we will get a polytope made of of points, lines, faces, hyperfaces up to (n-k)-hypervolume. Like polyhedra the positive Grassmannians have a discrete and combinatorial structure. And as it is with platonic solids, when we understand this structure we have also understood something about the symmetries of the space they live in!

Now let's take a look at Gr+(2,4) shall we?


Example: Gr+(2,4) looks like an octahedron! There are six ways to strike out two of the four columns of a 2×4 matrix, so we get six Plücker symbols.

Δ13
Δ12 Δ23 Δ34 Δ14
Δ24

Here, let me show you in binary which indices are mentioned in each symbol

1010
1100 0110 0011 1001
0101

There is a cycle of four where the ones and zeros are adjacent, and two symbols have no such pairs. It looks like an octahedron! There are two more internal faces, one is the square 1010-1100-0101-0011, the other the square 1010-0110-0101-1001, both are suspended vertically from the pair-free top and bottom vertices.

There are also four more 3-cells (volumes), I remember those been called tetrahedra, but I'm not quite sure.


This is part 1 of a series of three blog posts to explain in detail how the Amplituhedron simplifies the calculation of scattering amplitudes (Feynman diagrams and stuff). I should have everything, but I still need to clean up my notes a bit so they are more fun to read. Here are the titles, I'll add the links here when they become available:

Permutation 2: Locating the Amplituhedron, decompositions and projections
Part 3: Shooting and bagging Amplituhedra for profit


Links:

This post relates to about the first third to half of what we'll need from Alexander Postnikov's paper "Total positivity, Grassmannians, and networks":
https://arxiv.org/abs/math/0609764

That is, if you need more detail or can't wait for the next post. Or try this fun 4-part lecture series by Alexander Postnikov, "Combinatorics of the Grassmanian":
https://youtube.com/watch?v=5m6j_yiepFM
https://youtube.com/watch?v=gIHF8N6QjAI
https://youtube.com/watch?v=Qi9GgAuzWPY
https://youtube.com/watch?v=qd-Tro6ibEM

https://ncatlab.org/nlab/show/positive+Grassmannian

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

Once again, hat tip to +John Baez for making me want to write about this in detail!
https://plus.google.com/+johncbaez999/posts/A8sQSPmETHk