linear algebra - Geometric proof of the Vandermonde determinant?

I would have posted this as a comment, but it's too long for comment, so I post it here. Here is my version of a sketch a geometric proof for nonsingularity of it when $x_i$'s are distinct, but I don't think that I can improve it to find the determinant:

Two manifolds $P$ and $S$ are called to intersect transversally at a point $A$, if the tangent spaces of $P$ and $S$ together span the whole ambient space. Let $lambda_1,ldots, lambda_n$ be distinct real numbers, and let $A$ be the diagonal matrix $A=rm{diag}(lambda_1,ldots,lambda_n)$, and let $S$ be the set of all matrices with the same spectrum as $A$. In a small neighborhood around $A$, $S$ becomes a manifold. Let $P$ be the manifold of all the diagonal matrices of size $n$. The tangent space of $S$ and $P$ at $A$ can be computed and shown that the intersection is transversal.

On the other hand, define a function $f$ that maps any diagonal matrix $B$ (with $x_i$'s on its main diagonal) to $(frac{rm{tr}B}{1},frac{rm{tr}B^2}{2},dots,frac{rm{tr}B^n}{n})$. It can be seen that the Jacobian of $f$ evaluated at $A=rm{diag}(lambda_1,ldots,lambda_n)$ is the Vandermonde matrix, which is nonsingluar if and only if $lambda_i$'s are distinct.

Putting the two pieces above together, and with a little bit of discussion, one can show that $rm{Jac}(f) big|_A$ being nonsingular is equivalent to having $P$ and $S$ intersect transversally at $A$.

One can see the relations of the above approach to the Terry Tao's answer by noting that the tangent space to $S$ at $A$ is the set ${[B,A] : B text{ is a skew-symmetric matrix}}$.

One relation to the powers of $A$ comes form a way of showing the above Jacobian matrix is nonsingular. Note that $$rm{Jac}(f)big|_A = left[ begin{array}{} I_{11} & I_{22} & cdots & I_{nn}\ A_{11} & A_{22} & cdots & A_{nn} \ vdots & vdots & ddots & vdots\ A^{n-1}_{11} & A^{n-1}_{22} & cdots & A^{n-1}_{nn}end{array} right]$$
In order to show the nonsingularity above assume that $left[begin{array}{} alpha_1, ldots, alpha_n end{array} right] rm{Jac}(f)big|_A = 0$. This means if you consider the polynomial $p(x) = sum_{i=1}^{n} alpha_i x^{i-1}$ and let $X = p(A)$, you want to show if $Xcirc I = O$ ($circ$ is the Schur product) then $X=O$, but that is easy to show, since $A$ is diagonal, hence $p(A)$ is, and so $p(x)$ has $n$ distinct roots, but $rm{deg}(p(x))=n-1$, thus $p(x)$ is the zero polynomial.

set theory - Controlling Ultrapowers

Say I start with some a transitive model of a large fragment of ZFC (say enough to run Łoś' Theorem externally) and a specific set x∈M. Now let's say I'm going to pick some M-ultrafilter U on x. By M-ultrafilter, I mean that U measures the subsets of x which are in M.

My question is, by varying U, how much can I affect the ultrapower of M by U? Let's say I limit myself to a U which are countably complete, so that the ultrapower will be wellfounded. If this question is too vague or broad, I'd welcome any interesting examples of things that are possible or impossible.

cv.complex variables - Analogue of the Chebyshev polynomials over C?

I've been driven up a wall by the following question: let p be a complex polynomial of degree d. Suppose that |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ (for some small δ>0). Then what's the best upper bound one can prove on |p(1)|? (I only care about the asymptotic dependence on d and δ, not the constants.)

For the analogous question where p is a degree-d real polynomial such that |p(x)|≤1 for all 0≤x≤1-δ, I know that the right upper bound on |p(1)| is |p(1)|≤exp(d√δ). The extremal example here is p(x)=T_d((1+δ)x), where T_d is the d^th Chebyshev polynomial.

Indeed, by using the Chebyshev polynomial, it's not hard to construct a polynomial p in z as well as its complex conjugate z*, such that

(i) |p(z)|≤1 for all z such that |z|=1 and |z-1|≥δ, and

(ii) p(1) ~ exp(dδ).

One can also show that this is optimal, for polynomials in both z and its complex conjugate.

The question is whether one can get a better upper bound on |p(1)| by exploiting the fact that p is really a polynomial in z. The fastest-growing example I could find has the form p(z)=C_d,δ(1+z)^d. Here, if we choose the constant C_d,δ so that |p(z)|≤1 whenever |z|=1 and |z-1|≥δ, we find that

p(1) ~ exp(dδ²)

For my application, the difference between exp(dδ) and exp(dδ²) is all the difference in the world!

I searched about 6 approximation theory books---and as often the case, they answer every conceivable question except the one I want. If anyone versed in approximation theory can give me a pointer, I'd be incredibly grateful.

Thanks so much!
--Scott Aaronson

PS. The question is answered below by David Speyer. For anyone who wants to see explicitly the polynomial implied by David's argument, here it is:

p_d,δ(z) = z^d T_d((z+z^-1)(1+δ)/2+δ),

where T_d is the d^th Chebyshev polynomial.

Ultra Low Cost Telescope Project

If you're interested in a low cost, but usable telescope that provides decent performance, look into the Galileoscope.

http://galileoscope.org/

It provides 50 mm (2 in) of aperture, at a focal length of 500 mm - so it's an f/10 instrument.

The kit is very easy to assemble, and instructions are provided on the Internet in great detail. Even kids can assemble it, with some supervision.

The objective lens (the primary optics) is an achromat doublet - it's made of two actual lenses combined to reduce chromatic aberration. I was impressed by the quality of that lens, at least when compared to its price - it seems to work as well as optics in instruments that cost several times more. This is one of the main reasons why I recommend this instrument to beginners.

It uses standard 1.25" eyepieces. That means you can either use the eyepieces that come with the kit, or you could borrow any 1.25" eyepiece from another telescope and just use it. This is rare in an instrument this size.

The kit gives you the option to build either a galilean eyepiece, or a 2x barlow. My advice: don't bother with the galilean eyepiece; its field is far too narrow. Instead, build the 2x barlow.

There's another eyepiece in the kit, use it either alone or combined with the barlow.

In theory, the maximum magnification is 100x, but in practice things will get a bit difficult when you push magnification much over 50% of the max value. This is typical for many telescopes.

Important: Like any telescope used for astronomy, this scope will not work well if you just hold it with your hands. You must use some kind of mount. The scope comes with a standard mounting nut that fits any tripod used for photography or astronomy. Use the sturdiest, most rigid tripod you can find. Even a very little amount of wobble will make it very difficult for you to use the scope.

Alternatively, make and use a DIY mount such as this one (results will vary depending on skill and materials used):

https://aquillam.wordpress.com/2010/01/04/a-cheap-tripod-alternative/

At the very least, support one end of the telescope on a fixed, tall object such as a fence; this will make things a little better, but it's not a replacement for a true mount.

Einstein General relativity and bending of space

As Einstein's General relativity suggests that the space
time curves in the presence of matter, which is experimentally
proven too, this phenomenon also explains orbits. My question
is if the presence of mass curves space-time, in which direction
does it bend, if we assume that it bends in one of all possible
directions then what makes the body to choose that particular
direction. Does all the bodies bend the space-time in the same
direction ?

mg.metric geometry - Can any triangle be inscribed in any convex figure?

A more general result is known: if $C$ is any Jordan curve and $T$ is a triangle then there exists a triangle similar to $T$ inscribed in $C.$ Moreover, the vertices of such triangles are dense in $C.$ See the references in the Wikipedia article on the Inscribed Square Problem.

co.combinatorics - Transitivity-related property of finite permutation groups

Let $cal F$ denote the group of all finitely-supported permutations of $mathbb N$.
Say that a finite subgroup $G$ of $cal F$ is singular if $G$ acts transitively on
$lbrace 1,2,3 rbrace$ but no cyclic subgroup
of $G$ acts transitively on $lbrace 1,2,3 rbrace$ (this is equivalent
to saying that some element in $G$ sends $1$ to $2$, another sends $1$ to $3$
but no element of $G$ has all of $1,2$ and $3$ in a single orbit).

The Klein group (products of disjoint transpositions on $lbrace 1,2,3,4 rbrace$)
is in example of such a subgroup.

Question 1 : are there other simple examples of minimal singular subgroups ?
Is there a parametric description of all of them up to isomorphism ?

Question 2 : Denote by ${cal F}(i to j)$ the set of all permutations in $cal F$
sending $i$ to $j$. Say that a permutation $sin {cal F}(1 to 2)$
and a permutation $tin {cal F}(1 to 3)$ are related iff
the subgroup generated by $s$ and $t$ is a minimal singular subgroup of $cal F$.
Given $s$, let $R(s)$ denoted the set of all $t$'s such that $s$ and $t$ are related.
Does $R(s)$ admit a simple description ?

Of course, any answer to question 2 automatically provides an answer to
question 1.

ra.rings and algebras - equivalence of submodules

It is enough to show that

if $Msubseteq mathbb Z^3$ be a subgroup such that $mathbb Z^3/M$ is a cyclic group of order $k$, then there exists $ginmathrm{SL}(3,mathbb Z)$ such that $g(M)=langle (1,0,0),(0,1,0),(0,0,k)rangle$.

Let $Msubseteq mathbb Z^3$ be a subgroup such that $mathbb Z^3/M$ is a cyclic group of order $k$. Then $M$ is free of rank $3$, and there exists $Ain M(3,mathbb Z)$ such that $M=Acdotmathbb Z^3$. Using the Smith normal form, we know that there exists $3times 3$ matrices $P$ and $Q$, invertible over $mathbb Z$, such that $PAQ=D$ with $D=left(begin{smallmatrix}a\&b\&&cend{smallmatrix}right)$ and $amid bmid c$.
Then $PM=PAQmathbb Z^3=Dmathbb Z^3$.

It follows that $Pinmathrm{SL}(3,mathbb Z)$ is such that $PM$ is generated by $(a,0,0)$, $(0,b,0)$ and $(0,0,c)$ with $amid bmid c$. Since $mathbb Z^3/g(M)$ is cyclic of order $k$, we must have $a=b=1$ and $c=k$. This tells us that the claim above is true.

(I've done everything at the level of generality which your problem needs, and I'll leave the fun of finding the correct general statement for you...)

ag.algebraic geometry - Covers of the projective line over Z and arithmetic Grauert-Remmert

This question is the two-dimensional analogue of Etale coverings of certain open subschemes in Spec O_K

There I asked if one could characterize in a way certain covers of $textrm{Spec} O_K$. As Cam Mcleman answered, this is basically done by the Galois group of the maximal extension unramified outside $D$. A covering of $U$ is of the form $O_L[frac{1}{D}]$, where $L$ is any extension of $K$.

Here I would like to ask the same question, only now for $X=mathbf{P}^1_{mathbf{Z}}$.

Let $D$ be a normal crossings divisor on $mathbf{P}^1_{mathbf{Z}}$ and let $U$ be the complement of its support.

Q1. Is there an "equivalence of categories" as Georges Elencwajg mentions in his answer for the analytic case. (See above link.) Basically, is there an arithmetic Grauert-Remmert theorem?

Q2. What is known about the etale fundamental group in this case? Is it "finitely generated"? Has anybody studied the maximal pro-p-quotients of these groups?

Q3. The analytic analogue would be to consider the same question for $mathbf{P}^1_{mathbf{C}} times mathbf{P}^1_{mathbf{C}}$.

Q4 Lars (see above link) mentions a result for $mathbf{P}_{mathbf{Q}}^1$. Is there something similar for $mathbf{P}^2_{mathbf{Q}}$?

When do fibre products of smooth manifolds exist?

It suffices for one of the two maps to be a submersion for the fibre product to exist. In fact, if $f$ and $g$ are maps from $X$ to $Y$, the fibre product is the inverse image of the diagonal in $Y times Y$ under the map $f times g : Xtimes X to Y times Y$. So a sufficient condition to have a nice fibre product is that $f times g$ be transverse to the diagonal.

The next best thing to transverse intersection is clean intersection, as Ben has pointed out. Another definition of clean intersection of $A$ and $B$ in $X$ (more easily checked than the one about the local normal form) is that the intersection of $A$ and $B$ is a submanifold $C$, and that the tangent bundle to $C$ is the intersection of the tangent bundles to $A$ and $B$.

star - What is "Median Stellar Mass"?

just check what 'median' means. the median stellar mass is such that half of all stars have lower and the other half hihger mass. it has nothing to do with the logarithm. I havn't seen the term (median stellar mass) in the scientific literature.

---

Looking at the paper you're referring to, they never use the expression "median stellar mass". Moreover, this study is not concerned about the masses of individual stars, but with the total stellar mass of a galaxy, which they donote by $M$. They then consider the typical or characteristic value for $M$ (as obtained by fitting a Schechter function to the observed distribution of $M$), and denote it by $M^ast$. Since this is a rather large number, of the order of $sim10^{10}$M$_odot$, they prefer to use its logarithm to base ten: $$logleft(frac{M^ast}{mathrm{M}_odot}right).$$
I think the paper explains all that quite clearly and am puzzled by your difficulty.

Is there a name for this topology?

This topology is rather combinatorial in nature. You certainly could call it well-understood, but I would not expect it to have a name in the context of topology. It is essentially a special case of a concept in combinatorics known as an order ideal, or if you like the set of all order ideals. The set of all order ideals of a partially ordered set is indeed a topology, but not one that looks very topological. On the one hand, it does not satisfy any of the separation axioms other than $T_0$. On the other hand, it satisfies an even stronger axiom than finite intersection: The intersection of any collection of open sets is open. The set of order ideals is better understood as a distributive lattice than as a topology, even though it is both.

We can clean things up a little as follows. First, if we switch to the image of $f$, we can let $B_S$ instead be the union of images of $S$ including $S$ itself. Second, in each periodic orbit of $f$, any open set that contains one point contains the other one, so we might as well collapse that orbit to a point. After that, we can say that $x prec y$ when $x = f^n(y)$. This defines a partial ordering on $X$ with respect to which the topology is just the set of order ideals.

orbit - Does Sun have a reflection on Earth?

The new Google Maps¹ presents an actual view of Earth, with the current position of Sun illuminating half part of Earth in real time. It is quite an exquisite view.

My question is based on the following image:

As you can see, Sun's reflection from southern Atlantic Ocean looks very charming. But does it really happen like that? If we travel far² from Earth, can we actually see Sun's reflection or it is just something Google added for aesthetics?

---

¹: Yes, I don't like most of its new features too, especially when they broke several of the old ones. But that's a separate discussion.

²: Although, the camera's supposed position would be close to geostationary orbit, I guess the reflection can also been seen as close as ISS or Hubble.

soft question - Why do so many textbooks have so much technical detail and so little enlightenment?

I also suffer from this problem -- I used to learn best from books, but in grad school, I'm having real trouble finding any book I can learn from in some subjects. There are a few reasons for this sad state of affairs that come to my mind. I'll list them first and expand on them below.

1. Providing real enlightenment well is very, very hard, and requires a very intimate relationship with a subject.




2. Different mathematicians need vastly different motivations for the same subject.




3. Mathematics needs to age before it can be presented well.




4. Good writing is not valued enough in the mathematical community.

The first of these is true to such a strong degree that it surprises me. Even for well-established subjects, like undergraduate mathematics, where there are a million mathematicians who know the subject very well, I find that all the really good books are written by the true titans of the field -- like Milnor, Serre, Kolmogorov, etc. They understand the underlying structure and logical order of the subject so well that it can be presented in a way that it basically motivates itself -- basically, they can explain math the way they discovered it, and it's beautiful. Every next theorem you read is obviously important, and if it isn't then the proof motivates it. The higher-level the subject, the fewer the number of people who are so intimate with it that they can do this.
It's interesting how all the best books I know don't have explicit paragraphs providing the motivation - they don't need them. (Of course there are exceptions -- some amazing mathematicians are terrible writers, and there are people with exceptional writing ability, but the point stands).

Regarding the second point, different people want completely different things for motivation. The questions that pop into our heads when we read the theorems, the way we like to think, the kind of ideas we accept as interesting, important, etc., is different for all of us. For this reason, when people try to explicitly describe the motivation behind the subject they almost always fail to satisfy the majority of readers. Here, I'm thinking of books like Hatcher, Gullemin & Polluck, Spivak, etc., where some people find that they finally found the book that explains all the motivation perfectly, and others are surprised at the many paragraphs of text that dilute the math and make finding the results/proofs they want harder and reading slower.
At the same time, the amount of effort each of these authors must have spent on organization of their book seems absolutely immense. For this reason, unless there are 50 books written on a subject, the chances that you will find a book that seems well-motivated for you are low.

The third reason is simple: it takes time for a new subject to stop being ugly, for people to iron out all the kinks, and to figure out some accepted good way to present it.

Finally, it seems to me that good writing, especially expository writing, is not particularly valued in the community, and is valued less now than it was before. Inventing new results seems to be the most respectable thing to do for a mathematician, teaching is second-best, and writing has the third place. People like Hatcher & co. seem to be rare, and I don't know of many modern titans of mathematics who write any books at all, especially on a level more elementary than their current research.

---

So what do we do? I think what algori said in his answer is the only way to go.

fa.functional analysis - Bibliography for topologies defined by a family of seminorms

I just had a look at

Topological Vector Spaces,
Distributions and Kernels

by Francois Treves. It is divided into three parts:

I Topological Vector Spaces. Spaces of Funtions

• covering: basic material about locally convex spaces and Frechet spaces (with a lot of examples)

II Duality, Spaces of Distributions

• topologies on Duals, transposes of linear maps, convolution, barreled spaces

III Tensor Products. Kernels

• injective and projective tensor products and their relation to bilinear forms, nuclear spaces, nuclear mappings, Schwartz kernel theorem and applications

From the first sight, this looks like a good place to start if you are already familiar with functional analysis on Banach and Hilbert spaces.

lo.logic - Set theory and Model Theory

Your worries arise from asymmetry between how you view ordinary mathematics and how you view logic and model theory.

If it is the business of logic and model theory to provide foundations for the rest of mathematics then, of course, logicians and model theorists will not be allowed to use mathematical methods until they have secured them. But how might they accomplish this? The more we think about it, the more it becomes obvious that "securing the foundations of mathematics", whatever that means, is a task for philosophers at best and a form of mysticism at worst.

It is far more fruitful to think of logic and model theory as just another branch of mathematics, namely the one that studies mathematical methods and mathematical activity with mathematical tools. They follow the usual pattern of "mathematizing" their object of interest:

• observe what happens in the real world (look at what mathematicians do)


• simplify and idealize the observed situation until it becomes manageable by mathematical tools (simplify natural language to formal logic, pretend that mathematicians only formulate and prove theorems and do nothing else, pretend that all proofs are always written out in full detail, etc.)


• apply standard mathematical techniques

As we all know well, the 20th century logicians were very successful. They gave us important knowledge about the nature of mathematical activity and its limitations. One of results was the realization that almost all mathematics can be done with first-order logic and set theory. The set-theoretic language was adopted as a universal means of communication among mathematicians.

The success of set theory has lead many to believe that it provides an unshakeable foundation for mathematics. It does not, at least not the mystical kind that some would like to have. It provides a unifying language and framework for mathematicians, which in itself is a small miracle. Always remember that practically all classical mathematics was invented before modern logic and set theory. How could it exist without a foundation so long? Was the mathematics of Euclid, Newton and Fourier really vacouous until set theory came along and "gave it a foundation"?

I hope this explains what model theorists do. They apply standard mathematical methodology to study mathematical theories and their meaning. They have discovered, for example, that however one axiomatizes a given body of mathematics in first-order logic (for example, the natural numbers), the resulting theory will have unintended and surprising interpretations (non-standard models of Peano arithmetic), and I am skimming over a few technical details here. There is absolutely nothing strange about applying model theory to the axioms known as ZFC.

Or to put it another way: if you ask "why are model theorists justified in using sets?" then I ask back "why are number theorists justified in using numbers?"

soft question - books well-motivated with explicit examples

It is ultimately a matter of personal taste, but I prefer to see a long explicit example, before jumping into the usual definition-theorem path (hopefully I am not the only one here). My problem is that a lot of math books lack the motivating examples or only provide very contrived ones stuck in between pages of definitions and theorems. Reading such books becomes a huge chore for me, even in areas in which I am interested. Besides I am certain no mathematical field was invented by someone coming up with a definition out of thin air and proving theorems with it (that is to say I know the good motivating examples are out there).

Can anyone recommend some graduate level books where the presentation is well-motivated with explicit examples. Any area will do, but the more abstract the field is, the better. I am sure there are tons of combinatorics books that match my description, but I am curious about the "heavier" fields. I don't want this to turn into discussion about the merits of this approach to math (i know Grothendieck would disapprove), just want to learn the names of some more books to take a look at them.

Please post one book per answer so other people can vote on it alone. I will start:

Fourier Analysis on Finite Groups and Applications by Terras

PS. this is a similar thread, but the main question is different.
How to sufficiently motivate organization of proofs in math books

ca.analysis and odes - Is there a notion of "Morse index" for geodesics in a manifold with indefinite metric that is well-behaved under cutting and gluing?

More generally, I'm interested in the situation of Lagrangian mechanics. And actually my question is local, so you can work on $mathbb R^n$ if you like. I will begin with some background on Lagrangian mechanics, and then recall the notion of Morse index in the positive-definite case. My final question will be whether there is a similar story in the indefinite case.

Background on Lagrangian mechanics

Let $mathcal N$ be a smooth manifold. A Lagrangian on $mathcal N$ is a function $L: {rm T}mathcal N to mathbb R$, where ${rm T}mathcal N$ is the tangent bundle to $mathcal N$. I will always suppose that the Lagrangian is nondegenerate, in the sense that when restricted to any fiber ${rm T}_qmathcal N$, the second derivative $frac{partial^2 L}{partial v^2}$ is everywhere invertible (for $vin {rm T}_qmathcal N$, $frac{partial^2 L}{partial v^2}(v)$ makes sense as a map ${rm T}_v({rm T}_qmathcal N) to {rm T}_v^*({rm T}_qmathcal N)$). In this case, the Euler-Lagrange equations
$$ frac{partial L}{partial q}bigl(dotgamma(t),gamma(t)bigr) = frac{rm d}{{rm d}t}left[ frac{partial L}{partial v}bigl(dotgamma(t),gamma(t)bigr)right]$$
define a nondegenerate second order differential equation for $gamma$ a parameterized path in $mathcal N$.

An important example is when $mathcal N$ is equipped with a (semi-)Riemannian metric, in which case the Euler-Lagrange equations specify that $gamma$ is an arc-length-parameterized geodesic.

By nondegeneracy, a solution to the Euler-Lagrange equations is determined by its initial conditions, a point $bigl(dotgamma(0),gamma(0)bigr) in {rm T}mathcal N$. Thus, the Euler-Lagrange equations determine a smooth function $text{flow}: {rm T}mathcal N times mathbb R to mathcal N times mathcal N times mathbb R$, which sends an initial condition and a duration to the triple (initial location, final location, duration). Actually, the function is only defined on an open neighborhood of the zero section of ${rm T}mathcal N times mathbb R$.

I will use the shorthand path to mean a smooth function $[0,T] to mathcal N$, i.e. a parameterized path of finite duration. A path is classical if it solves the Euler-Lagrange equations, so that classical paths are in bijection with points in (that open neighborhood of) ${rm T}mathcal N times mathbb R$. A classical path is nonfocal if near it the function $text{flow}: {rm T}mathcal N times mathbb R to mathcal N times mathcal N times mathbb R$ is a local diffeomorphism. Thus a choice of nonfocal classical path of duration $T$ determines a function $gamma: mathcal O times [0,T] to mathcal N$, where $mathcal O subseteq mathcal N times mathcal N$, such that for each $(q_0,q_1) in mathcal O$, the path $gamma(q_0,q_1,-)$ is classical with $gamma(q_0,q_1,0) = q_0$ and $gamma(q_0,q_1,T) = q_1$. Given a nonfocal classical path $gamma$, the corresponding Hamilton principal function $S_gamma: mathcal O to mathbb R$ is given by:
$$S_gamma(q_0,q_1) = int_{t=0}^T Lleft( frac{partial gamma}{partial t}(q_0,q_1,t), gamma(q_0,q_1,t)right){rm d}t$$

Finally, given a classical path $gamma: [0,T] to mathcal N$, there is a well-defined second-order linear differential operator $D: gamma^*{rm T}mathcal N to gamma^*{rm T}^*mathcal N$, given by:
$$ D_gamma[xi] = -frac{rm d}{{rm d}t} left(frac{partial^2 L}{partial v^2} frac{{rm d}xi}{{rm d}t}right) - frac{rm d}{{rm d}t}left( frac{partial^2 L}{partial q partial v}xiright) + frac{partial^2 L}{partial v partial q} frac{{rm d}xi}{{rm d}t} + frac{partial^2 L}{partial q partial q}xi$$
The second derivatives of $L$ are evaluated at $(dotgamma(t),gamma(t))$ and act as "matrices"; in particular, $frac{partial^2 L}{partial v partial q}$ and $frac{partial^2 L}{partial q partial v}$ are transpose to each other. These individual matrices require local coordinates to be defined, but $D_gamma$ is well-defined all together if $gamma$ is classical.

Then $gamma$ is nonfocal iff $D_gamma$ has trivial kernel among the space of sections of $gamma^*{rm T}mathcal N to [0,T]$ that vanish at $0,T$.

Of course, really what's going on is that for $(q_0,q_1) in mathcal N times mathcal N$, the space of paths of duration $T$ that start at $q_0$ and end at $q_1$ is an infinite-dimensional manifold. Using the Lagrangian $L$ we can define an action function on this manifold. The Euler-Lagrange equations assert that a path $gamma$ is a critical point for this function, $S_gamma$ is the value of the function, and the operator $D_gamma$ is the Hessian at that point.

The Morse index of a classical path

Recall the following fact. Let $mathcal V$ be a vector space and $D: mathcal Votimes mathcal V to mathbb R$ a symmetric bilinear form on $mathcal V$. Then any subspace $mathcal V_- subseteq mathcal V$ that is maximal with respect to the property that $D|_{mathcal V_-}$ is negative-definite has the same dimension as any other such subspace. This dimension is the Morse index $eta$ of the operator $D$ acting on $mathcal V$.

Recall that there is a canonical pairing between sections of $gamma^*{rm T}mathcal N$ and sections of $gamma^*{rm T}^*mathcal N$ (pairing the vectors and covectors for each $tin [0,T]$ gives a function on $[0,T]$, which we then integrate). By composing with this pairing, we can think of the operator $D_gamma$ as a bilinear form on $gamma^*{rm T}mathcal N$, and it is symmetric on the space of sections of $gamma^*{rm T}mathcal N$ that vanish at the endpoints $0,T$.
Given a nonfocal classical path $gamma$, its Morse index $eta(gamma)$ is the Morse index of $D_gamma$ acting on such endpoint-zero sections.

Let $L$ be a Lagrangian on $mathcal N$, and assume moreover that the matrix $frac{partial^2 L}{partial v^2}$ is not just everywhere invertible but actually everywhere positive-definite (this is a coordinate-independent statement, even though the value of the matrix depends on coordinates). Then the Morse index of any nonfocal classical path is finite. (And conversely: if $frac{partial^2 L}{partial v^2}$ is not positive-definite along $gamma$, then the Morse index as defined above is infinite.)

Moreover, suppose that $gamma: [0,T] to mathcal N$ is classical and nonfocal and choose $T' in [0,T]$ such that the obvious restrictions $gamma_1: [0,T'] to mathcal N$ and $gamma_2: [T',T] to mathcal N$ are both nonfocal. Let $S_{gamma_1}$ and $S_{gamma_2}$ be the corresponding Hamilton-principle functions. Then $q = gamma(T')$ is a nondegenerate critical point for $S_{gamma_1}(gamma(0),-) + S_{gamma_2}(-,gamma(T))$. Define its Morse index of $eta(q)$ to be the number of negative eigenvalues of the Hessian of $S_{gamma_1}(gamma(0),-) + S_{gamma_2}(-,gamma(T))$ at $q$. Then the following is a fact:

$eta(gamma) = eta(gamma_1) + eta(q) + eta(gamma_2)$

My question

Is there a similar story when $frac{partial^2 L}{partial v^2}$ is invertible but indefinite? More precisely:

Suppose that you are given a nondegenerate (but not convex on fibers) Lagrangian $L$ on a manifold $mathcal N$. Is there a way to assign a (finite) number $eta(gamma)$ to each classical nonfocal path $gamma$ such that $eta(gamma) = eta(gamma_1) + eta(q) + eta(gamma_2)$, where $gamma,gamma_1,gamma_2$ are as above, and $eta(q)$ is the usual Morse index of $S_{gamma_1}(gamma(0),-) + S_{gamma_2}(-,gamma(T))$?

The starting idea would be to recall the fact that in the positive-definite case, $eta(gamma)$ counts with multiplicity the number of times $T' in [0,T]$ such that the restriction $gamma|_{[0,T']}$ is focal. (The multiplicity is given by the rank of the differential of the flow map.) This counting still makes sense in the indefinite case. So perhaps it can be used, and the signature of $S_{gamma_1}(gamma(0),-) + S_{gamma_2}(-,gamma(T))$ can be added in by hand?

dg.differential geometry - Deriving symmetries of a Gauge theory

Hello,
I don't know if this is a good place for exposing my problem but I'll try...

I have a gauge theory with action:

$S=int;dt L=int d^4 x ;epsilon^{munurhosigma} B_{munu;IJ} F_{munu}^{;;IJ} $

Where $B$ is an antisymmetric tensor of rank two and $F$ is the curvature of a connection $A$ i.e: $F=dA+Awedge A$, $mu,nu...$ are space-time indices and $I,J...$ are Lie Algebra indices (internal indices) I would like to find its symmetries. So I rewrite the Lagrangian by splitting time and space indices ${mu,nu...=0..3}equiv {O; i,j,...=1..3}$ I find:

$L = int d^3 x;(P^i_{;IJ}dot{A}_i+B_i^{,IJ}Pi^i_{,IJ}+A_0^{;IJ}Pi_{IJ})$

Where $dot{A}_i = partial_0 A_i$, $P^i_{;IJ} = 2epsilon^{ijk}B_{jk,IJ}$ is hence the conjugate momentum of $A_i^{,IJ}$

$B_i^{,IJ}$ and $A_0^{;IJ}$ being Lagrange multipliers we obtain respectively two primary and two secondary constraints:

$Phi_{IJ} = P^0_{;IJ} approx0$

$Phi_{;;IJ}^{munu} = P^{munu}_{;;IJ} approx0$

$Pi^i_{,IJ} = 2epsilon^{ijk}F_{jk,IJ} approx0$

$Pi_{IJ}=(D_i P^i)_{IJ} approx0$

Where $P^0_{;IJ}$ are the conjugate momentums of $A_0^{,IJ}$ and $P^{munu}_{;;IJ}$ those of $B_{munu}^{;;IJ}$. Making these constraints constant in time produces no further constraints.

Whiche gives us a general constraint:

$Phi = int d^3 x ;(epsilon^{IJ}P^0_{,IJ}+epsilon_{munu}^{IJ};P^{munu}_{;;IJ}+eta^{IJ}Pi_{IJ}+eta_i^{IJ}Pi^i_{;IJ})$

Each quantity $F$ have thus a Gauge transformation $delta F = {F,Phi}$ where ${...}$ denotes the Poisson bracket.

Knowing that this theory have the following Gauge symmetry:

$delta A = Domega$

$delta B = [B,omega]$

Where $omega$ is a 0-form, I would like to retrieve these transformations using the relation below. (where $Phi$ is considered as the generator of the Gauge symmetry) but my problem is that I don't know how to proceed, I already did this with a Yang-Mills theory and it worked... but for this theory it seems to le intractable! Someone to guide me?

Can anyone give me a good example of two interestingly different ordinary cohomology theories?

For any space that has the homotopy type of a CW complex, its cohomology is determined purely formally by the Eilenberg-Steenrod axioms, so a counterexample is necessarily some reasonably nasty space. Here's an example you can see with your bare hands: consider the space $X={1,1/2,1/3,1/4,...,0}$. Now 0th singular cohomology is exactly the group of $mathbb{Z}$-values functions on your space which are constant on path-components, so $H^0(X)=X^mathbb{Z}$ (an uncountable group) naturally for singular cohomology. On the other hand, 0th Cech cohomology computes global sections of the constant $mathbb{Z}$ sheaf, i.e. locally constant $mathbb{Z}$-valued functions on your space. These must be constant in a neighborhood of 0, so the Cech cohomology $H^0(X)$ is actually free of countable rank, generated (for example) by the functions $f_n$ that are $1$ on $1/n$, $-1$ on $1/(n+1)$, and $0$ elsewhere, plus the constant function $1$.

I should add that topologists don't actually care about such examples. The point of the Eilenberg-Steenrod axioms is to show that cohomology of reasonable spaces is determined by purely formal properties, and these formal properties are actually much more useful than any specific definition you could give (the only point of a definition is to show that the formal properties are consistent!). What is of interest is when you remove the dimension axiom to get "extraordinary" cohomology theories, which Oscar talks about in his answer.

ag.algebraic geometry - Understanding different Q-models of a curve over C

Yes.

Roughly, the idea is that once your base object $X/B$ is fixed then for any object $Y$ you can consider the sheaf $Iso(X,Y)$ of isomorphisms from $X$ to $Y$. This has an action of the sheaf $Aut(X)$ and on any open cover $U$ where $Y|_U cong X|_U$, making a choice of such an isomorphism gives an isomorphism of $Aut(X)(U)$-sets $Iso(X,Y)(U) cong Aut(X)(U)$.

The difference from the standard argument in this case is that you need to use the etale topology instead of the Zariski topology. The etale covers of $mathbb{Q}$ are finite field extensions $F$ and so the statement that $Y$ is "locally" equivalent to $X$ means that $Y_F cong X_F$ for a large enough field extension $F$.

puzzle - Ulam spiral coordinate system

Inspired by a Project Euler problem, I recently started playing around with Ulam spirals. My first thought was that an Ulam spiral could be a (rather useless) coordinate system, and how I might be able to convert from "Ulam coordinates" (i.e. just the number on the spiral at a given point) to rectangular coordinates and vice versa.

So the question is: Is there a single function that would return the number on the Ulam spiral given (x,y)? And also (and perhaps more of a challenge), one to convert back?

I'm just a sophomore computer science student, and not terribly competent in math compared to the people on these site, sadly (I'm in Calc III right now). I was able to work out four separate equations to find the number on the spiral given an ordered pair, but that's one for each of four sections in between the 'diagonal axes' (I'm not sure what to call them). Unfortunately, I have yet to make this more simple. Any ideas?

reference request - Is simple non-holonomic D-module a local concept?

For the last question [edit: this part of the question has now been removed..], the relation of constructible sheaves with the Fukaya category is the subject of the paper Microlocal branes are constructible sheaves by David Nadler, and its predecessor Constructible Sheaves and the Fukaya Category by Nadler and Eric Zaslow. As for D-modules in general, there are proposals in the physics dictionary, starting in work of Anton Kapustin (see e.g. here) and perhaps best summarized in the seminal paper of Kapustin-Witten. One has to be careful though what exactly you mean: to get nonholonomic D-modules, the idea is to look at coisotropic but non-Lagrangian A-branes, ie not the Fukaya category but an enlarged version that has yet to be defined mathematically. Also even in the Lagrangian case there are two really different things one can mean by Fukaya category of the cotangent bundle, depending on how you treat behavior at infinity. Nadler uses a refined version that corresponds precisely to constructible sheaves. The one that arises in the physics, and in most of the math literature (eg work of Abouzaid) is the "wrapped Fukaya category" -- this corresponds more closely to modules over infinite order differential operators - in particular delta functions at distinct points on the base become identified (by exponentiating translation)! so this is quite far from what you might want in say representation theory.

As to your other question, I'm not sure I understand the issue -- D-modules form a perfectly nice stack, i.e. they glue together, independently of holonomicity (one doesn't need Riemann-Hilbert to see this, it's immediate from the definition as modules over a sheaf of algebras - in fact from my, perhaps naive, point of view it's easier to see perverse sheaves form a stack by thinking of them as D-modules, but that's certainly not necessary either). So the gluing formalism you ask for is just sheaf theory if I understand correctly: D-modules on a cover plus gluing data define a D-module on the total space. Same holds on the (dg) derived category level. This is in fact how you even define what a D-module on a stack is (in terms of smooth covers), as is discussed at great length in the last "chapter" of the text by Beilinson-Drinfeld on Quantization of Hitchin's Hamiltonians.

Now for the question of describing simple modules, that's a seriously tricky issue that I know nothing about -- I believe Toby Stafford was the first to show that there even are simple but nonholonomic modules over rings of differential operators, and I would look at his papers for insight.

Jupiter FM - What are practical and inexpensive ways for the amateur detection of signals from Jupiter, especially of the transit of her moons?

@Arne is right in his answer about two things, that the most suitable frequency for Jovian amateur radio is 20.1 MHz, and that this is a 15 meter wavelength. However, the antenna can actually be half the wavelength, and amateur radio astronomers have had good results listening to all kinds of Jovian sounds, including detecting occultations of its many moons as they produce change in frequency due to the Doppler shift when passing in front of it, amplification due to echo of Jupiter's own radio wave signature when close beside it from the observer's vantage point, and other effects causing radio wave frequency and amplitude changes with a simple dual dipole array that might look something like this:

   

        ^{The Jove dual dipole array. Dipoles are suspended between PVC masts. Signals from the dipoles go to a power combiner and
        then to the receiver. Source: Amateur Radio Astronomy Projects — Radio Signals from Jupiter (PDF)}

Now this dual dipole antenna is more or less a simple two-piece of roughly half the wavelength of coaxial cable, stripped of insulation at the dipole ends and run dipoles parallel to each other at roughly 6.1 meters (20 ft) apart suspended over PVC masts. That is perfectly acceptable for nearly any amateur radio astronomer in both space and the price of the parts needed. There are also ways to help oneself with a lot smaller antenna, which I'll mention a bit later. Let's first clear the frequency range discussion with this really informative quote from Radio Receiver for Jupiter webpage (based on NASA's Radio JOVE project):

The peak of Jupiter signals occur around 10 MHz. Still this frequency
is not that suitable, as it is very near to Ionospheric cutoff. The
best suitable frequencies are in between 18 to 22 MHz, as the chances
of getting emissions are more. In practice 18.7 MHz, 20.1 MHz, 22.3
MHz are common. The frequencies above 30 MHz are not suitable because
of less strength. The frequency of 20.1 MHz is used for this project,
as the probability of getting emission is high. So the entire receiver
is designed considering 20.1 MHz as the operating frequency.

Antenna is of course merely one part of the story. Second thing that is required is the receiver. NASA is sponsoring Radio JOVE project with an assembly manual for roughly 100 electronic components and pieces of hardware worth RJ1.1 Receiver (Radio Jove 1.1 Receiver) that can be almost completely assembled out of parts ordered from Radio Shack (it even includes RS part numbers). I'll add a photograph and a few other links, and you'll have to be on your own then:

   

   ^{Self-assembled Radio Jove Receiver's front plate with two turning knobs for volume and tuning. Source: KB0LQJ Observatory}

Some relevant links for building Radio Jove (or Jupiter FM, if you like, it will be your own receiver so name it as you wish), starting with already mentioned ones:

• Amateur Radio Astronomy Projects — Radio Signals from Jupiter (PDF)

  
  
  

  
  

  Ten years ago a group of (mostly)
  University of Florida graduates working at
  NASA conceived an educational outreach
  program known as Radio Jove. The idea was
  to build an inexpensive radio telescope kit
  suitable for detecting signals from Jupiter.
  The Jove receiver (Figure 2) is a simple
  direct-conversion design operating over a
  few hundred kilohertz range centered at
  20.1 MHz.

  
  




• NASA's Radio JOVE project (PDF)

  
  
  

  
  

  The site is currently down due to US government shutdown, so here's a Google cached version that sadly doesn't come with in-document images

  
  




• KB0LQJ Observatory Amateur Radio Astronomy — Radio Jove observations

  
  
  

  
  

  For my home observatory, I started with the Radio Jove receiver from NASA’s Radio Jove Project. This was a fairly easy-to-build kit with great directions, not just for the receiver but also the antenna setup and installation. Unfortunately, I don’t have enough space to put up a phased antenna array. Also, I’m bordered by power lines on the north and south sides of my property. Also, since I’m in an urban area, I knew that I’d pick up a lot of noise. Not to worry. Solar observations are also quite interesting, and since the Sun is such a good signal source (especially so far this year), I opted for an antenna in my attic.

  
  




• Radio Receiver for Jupiter

  
  
  

  
  

  This website contains the Technical details of the antenna and receiver used for receiving natural radio emissions from Jupiter at 20.1MHz. The antenna and receiver discussed in this site is based on the design given by NASA Radio Jove Program.
  Natural radio emissions from Jupiter or from Sun are detected by using a dual dipole array as an antenna and with a sensitive receiver. The RF voltage developed at the antenna terminals is amplified with the RF amplifier and converted in to audio frequencies using a mixer. The audio signal thus generated is recorded on PC through a sound card in ‘wav’ format. Also a strip-chart recorder software is available to generate the strip-chart of data coming through sound card.

  
  

So there are indeed many ways to build your own antenna and receiver out of affordable and easily obtainable electronic parts, and some of these websites listed above will help you through the process of self-assembly, even providing some tricks how to do it easier, like for example assembling a smaller, in-house antenna array.

Now, one other thing that all these websites mention is also using various PC software that lets you analyze through computer's sound card received Jovian radio sounds, but since there's many different and free solutions out there, including NASA's own that runs on Windows PCs, I'll let you discover those on your own. Here's one page listing many links, to get you started. More computer savvy of you might even write your own software for this purpose, this is afterall.

And if anyone is wondering what such self-assembled radio receivers and antennas are capable of, here's a link to a collection of various Jovian sounds in the radio wavelengths on Astrosurf.com, and another one as a collection of solely amateur radio astronomy sounds of Jupiter and its moons (scroll a bit down to the list of recordings). And here's a short description of various sound types one might hear:

• Chorus (quote source: The University of Iowa Department of Physics and Astronomy)

  
  
  

  
  

  Chorus consists of brief, rising-frequency tones that sound like the chorus of birds singing at sunrise, hence the name "chorus" or "dawn chorus." Chorus at Earth is generated by electrons in Earth's Van Allen radiation belts. Once generated, the chorus waves affect the motions of the electrons through a process called a wave-particle interaction. Wave-particle interactions disturb the trajectories of the radiation belt electrons and cause the electrons to hit the upper atmosphere.

  
  




• Decametric noise storms (quote source: Radio-Jupiter Central on RadioSky.com)

  
  
  

  
  

  The emissions we can hear are often referred to as decametric noise storms, because the waves are tens of meters long. Okay, it is possible to hear Jupiter from 15 to 38 MHz, but what are the optimal frequencies? The consensus seems to be that 18 MHz up to about 28 MHz is a good place to listen. A good rule would be to pick the lowest frequency in this range which was not being hindered by ionospheric refraction.

  
  




• Whistlers (quote and image source: Wikipedia):

  
  
  

  
  

  A whistler is a very low frequency or VLF electromagnetic (radio) wave generated by lightning.1 Frequencies of terrestrial whistlers are 1 kHz to 30 kHz, with a maximum amplitude usually at 3 kHz to 5 kHz. Although they are electromagnetic waves, they occur at audio frequencies, and can be converted to audio using a suitable receiver. They are produced by lightning strokes (mostly intracloud and return-path) where the impulse travels along the Earth's magnetic field lines from one hemisphere to the other.

  

  
  




• Auroral Radio Emissions (quote source: Wikipedia)

  
  
  

  
  

  Auroral radio emissions from giant planets with plasma sources such as Jupiter's volcanic moon Io could be detected with radio telescopes.

  
  

And so on. The last two on the list are probably a bit of a stretch for amateur astronomers, with auroral radio emissions average frequency range at 100 to 500 kHz, and whistlers cutoff at usually around 30 kHz, and both requiring too big antennae, but I wouldn't bet against it being possible with smaller antennae either, although bow frequency cutoffs might prevent identification of actual aural events. But there are many other sounds to listen to both from Jupiter, as well as it's many moons, mostly though the bigger and closer ones.

Good luck tuning in to your own Jupiter FM and happy hunting for rare radio frequency events!

Monoidal structures on von Neumann algebras

The category of von Neumann algebras W* admits a variety of monoidal structures of three distinct flavors.

(1) W* is complete and therefore you have a monoidal structure given by the categorical product.

(2a) W* is cocomplete and therefore you also have a monoidal structure given by the categorical coproduct.

(2b) I suspect that there is also a “spatial coproduct”, just as there is a categorical
tensor product and a spatial tensor product (see below).
The spatial coproduct should correspond to a certain central projection in the categorical coproduct.
Perhaps the spatial coproduct is some sort of coordinate-free version
of the free product mentioned in Dmitri Nikshych's answer.

(3a) For any two von Neumann algebras M and N
consider the functor F from W* to Set that sends a von Neumann algebra L
to the set of all pairs of morphisms M→L and N→L with commuting images.
The functor F preserves limits and satisfies the solution set condition, therefore it is representable.
The representing object is the categorical tensor product of M and N.

(3b) There is also the classical spatial tensor product.
I don't know any good universal property that characterizes it except
that there is a canonical map from (3a) to (3b) and its kernel corresponds
to some central projection in (3a). Perhaps there is a nice description of this central projection.

Since your monoidal structure is of the third flavor and you don't want a monoidal
structure of the first flavor, I suggest that you try a monoidal structures of the second flavor.
I suspect that the spatial coproduct of two factors is actually a factor.
You are lucky to work with factors, because in the commutative case 2=3, in particular 2a=3a and 2b=3b.

ca.analysis and odes - Integral determines function behaviour

Let us define:

$f(t) = t^{-1} int_{mathbf{R}^{3}} Exp[-frac{x^2}{2t}] h(x) dx,$

for a real function h. What can I say about this function if I know that

$f(t) rightarrow 1$.

I think that the convergence implies that

$t rightarrow int_{B(t)} h(x) dx$ (where B(t) is a ball of radius t)

grows like $t^{1/2}$ but I do not know how prove it formally. Moreover I have a feeling that this question is so simple (at least in formulation) that must have been answered somewhere.

(my question seems to be analitic, I put it also on the probability forum as may be it is possible to use some tools for Gaussian distribution/processes).

linear algebra - matrices self-adjoint with respect to some inner product

Is it possible to give a nice characterization of matrices $A in R^{n times n}$ which are self-adjoint with respect to some inner product?

These matrices include all symmetric matrices (of course) and some nonsymmetric ones: for example, the transition matrix of any (irreducible) reversible Markov chain will have this property.

Naturally, all such matrices must have real eigenvalues, though I do not expect that this is a sufficient condition (is it?).

About the only observation I have is that since any inner product
can be represented as $langle x,y rangle = x^T M y$ for some positive definite matrix $M$, we are looking for matrices $A$ which satisfy $A^T M = M A$ or $M^{-1} A^T M = A$. In other words, we are looking for real matrices similar to their transpose with a positive definite similarity matrix.

pr.probability - for a natural exponential family, A is the cumulant function of h?

Reading "Monte Carlo Statistical Methods" by Robert and Casella, they mention that if
$f(x) = h(x) exp(langle theta, x rangle - A(theta))$
defines a family of distributions for $X$, parametrized by $theta$, then $A$ is the cumulant generating function of $h(X)$. It seems like this should be easy to prove if it's true, but I don't see how to proceed. Any ideas/references?

qa.quantum algebra - Why do my quantum group books avoid homotopical language?

The question is rather rambling and it is more about not so well-defined appetites (do you have a more conrete motivation?).

There is one thing which however makes full sense and deserves the consideration. Namely it has been asked what about higher categorical analogues of (noncommutative noncocommutative) Hopf algebras. This is not a trivial subject, because it is easier to do resolutions of operads than more general properads. Anyway the infinity-bialgebras are much easier than the Hopf counterpart. There is important work of Umble and Saneblidze in this direction (cf. arxiv/0709.3436). The motivating examples are however rather different than quantum groups, coming from rational homotopy theory, I think.

Similarly, there is no free Hopf algebra in an obvious sense what makes difficult to naturally interpret deformation complexes for Hopf algebras (there is a notion called free Hopf algebra, concerning something else). Boris Shoikhet, aided with some help from Kontsevich, as well as Martin Markl have looked into this.

Another relevant issue is to include various higher function algebras on higher categorical groups, enveloping algebras of higher Lie algebras (cf. baranovsky (pdf) or arxiv 0706.1396 version), usual quantum groups, examples of secondary Steenrod algebra of Bauese etc. into a unique natural higher Hopf setting. I have not seen that.

The author of the question might also be interested in a monoidal bicategorical approach to general Hopf algebroids by Street and Day.

cosmology - "How the Universe Works": is it scientifically accurate?

The documentary/mini-series How the Universe Works is based on scientific evidence and according to what is known nowadays, or sheer bunk?

Throughout the series, are there any particular topics which are still disputed among the scientific community and that were portrayed in a different light than they should've had been?

---

I've only watched a few episodes but they always talk in a way that gives you the feeling everything is well-established and accepted but that might not be the case. I figured that an astronomer that watched the series would have remember if this happened or not.

I understand the subjects were researched and there is footage of known scientists in the episodes, but that doesn't mean everyone agrees with them. (By everyone I mean the most part of scientists working on those subjects, and not literally every single one of them.)

To give you an example, on the second episode they state that there are massive black holes at the center of galaxies. From my perspective (i.e. of someone which is not an astronomer) if they say so it's because it's probably true. However, in the real world maybe this is a highly disputed theory and a lot of astronomers disagree with this stance, and they're note telling the audience about it. Why would they do that? I don't know. I don't know if they did, I'm just asking if someone knows whether it happened, or if anyone saw something in the series with which they disagree.

linear algebra - For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?

This is a linear program. Put down the contraints $r leq b_i forall i$, together with all the linear constraints you have above, and maximize $r$. All the constraints are linear, so linear programming will do this. There are tons of linear programming packages (Mathematica and MATLAB have decent ones), and there should be some good introductory material on linear programming on the web that is better than anything I would write here, so I'll let you look for that.

Linear programs can in general have a lots of solutions (although they're all on one face of a polytope), or they may have no solutions. In this case, it certainly has at least one solution (see the compactness argument in the comments), but it mgiht have many.

complex geometry - gauge theory construction of vector bundles on singular varieties

There are subtleties even in the simplest case - $C$ a compact, irreducible complex curve with one node, $Pic_0(C)$ the Picard variety of line bundles of degree $0$ - so why not start there?

Pulling back line bundles via the normalisation map $nucolon tilde{C}to C$ defines a map $Pic_0(C)to Pic_0(tilde{C})$. The latter space is a complex torus of dimension $g(tilde{C})$. To recover a line bundle $L$ from $nu^*L$ we also need an isomorphism $nu^ast L_pto nu^ast L_q$, where $p$ and $q$ are the two points of $tilde{C}$ that lie over the node of $C$. In this way, one sees that $Pic_0(C)$ is a $mathbb{C}^{ast}$-bundle over $Pic_0(tilde{C})$.

So far as I can see, it's straightforward to give a gauge-theoretic description of $Pic_0(C)$: it consists of pairs consisting of a flat, unitary connection (or equivalently a Cauchy-Riemann operator, or equivalently a holomorphic structure) in a complex line bundle of degree zero over $tilde{C}$, with an isomorphism $I$ of the fibres over $x$ and $y$, modulo gauge transformations respecting $I$.

One can compactify $Pic_0(C)$ to a $mathbb{C}P^1$ bundle over $Pic_0(tilde{C})$ in a natural way, which also makes good sense gauge-theoretically. By gluing the zero-section to the infinity-section, covering the map on the base given by translation by the divisor $q-p$, one constructs a variety with normal crossing singularities which is isomorphic to the complex points of the compactified Picard scheme, parametrizing torsion-free sheaves of rank 1 on $C$. You could think about what this means in gauge-theory terms. There's a large literature on such compactified Picard (and Jacobian) varieties and their behaviour in families; see e.g. L. Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994), no. 3, 589--660; MR1254134.

For higher rank stable bundles, there's a bewildering array of papers on assorted compactifications (some of which are moduli spaces, some not). I'll refer you only to an extraordinary paper by Donaldson that might give you some clues as to what to expect: Gluing techniques in the cohomology of moduli spaces, Topological methods in modern mathematics (Stony Brook, NY, 1991), 137--170, Publish or Perish, Houston, TX, 1993; MR1215963.

Does any iron fuse in stars before they go supernova?

The "iron core" in a supernova is actually the end product of a nuclear statistical equilibrium that begins when the silicon core begins to fuse with alpha particles (helium nuclei). Exothermic reactions are possible right up to Nickel-62 (which is actually the nucleus with the highest binding energy per nucleon). In fact, successive, rapid alpha captures produce nuclei with the same number of protons and neutrons, but at the same time, the competing processes of photodisintegration and radioactive decay work in the other direction. The process is thought to mostly stop at Nickel-56 which, because heavier nuclei are more stable with $n/p>1$, then undergoes a couple of $beta^{+}$ decays via Cobalt-56 to Iron-56. However, the core of a supernova just before it explodes is likely to contain a bit of a mixture of iron-peak isotopes.

Before all this happens it is possible for iron and nickel to undergo nuclear reactions if there is an appropriate source of free neutrons. The elements beyond iron in our universe are predominantly created by neutron-capture in either the r-process or the s-process.

The r-process is thought to occur after a core-collapse supernova (or a type Ia supernova) has been initiated. The neutron flux is created by the neutronisation of protons by a dense, degenerate electron gas in the collapsing core.

However, the s-process can occur outside the core of a massive star before it explodes. It is a secondary process because it needs iron nuclei to be present already - that is, the iron that is used for the seed nuclei is not produced inside the star, it was already present in the gas from which the star formed. The s-process in massive stars uses free neutrons produced during neon burning (so at advanced nuclear burning stages beyond helium, carbon and oxygen burning) and results in the addition of neutrons to iron nuclei. This builds up heavy isotopes, which may either be stable or undergo $beta$ decay and/or further neutron captures to build up a chain of "s-process elements" (e.g. Sr, Y, Ba) all the way up to lead. The overall process is endothermic, but the yields and reaction rates are so small that it has no major influence on the overall energetics of the star. The newly-minted s-process elements are easily blasted into the interstellar medium shortly afterwards when the supernova explodes.

ag.algebraic geometry - Determining the field from the topology of the affine space over it.

This is false. Let $K=bar{mathbb{Q}}$ and $L=bar{mathbb{F}}_2$. These are clearly both algebraically closed, of different characteristics, so $Knotcong L$. However, if we ONLY look at the topology, $mathrm{Spec}(K[x])$ and $mathrm{Spec}(L[x])$ will be be countable sets with the finite complement topology on the closed points, with a single generic point, so they're homeomorphic. For algebraically closed fields, the TOPOLOGY on the affine line over the field is determined by the cardinality.

For higher dimensions, it's less clear to me, because you might be able to recover characteristic (I'm a char 0 kind of person, so I don't know) from how the various curves/hypersurfaces sit inside it.

Is any meteorite known to come from a comet?

This page -- http://www.amsmeteors.org/fireballs/faqf/ -- suggests that it's very unlikely to find cometary-origin meteorites, even though the majority of observed meteors are cometary in origin, because the latter are too fragile to survive all the way to the ground:

Based upon photographic fireball studies, cometary meteoroids have
extremely low densities, about 0.8 grams/cc for class IIIA fireballs,
and 0.3 grams/cc for class IIIB fireballs. This composition is very
fragile and vaporizes so readily when entering the atmosphere, that it
is called “friable” material. These meteoroids have virtually no
chance of making it to the ground unless an extremely large piece of
the comet enters the atmosphere, in which case it would very likely
explode at some point in its flight, due to mechanical and thermal
stresses.

It goes on to claim that cometary meteoroids make up about 95% of observed meteors, 38% of observed fireballs, and 0% of fresh meteorites. (Since a hypothetical cometary-origin meteorite would weather faster than meteorites of asteroidal origin, they would be even less likely to be found later, and so it's not surprising that the page says 0% of all known meteorites are of cometary origin.)

First meteor(s) of Lyrid meteor shower?

So I am a newcomer to Astronomy - only aged 15 - and am currently looking upon the sky in hopes of witnessing several meteors. This is my first time ever - maybe - seeing them. I just went inside for food and drink, come out and i saw a trail (this trail was not here before I went in). I do have a few doubts, one it was seen below the moon, when it was supposedly meant to be near Vega - however I read about them having a tendency to be different to predictions. Also, considering it is so close to the Moon and I can't currently see Vega from this position I highly doubt it.

please view the below image in the hopes of identifying the object (please excuse the camera quality, I am currently setting up the DSLR and this was photographed on my Nexus.)

Algorithms on graphs of bounded degeneracy/arboricity

There is one more approach to solve problems like Max Clique on graphs of bounded degeneracy.
You can look at the complement graph of a graph $G$ (i.e. every edge is a non-edge and every non-edge is an edge).
Solving Max Clique on $G$ is the same as solving Max Independent set on the complement.

For the complement of bounded degeneracy graphs algorithms for many problems are known.
E.g. Maximum Independent Set, Minimum Dominating Set, Perfect Code, k-Coloring, H-
Cover, H-Homomorphism and H-Role Assignment are FPT parameterized by the degeneracy of the complement.
See http://www.ii.uib.no/~martinv/Papers/Logarithmic_booleanwidth.pdf (submitted to journal)

Some of these problems make sense to translate to the complement graph, such as:

Can G be colored with $k$ colors -> can the complement be covered by $k$ cliques? (fixed $k$)

Is there an $3$-regular induced subgraph of $G$ -> is there an induced $k$ regular subgraph of the complement on $k+4$ vertices?

exoplanet - Planned telescope to detect alien waste-heat

This would be the SETI's Colossus telescope project, that aims to build a high-resolution, multiple-mirror instrument with ability to directly image the heat generated by other civilizations on planets orbiting stars near us:

    

    ^{Artist's impression of the proposed SETI's Colossus Telescope (Credit: Innovative Optics/Colossus Corporation)}

As Earth-like civilizations evolve, they use more power. For example,
in Roman times, we estimate Ω [the ratio of a civilization’s power production to the amount of solar power it receives] was about 1/1000 what it is today.
Humans’ global power consumption is growing by about 2.5 percent per
year, even though the world’s population is growing at less than half
this rate. In contrast, our knowledge base (the combined total of all
recorded information) doubles in just two years. As cultures advance,
their information content also must grow, and the power required to
manipulate this knowledge eventually dominates a civilization’s total
power use.

Using a sensitive coronagraph to remove scattered light that would
obscure an exoplanet, Colossus would be able to find hundreds of
Earth-sized or larger planets in the habitable zone including any
civilizations on their surfaces.

^{Source: Paul Gilster for Centauri Dreams quoting Jeff Kuhn (University of Hawaii), Svetlana V. Berdyugina (University of Freiburg), David Halliday (Dynamic Structures, Ltd., in British Columbia), and Caisey Harlingten (Searchlight Observatory Network, Norwich, England)}

Here is a Jeff Kuhn's, Institute for Astronomy, University of Hawaii, talk on The Colossus Telescope and the search for ET, and a quite descriptive blog post by Brian Wang on Technology for 77 meter primary mirror ground based telescope exists today and it could survey earth sized exoplanets out to 60 light years advocating the project from a technology perspective.

A bit more can also be read on the Innovative Optics Ltd. web pages, that are sadly under construction, but they do give a hint that an article about it was written in June 2013 edition of the Astronomy Magazine, titled How to find ET with infrared light (PDF), by Jeff R Kuhn, Svetlana V Berdyugina, David Halliday, Caisey Harlingten. A nice digest is also available in this Centauri Dreams blog post.

ca.analysis and odes - Examples of loss of regularity by "creation of topology"

I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered) fails, giving rise to interesting topological classifications. What comes to my mind are the following famous facts:

1) The condition for $C^infty(mathbb R^k, M^n)$ to be dense in the manifold-valued Sobolev spaces $W^{1,p}(R^k, M^n)$, is that the homotopy group $pi_{[p]}(M)$ should be trivial.(Hang-Lin)

[this was kind of general, but gives the idea of what I'm looking for, maybe!]

2) A map $u$ in $W^{1,2}(B^3,S^2)$ is in the closure of $C^infty(B^3, S^2)$ if and only if for any $2$-form $omega$ on $S^2$ such that $int_{S^2}omeganeq 0$ one has $d(u^*omega)=0$.(Bethuel-Coron-Demengel-Helein)

3) In $4$ dimensions, if the Yang-Mills functional is finite on a connection $Ain L^2(M^4)$, then the curvature $F_A$ of $A$ realizes an integral Chern class (i.e. the number $c_2(A):=1/(8pi^2)int_{M^4}Tr(F_Awedge F_A)$ is an integer).(Uhlenbeck)

(Maybe I could also formulate the question differently, asking for mathematical situations having the "loss of differentiability" via "creation of new topology" analogous to the list above.)

orbit - Orbiting and landing on non-sperical objects

If we orbit a non-spherical asteroid or moon at a sufficient distance I believe that we can consider it to be a point mass. Therefore we can take up a conventional orbit.

Assuming our lander craft is of insignificant mass compared with the body we wish to orbit and land on, how do we cope when we get near to the surface?

Presumably an orbit will become more and more chaotic the nearer we get. As we try to land, unless we somehow synchronise with the body's rotation things will be equally tricky.

What implications did this have for Rosetta's Philae and what implications will it have for much larger and maybe more irregular bodies?

Is there some rule-of-thumb method to get an approximate answer or do spacecraft simply have to make minute by minute adjustments?

Note

I understand mathematics to a reasonable level but really I'm more interested in the practicalities of such landings. To what extent can they be pre-calculated and to what extent must they be adjusted on the fly.

meteor - What are shooting stars and how are they formed?

A shooting star is simply the visible path a meteriod takes burning up in the atmosphere. As it flies through the atmosphere, it leaves a trail of fire, creating the streak of light you see.

Before they enter the atmosphere, they are simply chunks of rock. They could form in one of many ways - be broken off of a planet, etc. After they do this:

they look like this:

As for the reasons for meteor frequency varying, it's because of where the Earth is in it's orbit around the sun. Since almost all meteors are in orbit around the sun, and there tend to be clumps of them, this makes a 'donut' of meteors in orbit around the sun.

When the earth intersects one of these donuts, there is a spike in meteor activity. Ultimately, it's about how many space-rocks happen to hit Earth.

One annual 'high', for example, is the Perseid peak, which tends to occur mid-late summer each year. Sometimes, this even delays launches of space vehicles.

dg.differential geometry - A question about a formula of Pfaffian

The supertrace can be evaluated either by Berezin Gaussian integration, or equivalently by
summation over a Clifford algebra. Here is a description of the second method.

Let $omega$ be a skew-symmetric 2n by 2n matrix. Let ${{ e_1, e_2, . . . . e_{2n} }}$ be a real
2n-dimensional Clifford algebra. Then:

$exp(Sigma_{k,l=1}^{2n} omega_{kl} e_k e_l) = Sigma_{|K| even} Pf(omega_K) hat{e}_K$

where: $K$ is a subset of ${{ 1, 2, . . . . 2n }}$ and $hat{e}_K$ is the corresponding wedge product of the Clifford generators and $omega_K$ is the skew-symmeterized submatrix containg only the rows and columns in $K$.

Now, the supertrace selects the coefficient of the top form, giving you the required formula.

A proof of this result can be found for example in the following book by: José Gracia Bondía, Joseph C. Várilly, Héctor Figueroa.

rt.representation theory - D-modules supported on the nilpotent cone

I think saying "constant sheaf" isn't quite right. You want to take the functions on each orbit, and do the intermediate extension of that D-module.

Now, that's a bit unexplicit, so let me try a different description. Recall that there is a Grothendieck simultaneous resolution, a map from G x_B b (the adjoint bundle for b on G/B) to g extending the inclusion of b in the obvious way. If one takes functions on G x_B b, and pushes them forward by this map, then one gets a D-module on g. The isotypic summands of this D-module are in bijection with the representations of S_n. Over regular semi-simple elements, the map G x_B b -> g is a Galois S_n-cover, and so we're just decomposing the local system. By some geometric magic, this decomposition extends over the rest of the Lie algebra.

Now, we have to do Fourier transform, which for D-modules, just means change your mind about which variables in the Weyl algebra are coordinate and which ones are differentiations. Then the summand corresponding to the S_n representation for a Young diagram becomes exactly the intermediate extension for the orbit with corresponding Jordan type (or is it the transpose? I always forget).

ag.algebraic geometry - Kodaira-Spencer Theory and moduli of curves

I was looking at a paper of Farkas and the following confusing point came up.

Let $mathscr{M}_g$ be the moduli stack of smooth genus $g$ curves and let $pi: mathscr{C} to mathscr{M}_g$ be the universal curve. Let $mathscr{F}$ be $Omega^1_pi otimes Omega^1_pi$, where $Omega^1_pi$ is the sheaf of relative differentials of $pi$. Then the pushforward $pi_* mathscr{F}$ is isomorphic $Omega^1_{mathscr{M}_g}$.

Why is this true? Farkas says this follows from Kodaira-Spencer theory. I googled for a while and asked a few students, but couldn't figure this out.

gr.group theory - Which graphs are Cayley graphs?

I've managed to answer a few of the latter questions, and please look below for a solution in the finite-degree case.

Theorem. There is an uncountable graph Γ that is not a Cayley graph in the set-theoretic universe V, but which is a Cayley graph in a larger set-theoretic universe, obtained by forcing.

Proof. Let Γ be a directed graph that is a tree, such that all vertices have infinite in-out degree, but such that some of these degrees are countable and some uncountable. For example, perhaps all the in-degrees are countable and all out-degrees are uncountable. It is not difficult to construct such a graph. Clearly, Γ cannot be a Cayley graph, since the degrees don't match. Denote the (current) set-theoretic universe by V, and perform forcing to a forcing extension V[G] in which Γ is countable. (It is a remarkable fact about forcing that any set at all can become countable in a forcing extension.) In the extension V[G], the graph Γ is the countable tree in which every vertex has countably infinite in-out degree. Thus, in the forcing extension, Γ is the Cayley graph of the free group on countably many generators.QED

Theorem. There is a graph Γ, which is not a Cayley graph, but every finite subgraph of Γ is part of a Cayley graph. Indeed, every countable subgraph of Γ is part of a Cayley graph. Every countable subgraph of Γ can be extended to a larger countable subgraph of Γ that is a Cayley graph.

Proof. The same graph Γ as above works. Every countable subgraph of Γ involves only countably many edges, and can be placed into a countable subgraph of Γ that is the Cayley graph of the free group on countably many generators. QED

The conclusion is that you cannot tell if an uncountable graph is a Cayley graph by looking only at the finite subgraphs, or indeed, by looking only at its countable subgraphs.

These observations show that in the uncountable case, one should just restrict the question at the outset to graphs satisfying the degree-matching condition.

---

Let me also give the fuller details for the finite-degree
case, following the suggestion of François Dorais (and my
tree-of-attempts argument).

Theorem. There is a finitistic condition on
countable graphs Γ that holds exactly of the
finite-degree Cayley graphs. Specifically, the set of
finite-degree countable Cayley graphs has complexity at
most Σ⁰₄ in the arithmetic
hierarchy.

Proof. Let us suppose that the graph Γ is given to us
simply as a binary relation on ω, that is, an element
of 2^ωxω. The assertion that the
graph is connected has complexity
Π⁰₂, since you must only say that
every two vertices are connected by a finite path. The
assertion that the graph has finite degree and all in-out
degrees are the same has complexity
Σ⁰₄, since you can say "there
is a natural number k such that for all vertices v there
are k vertices w₁,...,w_k pointing at
v, such that no other vertex points at v (and similarly for
pointing out).

Now, suppose that Γ is a connected directed graph and
all in-out degrees are k. Fix a node e that will represent
the group identity (if Γ is a Cayley graph, any node
will do since they all look the same, so take e=0). Let us
refer to the k nodes pointed at from e as the set of
generators (since if Γ really is a Cayley graph,
these will be the generators). Define that p is a partial
labeling of the edges of Γ with generators, if p is
a function defined in a finite distance ball of e in
Γ, where p labels each arrow in that ball with a
generator. Such a labeling is coherent if first, for
every node every generator is used once going out from that
node and once going into that node, and second, if for
every node v, if there is a loop starting and ending at v,
then the word w obtained from that loop works as a loop
from every node to itself. We only enforce the requirements on coherence of p as far as p is defined (since it is merely a partial function). Thus, a coherent labeling is an attempt to
make a labeling of the the graph into a Cayley graph, that
has not run into trouble yet.

Let T be the collection of finite coherent labelings,
labeling all edges on the first n nodes for some n. This is
a finitely branching tree under inclusion.

I claim that Γ is a Cayley graph if and only if there
are arbitrarily such large coherent labelings. The forward
direction is clear, since we may restrict a full coherent
labeling to any finite subgraph. Conversely, if we can find
a coherent labeling for any finite subgraph of Γ,
then the tree T is infinite and finitely branching. Thus,
by Konig's lemma there is an infinite branch. Such a branch
labels all the edges in Γ and satisfies the coherence
condition. Such a labeling exactly provides a group
presentation with Cayley graph Γ.

The complexity of saying that every initial segment of the
nodes admits a coherent labeling is
Π⁰₂, since you must say for every
n, there is a labeling p of the first n nodes such that p
is coherent. Being coherent is a
Δ⁰₀ requirement on p. QED

In particular, to recognize when a graph is a finitely
generated Cayley graph does not require one to quantify
over infinite objects, and so for finite degree graphs,
this is an answer to the main question.

I was surprised that the part of the description driving
the complexity is the assertion that the degrees must all
match, rather than the assertion that there are coherent
labelings on the finite subgraphs. Can this be simplified?

This still leaves the case of countably many generators
wide open. Also, this still doesn't answer the question
about having a computable graph Γ, which is a Cayley
graph, but which has no computable presentation. Such an
example would be very interesting. Even in the finite
degree case, as I mentioned in the question, the best the
argument above produces is a low branch.

gr.group theory - An example of a non-amenable exact group without free subgroups.

Owen, I'm a bit late to the party, but I think the answer to your question is ``no", to the best of my knowledge. To phrase it properly, I believe it is not known whether any of the known counterexamples to von Neumann's conjecture is exact.

Jon, one has to be careful with limits hyperbolic groups, for example Gromov's random groups which are not exact are such limits (they are lacunary hyperbolic, in the sense of Olshanskii, Osin and Sapir).

sg.symplectic geometry - Compact Symplectic Fano (strongly monotone) manfiolds

What are known examples of compact symplectic Fano manifolds, apart from those that come from algebraic geometry?

We define symplectic Fano manifold as a symplectic manifold $(M,w)$, such that
$[c_1(M)]=[w]in H^2(M)$. Note that $c_1(M)$ is canonically defined, because up to homothopy there is a unique almost complex structure on $M$ tamed by $w$. Such manifolds often called monotone, or strongly (strictly) montone (but I don't think that the terminology is fixed).

For an algebraic Fano manifold, defined over $mathbb {C}$ there is a Kahler form in the class of $[c_1(M)]$, since $-K(M)$ is ample by definition. So, as Tim pointed out in his remark such a manifold is obviously a "symplectic Fano". I am aware of just one alternative construction that produces "symplectic Fanos", it is given in http://arxiv.org/abs/0905.3237 in section 5. It produces non-algebraic symplectic Fano manifolds strating from real dimension 12. The construction goes via non-compact coadjoint orbits. Is there any other construction in the literature?

Note, that in dimension 4 all "symplectic Fanos" come from Algebraic geometry (Gromov+Taubes+Mcduff), i.e. there are 10 examples, $mathbb{C}P^1times mathbb{C}P^1$ and
$mathbb {C}P^2$ blown up in at most $8$ points.

soft question - What are important examples of filtered/graded rings in physics?

It is debatable that a physicist would use those very words, and if they did one would hope their meaning would be the same as for a mathematician, since it means that they are trying to speak the same language.

Having said, and coming from a Physics background, when I first learnt about filtered objects and associated graded objects, I immediately recognised the following examples from Physics. They all have to do with quantisation/classical limit in one way or another.

1. The Clifford algebra is filtered and its associated graded algebra is the exterior algebra. Under the "classical limit" map which takes the Clifford algebra to the exterior algebra, the first nonzero term in the graded commutator of two elements defines a Poisson structure on the exterior algebra. You can then view the Clifford algebra as the quantisation of this Poisson superalgebra. In Physics the exterior algebra is the "phase space" for free fermions and Clifford modules (=representations of the Clifford algebra) are Hilbert spaces for quantized fermions. Things get more interesting when the underlying vector space is infinite-dimensional, since not all Clifford modules are physically equivalent. (The relevant buzzword is Bogoliubov transformations; although you would not guess this from the wikipedia page.)




2. The algebra of differential operators on $mathbb{R}^n$, say, is also filtered and the associated graded algebra is the algebra of functions on $T^*mathbb{R}^n cong mathbb{R}^{2n}$ which are polynomial in the fiber coordinates (=the "momenta"). Again the first nonzero term in the commutator of two differential operators defines the standard Poisson bracket on $T^*mathbb{R}^n$ and one can view the algebra of differential operators as a quantisation of this Poisson algebra. In Physics, this corresponds to quantising $n$ free bosons.

In both cases there is no unique section to the map taking a filtered algebra to the associated graded algebra, but one has to make a choice. There are number of more or less standard ones: Weyl ordering for the bosons, complete skewsymmetrisation for the fermions,...

By the way, this (and a lot more) is explained in the fantastic paper Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras by Kostant and Sternberg.

---

Edit (inspired by Mariano's answer)

Kontsevich's deformation quantization is not just of interest to physicists, but has a quantum field theoretical reformulation due to Cattaneo and Felder. It is basically the perturbative computation of the path integral of the Poisson sigma model. (This is analogous to how the perturbative evaluation of the path integral of Chern--Simons theory gives the Vassiliev invariants of (framed) knots.)

The picture that seems to be emerging is that indeed quantisation (be it deformation or path-integral or what have you) of a classical physical system gives rise to a filtered object, filtered by powers of $hbar$.

nt.number theory - reduction of CM elliptic curves

I'm not sure what definition of supersingular you're taking; I'll assume you mean that the endomorphism ring is an order in a quaternion algebra.

Now, suppose $E$ is a supersingular elliptic curve over $mathbb{F}_{q}$ of characteristic $p$, and $varphi$ is the Frobenius endomorphism. You can deduce from noncommutativity of the endomorphism ring (over the algebraic closure) that $[p^n] = varphi^m$ for some $m,n in mathbb{Z}$. The argument goes along the lines of: if $[p^n]$ is never equal to $varphi^m$, this forces the endomorphism ring to commute as every endomorphism commutes with some power of the Frobenius. You can find more details in Elliptic Curves by Husemoller, including a proof of the converse.

From this, we deduce that multiplication by $p$ is purely inseparable on a supersingular elliptic curve, and hence there is no $p$-torsion.

We can also deduce the following, by degrees of endomorphisms. Let $alpha,beta$ be two endomorphisms, then define $langle alpha, beta rangle = frac{1}{2} left( deg(alpha + beta ) - deg(alpha) - deg(beta) right)$, this defines an inner product on the endomorphism ring. Using the above property that $[p^n] = varphi^m$, some basic algebra shows that $p mid t$, where $t$ is the trace of the Frobenius, which is equal to $langle varphi, 1 rangle = #E(mathbb{F}_q) - (q+1)$. It is however not necessarily the case that $t=0$; this happens for $q=p$, $p geq 5$ by Hasse's inequality, but there are examples in which $t neq 0$. Going the other way, we see that $#E(mathbb{F}_q) equiv 1 pmod{p}$, so by elementary group theory $E(mathbb{F}_q)$ has no $p$-torsion, which is equivalent to supersingularity as shown above.

For the question about reduction modulo $p$, the full criterion of Deuring is as follows:

Theorem: Let $L$ be a number field, and $E$ an elliptic curve with complex multiplication by an order in the imaginary quadratic field $K$.
Let $p$ be a (rational) prime, and $P$ a prime above $p$ at which $E$ has good reduction.
Then $E$ has supersingular reduction at $P$ iff there is a unique prime of $K$ above $p$. Otherwise, write $c$ for the conductor of the endomorphism ring of $E$ in $K$, and let $c = c_0 p^k$ with $p nmid c_0$. Then the ring of endomorphisms of the reduction mod $p$ is $mathbb{Z} + c_0 mathcal{O}_K$, the order of $K$ with conductor $c_0$.

I have taken this from Lang's book Elliptic Functions, which contains a proof (page 182 in my edition).

cmb - Galaxy Cluster Temperature

The answer to your first question is "Yes, the temperature referred to is the 'normal' temperature, reflecting the average kinetic energy of the gas particles".

The answer to your second question is a bit more complex:

Cooling function

Gas cools by various processes, with an efficiency depending on the temperature, the density, and the composition of the gas. At "low" temperatures, most cooling happens because particles collide and their kinetic energy is used to exite or even ionize atoms. The atoms subsequently recombine or de-excite, emitting radiation which can carry energy away from the system.

Most of the gas in the Universe is hydrogen, which cools most efficiently around $10^4,mathrm{K}$. At higher temperatures, helium becomes the major contributor. Additionally, various metals contribute at various temperatures depending on their density and ionization state. At very high temperatures ($gtrsim10^7,mathrm{K}$) where the gas is highly ionized, cooling mainly happens through Bramsstrahlung, i.e. deceleration of charged particles by other charged particles.

The figure below (from Mo, Bosch & White 2010) shows the cooling function for various metallicities (with my own annotations):

Collapse of gasous halos

Now what makes the intracluster medium (ICM) different from the interstellar medium (ISM)? In the expanding Universe, overdensities try to contract. These overdensities can reach hydrostatic equilibrium only if radiative cooling is small. If the cooling timescale $t_mathrm{cool}$ is much smaller than the free-fall timescale $t_mathrm{ff}$, it can collapse and form stars. From the virial theorem, which gives the relation between potential and kinetic energy of the system, you can calculate the relation between the two timescales. Its a bit too much math for this post, but not difficult. I recommend reading chapter 8.4 in Mo, Bosch & White (2010). Their Fig. 8.6 shows a cooling diagram with the locus of $t_mathrm{cool} = t_mathrm{ff}$ in the density-temperature plane:

Above the locus, cooling is effective, and the cloud can collapse. The tilted dashed lines are lines of constant gas mass. Now you see that even for Solar metallicity (the $Z = Z_odot$ line), clouds of gas masses larger that $sim10^{13},M_odot$ are unable to cool. This is the fundamental difference between a galaxy and a cluster. They are both overdensities that resisted the expansion, but the gas in the cluster is not able to cool and form stars.

Voids

When people talk about "voids", the ususally refer not to the ICM, which is the hot and dilute gas between the galaxies of a cluster, but to the immense regions of even more dilute gas which are… well, void of galaxies. Take a look at this image (from here). Here, I marked clusters with green and voids with purple. I also marked some filaments with cyan.

The cosmic microwave background

As for you last question, the ICM doesn't interact much with the CMB, so they're not in thermodynamic equilibrium, and there's no conflict. A small fraction (<10%) of the CMB photons does interact. This doesn't change the state of the gas, but it tends to polarize a fraction of the CMB, and this we can use to study the ionization history of the Universe.

EDIT: I forgot about the Sunyaev-Zel'dovich effect, which is the interaction of hot electrons with CMB photons. Rob Jeffries discusses this in his answer.