ds.dynamical systems - A regularity property of transition matrices for the cat map

I've noticed a rather strange phenomenon (not important for my particular research, but interesting) and wouldn't be surprised if someone more versed in symbolic dynamics (i.e., just about anyone who knows what those words mean together) could easily explain it.

Consider the cat map $A$ and the Markov partition $mathcal{R} =$ {$R_1,dots,R_5$} shown below:

The rectangles in the partition are numbered from 1 (darkest) to 5 (lightest).

Now for a given initial point $x$ with rational coordinates (so that the period $t(x)$ of the sequence $A^ell x$ is finite) consider the matrix $T(x)$ with entries $T_{jk}(x)$ equal to the cardinality of {$ell < t(x): A^ell x in R_j land A^{ell + 1}x in R_k$}, i.e., the number of times per period that the trajectory goes from the $j$th rectangle to the $k$th rectangle. Clearly the sparsity pattern of $T(x)$ is inherited from the matrix defining the corresponding subshift of finite type.

Let $L_q$ denote the set of rational points in $[0,1)^2$ with denominator $q$. When I compute the sum $T_{(q)} := sum_{x in L_q} T(x)$ I get some surprising near-equalities. For instance, with $q = 240$ I get

  301468           0      301310      186567           0
  186567           0      186407      114903           0
  301310           0      301251      186407           0
       0      301470           0           0      186407
       0      186407           0           0      115060

and when $q = 322$ I get

  262625           0      262624      162291           0
  162291           0      162312      100312           0
  262624           0      262632      162312           0
       0      262603           0           0      162312
       0      162312           0           0      100312

The entries of each matrix are bunched around 3 values. What's more, the stochastic matrices obtained by adding unity to each entry and then row-normalizing agree to one part in a thousand.

Is there a (simple) explanation for this?

ag.algebraic geometry - Is $mathbb{A}²$ the universal smooth scheme which is a finite cover of $mathbb{A}²/μ₂$?

One thing that confused me about Francesco's answer was how to actually construct the branch covers $f_k:Y_kto S$ which are branched over the vertex and a given curve. Since I was sheepish enough not to ask, perhaps somebody else (maybe future me) will benefit from a description.

Let $g(x,y,z)$ be a polynomial which does not vanish at the origin. We then have two interesting degree 2 maps to $S=Spec(k[a^2,ab,b^2])$:

• $mathbb A^2to S$, corresponding to the inclusion $k[a^2,ab,b^2]to k[a,b]$. Think of $S$ as $mathbb A^2/mu_2$, where $mu_2$ acts by $(a,b)mapsto (-a,-b)$. This is branched only over the vertex, since $(0,0)$ is the only point with a non-trivial stabilizer.


• $S[sqrt{g}]to S$ (almost certainly non-standard notation since I just made it up), corresponding to the inclusion of rings $k[a^2,ab,b^2]to k[a^2,ab,b^2,sqrt{g(a^2,ab,b^2)}]$. Think of $S$ as $S[sqrt g]/mu_2$, where $mu_2$ acts by $sqrt gmapsto -sqrt g$. This is branched over the vanishing locus of $g$, since that's exactly where you have non-trivial stabilizer.

We can then define a sort of common refinement, $tilde Y=Spec(k[a,b,sqrt{g(a^2,ab,b^2)}]$, which has an action of $mu_2times mu_2$. Quotienting by the first $mu_2$ gives us $S[sqrt g]$. Quotienting by the second $mu_2$ gives us $mathbb A^2$. Quotienting by both gives you $S$. Define $Y$ as the quotient by the diagonal $mu_2$ action, $(a,b,sqrt g)mapsto (-a,-b,-sqrt g)$.^† This action is free since $g(0,0,0)neq 0$, so $tilde Yto Y$ is actually an etale cover. If $V(g)cap S$ is smooth, $tilde Y$ is smooth, so $Y$ is smooth. We have a remaining $mu_2$ action on $Y$ with $Y/mu_2 = S$.

$$begin{array}{cccccc}
& & tilde Y\
& swarrow & downarrow & searrow\
mathbb A^2 & & Y & & S[sqrt g]\
& searrow & downarrow & swarrow \
& & S
end{array}$$

^† You can very explicitly describe the ring of invariants under this action. $Y$ is the spectrum of $k[a^2,ab,b^2,asqrt g,bsqrt g]$. The $mu_2$ action on $Y$ is $(a^2,ab,b^2,asqrt g,bsqrt g)mapsto (a^2,ab,b^2,-asqrt g,-bsqrt g)$.

nt.number theory - Polynomial with the primes as coefficients irreducible?

I will prove that $A_n$ is irreducible for all $n$, but most of the credit goes to Qiaochu.

We have
$$(x-1)A_n = b_{n+1} x^{n+1} + b_n x^n + cdots + b_1 x - p_n$$
for some positive integers $b_{n+1},ldots,b_1$ summing to $p_n$. If $|x| le 1$, then
$$|b_{n+1} x^{n+1} + b_n x^n + cdots + b_1 x| le b_{n+1}+cdots+b_1 = p_n$$
with equality if and only $x=1$, so the only zero of $(x-1)A_n$ inside or on the unit circle is $x=1$. Moreover, $A_n(1)>0$, so $x=1$ is not a zero of $A_n$, so every zero of $A_n$ has absolute value greater than $1$.

If $A_n$ factors as $B C$, then $B(0) C(0) = A_n(0) = p_n$, so either $B(0)$ or $C(0)$ is $pm 1$. Suppose that it is $B(0)$ that is $pm 1$. On the other hand, $pm B(0)$ is the product of the zeros of $B$, which are complex numbers of absolute value greater than $1$, so it must be an empty product, i.e., $deg B=0$. Thus the factorization is trivial. Hence $A_n$ is irreducible.

rt.representation theory - Examples of applications of the Borel-Weil-Bott theorem?

I'll just elaborate for you the example mentioned by Scott:
In the stereographic projection coordinates of S², the symplectic 2-form is given by:

ω = dz^dz̄/(1+zz̄)²

Classically, one can construct three hamiltonian functions representing the generators of the Lie algebra su(2) which constitute of a subalgebra of the Poisson algebra corresponding to ω

T_X = (z+z̄)/(1+zz̄)

T_Y = -i(z-z̄)/(1+zz̄)

T_Z = (1-zz̄)/(1+zz̄)

Quantum mechanically, the representation of spin j of SU(2) is realized on a (reproducing kernel) Hilbert space generated by holomorphic sections of a line bundle whose expressions in the stereographic coordinates are 1, z, . . . , z^2j. The su(2) Lie algebra can be realized on this space by means of the differential operators:

s_X = -(1-z²)∂/∂z + 2jz

s_Y = -i(1+z²)∂/∂z - 2ijz

s_Z = -2z∂/∂z -2j

Theories of geometric quantization offer systematic methods to make these constructions for a general compact Lie group for a concrete realization of the Bore-Weil-Bott theorem.

I would like to mention that many representation theoretical computations can be made using this realization of the representation theory of compact Lie groups. Also, this realization is connected to Perelomov's generalized coherent states.

There are some generalizations to representations on non-compact Lie groups. Also, the Borel-Weil-Bott theorem can be connected in many ways to supersymmetry.

The "linearization' of the classical mechanics is achievd through the realization of the quantum Hilbert space by sections of a "line" bundle. These sections also relates this realization to projective geometry via Kodaira's embedding thorem.

co.combinatorics - Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.

First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything novel, I decided to ask here.

Alright, let $S$ be a multiset of $n$ rational numbers mod 1. Assume that $0in S$. Define a additive decomposition of the set $S$ as two sets $A$ and $B$ such that

1. Both have elements rational numbers mod 1 and contain 0.


2. For all $ain A$ and $bin B$ the sum, $(a+b)mod{1}in S$


3. Every $sin S$ is the sum of an element from each of $A$ and $B$.

Just to be perfectly clear, lets consider an example. Let $S:=lbrace 0,dfrac{1}{2}, dfrac{1}{3}, dfrac{5}{6} rbrace$, then the only additive decompositions are

1. $A=lbrace 0rbrace$, $B=lbrace 0,dfrac{1}{2}, dfrac{1}{3}, dfrac{5}{6} rbrace$


2. $A=lbrace 0, dfrac{1}{2}rbrace$, $B=lbrace 0,dfrac{1}{3} rbrace$


3. $A=lbrace 0, dfrac{1}{2}rbrace$, $B=lbrace 0,dfrac{5}{6} rbrace$

Second Example:

If $A=lbrace 0, dfrac{1}{2}, dfrac{1}{3}rbrace$, $B=lbrace 0,dfrac{1}{2}, dfrac{1}{3} rbrace$, they would be a decomposition of the set $S=lbrace 0, 0, dfrac{1}{2}, dfrac{1}{2}, dfrac{1}{3}, dfrac{1}{3}, dfrac{2}{3}, dfrac{5}{6}, dfrac{5}{6}rbrace$

At this point there are a few things to mention. First, we quickly reduce the problem to looking at subsets whose orders are $alpha$ and $beta$ s.t. $alphabeta=n$. Additionally, we can see that by the additive structure splitting into these two subsets is adequate in the sense that we can get a complete decomposition recursively by breaking the set into two.

Question:

What is the fastest algorithm you can come up with to find all additive decompositions of a multiset $S$ of order $n$?

A computer has already been used to attack the problem. In small cases, the problem is not too bad. The situation arises in the fact that in the largest cases necissary $nsim 10^5$. The professor said that an algorithm of polynomial time with respect to $n$ would be a great improvement from this current.

A word on the current algorithm. Look at the factorization of $n$. Pick $alphamid n$. Select $alpha$ elements of $S$ and let them be $A$. Then for each $s_iin S$ remove $A+s_imod{1}$ from $S$. After running through $s_i$, the remaining elements for a candidate for $B$. If the cardinality of $B$ is $beta$ for $alphabeta=n$ then we have a decomposition.

In addition to searching for a solution, I want to encourage discussion of other aspects of this problem as they may yield some interesting observations not noticed before.

Thanks in advance!

pr.probability - Random values and their probability of reoccuring

Call your three questions A, B, C.

The probability that A gets chosen twelve times in a row is 1/(3^12), or 1 in 531441; similarly for B and C.

The probability that some question gets chosen twelve times in a row is thus 3/(3^12), or 1/(3^11), or 1 in 177147.

Personally, I think this seems like low enough a probability that if you knew for sure it actually happened, I'd be suspicious that your code isn't doing what you think it does. But it's also possible that your user really didn't get this question 12 times in a row and is just misremembering.

rt.representation theory - Introduction to W-Algebras/Why W-algebras?

W-algebras appear in at least three interrelated contexts.

1. Integrable hierarchies, as in the article by Leonid Dickey that mathphysicist mentions in his/her answer. Integrable PDEs like the KdV equation are bihamiltonian, meaning that the equations of motion can be written in hamiltonian form with respect to two different Poisson structures. One of the Poisson structures is constant, whereas the other (the so-called second Gelfand-Dickey bracket) defines a so-called classical W-algebra. For the KdV equation it is the Virasoro Lie algebra, but for Boussinesq and higher-order reductions of the KP hierarchy one gets more complicated Poisson algebras.




2. Drinfeld-Sokolov reduction, for which you might wish to take a look at the work of Edward Frenkel in the early 1990s. This gives a homological construction of the classical W-algebras starting from an affine Lie algebra and a nilpotent element. You can also construct so-called finite W-algebras in this way, by starting with a finite-dimensional simple Lie algebra and a nilpotent element. The original paper is this one by de Boer and Tjin. A lot of work is going on right on on finite W-algebras. You might wish to check out the work of Premet.




3. Conformal field theory. This is perhaps the original context and certainly the one that gave them their name. This stems from this paper of Zamolodchikov. In this context, a W-algebra is a kind of vertex operator algebra: the vertex operator algebra generated by the Virasoro vector together with a finite number of primary fields. A review about this aspect of W-algebras can be found in this report by Bouwknegt and Schoutens.

There is a lot of literature on W-algebras, of which I know the mathematical physics literature the best. They had their hey-day in Physics around the late 1980s and early 1990s, when they offered a hope to classify rational conformal field theories with arbitrary values of the central charge. The motivation there came from string theory where you would like to have a good understanding of conformal field theories of $c=15$. The rational conformal field theories without extended symmetry only exist for $c<1$, whence to overcome this bound one had to introduce extra fields (à la Zamolodchikov). Lots of work on W-algebras (in the sense of 3) happened during this time.

The emergence of matrix models for string theory around 1989-90 (i.e., applications of random matrix theory to string theory) focussed attention on the integrable hierarchies, whose $tau$-functions are intimately related to the partition functions of the matrix model. This gave rise to lots of work on classical W-algebras (in the sense of 1 above) and also to the realisation that they could be constructed à la Drinfeld-Sokolov.

The main questions which remained concerned the geometry of W-algebras, by which one means a geometric realisation of W-algebras analogous to the way the Virasoro algebra is (the universal central extension of) the Lie algebra of vector fields on the circle, and the representation theory. I suppose it's this latter question which motivates much of the present-day W-algebraic research in Algebra.

Added

In case you are wondering, the etymology is pretty prosaic. Zamolodchikov's first example was an operator vertex algebra generated by the Virasoro vector and a primary weight field $W$ of weight 3. People started referring to this as Zamolodchikov's $W_3$ algebra and the rest, as they say, is history.

Added later

Ben's answer motivates the study of finite W-algebras from geometric representation theory and points out that a finite W-algebra can be viewed as the quantisation of a particular Poisson reduction of the dual of the Lie algebra with the standard Kirillov Poisson structure. The construction I mentioned above is in some sense doing this in the opposite order: you first quantise the Kirillov Poisson structure and then you take BRST cohomology, which is the quantum analogue of Poisson reduction.

soft question - Most helpful math resources on the web

Sci-Hub is pretty helpful in accessing articles, even for those researchers who already have access to several journals. The interface is great, the site is pretty fast, and the database is huge. See this article and other linked articles there for a nice overview of who all are downloading pirated papers.

Edit: as pointed out in the comments, it should be noted that there is an ongoing lawsuit against the website.

pr.probability - Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.

My guess is that they are more useful in probability than in analysis. Many people have the impression that probability is just analysis on spaces of measure 1. However, this is not exactly true. One way to tell analysts and probabilists apart: ask them if they care about independence of their functions.

Suppose that $mathcal{F}_1,mathcal{F}_2,...,mathcal{F}_n$ are families of subsets of some space $Omega$. Suppose further that given any $A_iin mathcal{F}_i$ we know that $P(A_1cap A_2 cap ...cap A_n)=P(A_1)P(A_2)...P(A_n)$. Does it follow that the $sigma(mathcal{F}_i)$ are independent? No. But if the $mathcal{F}_i$ are $pi$-systems, then the answer is yes.

When proving the uniqueness of the product measure for $sigma$-finite measure spaces, one can use the $pi$-$lambda$ lemma, though I think there is a way to avoid it (I believe Bartle avoids it, for instance). However, do you know of a text which avoids using the monotone class theorem for Fubini's theorem? This, to me, has a similar feel to the $pi$-$lambda$ lemma. Stein and Shakarchi might avoid it, but as I recall their proof was fairly arduous.

Here is a direct consequence of the $pi$-$lambda$ lemma when you work on probability spaces:

Let a linear space H of bounded functions contain 1 and be closed under bounded convergence. If H contains a multiplicative family Q, then it contains all bounded functions measurable with respect to the $sigma$-algebra generated by Q.

Why is this useful? Suppose that I want to check that some property P holds for all bounded, measurable functions. Then I only need to check three things:

1. If P holds for f and g, then P holds for f+g.


2. If P holds for a bounded, convergent sequence $f_n$ then P holds for $lim f_n$.


3. P holds for characteristic functions of measurable sets.

This theorem completely automates many annoying "bootstrapping from characteristic functions" arguments, e.g. proving Fubini's theorem.

fa.functional analysis - Nonlinear Nuclear Operators ?

Funny you should ask. My former student, Bentuo Zheng, and my visitor, Dongyang Chen, are in the process of developing this theory. The "right" definition involves the Pietsch factorization diagram, and is an off-shoot off what Farmer and I did for p-summing and p-integral operators (get the paper from my home page). Send me an email and I'll put you in contact with them.

On a related topic, my current student Alejandro ChavezDominguez is developing the operator ideal theory connected to non-linear p-summing operators and related mappings. This is something I have been interested in for a long time but did not see what to do (even the duality theory). In Javier's hands, the theory is developing very well.

dg.differential geometry - Killing Fields to Laplacian

One is familiar from Quantum Theory that each of the angular momentum generators $L_{x,y,z}$ are Killing Fields for the standard metric on $S^2$ and the sum of the squares of these generators gives the Laplacian on R^3.

It seems from some literature that this idea in some sense generalizes.

Vaguely what it seems to me is that for a homogeneous spaces $G/H$ if $K_i$ are the killing fields (on $G/H$ ?) then $sum_i K_i K_i$ is the Laplacian on $G/H$.

It would be helpful if one can tell me what is the precise statement that contains the above idea and also what are the caveats and the proof of why it should be so.

In this context people also talk of the "Casimir Laplacian". What precisely is that?

Casimir Laplacian comes about in this way,

If $T_a$ happen to be the Killing Fields on $G$ and $X_b$ be the Killing fields on $H$ then in some cases (when the algebra is reductive?) a relation of this kind holds,

$$K_i K_i = T_aT_a - X_b X_b$$

(sum over repeated indices implied)

Here too I don't know the precise statement or the proof, but just am seeing allusions to it in the papers.

There is also this issue of 3 `different' ways of defining the laplacian, either as the ordinary one $nabla ^{mu}nabla _{mu}$

$$or$$

as $sum_i nabla_{X_i^*}nabla _ {X_i ^*}$ where $nabla _ {X_i ^*}$ is defined as the so called $H$-connection on $G/H$."

This is how apparently the $H$ connection's evaluation on a section $psi: G/H rightarrow G$ along the vector filed $X$ (section of a homogeneous vector bundle over $G/H$) is defined,

$$nabla_{X^{*}} psi (x_{0}) = lim_{trightarrow 0} frac{exp(-tX)psi(gamma_X(t))-psi(x_0)}{t}$$

where $gamma_X(t) = exp(tX)x_o$ is the integral curve of $X$ through the origin of $G/H$ i.e $x_0$

$$or$$

as through the Lie derivative as $sum_i L_{K_i} L_{K_i}$

How to understand the difference and the relations between these notions of Laplacians?

---

As an example of the kind of relationships I am trying to understand let me quote $3$ of such equations,

• $$nabla_{beta} nabla ^{beta} V^{alpha} = left ( sum _i L_{K_i}L_{K_i} V right )^{alpha} + R^{alpha} _{beta} V^{beta} + f^{beta}_{gamma}{^{alpha}} nabla _{beta} V^{gamma}$$

where the structure constant $f$ and the Ricci Tensor are of the $G/H$ and $V$ is a vector field on $G/H$ (written here in the vielbein basis) and the connection is on $G/H$ but the Killing fields are of $G$. The Ricci Tensor can for such spaces be written in terms of the structure constants or the Casimir operator of the representation of $H$ which defines the vector bundle in concern.

• $$sum _i L_{K_i}L_{K_i} = -sum _{lambda} C_2(lambda)$$ where the right hand side is a sum over Casimirs of all irreducible representations of $G$ and the Lie derivatives on $G$ acting in a `natural' way on the fields of $G/H$




• For the `H-connection" the first equation reduces to,

  
  
  

  $$nabla_{beta} nabla ^{beta} V^{alpha} = left ( sum _i L_{K_i}L_{K_i} V right )^{alpha} - (f_p f^{p})^{alpha}_{gamma} V^{gamma} $$

  
  
  

  where the $f$ are the generators of the representation of $H$. Basically the new terms is the components of the Casimir of that representation of $H$ along the $G/H$ components.

fa.functional analysis - Bounded and weakly bounded sets in top. vector spaces

This is direct consequence of the Mackey Theorem: Having a dual pair (V,V') with V' as the dual of the locally convex space V, the bounded sets on V under any dual topology are identical. A dual topology on V is a locally convex topology $tau$ such that (V,$tau$)' = V'.

As the original and the weak topology give the same dual, the bounded sets are identical.

pr.probability - Harmonic mean of random variables

Any class of distributions which is closed under independent sums and almost surely nonzero will work here (and of course will also give an example for geometric means corresponding to the log-normal) the same way as in Michael's example. So besides normal (p=2) and Cauchy (p=1) random variables, the reciprocals of any p-stable random variables work. Of course, only Cauchy have the property of being distributed the same as their reciprocals, so John's answer doesn't generalize this far.

gn.general topology - The continuous as the limit of the discrete

I'd like to clear up something that came up in the comments. There are two natural ways to fit the finite cyclic groups together in a diagram. One is to take the morphisms $mathbb{Z}/nmathbb{Z} to mathbb{Z}/mmathbb{Z}, m | n$ given by sending $1$ to $1$. This gives a diagram (inverse system) whose limit (inverse limit) is the profinite completion $hat{mathbb{Z}}$ of $mathbb{Z}$. This diagram also makes sense in the category of unital rings, since they also respect the ring structure, giving the profinite integers the structure of a commutative ring.

This is not the diagram relevant to understanding the circle group. Instead, one needs to take the morphisms $mathbb{Z}/nmathbb{Z} to mathbb{Z}/mmathbb{Z}, n | m$ given by sending $1$ to $frac{m}{n}$. This is the diagram relevant to understanding the cyclic groups as subgroups of their colimit (direct limit), which is, as I have said, $mathbb{Q}/mathbb{Z}$. And this group, in turn, compactifies to the circle group in whichever way you prefer.

(These two diagrams are "dual," though, something which I learned recently when I was asked to prove on an exam that $text{Hom}(mathbb{Q}/mathbb{Z}, mathbb{Q}/mathbb{Z}) simeq hat{mathbb{Z}}$. Just observe that $text{Hom}(mathbb{Z}/nmathbb{Z}, mathbb{Q}/mathbb{Z}) simeq mathbb{Z}/nmathbb{Z}$ and that contravariant Hom functors send colimits to limits!)

Edit: Let me also say something about the precise meaning of "compactification" here. A compactification of a space $T$ is an embedding $T to X$ into a compact Hausdorff space $X$ with dense image. The embedding being considered here is the obvious one from $mathbb{Q}/mathbb{Z}$ to $mathbb{R}/mathbb{Z}$, and the fact that it has dense image is essentially what the word "completion" also means. Compactifications are not unique, but it's possible that there is a sense in which as a topological group $mathbb{R}/mathbb{Z}$ is the "most natural" compactification of $mathbb{Q}/mathbb{Z}$. But I don't know too much about topological groups.

ag.algebraic geometry - What do the local systems in Lusztig's perverse sheaves on quiver varieties look like?

In "Quivers, perverse sheaves and quantized enveloping algebras," Lusztig defines a category of perverse sheaves on the moduli stack of representations of a quiver. These perverse sheaves are defined as summands of the pushforwards of the constant sheaves on stacks of quiver representations along with a choice of invariant flag (and thus, by definition are supported on the nilpotent locus in the moduli stack). They're mostly of interest since they categorify the canonical basis.

My question is: Is there a stratum in this stack where the pull-back of one of these sheaves is not the trivial local system?

Now, in finite type, this is not a concern, since each stratum is the classifying space of a connected algebraic group, and thus simply connected. But I believe in affine or wild type this is no longer true; this was at least my takeaway from the latter sections of "Affine quivers and canonical bases." However, I got a little confused about the relationship between the results of the two papers mentioned above, since they use quite different formalisms, so I hold out some hope that the local systems associated to symmetric group representations aren't relevant to the perverse sheaves for the canonical basis. Am I just hoping in vain?

mathematical modeling - Resultant probability distribution when taking the cosine of gaussian distributed variable

Given a normal distribution with mean $mu$ and variance $sigma^2$, $X = mathcal{N}(mu,sigma^2)$, if you pass it through trigonometric functions, you can approximate the result with the new normal distributions below

1) normal distribution passed through Cosine function:

$X_{cos} = mathcal{N}(cos(mu),sigma^2sin^2(mu))$

so the new average is $cos(mu)$ and the new standard deviation is $|sigmasin(mu)|$.

2) normal distribution passed through a Sine function:

$X_{sin} = mathcal{N}(sin(mu),sigma^2cos^2(mu))$

so the new average is $sin(mu)$ and the new standard deviation is $|sigmacos(mu)|$.

The Matlab script that I used to find these relations is below.

%% Cody Martin
% 9/2/2010
% m-file used to discover the mean and variance of a normal distribution
% passed through cosine and sine functions...results:
%   - N(mu,sig^2) -> cos(N(mu,sig^2)) = N(cos(mu),sig^2*sin^2(mu))
%   - N(mu,sig^2) -> sin(N(mu,sig^2)) = N(sin(mu),sig^2*cos^2(mu))

%% distribution of cosine and sine of a normal distribution?
cresults = zeros(0,5);
sresults = zeros(0,5); 
% loop from an average angle -90 degrees to +90 degrees
for theta = -pi/2:pi/180:pi/2
    theta1sig = pi/36;                          % standard deviation of orinigal normal distribution
    vtheta = theta + theta1sig*randn(99999,1);  % create 99999 points using this avg and std
    vctheta = cos(vtheta);                      % take the cosine of those points
    vstheta = sin(vtheta);                      % take the sine of those points
    theta_ = min(vtheta):0.01:max(vtheta);      % for plotting ideal distributions
    ctheta_ = min(vctheta):0.01:max(vctheta);   % for plotting
    stheta_ = min(vstheta):0.01:max(vstheta);   % for plotting

    figure(1); clf;
    subplot(211); hold on;
    plot(theta_,cdf('normal',theta_,theta,theta1sig),':');  % plot cdf of normal distribution with avg and std
    plot(sort(vtheta),[1:length(vtheta)]/length(vtheta));   % plot cdf of 99999 points
    plot(sort(vctheta),[1:length(vctheta)]/length(vctheta),'k','LineWidth',2); % plot cdf of cos(99999 points)
    plot(ctheta_,cdf('normal',ctheta_,cos(theta),...        % plot cdf of norm dist with new avg and std after being passed through cos()
         sqrt(theta1sig^2*sin(theta)^2)),'r:');
    plot(cos(theta)*[1 1],[0 1],'k:');                      % vertical line @ cos(theta) - shows new average matches cos(old avg)
    title('Cosine of a Normal Distribution (for Different Initial Averages)');
    legend('Norm CDF Theory','Norm CDF 99999','Cos(Norm CDF 99999)','Cos(Norm CDF) Theory');
    axis([-pi/2 pi/2 0 1])

    subplot(212); hold on;
    plot(theta_,cdf('normal',theta_,theta,theta1sig),':');
    plot(sort(vtheta),[1:length(vtheta)]/length(vtheta));
    plot(sort(vstheta),[1:length(vstheta)]/length(vstheta),'k','LineWidth',2);
    plot(stheta_,cdf('normal',stheta_,sin(theta),...
         sqrt(theta1sig^2*cos(theta)^2)),'r:');
    plot(sin(theta)*[1 1],[0 1],'k:');
    title('Sine of a Normal Distribution (for Different Initial Averages)');
    legend('Norm CDF Theory','Norm CDF 99999','Sin(Norm CDF 99999)','Sin(Norm CDF) Theory');
    axis([-pi/2 pi/2 0 1])

%   fprintf('theta: %3.0ftstd: %5.3ftsin(theta): %5.3ftavg: %5.3ftstd: %5.3fn',theta*180/pi,theta1sig,sin(theta),mean(vstheta),std(vstheta));
    cresults = [cresults; theta theta1sig cos(theta) mean(vctheta) std(vctheta)];
    sresults = [sresults; theta theta1sig sin(theta) mean(vstheta) std(vstheta)];
end

figure(2); clf;
subplot(211); hold on;
plot(cresults(:,1),cresults(:,end));
plot(cresults(:,1),abs(theta1sig*sresults(:,3)),'r:');
title('Standard Deviation of Cosine of a Normal Distribution as a Function of the Original Average');
legend('From 99999 Points','Fit: std = |sigmasin(mu)|');
ylabel('std(cos(theta_{vector})) [rad]');
xlabel('theta [rad]');

subplot(212); hold on;
plot(sresults(:,1),sresults(:,end));
plot(sresults(:,1),abs(theta1sig*cresults(:,3)),'r:');
title('Standard Deviation of Sine of a Normal Distribution as a Function of the Original Average');
legend('From 99999 Points','Fit: std = |sigmacos(mu)|');
ylabel('std(sin(theta_{vector})) [rad]');
xlabel('theta [rad]');

figure(3); clf;
subplot(211); hold on;
plot(cresults(:,1),abs(theta1sig*sresults(:,3))-cresults(:,end));
title('Error Between sigma^2sin^2(mu) and std of 99999 Draws of cos(theta)')
ylabel('Residual [rad]');
xlabel('theta [rad]');


subplot(212); hold on;
plot(sresults(:,1),abs(theta1sig*cresults(:,3))-sresults(:,end));
title('Error Between sigma^2cos^2(mu) and std of 99999 Draws of cos(theta)')
ylabel('Residual [rad]');
xlabel('theta [rad]');

As others have pointed out, this fails where $cos(mu)$ and $sin(mu)$ are near 0. Residuals between my proposed solution and the empirical results from 99999 draws are shown below.

ag.algebraic geometry - Homeomorphism onto a closed subset of a scheme that isn't a closed immersion

Yes, for example if $K subset L$ is an inclusion fields, then the induced map
Spec $L to $ Spec $K$ is a homeomorphism (both source and target are single points),
but the induced map on sheaves is the given inclusion of $K$ into $L$, which is
surjective only if $K = L$.

For another example, let $X'to Y$ be a closed immersion of schemes over ${bar{mathbb F}}_p$, and let $X to X'$ be the relative Frobenius morphism.
Then $Xto X'$ is a homeomorphism on underlying topological spaces but is not an isomorphism of schemes, and so the composite $Xto Y$ is a closed embedding on underlying spaces but not a closed immersion of schemes.

As one last example, let $X'$ be the cuspidal cubic given by $y^2 = x^3$ in the affine
plane $Y$ (over $mathbb C$, say), and let $X$ be the normalization of $X'$ (which is just
the affine line). Then $X to X'$ is a homeomorphism on underlying spaces, but is not
an isomorphism of schemes. The composite $X to Y$ is thus not a closed immersion,
but induces a closed embedding of underlying topological spaces.

nt.number theory - homogeneous forms as norms

Motivation/example. Consider $K = mathbb{Q}(sqrt[3]{2})$. This is a number field with ring of integers $O_K = mathbb{Z}[sqrt[3]{2}]$. We have a norm map $N_{K/mathbb{Q}}$ which maps $x + ysqrt[3]{2} + zsqrt[3]{4}$ to $x^3 + 2y^3 + 4z^6 - 6xyz$; restricting to $mathcal{O}_K$ gives of course the same form. Using standard results about factorization of prime ideals, it is not too hard to see which integers are norms of elements in $mathcal{O}_K$. Therefore we can get the values of $n$ (with some work...) for which $x^3 + 2y^3 + 4z^3 - 6xyz = n$ has integral solutions, and I guess it is also possible to say something about the solutions for a fixed $n$ - although it is not obvious to me how to do this in general.

The same is of course true for many interesting quadratic forms - to cite a famous example: we can get all positive integers $n$ which can be written as the sum of two perfect squares, or more generally as $x^2 + alpha y^2$ (for some interesting values of $alpha$).

Questions. Is this a fruitful method to study diophantine equations? Are there interesting "large" classes of higher degree polynomials/diophantine equations which can be treated by this sort of argument? What is known in general about such "norm forms"? How to decide whether a polynomial is a norm form? Et cetera :)

(I know that it is a bit vague... I didn't find any useful references.)

reference request - Derivators (in English)

For a few references in English, there are the papers of Heller, the main one being:

A. Heller, Homotopy theories, Mem. Amer. Math. Soc. 71 (383) (1988)

There is also a paper I wrote with A. Neeman, in which there is a little introduction to derivators in the second half of:
Additivity for derivator K-theory, Adv. Math. 217 (2008), no. 4, 1381-1475

One can see derivators in action in the work of G. Tabuada (he explains Bousfield localization and stabilization in this setting, and compares with the model category point of view):
Higher K-theory via universal invariants, Duke Math. J. 145 (2008), no. 1, 121–206
(availabe as arXiv:0706.2420).

teaching - How to motivate the skein relations?

Alexander realized they were useful, then Conway. However, Jones clearly was the one
who really made a big bang with a skein relation. This allowed him to see a connection between the Jones polynomial and state sums in statistical mechanics. This was followed by HOMFLYPT, which might be the first time a skein relation was used to define an invariant rather than encapsulate some of its properties. I would say that Kauffman really simplified the study of the Jones polynomial by way of his bracket relation. Witten used the skein relation to build a hypothetical connection with TQFT. This connection was made rigorous by Reshitkhin and Turaev from a quantum groups perspective. The work of Habeger, Masbaum, Vogel, and Blanchet made it rigorous from a skein theoretic viewpoint. The connection between skein relations and Lie groups probably appeared first in the work of Turaev and Wenzl, where they classified skein relations by what family of Lie groups you were working with. There was work of Blanchet and Beliakova that built on this, especially understanding B-type Lie algebras. Xiao-Song Lin built a connection between the Jones polynomial and finite type invariants with skein relations. Effie Kalfagianni used them to extend the Jones polynomial to knots in other three-manifolds. Bar-Natan used skein relations to powerful effect to build connections between finite type invariants and Lie algebras. Bullock's work was the first to build the direct connection between skein relations and trace identities. This was built on in the work of Sikora who did it for many more Lie groups. Kuperberg's spiders were also actually about skein relations, and his work is starting have impact in the study of flag varieties. The skein relation short exact sequence of chain complexes first appeared in the work of Khovanov, though it had been simmering in Floer theory for a long time in the related surgery triangle.

I thought I would give a motivating example.

Recall that matrices satisfy their characteristic polynomial. If $A$ is a two by two matrix of determinant one, then its characteristic polynomial is $lambda^2-tr(A)lambda +1$. Hence we know $A^2-(tr(A)A+Id=0$ where the zero on the right is a two by two matrix of zeroes.
Multiply this through by $A^{-1}$ to symmetrize it as $A+tr(A)Id +A^{-1}=0$. Not multiply by any two by two matrix $B$ to get $BA-tr(A)B+BA^{-1}=0$. Finally, take the trace to get
$tr(BA)-tr(A)tr(B)+tr(BA^{-1})=0$. This is the fundamental trace identity for $SL_2(mathbb{C})$. Let $X_i$ be variables which you can think of as $2times 2$ matrices.
Form all words in the $X_i$ and then take all polynomials in traces of these words. Every identity between these polynomials is a consequence of $tr(A)=tr(A^{-1})$, $tr(AB)=tr(BA)$,
$tr(Id)=2$,
and the fundamental trace identity.

Let $M$ be a topological space. Take polynomials on free homotopy classes of loops in $M$ and mod out by the relations coming from replacing a loop by its inverse, replacing the null homotopic loop by $-2$, and the Kauffman bracket skein relaton with $A$ set equal to $-1$. In this setting the Kauffman bracket skein relation is the fundamental trace identity.

This ring is the coordinate ring of the unreduced affine scheme of characters of $SL_2(mathbb{C})$ representations of the fundamental group of $M$.

If $M$ is a surface then you can fatten the surface up to $Mtimes [0,1]$ and instead use framed links in $Mtimes [0,1]$. You multiply by stacking framed links. Use the Kauffman bracket skein relations at $A=e^h$. This algebra is a quantization of the $SL_2(mathbb{C})$ representations of the fundamental group of the surface with respect to Goldman's Poisson structure. That fact is established by proving that Goldman's Poisson bracket can be computed skein theoretically.

A good starting point for the above material might be the article "Understanding the Kauffman Bracket Skein Relation" by Bullock, F, and Kania-Bartoszynska.

The importance of skein relations is that it allows you to make teleological connections between knot polynomials and other, seemingly distant areas of mathematics like representation theory, quantization, algebraic geometry, gauge theory, and low dimensional geometry and topology.

lo.logic - How do we construct (in a vector space) a chain of countable dimensional subspaces that can only be bounded by an subspace of uncountable dimension?

The other answers asked you first to well-order the whole vector space (or a basis for it), and those answers are perfectly correct, but perhaps you don't like well order the whole space. So let me describe a construction that appeals directly to the Axiom of Choice.

Let V be your favorite vector space having uncountable dimension. For each countable dimension subpace W, let a_W be an element of V that is not in W. Such a vector exists, since W is countable dimensional and V is not, and we choose such elements by the Axiom of Choice.

Having made these choices, the rest of the construction is now completely determined. Namely, we construct a linearly ordered chain of countable dimensional spaces, whose union is uncountable dimension. Let V₀ be the trivial subspace. If V_α is defined and countable dimensional, let V_α+1 be the space spanned by V_α and the element a_Vα. If λ is a limit ordinal, let V_λ be the union of all earlier V_α. It is easy to see that { a_Vβ | β < α} is a basis for V_α. Thus, the dimension of each V_α is exactly the cardinality of α. In particular, if ω₁ is the first uncountable ordinal, then V_ω1 will have uncountable dimension, yet be the union of all V_α for α < ω₁, which all have countable dimension, as desired.

If you forbid one to use the Axiom of Choice, then it is no longer true that every vector space has a basis (since it is consistent with ZF that some vector spaces have no basis), and the concept of dimension suffers in this case. But some interesting things happen. For example, it is consistent with the failure of AC that the reals are a countable union of countable sets. R = U A_n, where each A_n is countable. (The irritating difficulty is that although each A_n is countable, one cannot choose the functions witnessing this uniformly, since of course R is uncountable.) But in any case, we may regard R as a vector space over Q, and if we let V_n be the space spanned by A₁ U ... U A_n, then we can still in each case make finitely many choices to witness the countability and conclude that each V_n is countable dimensional, but the union of all V_n is all of R, which is not countable dimensional.

ra.rings and algebras - What is an exponential?

Is there a notion of exponentiation that subsumes the well known versions, and in particular the versions on

• tangent spaces (e.g., of Lie groups and Riemannian manifolds), in which the exponential map sends a vector to a point on a curve naturally defined in terms of the vector;


• unital Banach algebras?

(NB. I am not conversant with category theory beyond the words "morphism" and "functor". But a categorically flavored answer that takes my limited knowledge base into account would be preferable. An internet search led me to the notion of a "Cartesian closed category", which doesn't seem to be the sort of thing I have in mind.)

ag.algebraic geometry - Mirror symmetry mod p?! ... Physics mod p?!

For fixed integers $g,n$, any projective scheme $X$ over a field $k$, and a linear map $beta:operatorname{Pic}(X)tomathbb Z$, the space $overline{M}_{g,n}(X,beta)$ of stable maps is well defined as an Artin stack with finite stabilizer, no matter the characteristic of $k$. You can even replace $k$ by $mathbb Z$ if you like.

Now if $X$ is a smooth projective scheme over $R=mathbb Z[1/N]$ for some integer $N$, then $overline{M}_{g,n}(X,beta) times_R mathbb Z/pmathbb Z$ is a Deligne-Mumford stack for almost all primes $p$. For such $p$, $overline{M}_{g,n}(X,beta) times_R mathbb Z/pmathbb Z$ has a virtual fundamental cycle, and so you have well-defined Gromov-Witten invariants. This holds for all but finitely many $p$. Nothing about $mathbb C$ here, that is my point, the construction is purely algebraic and very general.

It is when you say "Hodge structures" then you better work over $mathbb C$, unless you mean $p$-Hodge structures.

As far as mirror symmetry in characteristic $p$, much of it is again characteristic-free. For example Batyrev's combinatorial mirror symmetry for Calabi-Yau hypersurfaces in toric varieties is simply the duality between reflexive polytopes. You can do that in any characteristic, indeed over $mathbb Z$ if you like.

rt.representation theory - Embeddings between p-adic linear groups?

Let p be a prime and let $mathbb Z_p$ denote the p-adic integers.

If n<m, then what are the embeddings $SL_n(mathbb Z_p)rightarrow SL_m(mathbb Z_p)$? I am particularly interested in those which carry $SL_n(mathbb Z)$ into $SL_m(mathbb Z)$.

There are obvious "block" embeddings, e.g., carrying a matrix to the upper-left hand corner of a larger matrix. There are also certain conjugates of these. In general, the embeddings should come from representations of $SL_n(mathbb Z_p)$, but I do not know where they are catalogued or what exactly to do with the catalog.

gr.group theory - Distinguishing finite-orbit permutation groups by action on tuples

Here's a case where $G$ and $H$ can be conjugate. First some notation: given a sequence ${k_n}$ of positive integers, let $[k_1,k_2,ldots]$ denote the permutation

$$(1,ldots,k_1)(k_1+1,ldots,k_1+k_2)(k_1+k_2+1,ldots,k_1+k_2+k_3)cdots$$

with cycles of size $k_1,k_2,k_3ldots$. For example, $[1,1,1,1,ldots]$ denotes the identity, $[2,2,2,2,ldots]$ denotes $(1,2)(3,4)(5,6)(7,8)cdots$, and $[2,3,2,3ldots]$ denotes $(1,2)(3,4,5)(6,7)(8,9,10)cdots$.

Let
$$g = [1,2,;;1,2,4,;;1,2,4,8,;;ldots],$$
let
$$h = [1,1,1,;;1,1,1,2,2,;;1,1,1,2,2,4,4,;;ldots],$$
and let $G$ and $H$ be the cyclic subgroups generated by these elements. Since $g$ and $h$ have the same cycle structure, they are conjuagte in $Sym(mathbb{N})$, so $G$ and $H$ are conjugate subgroups.
However, for sufficiently large $n$, the orbit of $(pi(1),pi(2),ldots,pi(n))$ under $G$ will be precisely twice the size of the orbit under $H$.

Of course, in this example $G$ and $H$ both have infinitely many orbits of size $2^k$ for every $k$, so this does not answer the more restrictive version of the question.

invariant theory - Generalized symmetric algebras and Dickson algebras over ${mathbb F}_p$.

Four quick references that contain substantial info on your questions (for more, it'd be good to know what exactly you would like to know...):

de Concini, C.; Procesi, C.
A characteristic free approach to invariant theory.
Advances in Math. 21 (1976), no. 3, 330--354.

Grosshans, F. D.
Vector invariants in arbitrary characteristic.
Transform. Groups 12 (2007), no. 3, 499--514.

Stepanov, S. A.
Vector invariants of symmetric groups in the case of a field of prime characteristic. Discrete Math. Appl. 10 (2000), no. 5, 455--468

Stepanov, Serguei A.
Orbit sums and modular vector invariants. Diophantine approximation, 381--412,
Dev. Math., 16, Vienna, 2008.

knot theory - 4-genus of a 2-bridge link

Rasmussen and Lee's results say that the $s$ invariant of a 2-bridge knot will be just equal to the signature of the knot. So you can compute the signature of your knot to get a lower bound (there are extremely rapid ways of doing this from an alternating diagram). Unfortunately the only decent way to get an upper bound that I know of is by spotting a smooth surface! Good luck.

Remember that $s$ might not be the best you can do. In particular, among alternating knots, the figure 8 knot ($4_1$ in Rolfsen) has vanishing $s$ invariant (for example, because it is torsion in the concordance group) and yet it is not even slice if you allow your surfaces to be locally flat, let alone smooth.

career - Teaching statements for math jobs?

Having been on both sides of the issue, I might say that having considered it for some time, I really don't know! But in reality if you are looking for a position at a research university, the Dean will want to have evidence (or the non-research faculty will want to have evidence) that you care about teaching. More precisely, some subset of your peers might have a very specific teaching philosophy although they may not be able to articulate it. Those peers want to know if your teaching philosophy coincides with theirs.

A few years back everyone was "hot" on the use of technology in the classroom. I don't know what that means, but suppose that it means using TI calculators, power point (the horror, the horror) or a course blog. If you have a point of view on the positive value of these things then you should say so.

The problem is that each department has its own mix of bozos. I am pretty much a chalk on slate kind of guy, and when someone tells me they like clickers in large classes, I wonder do they turn around to look at their students faces. So in an ideal world you would tailor your teaching statement to the place you want to go, or to the place that you are applying. Of course, you don't want to write 200 teaching statements, so that won't work.

So I am back to the original premise. They want to know that you have thought about teaching.

lo.logic - Elementary equivalence of ordinals

The first-order theory of well-orderings was studied in great detail in a paper of Doner, Mostowski, and Tarski, "The elementary theory of well-ordering -- a metamathematical study" [Logic Colloquium '77, edited by A. Macintyre, L. Pacholski, and J. Paris, North-Holland (1978) pp. 1-54]. In particular, their Corollary 44 characterizes (unless their notation is very non-standard --- I haven't checked carefully) when two ordinals are elementarily equivalent. Modulo an apparent typo in the definition just before the corollary (one of the strict inequalities should be non-strict), it seems that the first pair of distinct but elementarily equivalent ordinals is $omega^omega$ and $omega^omegacdot2$. A thorough reading of the paper (which I don't have time for right now) should also reveal the answer to your second question, about elementary submodels (probably the same pair of ordinals).

pr.probability - Probability Question

You have $N$ boxes and $M$ balls. The $M$ balls are randomly distributed into the $N$ boxes. What is the expected number of empty boxes?

I came up with this formula:

$sum_{i=0}^{N}ibinom{N}{i}left(frac{N-i}{N}right)^{M}$

This seems to yield the right answer. However, it requires calculating large numbers, such as $binom{N}{frac{N}{2}}$. Is there a more direct way, perhaps using a probability distribution? It seems that neither the binomial nor the hypergeometric distributions fit the problem.

ag.algebraic geometry - Gauge theory construction of moduli of vector bundles

Hi Botong, I guess I could say this in person but this is faster.

Given a reductive group acting a finite dimensional Euclidean space, Kempf and Ness proved that the orbits of stable points are precisely the ones on which the norm attains a minimum. This gives a hint of the role of stability in the analytic construction,
where now one would like to minimize a suitable energy on the orbits of an infinite dimensional gauge group. This is spelled out in more detail in
[DK] "The geometry of four manifolds" by Donaldson and Kronheimer, and
[C] "Flat G-bundles with canonical metrics" by Corlette in addition to the references given above.

Regarding your second question, I suspect that the most meticulous comparison of the
algebraic and analytic constructions can be found in Simpson's papers "Moduli
of representations of the fundamental group I & II".

Addendum: Perhaps I should expand the my answer a bit, since it wasn't terribly enlightening. In very rough terms on the algebraic side one proceeds as follows.
One first needs to prove that the set of stable vector bundles with fixed topological type $V$ form a bounded family. This already uses the stability condition in an essential way.
From this one deduces the existence of a scheme $Q$, usually a subscheme of a Quot scheme, such that $Xtimes Q$ carries a vector $E$ such that all the stable bundles in question occur among the fibres $E_q$. The moduli space $M$ would then be quotient $M=Q/G$ for some appropriate reductive $G$ acting by "change of basis". At this point, to apply GIT,
one needs to know that stability in the abstract is related to to stability of the
$G$-action. Note that $E$ typically won't descend i.e. $M$ need not be fine.

On the analytic side, one can form the quotient $N$ of the set of stable
complex structures $S$ on $V$ modulo gauge equivalence. Again stability comes into play
to guarantee that $N$ is reasonable. So then one gets a map of
spaces $Q^{an}to S$ induced by $E$. This should descend to a map $M^{an}to N$. To check that this is an analytic equivalence, one would need a description of the local analytic structures on both sides. Fortunately one has this this, see theorem 4.5.1 of [Huybrechts-Lehn]
and prop 6.4.3 of [DK]. Making this into a proof would take a lot of work of course.

Formal Geometry

I am writing to post an answer to my own question. The answer below consists of what I was able to jot down from a seminar talk by Jacob Lurie, and a patient followup explanation by Roman Travkin, followed by a correction by David Ben-Zvi. Of course, mistakes and naivete in the translation are solely attributed to me. Since the answer was given to me in response to my asking on MO, it seems that karma dictates that I record it here.

The deRham stack, $X_{dR}$, of a scheme $X$ is defined as a functor on test schemes S by

$$X_{dR}(S) := {textrm{Maps}: S_{red} to X }.$$

where $S_{red}$ is the reduced scheme associated to a scheme $S$. It is representable as a stack - I think always, but at least when $X$ is smooth - which we should assume for later purposes anyways (maybe everything should be based over $mathbb{C}$ also? I welcome corrections, which I'll incorporate).

Okay so $X$, viewed as the functor $X(S)={textrm{Maps}:Sto X}$ has a natural map $pi:Xto X_{dR}$, by pre-composing with $S_{red} to S$.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(xin X, t_x: X_{(x)}cong hat{D_n})$, where $hat{D_n}$ is the formal disc, $mathcal{O}(hat{D_n}):=mathbb{C}[[x_1,ldots,x_n]]$. The fiber over each point $xin X$ is the group $Aut^0(X_{(x)})$ of automorphisms of the formal neighborhood of $x$ preserving the maximal ideal, which is in turn (non-canonically) isomorphic to $Aut^0(hat{D_n})$, the group of automorphisms of $hat{D_n}$ preserving the origin.

Now, the scheme $Aut_X$ is actually a $hat G$-torsor over $X_{dR}$, where (I gather from David Ben-Zvi's comments) $hat G$ is something more like an inductive limit of the groups of automorphisms on $n$th infinitessimal neighboroods of the origin in $D_n$.

Now, suppose that $M$ is a $W_n$ module, which has the property that the operators $x_ipartial_i$ for all $i$ are diagonalizable with finite dimensional eigenspaces. (These we should think of as being the diagonals $h_i$ for a copy of $mathfrak{gl}_n$ sitting inside $W_n$). In this case, one can prove some nice things.

First, in representation theory: $M$ lies in the category generated by modules $mathcal{F}_lambda$ which are coinduced from $V_lambda$, the irreducible of $mathfrak{gl}_n$.` That is, as vector spaces the $mathcal{F}_lambda$s are just $V_lambda otimes C[x_1,ldots, x_n]$, and the action is given by natural formulas. So $M$ is a (possibly infinite) extension of such things. But there is a theory of weights which control the representation theory somewhat. See A. N. Rudakov. Irreducible representations of infinite-dimensional Lie algebras of Cartan type. Math. USSR Izv. Vol. 8, pgs. 836-866, which has been translated to English.

Second, in geometry: The module $M$ is a Harish-Chandra module for the pair $(W_n,Aut_0(C[[x_1,...,x_n]]))$. This means: $W_n$ has a Lie sub algebra $W^0$ of vector fields which vanish at the origin (so they have no constant vector field terms like $partial_i$). $W^0$ (or perhaps its completion w.r.t order of vanishing at origin) is the Lie algebra of $Aut_0(C[[x_1,...,x_n]])$, as it consists of derivations which preserve the maximal ideal (not sure exactly how you make precise that it is "the Lie algebra", but anyways, there is an exponential map turning an integrable $W^0$ module into an $Aut_0(C[[x_1,...,x_n]]$)-module ). The assumptions on $M$ were precisely those that make $M$ an integrable $W^0$ module, and so we can regard $M$ as a $G$-module. Well now we have an associated bundle construction for the $G$-torsor $Aut_Xto X$, and we can use this to produce a sheaf of vector spaces (not quasi-coherent!) over $X$ with fiber $M$.

Better $M$ has an action of the operators $partial_i$, and we can exponentiate the action of all of $W_n$ to the group $hat G$. That means that we can instead construct the associated bundle for the $hat G$ torsor $Aut_Xto X_{dR}$, which will give us a bundle over $X_{dR}$ with fiber $M$ again. Now, we can pullback this bundle via $pi$ to get a bundle on $X$ with fiber $M$. This time, however, it's a pullback of a sheaf of vector spaces on the deRham stack, which by (some people's) definition is a crystal of vector spaces on $X$.