fa.functional analysis - Dual Spaces of Sobolev Spaces

I am going to stick with the standard terminology $H^m$ here. Taking Fourier transforms one finds that $$langle u,vrangle_m=inthat u(xi)bar{hat v}(xi)(1+|xi|^2)^m,dxi$$ (give or take the odd multiplicative constant), where $H^m$ consists precisely of those $uin L^2$ for which $langle u,urangle<infty$. This works even for $m<0$, if you allow distributions whose Fourier transforms are functions. Everything follows from this, including the fact that $H^{-m}$ acts as the dual of $H^m$ simply by the distribution $u$ acting on the function $v$, which corresponds to the integral $$langle u,vrangle=int hat u(xi)bar{hat v}(xi),dxi=int hat u(xi)(1+|xi|^2)^{-m/2}cdothat{bar v}(xi)(1+|xi|^2)^{m/2},dxi$$ where I have split up the integrand into a product of two $L^2$ functions.

For this reason, it seems more natural to identify $H^{-m}$ with the dual of $H^m$ than to identify $H^m$ with its own dual. However, you can go ahead and identify any Sobolev space with the dual of any other just by inserting a suitable power of $1+|xi|^2$ in the integral defining the pairing between the two.

Rather than coming straight out and answering your question, I'll leave it to you to ponder the consequences of the above. In particular, note that you we embed and identify you have to keep careful track of what space you have identified with whose dual, or you will be endlessly befuddled.

Addendum: To spell out a more direct answer to your question, $langlecdot,cdotrangle_m$ can identify $H^{m+sigma}$ with the dual of $H^{m-sigma}$, since we can write $$langle u,vrangle_m=int hat u(xi)(1+|xi|^2)^{(m-sigma)/2}cdotbar{hat v}(xi)(1+|xi|^2)^{(m+sigma)/2},dxi$$ where I have split the integrand into a product of two $L^2$ functions.

Edit: Changed a couple $hat{bar v}$ into $bar{hat v}$.

gr.group theory - Maximal subgroups of abelian groups and Q-algebras

Assuming that by a $mathbb{Q}$ algebra you mean a $mathbb{Q}$ vector space, the abelian groups that admit a $mathbb{Q}$ - vector space structure are precisely the divisible torsion-free abelian groups, i.e. torsion-free abelian groups A such that $forall$ $xin A, nin mathbb{N}$, $exists$ $yin A$ s.t. $ny=x$. The condition is clearly necessary, and for sufficiency, consider the canonical mapping of $A$ into $mathbb{Q}otimes_{mathbb{Z}}A$. This will be an isomorphism precisely when $A$ satisfies the above mentioned condition.

As for the question regarding maximal subgroups, one can show that an abelian group has no maximal subgroups if and only if it is divisible, in the sense mentioned above. If a group is not divisible, then there will exist a prime $p$ such that $pA$ is a proper subgroup of $A$; we can then use the fact that $A/pA$ is a vector space over $mathbb{Z}/pmathbb{Z}$ to pick out a maximal subgroup. On the other hand, maximal subgroups in abelian groups must always be of finite (in fact prime) index. Thus, if we have a maximal subgroup $B$ in $A$, of index $p$, then $pAsubseteq B$. However, by divisibility, $pA=A$, a contradiction.

pr.probability - Analog of Chebyshev's inequality for higher moments

Tom is right: the proof of Chebyshev's inequality can be easily adapted to every nondecreasing nonnegative function. The proof of this generalization that I prefer has a principle worth remembering:

First find an inequality between random variables, then integrate it.

To apply the principle, let $g$ denote a nondecreasing nonnegative function defined on $[0,+infty)$ and $Z$ a nonnegative random variable. Let $zge0$ such that $g(z)>0$. Then,
$$
g(z)mathbf{1}_Ale g(Z) mbox{with} A=[Zge z].
$$
Proof: if $omeganotin A$, the assertion reduces to $0le g(Z(omega))$, which holds because $g$ is nonnegative everywhere; if $omegain A$, the assertion reduces to $g(z)le g(Z(omega))$, which holds because $Z(omega)ge z$ and $g$ is nondecreasing.

Integrating the inequality yields
$$
g(z)P(A)le E(g(Z)),
$$
and, dividing both sides by $g(z)$, we are done.

The usual case is when $Z=|X-E(X)|$ and $g(z)=z$. The case mentioned by Thomas is when $Z=|X-E(X)|$ and $g(z)=z^p$, for every positive $p$ (and not only for $pge2$). Another case, mentioned by Tom and at the basis of the whole field of large deviations principles, is when $Z=mathrm{e}^{rX}$ for a nonnegative $r$ and $g(z)=z$. But to deal with the $ulog(u)$ case mentioned by Tom requires to be more careful because the function $umapsto ulog(u)$ is non monotonous on $[0,1]$ and not of a constant sign on $[0,+infty)$ (but everything works fine for $g(z)=zlog(z)$ on $[1,+infty)$, that is, if $Zge1$ almost surely).

st.statistics - Sorting Based on Rating

I'm doing some machine learning stuff and I want to take some random samples and determine if a human agrees with the computer. To do this a user just votes up or down on a given item. Then I want to be able to sort by the items with the highest rating. I want to use something more complicated than simply up-down to get good results.

I've looked into the Wilson Interval Score and it seems like a decent solution, but I'm wondering if there are other alternatives.

I'm going to be using C# 4.0 if that matters but for now I'm strictly interested in the math.

Example below:

Lets suppose I have 3 items and multiple people have voted on them according to the table:

Item    Up    Down
1       6       1
2       60      11
3       100     40

In this example I would like Item 3 to be listed first, item 2 second and 3 third. This is a rough approximation of my expectations.

Item 3 has the most responses and highest relative approval. Item 2 has more responses than Item 1 despite having a lower percentage approval.

I'm trying to list the items in terms of some sort of relative metric and algrotithm without using something like percent approval or net score; something more complicated.

at.algebraic topology - Applications of topological and diferentiable stacks

I would like to point out that stacks are "just" higher analogues of sheaves - a very basic tool to arrange structure. The same is true for topological or differentiable stacks. So I think everybody who expects amazing applications of stacks should be able to name an equally amazing application of a sheaf. (I am not saying that those don't exist!)

That said, let me mention an application. In view of the fact that you didn't get any answers so far (apart from your own), I hope it's not too inappropriate to take one from my own research. It applies abelian gerbes with connection to lifting problems for principal bundles.

I hope the following specifications qualify the theorem below as application: its statement does not involve any stacks or gerbes, just "basic" differential geometry. Its proof, however, is a simple composition of two gerbe-theoretical theorems.

Theorem. Let $M$ be a connected smooth manifold, let $P$ be a principal $G$-bundle with connection over $M$, let $hat G$ be a central extension of $G$ by an abelian Lie group $A$, and let $rho in Omega^2(M,mathfrak{a})$ be a 2-form. Then, there exists a principal $A$-bundle $mathcal{L}_P$ over $LM$ with a connection and with a fusion product, and a bijection between

1. isomorphism classes of lifts of the structure group of $P$
   from $G$ to $hat G$ with compatible connection of scalar curvature $rho$,
   and




2. smooth sections of $mathcal{L}_P$ that preserve the fusion product and pull back the connection to the transgressed 1-form $Lrho in Omega^1(LM,mathfrak{a})$.

Of course some concepts that appear here would need some more explanation - but that's not the point. Let me better point out how gerbes with connection come into the picture. We employ two results from gerbe theory:

1. Associated to every lifting problem posed by a bundle $P$ is an $A$-gerbe over $M$, called the "lifting gerbe" and denoted $mathcal{G}_P$. This gerbe represents geometrically the obstruction against lifts. Moreover, the actual lifts are in equivalence with trivializations of $mathcal{G}_P$. The same works if one wants to include connections into the lifting problem. These are results of Murray and Gomi.




2. The category of $A$-gerbes with connection over $M$ is equivalent to a certain category of principal $A$-bundles with connection over $LM$ which are additionally equipped with "fusion products". The equivalence is established by a transgression functor, which has been introduced by Brylinski and McLaughlin. It takes trivializations of gerbes to sections of bundles.

Now, define $mathcal{L}_P$ as the transgression of $mathcal{G}_P$. Since transgression is an equivalence of categories, it is a bijections on Hom-sets, and this bijection is exactly the statement of the theorem.

Ok, in order to complete my claim that this is an application, I should probably mention an example where the theorem is useful. That's the case for $spin$ and $spin^c$ structures on manifolds, and I have learned about it from Stephan Stolz and Peter Teichner. In the case of $spin$ structures, $mathcal{L}_P$ is a $mathbb{Z}_2$-bundle over $LM$ and plays the role of the orientation bundle of $LM$. Since $mathbb{Z}_2$ is discrete, all the connections disappear and forms are identically zero. So, the theorem says that isomorphism classes of $spin$ structures on $M$ are in bijection to "fusion-preserving orientations" of $LM$. In the $spin^c$ case, a similar statement follows that additionally includes the scalar curvature of the $spin^c$-structures.

ag.algebraic geometry - Cohomology with compact support for coherent sheaves on a scheme

If X is a smooth scheme over complex numbers then you can consider $X_{an}$ as an complex analytic manifold and compute singular/ deRham/ simplicial cohomology with compact supports (this will be different from usual cohomology if X is not proper)
On the Algebraic side there is etale cohomology with compact supports (which is defined by embedding X into a proper scheme...).

Comparison theorems tell you that etale cohomology with torsion coefficients agree with singular cohomology (with torsion coeff).

Any reference on etale cohomology will discuss this.

Ref:SGA 4.5, Milne: Etale Cohomology.

dg.differential geometry - Does the Baker-Campbell-Hausdorff formula hold for vector fields on a (compact) manifold?

Consider a compact manifold M. For a vector field X on M, let $phi_X$ denote the diffeomorphism of M given by the time 1 flow of X.

If X and Y are two vector fields, is $phi_X circ phi_Y$ necessarily of the form $phi_Z$ for some vector field Z?

Since $Xmapsto phi_X$ can be thought of as the exponential map from the Lie algebra of vector fields to the group of diffeomorphisms, an obvious candidate is that Z should be given by the Baker-Campbell-Hausdorff formula $B(X, Y) = X+Y+frac{1}{2}[X,Y]+cdots$. But does this hold in this infinite-dimensional setting? If so, in which sense does the series converge to Z?

Also, I'm interested in the case where M is a symplectic manifold and we consider only symplectic vector fields (ie. vector fields for which the contraction with the symplectic form is a closed 1-form). Locally, X and Y are the Hamiltonian vector fields associated to smooth functions f and g, so I assume that asking whether B(X, Y) makes sense/is symplectic corresponds to asking whether B(f, g) makes sense/defines a smooth function (where, of course, we use the Poisson bracket in the expansion of B(f, g)). The right-hand side of B(f,g) consists of lots of iterated directional derivatives of f and g in the X_f and X_g directions; it is not clear to me that the coefficients in the BCH formula make the series converge (uniformly, say) for any choice of f and g.

big list - What is your favorite isomorphism?

Here is an example that Mel Hochster used to explain the notion of isomorphism to a group of talented high school students. I was one of the course assistants rather than one of the students, but I'm sure the insight was at least as valuable for me as for them.

Consider the following game. I'll write down the numbers 1 through 9 on a sheet of paper, and you and I will take turns selecting numbers from the list (crossing off each number once it has been selected). The winner is the first person to have chosen exactly three numbers which add up to 15. For example if I selected 9, 6, 2 and you selected 3, 8, 1, 4 then you would win because 3 + 8 + 4 = 15.

The interesting thing is that this game is isomorphic to tic-tac-toe. I don't know what I precisely mean by that, but I can explain why it is true. Simply consider a 3 x 3 magic square:

4 9 2

3 5 7

8 1 6

The rows, columns, and diagonals all add up to 15, and moreover every way of writing 15 as the sum of three numbers from 1 to 9 is represented. When you choose a number, draw an X over it; when I choose a number, circle it. Tic-tac-toe!

mg.metric geometry - Pushing convex bodies together

The sets ${ (A(t),t)|tin mathbb{R} } subset mathbb{R}^4$ and ${ (B(t),t)|tin mathbb{R} } subset mathbb{R}^4$ are convex, their intersection $K$ is a bounded convex set, and $f(t)$ is the volume of the slice of $K$ at height $t$. By Brunn-Minkowski inequality, this is log-concave, so definitely unimodal.

gn.general topology - p-adic noninvariance of dimension

$mathbb{Q}_p$ is homeomorphic to a countable direct sum of copies of the Cantor set $C$. Indeed, because the valuation is discrete, for each $n geq 1$ the "annulus"

$A_n =$ {$x in mathbb{Q}_p | p^{n-1} < ||x|| leq p^{n}$}

is closed and homeomorphic to the Cantor set $C$. (Take of course $A_0 = mathbb{Z}_p$.)

Since as you observed above, $C times C cong C$, it follows that $mathbb{Q}_p^n$ and
$mathbb{Q}_p^m$ are homeomorphic for all $m, n in mathbb{Z}^+$ (and the homeomorphism type is independent of $p$).

mp.mathematical physics - Where does a math person go to learn quantum mechanics?

I think there are some excellent recommendations above. I learned quantum mechanics for real from Shankar, I think it's a great choice. Griffiths is also a great physics text. I would also recommend these following less famous books:

Physical chemistry and materials science textbooks. I would also highly recommend newer textbooks in physical chemistry as a perhaps less obvious place to look for excellent introductions to quantum mechanics - as Dirac famously said once, it's really the foundation for all of chemistry. An excellent physical chemistry is Physical Chemistry by Berry, Rice and Ross. Presumably there are also good introductions in materials science books, although I don't have any to recommend.

Not Feynman. In my opinion Feynman's Lectures in Physics is great for insight, but it's a terrible idea to learn anything from it the first time - remember that when Feynman actually lectured, most of the freshmen and sophomores (the intended audience) dropped the course, and were replaced by senior students!

Weyl (group theory). I'm surprised no one's mentioned Hermann Weyl's textbook "Theory of groups and quantum mechanics". It's an oldie but goodie, and perhaps best appreciated with someone with a good background in group theory.

Lieb (analysis). I recommend Elliott Lieb's Analysis GSM textbook - on the surface, it looks like it's about functional analysis, but it's secretly also a text on quantum mechanics!

---

There are some subjects that none of the introductory quantum mechanics texts I've read ever do a satisfactory job of explaining, and I think are really worth following up after Shankar or another such book. The most important ones I think are:

• Many-body phenomena. This is really where some of the strangest predictions of quantum mechanics come from, like the EPR paradox and spin statistics. Levine's Physical Chemistry is an excellent place to start. Another great book is Blaizot and Ripka's Quantum Theory of Finite Systems, which does a superb job with boson and fermion statistics.




• Dynamics (time-dependent quantum mechanics. I cannot recommend Tannor's Introduction to Quantum Mechanics: A Time-dependent Perspective enough as a really fantastic resource for learning how practicing physicists and chemists actually do these calculations, beyond the really simplistic calculations presented in most introductory texts. That could also work as a first textbook.

---

You know, I'm in the building next to you. Maybe you should come by and talk sometime. :)

nt.number theory - Have any long-suspected irrational numbers turned out to be rational?

There are reasons that any modern example is likely to resemble the status of Legendre's constant. Most (but not all) interesting numbers admit a polynomial-time algorithm to compute their digits. In fact, there is an interesting semi-review by Borwein and Borwein that shows that most of the usual numbers in calculus (for example, $exp(sqrt{2}+pi)$) have a quasilinear time algorithm on a RAM machine, meaning $tilde{O}(n) = O(n(log n)^alpha)$ time to compute $n$ digits. Once you have $n$ digits, you can use the continued fraction algorithm to find the best rational approximation with at most $n/2-O(1)$ digits in the denominator. The continued fraction algorithm is equivalent to the Euclidean algorithm, which also has a quasilinear time version according to Wikipedia.

Euler's constant has been to computed almost 30 billion digits, using a quasilinear time algorithm due to Brent and McMillan.

As a result, for any such number it's difficult to be surprised. You would need a mathematical coincidence that the number is rational, but with a denominator that is out of reach for modern computers. (This was Brent and MacMillian's stated motivation in the case of Euler's constant.) I think that it would be fairly newsworthy if it happened. On the other hand, if you can only compute the digits very slowly, then your situation resembles Legendre's.

---

I got e-mail asking for a reference to the paper of Borwein and Borwein. The paper is On the complexity of familiar functions and numbers. To summarize the relevant part of this survey paper, any value or inverse value of an elementary function in the sense of calculus, including also hypergeometric functions as primitives, can be computed in quasilinear time. So can the gamma or zeta function evaluated at a rational number.

rt.representation theory - Are complex semisimple Lie groups matrix groups?

Actually, my question is a bit more specific: Does every complex semisimple Lie group $G$ admit a faithful finite-dimensional holomorphic representation? [As remarked by Brian Conrad, this is enough to prove that $G$ is a matrix group (at least when it's connected) because $G$ can be made into an (affine) algebraic group over $mathbb{C}$ in unique way which is compatible with its complex Lie group structure, and under which every finite-dimensional holomorphic representation is algebraic. Furthermore, one can show that the image of a faithful representation would then be closed.]

Of course the analogous question for real semisimple Lie groups has a negative answer -- "holomorphic" having been replaced by "continuous", "smooth" or "real analytic" -- with the canonical counterexample being a nontrivial cover of $mathrm{SL}(2,mathbb{R})$.

For a connected complex semisimple Lie group $G$ I believe the answer is "YES." The idea is to piggy back off a 'sufficiently large' representation of a compact real form $G_mathbb{R}$; here by "compact real form" I'm referring specifically to the definition which allows us to uniquely extend continuous finite-dimensional representations of $G_mathbb{R}$ to holomorphic representations of $G$. I know (e.g. from the proof of Theorem 27.1 in D. Bump's Lie Groups) that such a definition is possible if we require $G$ to be connected (and I'd like to know if it's possible in general).

The details of the argument for connected $G$ are as follows. Consider the adjoint representation $mathrm{Ad} colon G to mathrm{GL}(mathfrak{g})$. Since $G$ is semisimple, $mathrm{Ad}$ has discrete kernel $K$. Consider next the restriction of $mathrm{Ad}$ to $G_mathbb{R}$. Observe that the kernel of this map is also $K$, for otherwise its holomorphic extension is different from the adjoint representation of $G$. Thus $K$ is finite, being a discrete, closed subset of a compact space. So by the Peter-Weyl theorem, we can find a representation $pi_0$ of $G_mathbb{R}$ that is nonzero on $K$. Extend $pi_0$ to a holomorphic representation $pi$ of $G$ and put $rho = pi oplus mathrm{Ad}$. Notice that $rho$ is a holomorphic representation of $G$ with kernel $kerpi cap K = 0$, which is what we were after.

What can we say if $G$ is disconnected?

pr.probability - Probability of random permutation having certain cycles

The following paper may be what you're looking for:

Schramm, Oded. Compositions of random transpositions. Israel J. Math. 147 (2005), 221--243. MR2166362 (2006h:60024)

http://www.springerlink.com/content/f572513876635mj5/

This paper is concerned with the distribution of a random permutation in $S_n$ generated by $c n$ random transpositions (where $c>1/2$). Schramm calculates the limiting distribution of the cycle lengths (ordered from largest to smallest).

A recent paper of Nathanael Berestycki, Oded Schramm, and Ofer Zeitouni has extended the techniques of Schramm's earlier paper to the case where the random permutation is generated by random $k$-cylces. I don't know if this recent paper answers the same questions, but it might be relevant.

http://arxiv.org/abs/1001.1894

gt.geometric topology - Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?

A combinatorial way to construct examples consists of taking any (topological) ideal triangulation with k ideal vertices where all edges have valence >=7 (i.e. each edge meets at least 7 tetrahedra, counted with multiplicity: if you take at least 6 you might find toric cusps). After removing an open star of the vertices you get a 3-manifold with k boundary components which admits such a metric.

(To prove this, see http://arxiv.org/abs/math/0402339, or prove that there cannot be any normal surfaces with non-negative Euler characteristic in such a triangulation and use geometrization.)

You can constrcut concretely the metric if all edges have the same valence v>=7: in this case you realize every tetrahedron as a regular truncated hyperbolic tetrahedron with dihedral angles at all 6 edges equal to 2pi/v (such an object exists since this angle is smaller than pi/3).

The number of tetrahedra you need for such a construction of course grows with k. For k=1, two tetrahedra suffice (see Thurston's knotted Y from his lecture notes).

ag.algebraic geometry - injective objects

First of all, let me quote the famous paper by Grothendieck "Sur quelques points d'algebre homologique" (Tohoku Math part 1 & part 2). According to Theorem 1.10.1 there, every abelian category satisfying (AB5) (equivalent to exactness of filtered direct limits) and with a generator has enough injectives, For every scheme $X$ it is well known that $Qco(X)$ is abelian and satisfies (AB5), see, for instance, EGA I, new edition, Corollaire (2.2.2)(iv) where it is proved that such a limit preserves quasi-coherence.

So, the only issue is the existence of a generator. For some time it was known that over a quasi-compact quasi-separated scheme, $Qco(X)$ has a generator. For a nice geometric argument, consult the proof of Theorem (4) in Kleiman's "Relative duality for quasi-coherent sheaves". Note that any noetherian scheme $X$ is quasi-compact and quasi-separated, EGA I, (6.1.1) and (6.1.13).

Surprisingly, using techniques from relative homological algebra, Enochs and Estrada proved in 2005 the existence of a generator for any scheme. See their paper "Relative homological algebra in the category of quasi-coherent sheaves". (There was a previous unpublished proof by O. Gabber).

Summing up, for any scheme $X$, the category $Qco(X)$ has enough injectives.

graph theory - Comparing number of spanning subgraphs

Hi all,

Let be $G_n=(V_n,E_n)$ a finite graph, where
$V_n= {0,1,ldots, n} times{0,1,ldots,n}$

and $E_nsubset V_n^{(2)}$ is the edge set of the nearest neighbors in the $ell^1$ norm, that is,
$ E_n={ {z,w}subset V_n; sum_{i=1}^2 |z_i-w_i| =1 }.$

Fix a vertex $x=(x_1,x_2)in G_n$ such that $x_2>x_1$ (up-diagonal).
I would like to know if it is true the following inequality:

$sharp[m,p]_{x}leq sharp[p,m]_x$, whenever $p < m$

where $[m,p]_{x}$ is the set of all spanning subgraphs of $G_n$
satisfying the following properties:

1- the spanning subgraph has $m$ horizontal edges and $p$ vertical edges;

2- the vertices $(0,0)$ and $x=(x_1,x_2)$ are in the same connected component,

and $sharp A$ is the cardinality of $A$.

In other words, if I have avaliable more vertical edges than horizontal ones is it true that I can find more configurations connecting
$0$ and $x$ if $x$ is up-diagonal than in case that the quanties of horizontal and vertical are inverted ?

Thanks in advance for any idea or reference.

What subspaces of n-tuples of rational functions can be the solution space to a system of differential equations?

Let $K=mathbb{C}(x)$ denote the field of rational functions (in 1 variable), and let $K^n$ denote an $n$-dimensional $k$-vector space (with basis). For some integer $m$, let
$$delta_{11}, delta_{12},...delta_{1n};delta_{21},delta_{22}...delta_{2n};...;delta_{m1},delta_{m2},...delta_{mn}$$
be $mn$-many differential operators in $x$ with coefficients in $K$, so that
$$ delta_{11}(f_1)+...+delta_{1n}(f_n)=0$$
$$ delta_{21}(f_1)+...+delta_{2n}(f_n)=0$$
$$...$$
$$ delta_{m1}(f_1)+...+delta_{mn}(f_n)=0$$
defines a system of $m$-many differential equations on $(f_1,...f_n)in K^n$. Let $Ssubseteq K^n$ denote the space of solutions to this system of differential equations; by homogeneous linearity it is a $mathbb{C}$-subspace of $K^n$.

Usually, people are interested starting with a system of equations and finding the solution space. I have a weak inverse question. When is a given $mathbb{C}$-subspace $Vsubseteq K^n$ the solution space to such a system of differential equations? I don't care (yet) about finding the system of equations or how many equations there are, only whether they exist.

For $n=1$, the answer is appealingly simple. Either the system is degenerate, and the solution space is all of $K$; or it is not, and the solution space is finite $mathbb{C}$-dimensional. Therefore, an infinite $mathbb{C}$-dimensional proper subspace of $K$ is not the solution space to any system of equations, and every finite $mathbb{C}$-dimensional subspace is the solution space of some system (I believe, I have not checked).

I am interested in similar results for larger $n$. I suspect that there are similar `small or everything' type results, but I don't know what a good guess for what small might be. Note that any $K$-subspace of $K^n$ is a solution space, with defining equations given by a matrix over $K$ with that space as the kernel.

ag.algebraic geometry - Special divisors on hyperelliptic curves

I was reading a proof that used the following result

Let $C$ be a hyperelliptic of genus $ge 3$ and $tau colon C to C$ the hyperelliptic involution. If $D$ is an effective divisor of degree $g-1$ such that $h^0(D)>1$ then $D = x + tau(x) + D'$ where $D'$ is an effective divisor.

My question is, how is this result proved? It seems equivalent to showing that $|D|$ contains the unique $g^1_2$ and this made me think of Clifford's theorem but this didn't lead to much. For $g = 3$ the result holds because then $|D| = g^1_2$. But already for $g = 4$ I'm stuck. I tried playing around with the Riemann-Roch theorem but didn't get far.

qa.quantum algebra - Transmutation versus operads

A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In particular, there seems to be the claim that if $H$ is a quasi-triangulated quasi-Hopf algebra, then the braided endomorphisms of the identity functor on (a suitably large category of) $H$-comodules form a Hopf algebra object $H'$ in $H$-comodules, in a way that identifies $H'$-comodules with $H$-comodules. Furthermore, $H'$ is commutative and cocommutative with respect to the braided structure on the category of $H$-comodules, and under some nondegeneracy assumptions, it is self-dual. This is called "transmutation" because $H'$ appears to have nicer properties than $H$ (although it may live in a strange category). Some examples are given, e.g., $U_q(g)$ and quantum doubles of finite groups. Unfortunately, the arguments in the proof are given in a diagrammatic language that I was unable to fathom.

Why does this result seem problematic?

The first problem comes from reasoning by analogy. If I want to describe a Hopf algebra object in a monoidal category, I need some kind of commutor transformation $V otimes W to W otimes V$ to even express the compatibility between multiplication and comultiplication, e.g., that comultiplication is an algebra map. In operad language, I need (something resembling) an E[2]-structure on the category to describe compatible E[1]-algebra and E[1]-coalgebra structures. If you think of the spaces in the E[k] operad as configurations of points in $mathbb{R}^k$, this is roughly saying that you need two dimensions to describe compatible one-dimensional operations. In the above case, the category of $H$-comodules has an E[2]-structure, but I'm supposed to get compatible E[2]-algebra and E[2]-coalgebra structures. Naively, I would expect an E[4]-category to be necessary to make sense of this, but I was unable to wrestle with this successfully.

The second problem comes from a construction I've heard people call Koszul duality, or maybe just Bar and coBar. If we are working in an E[n]-category for n sufficiently large (like infinity, for the symmetric case), then there is a "Bar" operation that takes Hopf algebras with compatible E[m+1]-algebra and E[k]-coalgebra structures, and produces Hopf algebras with compatible E[m]-algebra and E[k+1]-coalgebra structures. There is a "coBar" operation that does the reverse, and under some conditions that I don't understand, composing coBar with Bar (or vice versa) is weakly equivalent to the identity functor. In the above case, I could try to apply Bar to $H'$, but the result cannot have an E[3]-coalgebra structure, since E[3] doesn't act on the category. Applying Bar then coBar implies the coalgebra structure on $H'$ is a priori only E[1], and applying coBar then Bar implies the algebra structure on $H'$ is a priori only E[1]. It is conceivable (in my brain) that the E[2]-structures could somehow appear spontaneously, but that seems a little bizarre.

Question

Am I talking nonsense, or is there a real problem here? (or both?)

soft question - Your favorite surprising connections in Mathematics

My favorite connection in mathematics (and an interesting application to physics) is a simple corollary from Hodge's decomposition theorem, which states:

On a (compact and smooth) riemannian manifold $M$ with its Hodge-deRham-Laplace operator $Delta,$ the space of $p$-forms $Omega^p$ can be written as the orthogonal sum (relative to the $L^2$ product) $$Omega^p = Delta Omega^p oplus cal H^p = d Omega^{p-1} oplus delta Omega^{p+1} oplus cal H^p,$$ where $cal H^p$ are the harmonic $p$-forms, and $delta$ is the adjoint of the exterior derivative $d$ (i.e. $delta = text{(some sign)} star dstar$ and $star$ is the Hodge star operator).
(The theorem follows from the fact, that $Delta$ is a self-adjoint, elliptic differential operator of second order, and so it is Fredholm with index $0$.)

From this it is now easy to proof, that every not trivial deRham cohomology class $[omega] in H^p$ has a unique harmonic representative $gamma in cal H^p$ with $[omega] = [gamma]$. Please note the equivalence $$Delta gamma = 0 Leftrightarrow d gamma = 0 wedge delta gamma = 0.$$

Besides that this statement implies easy proofs for Poincaré duality and what not, it motivates an interesting viewpoint on electro-dynamics:

Please be aware, that from now on we consider the Lorentzian manifold $M = mathbb{R}^4$ equipped with the Minkowski metric (so $M$ is neither compact nor riemannian!). We are going to interpret $mathbb{R}^4 = mathbb{R} times mathbb{R}^3$ as a foliation of spacelike slices and the first coordinate as a time function $t$. So every point $(t,p)$ is a position $p$ in space $mathbb{R}^3$ at the time $t in mathbb{R}$. Consider the lifeline $L simeq mathbb{R}$ of an electron in spacetime. Because the electron occupies a position which can't be occupied by anything else, we can remove $L$ from the spacetime $M$.

Though the theorem of Hodge does not hold for lorentzian manifolds in general, it holds for $M setminus L simeq mathbb{R}^4 setminus mathbb{R}$. The only non vanishing cohomology space is $H^2$ with dimension $1$ (this statement has nothing to do with the metric on this space, it's pure topology - we just cut out the lifeline of the electron!). And there is an harmonic generator $F in Omega^2$ of $H^2$, that solves $$Delta F = 0 Leftrightarrow dF = 0 wedge delta F = 0.$$ But we can write every $2$-form $F$ as a unique decomposition $$F = E + B wedge dt.$$ If we interpret $E$ as the classical electric field and $B$ as the magnetic field, than $d F = 0$ is equivalent to the first two Maxwell equations and $delta F = 0$ to the last two.

So cutting out the lifeline of an electron gives you automagically the electro-magnetic field of the electron as a generator of the non-vanishing cohomology class.

How to find the smallest flabby sheaf containing a given sheaf ?

There is the "flabbification" or "flasquification" functor used in the Godement resolution. Namely, given a sheaf $mathcal{F}$ on $X$, let $mathcal{F}_x$ be the stalk at a point $x$. Then we define the sheaf $Phi(mathcal{F})$ to have sections $$Gamma(U, Phi(mathcal{F}))=prod_{xin U} mathcal{F}_x$$
with the obvious restriction maps. (This is the same as endowing the etale space of $mathcal{F}$ with the trivial topology and considering the sheaf of sections.) There is a natural injection $mathcal{F}to Phi(mathcal{F})$ sending a section of $mathcal{F}$ to its stalks. Note that $Phi$ is functorial; indeed, it is the right adjoint to the natural inclusion of the full subcategory (Flasque sheaves on $X$) $hookrightarrow$ (Sheaves on $X$).

See for example this Wikipedia article or Godement's book on sheaf theory.

Edit: One can also mimic your construction of $C^k_{nd}$ as follows. Let $mathcal{F}_{nd}$ have global sections given by $bigcup Gamma(X-partial U, mathcal{F})/sim$ where the union is taken over all open sets $U$, and we say $(f, X-partial U)sim (f', X-partial U')$ if $f=f'$ when restricted to $X-(partial Ucup partial U')$. Then local sections will be restrictions of these global sections. This will always be flabby, but will not necessarily have a morphism $mathcal{F}to mathcal{F}_{nd}$ unless $mathcal{F}$ has enough sections (for example, if $mathcal{F}$ is fine, as in your example).

rt.representation theory - Conjugacy classes of reductive groups defined over local commutative rings

Background: I'm trying a problem on representations of reductive groups over various finite rings towards which this is very relevant (what I want to do is a very specialized case of this problem, and I want to know what background theory has been done for this situation in the literature and what is known about this). In characteristic $0$ over an algebraically closed field, and over finite fields, classifying conjugacy classes in reductive groups over field is a very well-known and well-studied problem (sometimes in preparation for studying representations of these).

Question: Let $R$ be a local commutative ring, either in characteristic $0$, algebraically closed, or in characteristic $p$ (algebraically closed OR finite field). If you want to complicate matters, and have an answer for non-commutative rings as well, I would be happy to see it, but I think the problem is non-trivial enough as is - and as far as I know, only $GL$ can be easily defined over non-commutative rings). Edit: Also specify that $R$ is an algebra over its residue field, and has an identity (which I believe is necessary to make the following argument work; again if you don't need this restriction feel free to not use it).

Let $G$ be a reductive group (if you want, feel free to restrict to just the classical groups, $GL$, $SL$, $Sp$, and $SO$) defined over $R$. What can be said about classifying conjugacy classes in $G$? What is clear is the Levi decomposition of $G$, as the semi-direct product of the reductive group defined over the residue field of $R$, and $N$, the set of all matrices that are congruent entry-wise to the identity matrix, modulo the maximal ideal of $R$ (the latter is the normal subgroup). Using the semi-direct product, one can say something implicit about the conjugacy classes; first by studying the conjugation action of the reductive group on the unipotent algebraic group $N$, then studying the conjugacy classes in $N$, then extending this to the whole group.

Are there any special cases of this problem that have been studied in the literature? Is there something more than can be said in general (further to what I have said above about the semi-direct product)?.

na.numerical analysis - Are the banded versions of a positive definite matrix positive definite?

Wouldn't that mean that the quadratic form $x^2+y^2+z^2+2xy+2yz$ must be nonnegative definite (as it is a band restriction of the quadratic form $x^2+y^2+z^2+2xy+2yz+2zx$, which is clearly nonnegative definite), which contradicts its value at $x=1$, $y=-1$, $z=1$ ?

(Note that I replaced your "positive definite" by "nonnegative definite" - feel free to add $epsilonleft(x^2+y^2+z^2right)$ to the form for some $epsilon>0$ to keep everything positive.)

EDIT: There's a bit more to this:

Let us denote by $Aast B$ the Hadamard product of two $ntimes n$ matrices $A$ and $B$ (defined by

$Aast B=left(a_{i,j}b_{i,j}right)_{1leq ileq n, 1leq jleq n}$,

where

$A=left(a_{i,j}right)_{1leq ileq n, 1leq jleq n}$

and $B=left(b_{i,j}right)_{1leq ileq n, 1leq jleq n}$).

Let $A$ be a symmetric matrix. Then, (the matrix $Aast B$ is nonnegative definite for every nonnegative definite matrix $B$) if and only if the matrix $A$ is nonnegative definite. The $Longrightarrow$ direction is more or less trivial (just take $B$ to be the matrix $left(1right)_{1leq ileq n, 1leq jleq n}$) and disproves your conjecture (by taking $A$ to be the matrix whose $left(i,jright)$-th entry is $1$ if $left|i-jright|leq d$ and $0$ otherwise). The $Longleftarrow$ direction is interesting and most easily proven by decomposing the matrix $A$ in the form $u_1u_1^T+u_2u_2^T+...+u_nu_n^T$, where $u_1$, $u_2$, ..., $u_n$ are appropriate vectors. Another proof reduces it to Corollary 2 in my answer to MathOverflow #19100 - do you see how?

game theory - The Worst Possible Winner

First a little background. In racing it is possible for a player to win a tournament without winning a single race, however, how bad can a tournament winner actually be? Can a player win a tournament without even doing better than coming third? Or even fourth? Obviously this depends on the scoring method used for awarding points for each race.

More formally, suppose $p$ players, named $alpha_1, alpha_2, ldots, alpha_p$, play a game consisting of $n$ races (with no possibility of ties for a position).

Suppse that player $alpha_i$ finishes race $j$ in position $beta_{i,j} in lbrace 1, 2, ldots p rbrace$ (with $beta_{i,j} = 1$ being the best possible result for player $alpha_i$). And that for each race the points scored by a player are given by a non-negative, strictly decreasing function called a scoring function $f : lbrace 1,2, ldots, p rbrace to mathbb{N}$, i.e. the player coming first receives $f(1)$ points, the player coming second receives $f(2)$ points and the player coming last receives $f(p)$ points.

Let $text{score}(alpha_i) = sum_{j = 1}^{n} f(beta_{i,j})$ be the total score obtained by player $alpha_i$.

Let $text{best}(alpha_i) = min_{1 leq j leq n} lbrace beta_{i,j} rbrace$, be the best position that player $alpha_i$ came in.

We say that player $alpha_i$ is a winner iff $forall j in lbrace 1, 2, ldots, p rbrace$ $text{score}(alpha_i) geq text{score}(alpha_j)$, note there may be more than one winner of a game.

Given a particular choice of scoring function $f$, if $alpha_i$ is a winner what is the maximum value $text{best}(alpha_i)$ can possibly be?

Or alternatively:

For what $k in lbrace 1, 2, ldots, p rbrace$, is there a choice of scoring function $f$ such that it is possible for $alpha_i$ to be a winner and $text{best}(alpha_i) geq k$?

If the general case is too hard, how about when $f(x) = p + 1 - x$?

ag.algebraic geometry - What are non-trivial examples of non-singular blow-ups of a non-singular variety?

This question arose from the responses to this question. The references to the comments of Karl Schwede and VA are to comments made there.

The blow-up of the variety $X=mathbb{A}^2$ along the closed subscheme $Z$ defined by $(x,y)^2$ is non-singular. As Karl Schwede points out, this example is trivial in the sense that the blow-up along the power of a maximal ideal is naturally isomorphic to the blow-up of the maximal ideal. VA's comment, on the other hand, suggests that perhaps singular closed schemes $Z$ with $operatorname{Bl}_{Z}(X)$ non-singular are ubiquitous.

This suggests a question: what are non-trivial examples of a singular closed subscheme $Z$ of a non-singular variety $X$ with $operatorname{Bl}_{Z}(X)$ non-singular. Here "non-trivial" means the ideal of $Z$ is not a power of the ideal of a non-singular subvariety.

Particularly interesting would be such a $Z$ such that

$operatorname{Bl}_{Z}(X)$

is not isomorphic (as a scheme over $X$) to $operatorname{Bl}_{Z'}(X)$ for any non-singular subvariety $Z'$ of $X$.

Edit: I have not been able to access the paper "On the smoothness of blow-ups" (MR1446135, by O'Carroll and Valla) yet, but the mathsci review states that they prove that the blow-up of a regular local ring $A$ along an ideal generated by a subset of a regular system of parameters is smooth. Let's also consider those examples to be trivial.

Edit: I added "of a non-singular variety" to the title to emphasize that I am interested in examples where the ambient space is non-singular.

dg.differential geometry - What is the origin of the formula for the Lie derivative along a Killing vector?

Background

Let $(M,g)$ be an $n$-dimensioal riemannian manifold. A vector field $X$ on $M$ is said to be a Killing vector if the flow it generates is an isometry; that is, it preserves the metric $g$. There are many ways of writing this. The one which is relevant for this question is the following. If we let $nabla$ denote the Levi-Civita connection, then $X$ is Killing if and only if the endomorphism $A_X : TM to TM$ defined by
$$A_X(Y) = - nabla_Y X$$
is skewsymmetric, so that for all vector fields $Y,Z$ on $M$ one has that
$$ g(A_X(Y),Z) = - g(Y,A_X(Z)).$$

In summary, if we let $mathfrak{so}(TM)$ denote the bundle of skewsymmetric endomorphisms of $TM$, then $X$ is Killing if and only if $A_X$ defines a section of $mathfrak{so}(TM)$.

Let $mathrm{SO}(TM)$ denote the bundle of oriented orthonormal frames of $TM$. It is a principal $mathrm{SO}(n)$ bundle over $M$. In the case I'm mostly interested in, $M$ is a spin manifold, so that there is a principal $mathrm{Spin}(n)$ bundle $mathrm{Spin}(TM)$ and a bundle surjection $mathrm{Spin}(TM) to mathrm{SO}(TM)$ which restricts fibrewise to the covering homomorphism $mathrm{Spin}(n) to mathrm{SO}(n)$.

If $rho : mathrm{Spin}(n) to mathrm{GL}(V)$ is a representation, then we can form the associated vector bundle
$$E := mathrm{Spin}(TM) times_rho V.$$
Attached to every Killing vector $X$ on $M$ we have a Lie derivative $mathcal{L}_X$ on sections of $E$. Explicitly, this Lie derivative takes the form
$$ mathcal{L}_X sigma = nabla_X sigma + rho(A_X) sigma,$$
where I am using $rho : mathfrak{so}(n) to mathfrak{gl}(V)$ also to denote the derivative map of the the representation. (I am also identifying $mathfrak{so}(TM)$ with $mathfrak{so}(n)$ via a choice of local frame.)

For example, in the case of the tangent bundle itself viewed as an associated bundle where $rho$ is the defining representation of $mathfrak{so}(TM)$, then as expected, we find
$$ mathcal{L}_X Y = nabla_X Y + A_X(Y) = nabla_X Y - nabla_Y X = [X,Y] .$$

Question

Although I quite often use the formula for the Lie derivative $mathcal{L}_X$ along a Killing vector, I do not feel I have a good conceptual understanding of it.

Could someone enlighten me?

mp.mathematical physics - Morphisms of supermanifolds

Unlike many schemes, but similar to ordinary manifolds, a map of super-manifolds $$(X, mathcal{O}_X) to (Y, mathcal{O}_Y)$$ determines and is completely determined by the map of superalgebras obtained by looking at global sections:
$$mathcal{O}_Y(Y) to mathcal{O}_X(X)$$
In the example at hand this is the graded ring map:
$$ x mapsto x' + a'b'$$
$$ a mapsto a'$$
$$ b mapsto b'$$

This map induces a map of rings after we mod out by nilpotents:
$$C^infty(Y) = mathcal{O}_Y/Nil to mathcal{O}_X / Nil = C^infty(X)$$
This map in turn induces a smooth map $X to Y$ (in fact it is equivalent to such a map). In this case, after modding out by nilpotents we get the map $x mapsto x'$, i.e. the identity on the underlying manifold $mathbb{R}$.

ag.algebraic geometry - Polarizations on intermediate Jacobians

If enough Hodge numbers vanish so that the Hodge structure $H^{2k+1}(X)$ has level one,
then $J^kX$ should be an abelian variety. This applies to Fano (e.g. cubic) 3-folds for example.

Later that day: Partly in response to Charles Siegel's comment/question, let me
sketch a proof of a slightly more general statement. Suppose X is a projective rather
than just Kaehler (which I forgot to say before), so $H$ has a polarization $Q$. Assume
further that
$$H^{2k+1}(X) = H= H^{pq}oplus H^{qp}$$
has only two terms.
Let $G$ be the same thing as $H$ viewed as a weight one structure. More precisely,
the lattices $G_Z=H_Z$ are the same, and $G^{10}=H^{pq}$.

Then one sees
that $J^kH= G^{01}/G_Z$, and that $pm Q$ gives a polarization on $G$. So this is abelian variety.

ct.category theory - When is the $(F_!,F^*)$ counit a natural isomorphism?

In general, the counit of an adjunction is an isomorphism if and only if the right-adjoint is fully faithful (dually the unit is an iso iff the left-adjoint is fully-faithful). So, your question is easily seen to be equivalent to asking "When is $F^{*}$ fully-faithful? In topos-theory lingo, when is the induced geometric morphism $mathbf{F}:Set^{C^{op}} to Set^{D^{op}}$ satisfies $F^*$ is faithful, then $mathbf{F}$ is said to be a SURJECTION of topoi. In this setting, this is equivalent to every object in $D$ being a retract of an object of the form $F(C)$.

Ok, so how about asking for $F^*$ to also be full? $F^*$ being faithful AND full means you are looking at what is called a CONNECTED geometric morphism of topoi. What properties $F$ do we need to ensure this? This is in general a hard problem. However, there are at least sufficient conditions. Given $F$, you first construct the category $Ext_{F}$ of "F-extracts"- these are quadruples $(U,V,r,i)$ with $U in C$, $V in D$, $r:FU to B$, and $i:V to FU$ such that $ri=1$, with the evident morphisms. There is a canonical functor $tilde F:Ext_{F} to D$ which sends $(U,V,r,i) mapsto V$. Denote by $Ext_F(V)$ the fiber over $V$ of this functor. Then if $tilde F$ is full and each $Ext_F(V)$ is a connected category, then $mathbf{F}$ is a connected morphism.

This is in "Sketches of an Elephant" C.3.3.

differential topology - Is there a Morse theory for sections of bundles or more generally for maps?

Ryan: my use of the phrasing 'classify functions' is both inaccurate and imprecise. But we're in danger of talking past one another, because the classification I am pursuing is on the basis of different physical processes. This is a physics goal, not a mathematical one.

The functions observational cosmologists consider are invariably smooth. The cosmological dark matter density is an example and the calculation of the Euler characteristic for level sets of that function is a nice application of Morse theory to physics. The velocity of the dark matter fluid, which a physicist would call a vector-valued function, is an example of an object to which we would like to apply the same apparatus, but are unsure how to proceed.

I think that you indicate that the concept of a level set is not well-formed for an object of that kind? Could you comment on the generality of that statement bearing in mind the relative triviality of the objects in question?

I would also like to clarify Jose's phrasing. Studies of this kind aim very much to study the properties of $f$, rather than $M$. The study of the topology of the spatial Universe is an excellent and interesting problem as well! But the aim here is to use techniques from Morse theory as a way of probing how physical effects alter the form of $f$.

Thanks for all the help so far! I hope there's more to discuss.

dg.differential geometry - Why did the word "exterior" get chosen for the idea of "exterior derivative"?

I) The term exterior multiplication ("äussere Multiplication") is due to Grassmann, who introduced the term in his book (written in 1844)

Die Wissenschaft der extensiven Grösse oder die Ausdehnungslehre, eine neue Mathematische Disciplin"

As you can check in the table of contents of the book (on page 276), paragraphs §§34,35 are called Grundgesetze der äusseren Multiplication (Basic laws of exterior multiplication).
Here is the scan of this book by Google .

II) The terminology exterior differential ("différentielle extérieure") was introduced in the 1930's by articles of Elie Cartan, inspired by Grassmann.

Here is a secondary reference from an Analysis course by Chatterji and another by Chern and Chevalley, in their analysis of Elie Cartan's mathematical contributions (cf. in particular pages 229 and 230 )

big picture - Does the presence of cocycle conditions indicate the existence of an underlying cohomology theory?

I had lots of thoughts on that kind of question, and feel uneasy to speak as my answer can range from a tautology, through systematic and positive, but somewhat ignorant toward not-well understood cases, to mere impressions and (seeming?) "counterexample" oriented answer. The basic question is what you mean by a cocycle. Usually one talks on expressions of some higher categorical coherence, or about some notion of homotopy behind it. In such cases the answer is normally yes: the equivalent or homotopic cocycles will form cohomology classes and this can be in all understood cases done naturally and systematically. Higher nonabelian cohomology can be done for all $n$, as now many frameworks know (Brown, Jardine, Toen, Street...) and cohomology boils down to take homotopy classes into certain suspension of the coefficient object. For one recent framework we can advertise our own work (pdf).

I slightly believe anyway that some algebraic cases can be outside of the current homotopy categorical framework and I discussed that much on the n-category cafe, nforum and elsewhere. Namely model categories treat on equal footing homology and cohomology, while the minimal conditions on a setup to be able to do cohomology of homology is less than both simultaneously (cf. work of Rosenberg on "right exact structures" on a category, pdf).

Finally, we can imagine more complicated category-like structures where one can do much of the usual combinatorics but can not properly do the equivalence classes when needed for cohomology. There is one example which is maybe repairable, due Shahn Majid, namely he has a notion of bialgebra cocycles for a noncommutative and noncocommutative bialgebra. Now in special cocommutative or commutative cases like Lie algebras and/or abelian coefficients he recovers some known cohomology theories like Chevalley-Eilenberg cohomology for Lie algebras. In low dimensional cases he also gets some interesting nonabelian cocycles of much usage like Drinfel'd 2-twist and Drinfel'd 3-associator which are used in the study of monoidal categories, CFT, knot theory and quantum groups. In this example the differential and cocycles are defined for every $n$ but 17 years after the discovery, there is still no known way to define well the cohomology classes, for dimension 3 or more, for general bialgebra, despite the special cases and despite the cocycles and the differential. See the nlab page bialgebra cocycle for the basics (and the references therein).

arithmetic geometry - Finiteness of Obstruction to a Local-Global Principle

"Has there been further progress in this area since 1993?"

So far as I know, there has been no direct progress. I feel semi-confident that I would know if there had been a big breakthrough: Mazur was my adviser, this is one of my favorite papers of his, and I still work in this field. Also, I just checked MathReviews and none of the citations to this paper makes a big advance on the problem, although two are somewhat relevant:

MR1905389 Thăńg, Nguyêñ Quoc On isomorphism classes of Zariski dense subgroups of semisimple algebraic groups with isomorphic $p$-adic closures. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 5, 60--62.

MR2376817 (2009f:14040) Borovoi, M.; Colliot-Thélène, J.-L.; Skorobogatov, A. N. The elementary obstruction and homogeneous spaces. Duke Math. J. 141 (2008), no. 2, 321--364.

I'm not sure what you mean by an effective bound on Shafarevich-Tate groups (henceforth "Sha"). It is certainly expected that the Sha of any abelian variety over a global field is finite. If this is true, then in any given case one can, "in principle", give an explicit upper bound on Sha by the method of n-descents for increasingly large n. (In practice, even for elliptic curves reasonable algorithms have been implemented only for small values of n.) I really can't imagine any algorithm having to do with Sha that has "a priori bounded running time". What do you have in mind here?

As to the final question, let me start by saying that it seems reasonable at least that the set of "companion varieties" (i.e., Q-isomorphism classes of varieties everywhere locally isomorphic to the given variety) of a projective variety V/Q is finite: as above, we believe this for abelian varieties, and Barry Mazur proved in this paper a lot of results in the direction that the conjecture for abelian varieties implies it for arbitrary varieties. (For instance, quoting from memory, I believe he proved the implication for all varieties of general type.)

Here is a key point: suppose you are given a variety V/Q and you are wondering whether it has rational points. If V is itself a torsor under an abelian variety (e.g. a genus one curve), then if you can compute Sha of the Albanese abelian variety of V, you can use this to determine whether or not V has a Q-rational point. In general, the connection between computation of sets of companion varieties of V and deciding whether V has a Q-rational point is less straightforward. If V is a curve, then there are theorem in the direction of the fact that finiteness of Sha(Jac(V)) implies that the Brauer-Manin obstruction is the only one to the existence of rational points on V. In particular, people who believe this (including Bjorn Poonen, I think), believe that there is an algorithm for deciding the existence of rational points on curves. But nowadays we know examples of varieties where the Brauer-Manin obstruction is not sufficient to explain failure of rational points.

So, in summary, it is a perfectly tenable position to believe that companion sets are always finite, even effectively computable, but still there is no algorithm to decide the existence of Q-points on an arbitrary variety.

pr.probability - probability puzzle - selecting a person

I have written an article about this game, and solved it numerically for the case of 10 person. I computed that the probability is the same for each of the 9 person.

In short, I have defined the following:

Let P(n,i,j,k) be the probability of at the n-th round, the k person is having the coin, while the people from i counting clockwise to k have all received the coin before.

And 3 recurrence equations are formulated:

$P (n+1,i,j,k)= frac{1}{2} P (n,i,j,k-1)+ frac{1}{2} P (n,i,j,k+1)$ ........(1)

$P (n+1,i,j,j)=frac{1}{2} P (n,i,j-1,j-1)+ frac{1}{2} P(n,i,j,j-1)$ ........(2)

....(3)

For details, please refer to:

Solving a probability game using recurrence equations and python

In this article, a(n,i) which denotes the probability of the i-th person being the head in the n-th round is found and plotted out as well.

nt.number theory - The product of n radii in an ellipse

(Not an answer.)

Let $r = frac{1}{a}, s = frac{1}{b}$. Reindex your points to $(x_0, y_0), ... (x_{n-1}, y_{n-1})$ and let $z_k^2 = x_k^2 + y_k^2$. Then $x_k = z_k cos frac{2pi k}{n}, y_k = z_k sin frac{2pi k}{n}$, and the intersection condition becomes

$$z_k^2 left( r^2 cos^2 frac{2pi k}{n} + s^2 sin^2 frac{2pi k}{n} right) = 1.$$

Together with the condition that $prod_{k=0}^{n-1} z_k = 1$, it follows that the desired conditions can be stated as

$$prod_{k=0}^{n-1} left( r^2 cos^2 frac{2pi k}{n} + s^2 sin^2 frac{2pi k}{n} right) = 1.$$

This is likely to be a hard Diophantine equation to solve in general. For $n = 3$, for example, the equation is

$$r^2 (r^2 + 3s^2)^2 = 16.$$

The curve $r(r^2 + 3s^2) = 4$ is an elliptic curve, and in general one must use computer algebra to rule out the existence of rational points on such curves. In this particular case we might be able to get away with some argument using unique factorization in $mathbb{Z}[omega]$, but this strategy will fail in general for the same reason it fails in Fermat's Last Theorem.

Deformations and the dual numbers

If you want to look at Deformation Theory from a complex-analytic point of view ( i.e. in the spirit of Kodaira-Spencer "deformations of complex structures" ), you need to solve the Maurer-Cartan equation

$bar partial varphi + frac{1}{2}[varphi, varphi]=0$,

were $varphi in mathcal{E}^{0,1}(T^{1,0})$. In order to do this, one first look at a solution which is a formal power series

$varphi(t)=varphi_1 t + varphi_2 t^2 + varphi_3 t^3 +...$

Collecting powers of $t$ we obtain equations

$bar partial varphi_1=0$

$bar partial varphi_2 + frac{1}{2}[varphi_1, varphi_1]=0$

...

The first equation states that $varphi_1$ is an harmonic form, that is an element of
$mathcal{H}^1(T^{1,0})$. By Hodge Theorem, this space can be identified with $H^1(X, T_X)$, which is exactly the space parametrizing "first-order" deformations.

The second equation states that you can extend the first order deformation to a second-order one (i.e., you can solve the Maurer-Cartan equation modulo $(t^3)$ ) if and only if the 2-cocycle $[varphi_1, varphi_1]$ is a coboundary. So the class of $[varphi_1, varphi_1]$ in $H^2(X, T_X)$ is the "primary obstruction" to your deformation problem.

In this way, you can try to solve modulo higher and higher powers of $t$. If all the higher order obstructions vanish and the series defining $varphi(t)$ converges, you obtain
a "genuine" deformation, namely a deformation over a small disk.

Now it should be clear that, in order to generalize this in the algebraic framework, you need a substitute for the step "solve the Maurer-Cartan equation modulo $(t^k)$ ". This substitute is roughly speaking obtained by considering deformations over Spec $k[epsilon]/(epsilon^k)$.

rt.representation theory - Uniqueness of local Langlands correspondence for connected reductive groups over real/complex field.

Dear Kevin,

Here are some things that you know.

(1) Every non-tempered representation is a Langlands quotient of an induction of a non-tempered twist of a tempered rep'n on some Levi, and this description is canonical.

(2) Every tempered rep'n is a summand of the induction of a discrete series on some Levi.

(3) The discrete series for all groups were classified by Harish-Chandra.

Now Langlands's correspondence is (as you wrote) completely canonical: discrete series
with fixed inf. char. lie in a single packet, and the parameter is determined from the
inf. char. in a precise way.

All the summands of an induction of a discrete series rep'n are also declared to lie
in a single packet. So all packet structure comes from steps (1) and (2).

The correspondence is compatible in a standard way with twisting, and with parabolic induction.

So:

If we give ourselves the axioms that discrete series correspond to irred. parameters,
that the correspondence is compatible with twisting, that the correspondence is compatible
with parabolic induction, and that the correspondence is compatible with formation of
inf. chars., then putting it all together, it seems that we can determine step 1, then
2, then 3.

I don't know if this is what you would like, but it seems reasonable to me.

Why no need for epsilon-factor style complications: because there are no supercuspidals,
so everything reduces to discrete series, which from the point of view of packets are described by their inf. chars. In the p-adic world this is just false: all the supercuspidals are disc. series, they have nothing analogous (at least in any simple way) to an inf. char., and one has to somehow identify them --- hence epsilon factors to the rescue.

[Added: A colleague pointed out to me that the claim above (and also discussed below
in the exchange of comments with Victor Protsak) that the inf. char. serves to determine
a discrete series L-packet is not true in general. It is true if the group $G$ is semi-simple, or if the fundamental Cartan subgroups (those which are compact mod their centre) are connected. But in general one also needs a compatible choice of central character to determine the $L$-packet. In Langlands's general description of a discrete series parameter, their are two pieces of data: $mu$ and $lambda_0$. The former is giving the inf. char., and the latter the central char.]

at.algebraic topology - Changing the orientation of a Landweber exact cohomology theory

Let the ring R be a MU_*-module via a ring homomorphism φ and suppose it satisfies the condition of the Landweber exact functor theorem such that we obtain a cohomology theory $R^*(-) := R otimes_{MU_*} MU^*(-)$. If ω denotes the complex orientation class in $widetilde{MU}^2(mathbb{C}P^infty)$, then R^* is oriented by the class $omega_R := 1 otimes omega$.

Any other complex orientation of R^* is obtainable by homogeneous power series θ with leading term x over R: θ(ω). These power series are in 1-1 correspondence with multiplicative natural transformations $t_thetacolon MU^*(-) to R^*(-)$.

Question: Which t_θ restrict to ring homomorphisms which satisfy the Landweber criterion on coefficients? For which theories is this true for any θ?

The place to start seems to be by noting that if the formal group law associated to R^* (with the orientation given by ω_R) is F, then t_θ classifies the FGL $F^theta(x,y) := thetabig(F(theta^{-1}(x),theta^{-1}(y))big)$ over R. Further, the p-series are related by $[p]_{F^theta}(x) = thetabig([p]_{F}(theta^{-1}(x))big)$, so it would suffice to show that the sequence of coefficients in the right degrees stay regular under this conjugation by θ.

This seems to be true for any θ as long as $[p]_F(x)$ is of the form $sum_{n geq1} a_n x^{p^n}$ modulo p. In general, it is of the form $sum_{kgeq1} a_kx^{kp^m}$, where m can be taken to be the height of the FGL (Ravenel's Green Book), but I don't see why it should be true in the general case.

I am sure this has been treated by someone, but have yet to see it on print. If anyone has seen question discussed somewhere, please let me know.

ca.analysis and odes - On linear independence of exponentials

I have some partial answers.

I. It is not hard to construct a Dirichlet series
$$f(z)=sum_{n=1}^infty a_ne^{lambda_n z}$$
which converges to $0$ absolutely and uniformly on the real line but does not converge at some points
of the complex plane.
It is constructed as a sum of 3 series $f=f_0+f_1+f_2.$ Let $f_1$ be a series with imaginary
exponents $lambda_n$ which converges to an entire function in the closed lower half-plane,
but not in the whole plane.
Such series is not difficult to construct, see V. Bernstein, page 34, (see the full reference below) and there are simpler examples,
with ordinary Dirichlet series. Then put $f_2=overline{f_1(overline{z})}$,
and $f_0=-f_1-f_2$. So all three functions are entire. Now, according to Leontiev, EVERY entire function
can be represented by a Dirichlet series which converges in the whole plane.
Thus we have a Dirichlet series $f_0+f_1+f_2$ which converges on the real line to $0$ but does not
converge in the plane.

A counterexample to the original question also requires real coefficients, this I do not know
how to do (for $f_0$).

II. It is clear from the work of Leontiev, that to obtain a reasonable theory,
one has to restrict to exponents of finite
upper density, $n=O(|lambda_n|)$, otherwise there is no uniqueness in $C$. In the result I cited
above the expansion of $f_0$ is highly non-unique.

Assuming finite upper density I proved that if a series is ABSOLUTELY and uniformly
convergent on the real line to zero, then all coefficients must be zero.
http://www.math.purdue.edu/~eremenko/dvi/exp2.pdf
I don't know how to get rid of the assumption of absolute convergence.

But there is a philosophical argument in favor of absolute convergence: the notion of "linear
dependence" should not depend on the ordering of vectors:-)

III. The most satisfactory result on my opinion, is that of Schwartz. Let us say that
the exponentials are S-linearly independent if none of them belongs to the closure of
linear span of the rest. Topology of uniform convergence on compact subsets of the real line. Schwartz gave a necessary and sufficient conditon of this:
the points $ilambda_k$ must be contained in the zero-set of the Fourier transform of a measure
with a bounded support in R.

(L. Schwartz, Theorie generale des fonctions moyenne-periodiques, Ann. Math. 48 (1947) 867-929.)

A complete explicit characterization of such sets is not known, but they have finite upper density,
and many of their properties are understood. These Fourier transforms are entire functions of
exponenitial type bounded on the real line. The link I gave above contains Schwartz's proof
in English. S-linear dependence is also non-sensitive to the ordering of functions, which is good.

IV. Vladimir Bernstein's book is "Lecons sur les progress recent de la theorie des series de Dirichlet", Paris 1933. This is the most comprehensive book on Dirichlet series, but unfortunately
only with real exponents.

V. The application to the functional equation mentioned by the author of the problem is not a good
justification for the study of the problem in such generality. The set of exponentials there is
very simple, and certainly we have $R$-linear independence for SUCH set of exponentials. Besides
the theorem stated as an application has been proved in an elementary way.

VI. Finally, I recommend to change the definition of $R$-linear independence by allowing complex
coefficients (but equality to $0$ on the real line). Again in the application mentioned in the original problem, THIS notion of $R$-uniqueness is needed: the function is real, but the exponentials
are not real, thus coefficients should not be real.

oa.operator algebras - Does a crossed product R⋊_α F_n of the hyperfinite factor of type II_1 and a free group have the QWEP?

Yes. If $a$ and $b$ are generators of $mathbb F_2$ then $mathcal R rtimes_alpha mathbb F_2$ decomposes as an amalgamated free product of $(mathcal R rtimes_alpha langle a rangle)$ and $(mathcal R rtimes_alpha langle b rangle)$ over $mathcal R$, where each of these are hyperfinite. Brown, Dykema, and Jung showed in http://arxiv.org/abs/math/0609080 that for separable finite von Neumann algebras being embeddable into $mathcal R^omega$ is stable under amalgamated free products over a hyperfinite von Neumann algebra. Thus $mathcal R rtimes_alpha mathbb F_2$ is embeddable into $mathcal R^omega$, which is equivalent to QWEP. Induction then gives the case when $2 leq n < infty$, and the case $n = infty$ then follows since QWEP is preserved under (the weak-closure of) increasing unions.

Related to this, Collins and Dykema in http://arxiv.org/abs/1003.1675 have recently shown that the class of Sophic groups is stable under taking amalgamated free products over amenable groups.

I believe this is an open problem however if we consider arbitrary residually finite groups instead of only $mathbb F_n$.

pr.probability - The probability for a sequence to have small partial sums

For $t$ fixed, the count is proportional to $lambda^n$, where $lambda = 2 cos fracpi{2t+2}$ is the principal eigenvalue of the adjacency matrix of the path with $2t+1$ vertices. The all-positive (Perron-Frobenius) eigenvector corresponding to $lambda$ is

$$bigg(sin frac{pi}{2t+2}, sin frac{2pi}{2t+2},sin frac{2pi}{2t+2},dots,sin frac{(2t+1)pi}{2t+2}bigg).$$

Since $-lambda$ is also an eigenvalue, the stable behavior of the distribution of endpoints of paths which stay in $[-t,t]$ is an oscillation between the odd entries

$$bigg(sin frac{pi}{2t+2}, 0,sin frac{3pi}{2t+2},0,dots,sin frac{(2t-1)pi}{2t+2},0,sin frac{(2t+1)pi}{2t+2}bigg).$$
and even entries
$$bigg(0,sin frac{2pi}{2t+2}, 0,sin frac{4pi}{2t+2},0,cdots ,0,sin frac{2tpi}{2t+2},0bigg).$$

The exact count of paths staying in $[-t,t]$ is a sum of signed binomial coefficients.

The number of paths from $0$ to $i$ is 0 if $n not equiv i ~mod 2$, and $n choose (npm i)/2$ when $n equiv i ~mod 2$.

The number of paths which never leave $[-t,t]$ from $0$ to $i in [-t,t]$ with $n equiv i ~mod 2$ is

$$ sum_{jin mathbb Z} (-1)^j {nchoose (n +i)/2 + j(t+1)}$$

by the reflection principle applied to the group of isometries of $mathbb R$ generated by reflecting about $t+1$ and $-t-1$.

If you sum over all $i in [-t,t]$, then when $n$ is even, you get a signed sum of binomial coefficients with $t+1$ positive signs in a row alternating with $t+1$ negative signs in a row. If $n$ is odd, then you get $t$ positive signs in a row, skip a term (give it a coefficient of $0$ instead of $pm 1$), then $t$ negative signs in a row, skip a term, etc.

For example, for $n=100, t=2,$ the number of paths is

$$ ... +{100choose 43} + {100choose 44} + {100 choose 45} - {100 choose 46} - {100 choose 47} - {100choose 48} + {100choose 49} + {100 choose 50} + {100choose 51} - ...$$

For $n=101, t=2,$ the number of paths is

$$ ... +{101choose 44} + {101choose 45} - {101choose 47} - {101 choose 48} + {101choose 50} + {101choose 51} - {101choose 53} - {101choose 54} + ...$$

These can be summed using the techniques in the answers to the Binomial distribution parity question.

A lot more can be said when $t$ varies, but the answers are more complicated. For $t$ slowly increasing, as $csqrt[3]n$, there is enough time for the distribution to stabilize (for each parity) at a given value of $t$, since the ratio between the magnitudes of the largest two eigenvalues and the magnitudes of the next two is about $1+c/t^2$, and the principal eigenvectors have a small $L^1$ distance for adjacent values of $t$. You should pick up a constant factor for each transition. In other words, the number of paths when you spend at least $n_t gt c t^2$ steps at a given $t$ should be

$$C prod_{t le t_{max}} (2 cos frac{pi}{2t+2})^{n_t}$$

where $C$ is between some functions $f_{lower}(t_{max}) lt C lt f_{upper}(t_{max})$ which does not depend on the values of $n_t$. I don't think the $n_t gt c t^2$ condition is sharp for this behavior. Something like $n_t gt c t^2/log t$ should work, too. The geometry of the eigenvectors for adjacent values of $t$ lets you estimate $f_{lower}$ and $f_{upper}$.

For $t$ more rapidly increasing, different behaviors occur. By the law of the iterated logarithm, if $t$ increases as $t(n) = sqrt {(2-epsilon) n loglog n},$ random paths will almost surely violate the constraint. I think there are precise versions of the law of the iterated logarithm which may tell you when a positive proportion of random paths do not violate the constraint. I would guess that if $t(n) = sqrt{(2+epsilon) n loglog n}$ then a positive percentage of random paths won't violate the constraint.

rt.representation theory - Reference for quantum Schur-Weyl duality

This goes back to Jimbo I think. A reference is:
"A q-difference analogue of $U(mathfrak g)$, Hecke algebra and the Yang-Baxter equation'', Lett. Math. Phys. 11 (1986).

It has been much studied though, so there are lots of subsequent papers, some of which might be closer to what you are looking for? For example this paper studies an analogue of Schur-Weyl duality for "walled Brauer algebras", and this paper studies a two-parameter version.

reference request - How to Compute the coordinate ring of flag variety?

$G/B$ is most naturally a multi-projective variety, embedding in the product of projectivizations of fundamental representations: $prod mathbb{P}
(R_{omega_i})$. So there is a multi-homogeneous coordinated ring on $G/B$. You mentioned that this ring is $bigoplus_{lambdain P_+}$ $R_lambda$. This is correct, and the grading is also apparent. It's given by the weight lattice. (To be more canonical, you should take the duals of every highest weight representation, but what you've written down is isomorphic to that.)

So all that remains is giving the multiplication law on $bigoplus_{lambdain P_+}$ $R_lambda$. You need to specify maps $R_lambda otimes R_mu mapsto R_{lambda+mu}$. There's a natural candidate: If you decompose $R_lambda otimes R_mu$ into a direct sum of irreducible representations, $R_{lambda+mu}$ will appear exactly once. The multiplication law is simply projection onto this factor.

@Shizuo: For $sl_n$, the situation is more explicit. The flag variety here is the set of flags $0 = V_0 subset V_1 cdots V_{n-1} subset V_n = mathbb{C}^n$ with $mathrm{dim} V_i = i$. So the flag variety is a closed subvariety of the product of Grassmannians $Gr(1,n) times cdots Gr(n,n) $. Each of these Grassmannians have a explicitly Plucker embedding into the projectivization of the exterior power of $mathbb{C}^n$. In particular, the homogeneous ideal is explicitly given by the Plucker relations. So the multi-homogeneous coordinate ring of $Gr(1,n) times cdots Gr(n,n) $ is just the tensor product of the known homogeneous coordinate rings.

Finally, to get the multi-homogeneous coordinate ring for the flag variety, we need to specify an incidence locus inside $Gr(1,n) times cdots Gr(n,n) $. Namely, we need to specify those tuples of subspaces that form a flag. But this is easy: it corresponds to certain wedge products being zero. Just impose those additional relations, i.e. mod out by the corresponding multi-homogenous ideal. Now you should have an explicit, albeit fairly long description of the homogeneous coordinate ring. The above answer for $SL_3$ looks like a special case of this construction.

Solving a noisy set of linear equations.

Suppose we have a square $ntimes n$ real matrix $A$ of full rank such that the squares of the elements in each row sum to 1, an $ntimes 1$ vector of variables $x$, and an $ntimes 1$ real vector $a$, such that $Acdot x = a$. We can of course take the inverse of $A$ to solve uniquely for $x$.

My question is as follows: suppose we do not know $a$ exactly, but only up to additive error epsilon: that is, we know $a'$ such that $a' = a + error$, where $error$ is a real $ntimes 1$ vector with each component in the range $[-epsilon,epsilon]$. Vector $x$ is no longer uniquely determined. However, we can solve for some $x'$ such that $x' = x + error'$. My question is, what can we say about the magnitude of the components of $error'$, and how they relate to $epsilon$?

co.combinatorics - Reference request: The stable Kronecker ring is isomorphic to the ring of symmetric polynomials

For $lambda$ any partition and $n$ a positive integer, write $lambda[n]$ for the sequence $(n - |lambda|, lambda_1, lambda_2, ldots, lambda_r)$. For $n$ large enough, this is a partition of $n$.

The irreducible representations of $S_n$ are indexed by partitions of $n$; we denote them by $S_{lambda}$. The Kronecker coefficients $g_{lambda mu}^{nu}$ are defined by the equality
$$S_{lambda} otimes S_{mu} cong bigoplus g_{lambda mu}^{nu} S_{nu}$$
of $S_n$ representations.

It is a theorem of Murnaghan that $g_{lambda[n] mu[n]}^{nu[n]}$ becomes constant as $n to infty$. This constant value is called the stable Kronecker coefficient, and denoted $overline{g}_{lambda mu}^{nu}$. It is also a result of Murnaghan that, for given $lambda$ and $mu$, there are only finitely many $nu$ for which $overline{g}_{lambda mu}^{nu} neq 0$.

Therefore, we can define a commutative, associative ring to be spanned by the generators $overline{S}_{lambda}$, with relations
$$overline{S}_{lambda} overline{S}_{mu} = sum overline{g}_{lambda mu}^{nu} overline{S}_{nu}.$$

I'll call this the stable Kronecker ring.

I can prove that the stable Kronecker ring is isomorphic to the ring of symmetric functions. Is this fact already in the literature?

pr.probability - Expected position of a card in a deck after repeating a procedure

Let me see if I understand the procedure. We have two piles, which I will call the deck and the pile. If KH is not in the pile, we pull three cards off the top of the deck and put them in the pile. If KH is in the pile, we riffle shuffle the deck and the pile together. I will assume the mathematically idealized riffle shuffle where the order of cards within the deck and the pile do not change, but all possible ways of shuffling them together with this constraint are equally likely.

Here's a heuristic answer to your question, assuming that I've understood the procedure right.

When we pull the cards off by threes, if the King of Hearts is the $k$th card down, we get roughly an average of $k+1$ cards in the second pile, assuming $k$ varies over a range which is substantially more than 3.
The KH will be one of the top three cards in the pile when the pile and the rest of the deck are shuffled. It should be roughly equally likely to be any of the top three. Suppose that when they are shuffled, the expected number of cards in the pile is $P$, and in the deck is $52-P$. If you assume a riffle shuffle, and that there are exactly $P$ cards in the pile, then the expected number of cards above the King of Hearts after the shuffle is $2frac{52-P}{P+1}+1$ (one card from the pile and $2frac{52-P}{P+1}$ cards from the deck). We have the heuristic equation
$$ P-1 =k = 2frac{52-P}{P+1}+1$$ or $P approx 9.8$.

One reason it's not exactly correct is that we can't average over $P$ if it's in the denominator. Another reason is that $k$ is probably too small to be equally likely to be one of the three residues mod 3. It shouldn't be hard to write a program to simulate this, if you want a more exact answer.

UPDATE: improved estimate slightly

soft question - Mathematicians who were late learners?-list

In May 2006, the AMS Notices printed a remembrance article for Serge Lang. Dorian Goldfield was one of the contributors, and as an undergraduate, he described himself as follows:

Of the many people who had serious
interactions with Serge, I am one of
those who came away with fierce
admiration and loyalty. In the
mid-1960s, I was an undergraduate in
the Columbia engineering school on
academic probation with a C–average.
In my senior year I had an idea for a
theorem which combined ergodic theory
and number theory in a new way, and I
approached Serge and showed him what I
was doing. Although I was only a
C–level student in his undergraduate
analysis class he took an immediate
interest in my work and asked Lorch if
he thought there was anything in it.
When Lorch came back with a positive
response, Lang immediately invited me
to join the graduate program at
Columbia the next year, September
1967.

Then again, Goldfield was not a "late learner" as he was 20 when he finished college and 22 when he earned his PhD. But...

linear algebra - Estimating the spectral radius of a matrix, noniteratively

Morris Marden's "Geometry of Polynomials" displays a number of formulae that allow one to estimate bounds on the largest root of a polynomial that do not involve actual rootfinding. Having been inspired by this, and since this particular problem crops up in one of the things I'm working on, I was wondering if one could get good estimates of the spectral radius of a general dense n-by-n matrix A that has been previously processed as follows:

1. a similarity transformation to upper Hessenberg form ($A=QHQ^T$, $Q$ orthogonal and $H$ upper Hessenberg); and


2. subtracting the identity multiplied by the mean of the eigenvalues from $H$ ($H'=H-frac{trace(H)}{n}I$, this corresponds geometrically to centering the eigenvalues around the origin of the complex plane).

As much as possible, I am trying to avoid having to resort to an eigenvalue method (e.g. QR (too much effort!), power method (the power method can misbehave when there is more than one eigenvalue whose modulus is equal to the spectral radius)) since I only need a quick 2-3 digit approximation of the spectral radius. I have considered actually expanding $H'$ to its characteristic polynomial (equivalently, a similarity transformation of $H'$ to a Frobenius companion matrix) so that the formulae listed in Marden can apply, but after reading Wilkinson's wonderful book "The Algebraic Eigenvalue Problem" where he details how unstable the computation of coefficients of the characteristic polynomial can get from a matrix, I suppose that idea is shot.

My other naïve attempts at estimating the spectral radius include using $||H'||_infty$ as an estimate, and deriving rough bounds using Gerschgorin's theorem; the problem I've seen is that both attempts tend to overestimate the spectral radius by a significant factor.

Is there a way to estimate the spectral radius more cheaply and noniteratively than actually computing eigenvalues?

dg.differential geometry - Diffeomorphism group of the unit sphere of complex n-space

I think that you are asking for the group of automorphisms of the CR-structure on $mathbb{S}^{2n-1}$ as hypersurface in $mathbb{C}^n$ (assuming $n>1$, since $n=1$ is another story). Even locally, any such automorphism is induced by an element of $PU(n,1)$, which is the subgroup of $PGL(n+1,mathbb{C})=mathrm{Aut}(mathbb{C}P^n)$ preserving the unit ball $mathbb{B}subsetmathbb{C}^nsubsetmathbb{C}P^n$. This result is due to S.S. Chern and J. Moser, Acta Math. 133 (1974), 219--271 (but the global version you are asking for is perhaps older).

Concretely, $PU(n,1)$ is the isometry group of the hermitian form $|z_1|^2+dots+|z_n|^2-|z_{n+1}|^2$, quotiented by its scalar subgroup $U(1)$.
The diffeomorphism of $mathbb{B}subsetmathbb{C}^n$ associated to a matrix $Ain U(n,1)$ is given by $vmapsto (a_{11}v+a_{12})/(a_{21}v+a_{22})$, where $a_{ij}$ are the blocks of $A$ for the obvious decomposition $mathbb{C}^{n+1}=mathbb{C}^noplusmathbb{C}$ (thus $a_{22}$ is scalar, for instance, and $a_{21}$ is an $1times n$ line matrix).

Hope this answers your question

soft question - What's your favorite equation, formula, identity or inequality?

My favorite equation is

$$frac{16}{64} = frac{1}{4}.$$

What makes this equation interesting is that canceling the $6$'s yields the correct answer. I realized this in, perhaps, third grade. This was the great rebellion of my youth. Sometime later I generalized this to finding solutions to

$$frac{pa +b}{pb + c} = frac{a}{c}.$$

where $p$ is an integer greater than $1$. We require that $a$, $b$, and $c$ are integers between $1$ and $p - 1$, inclusive. Say a solution is trivial if $a = b = c$. Then $p$ is prime if and only if all solutions are trivial. On can also prove that if $p$ is an even integer greater than $2$ then $p - 1$ is prime if and only if every nontrivial solution $(a,b,c)$ has $b = p - 1$.

The key to these results is that if $(a, b, c)$ is a nontrivial solution then the greatest common divisor of $c$ and $p$ is greater than $1$ and the greatest common divisor of $b$ and $p - 1$ is also greater than $1$.

Two other interesting facts are (i) if $(a, b, c)$ is a nontrivial solution then $2a leq c < b$ and (2) the number of nontrivial solutions is odd if and only if $p$ is the square of an even integer. To prove the latter item it is useful to note that if $(a, b, c)$ is a nontrivial solution then so is $(b - c, b, b - a)$.

For what it is worth I call this demented division.

graph theory - Can one make Erdős's Ramsey lower bound explicit?

As was mentioned in the previous answers, the answer is no. Or more accurately I'd say that the answer is currently no, but possibly yes.

Also, consider the related question of constructing a bipartite graph with parts of size $2^n$, which contains no $K_{k,k}$ and whose complement contains no $K_{k,k}$ where $k = O(n)$. Such an explicit construction will have as far as I can tell huge impact on derandomization of randomized algorithms, among other topics in theoretical computer science. See e.g. this paper, where such an explicit construction is given for $k = 2^{n^{o(1)}}$.

You might also be interested in the following accompanying paper (seems like I cannot post it, being a new user; you can google it though, its title is "Pseudorandomness and Combinatorial Constructions") to Luca Trvisan's talk at ICM '06. This may contain more connections between explicit constructions of combinatorial objects and applications in theoretical computer science.

gr.group theory - Mystery of the Monstrous Moonshine

I can give you half of the answer, but the other half is wide open. I will use the characterization of supersingular primes as those primes p for which the normalizer of Gamma₀(p) in SL(2,R) acts on the complex upper half plane to yield a genus zero quotient.

The monstrous moonshine conjecture asserted the existence of an infinite dimensional graded representation of the monster satisfying some exceptional properties. It was conjectured by Conway and Norton, and proved by Borcherds, using the representation constructed by I. Frenkel, Lepowsky, and Meurman. One can take the graded dimension of this representation to get a power series, and it is the q-expansion of the J-function. Furthermore, the graded trace of any element of order n in the monster is the q-expansion of a genus zero modular function that is invariant under Gamma₀(nh) for some h|(12,n). One can conclude somewhat abstractly that the normalizer of Gamma₀(p) in SL(2,R) has to be genus zero for any prime p dividing the order of the monster.

The proof I've seen that no other primes satisfy the genus zero condition does not seem to have anything to do with the monster. Instead, it is a delicate construction by Mazur involving the Eisenstein ideal, combined with some computations by H. Lenstra. I may be ignorant of more refined arguments developed in the last 30 years, though. [Edit: FC has pointed out that the proof of the bijection is a reasonably straightforward calculation. Still, I haven't seen any good arguments explaining the universality of the monster with respect to the genus zero property.]

Linear algebra lemma

First note that if a 2-form is degenerate, it is 0 on some 4-subspace (take a lagrangian subspace of the quotient by the kernel).

Now, assume not. Pick two elements that span $W$. If either of them has 4-d kernel, it is 0 on any 4-d subspace, and we can use whichever on the other vanishes on.

Thus, every element of $W$ has 2-d kernel. If two elements had different kernels, then one of their linear combinations would have no kernel. Thus, they all kill the same 2-d subspace. Thus, we've reduced to the statement that any two 2-forms on a 4-d space $Z$ have a common Lagrangian subspace. Pick any line $L$; this is isotropic for both, since all lines are. Consider the intersections of the symplectic orthogonals of $L$ for the two 2-forms. These are 3-d, so their intersection is a 2-space. Now you win.

Is there any direct approach to generate discrete 2 partitions of a set of having 2^n elements for a given n ?

Assume that you have a set S of having 2^2 elements first, let S={0,1,2,3}
Then the desired 2 partitions would be
1-{{0,1},{2,3}}
2-{{0,2},{1,3}}
3-{{0,3},{1,2}}

If S={0,1,2,3,4,5,6,7} having 2^3 elements then similarly the 2 partitions would be

1- {{0,1},{2,3},{4,5},{6,7}}
2- {{0,2},{1,3},{4,6},{5,7}}
3- {{0,3},{1,5},{2,6},{4,7}}
4- {{0,4},{1,6},{2,5},{3,7}}
5- {{0,5},{1,4},{2,7},{3,6}}
6- {{0,6},{1,7},{2,4},{3,5}}
7- {{0,7},{1,2},{3,4},{5,6}}

So it seems that for S having 2^n elements the initial table would like

1: {{0,1},{2,3},{4,5},{6,7},...,{(2^n)-2,(2^n)-1}}
2: {{0,2},... }
.
.
.
(2^n)-1: {{0,(2^n)-1},... }

I am just wondering whether there is an direct approach to generate the above table for a given n.

Thanks

st.statistics - Data Mining-- How do You Know Whether The Pattern You Extract is Valid?

Your question is a difficulty one which every scientist grapples with every day of their career. Broadly there are two mechanisms by which hypothesis testing can be biased.

The first is the obvious mechanism of biasing the data in favor an hypothesis. Sometimes this occurs through straightforward fraud (i.e. South Korean stem cell findings). But more frequently this source of bias is much more subtle; involving issues such as unrecognized selection bias in the data.

The second mechanism is far more difficult to prevent and is often seen in peer reviewed literature, this mechanism is the biasing of the hypothesis in favor of the data. Two typical examples are over fitting the data with a large number of extra parameters to obtain a better fit, or using an inappropriate statistical model on a large dataset to yield statistical significance (i.e. fitting a time dependent model of the speed of light on cosmological scales). Another example is confusing data analysis for hypothesis testing (i.e. low frequency heart rate modulation research).

There is no single algorithm to prevent this second form of bias, and using general data mining and data analysis tools will almost guarantee that you will over fit the data. The best practice available in the scientific field is an iterative practice: First look for the most obvious pattern that has the simplest explanation, test for the explanations fit with a reasonably course dataset. If the explanation fits, then refine your data and look at how your previous explanation fails on the refined data, propose and test a more nuanced hypothesis. Continue ad infinitum, or ad nausea which ever comes first. Thus a first step is to ask a simple question for which the extraneous factors and error sources can be controlled and don't analyze very detailed data, this will only lead you on a wild goose chase.

nt.number theory - Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.

Question. Is there a smooth proper scheme $Xtooperatorname{Spec}(mathbb{Z})$ such that $X(mathbb{Q}_v)neqemptyset$ for every place $v$ of $ mathbb{Q}$ (including $v=infty$), and yet $X(mathbb{Q})=emptyset$ ?

I believe the answer is No. The evidence is flimsy : $X$ cannot be a curve or a torsor under an abelian variety (Abrashkin-Fontaine) or a twisted form of $mathbb{P}_n$.

I haven't gone through a list of smooth projective $mathbb{Q}$-varieties which contradict the Hasse principle to check if any of them has good reduction everywhere, but the chances of such a thing are slim.

Colliot-Thélène and Xu give a systematic treatement of many known examples of quasi-projective $mathbb{Z}$-schemes $Y$ such that $Y(mathbb{Z}_p)neqemptyset$ for every prime $p$ and $Y(mathbb{R})neqemptyset$, but $Y(mathbb{Z})=emptyset$. Some of these schemes might even be smooth over $mathbb{Z}$, but none of them is proper.

Fontaine's letter to Messing (MR1274493) might be of some relevance here.

simplicial stuff - Is there any generalization of the Dold-Kan correspondence?

Associated to a simplicial group $G_n$, there is indeed a "chain complex". Namely, for each n one has the group $C_n$ of all elements in $G_n$ that map to zero under the boundary maps $d_1, ldots, d_n$ (but not necessarily $d_0$) that are part of the simplicial structure. The map $d_0$ is then a group homomorphism $C_n to C_{n-1}$ with $Im(d_0) subset ker(d_0)$, and the image is a normal subgroup. One could view this as a "chain complex" of nonabelian groups, and the homology groups have a nice interpretation: they're the homotopy groups of the geometric realization |G|.

However, this method of passing to a graded object isn't in any sense an equivalence of categories. Moreover, it is known that simplicial groups mod weak equivalence are a model for the homotopy category of pointed CW-complexes. In some ways, then, we don't expect much in the way of purely algebraic characterizations of simplicial groups.

For simplicial rings there is the slight issue that the normalized chain functor plays somewhat poorly together with the tensor product, and so the Dold-Kan correspondence doesn't preserve the symmetric monoidal structure. This is not as serious an issue for associative algebras or commutative algebras in characteristic zero if one is only working up to weak equivalence/quasi-isomorphism, but if you are interested in an actual equivalence of categories it means that a simplicial commutative $mathbb{Q}$-algebra is not quite the same as a nonnegatively graded commutative differential graded algebra.