algebraic number theory - sum of squares in ring of integers

To address the particularities of this question for number fields, the basic theorem is attributed to Hilbert, Landau and Siegel. First of all, any nonzero sum of squares in a number field has to be totally positive (that is, it is positive in all real embeddings). Hilbert (1902) conjectured that in any number field, a totally positive element is a sum of 4 squares in the number field. This was proved by Landau (1919) for quadratic fields and by Siegel (1921) for all number fields.

This sounds superficially like a direct extension of Lagrange's theorem, but there is a catch: it is about field elements, not algebraic integers as sums of squares of algebraic integers. A totally positive algebraic integer in a number field $K$ need not be a sum of 4 squares of algebraic integers in $K$. The Hilbert-Landau-Siegel theorem only says it is a sum of 4 squares of algebraic numbers in $K$.

For instance, in $mathbf{Q}(i)$ all elements are totally positive in a vacuous sense (no real embeddings), so every element is a sum of four squares. As an example,
$$
i = left(frac{1+i}{2}right)^2 + left(frac{1+i}{2}right)^2.
$$
This shows $i$ is a sum of two squares in $mathbf{Q}(i)$.
It is impossible to write $i$ as a finite sum of squares in ${mathbf Z}[i]$ since
$$
(a+bi)^2 = a^2 - b^2 + 2abi
$$
has even imaginary part when $a$ and $b$ are in $mathbf{Z}$. Thus any finite sum of squares in $mathbf{Z}[i]$ has even imaginary part, so such a sum can't equal $i$.
Therefore it is false that every totally positive algebraic integer in a number field is a sum of 4 squares (or even any number of squares) of algebraic integers.

Here are some further examples:

1. In $mathbf{Q}(sqrt{2})$, $5 + 3sqrt{2}$ is totally positive since
   $5+3sqrt{2}$ and $5-3sqrt{2}$ are both positive. So it must be a sum of at most four squares in this field by Hilbert's theorem, and with a little fiddling around you find
   $$
   5 + 3sqrt{2} = (1+sqrt{2})^2 + left(1 + frac{1}{sqrt{2}}right)^2 + left(frac{1}{2}right)^2 + left(frac{1}{2}right)^2.
   $$
   It is impossible to write $5 + 3sqrt{2}$ as a sum of squares in the ring of integers $mathbf{Z}[sqrt{2}]$ because of the parity obstruction we saw for $i$ as a sum of squares in $mathbf{Z}[i]$: the coefficient of $sqrt{2}$ in $5 + 3sqrt{2}$ is odd.




2. In $mathbf{Q}(sqrt{2})$, $sqrt{2}$ is not totally positive (it becomes negative when we replace $sqrt{2}$ with $-sqrt{2}$), so it can't be a sum of squares in this field. But in the larger field $mathbf{Q}(sqrt{2},i)$, everything is totally positive in a vacuous sense so everything is a sum of at most four squares in this field by the Hilbert-Landau-Siegel theorem. And looking at $sqrt{2}$ in $mathbf{Q}(sqrt{2},i)$, we find
   $$
   sqrt{2} = left(1 + frac{1}{sqrt{2}}right)^2 + i^2 + left(frac{i}{sqrt{2}}right)^2.
   $$

Hilbert made his conjecture on totally positive numbers being sums of four squares as a theorem, in his Foundations of Geometry. It is Theorem 42. He says the proof is quite hard, and no proof is included. A copy of the book (in English) is available at the time I write this as http://math.berkeley.edu/~wodzicki/160/Hilbert.pdf. See page 83 of the file (= page 78 of the book).

Siegel's work on this theorem/conjecture was done just before the Hasse-Minkowski theorem was established in all number fields (by Hasse), and the former can be regarded as a special instance of the latter.

Indeed, for nonzero $alpha$ in a number field $K$, consider the quadratic form $$Q(x_1,x_2,x_3,x_4,x_5) = x_1^2+x_2^2+x_3^2+x_4^2-alpha{x}_5^2.$$ To say $alpha$ is a sum of four squares in $K$ is equivalent to saying $Q$ has a nontrivial zero over $K$. (In one direction, if $alpha$ is a sum of four squares over $K$ then $Q$ has a nontrivial zero over $K$ where $x_5 = 1$. In the other direction, if $Q$ has a nontrivial zero over $K$ where $x_5 not= 0$ then we can scale and make $x_5 = 1$, thus exhibiting $alpha$ as a sum of four squares in $K$. If $Q$ has a nontrivial zero over $K$ where $x_5 = 0$ then the sum of four squares quadratic form represents 0 nontrivially over $K$ and thus it is universal over $K$, so it represents $alpha$ over $K$.) By Hasse-Minkowski, $Q$ represents 0 nontrivially over $K$ if and only if it represents 0 nontrivially over every completion of $K$.

Since any nondegenerate quadratic form in five or more variables over a local field or the complex numbers represents 0 nontrivially, $Q$ represents 0 nontrivially over $K$ if and only it represents 0 nontrivially in every completion of $K$ that is isomorphic to ${mathbf R}$. The real completions of $K$ arise precisely from embeddings $K rightarrow {mathbf R}$. For $t in {mathbf R}^times$, the equation
$x_1^2+x_2^2+x_3^2+x_4^2-t{x}_5^2 =0$ has a nontrivial real solution if and only if $t > 0$, so $Q$ has a nontrivial representation of 0 in every real completion of $K$ if and only if $alpha$ is positive in every embedding of $K$ into ${mathbf R}$, which is what it means for $alpha$ to be totally positive. (Strictly speaking, to be totally positive in a field means being positive in every ordering on the field. The orderings on a number field all arise from embeddings of the number field into $mathbf R$, so being totally positive in a number field is the same as being positive in every real completion.)

Siegel's paper is "Darstellung total positiver Zahlen durch Quadrate, Math. Zeit. 11 (1921), 246--275, and can be found online at http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PPN=PPN266833020_0011&DMDID=DMDLOG_0022.

knot theory - slice=ribbon generalization to higher genus + potential counterexamples to slice=ribbon.

I have two questions about the slice=ribbon conjecture.

(1) If a knot $K hookrightarrow S^3$ has smooth slice genus $g$, you can ask if it bounds a smooth genus $g$ surface in $S^3 times [0, -infty)$, with the function defined by restriction to $[0, -infty)$ being Morse on the surface without index=0 critical points (maximal points). When $g=0$ this is just asking if the slice knot $K$ has a ribbon disc. I was wondering if there are any knots known with $g geq 1$ for which such a surface cannot exist. If there are none such known, is there a topological reason why the truth of the slice=ribbon conjecture would also imply the existence of such surfaces?

(2) Are there any potential counterexamples to slice=ribbon (in the same way that there are potential counterexamples to smooth 4-d Poincare [until Akbulut kills them])?

Thanks,
Andrew.

at.algebraic topology - Algorithm or theory of diagram chasing

One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ideally also an algorithm?

To be precise about what I mean, a diagram is a directed graph $D$ whose vertices are labeled by objects in an abelian category, and whose arrows are labeled by morphisms. The diagram might have various triangles, and we can require that certain triangles commute or anticommute. We can require that certain arrows vanish, which can be used to ask that certain compositions vanish. We can require that certain compositions are exact. Maybe some of the arrows are sums or direct sums of other arrows, and maybe some of the vertices are projective or injective objects. Then a diagram "lemma" is a construction of another diagram $D'$, with some new objects and arrows constructed from those of $D$, or at least some new restrictions.

As described so far, the diagram $D$ can express a functor from any category $mathcal{C}$ to the abelian category $mathcal{A}$. This looks too general for a reasonable algorithm. So let's take the case that $D$ is acyclic and finite. This is still too general to yield a complete classification of diagram structures, since acyclic diagrams include all acyclic quivers, and some of these have a "wild" representation theory. (For example, three arrows from $A$ to $B$ are a wild quiver. The representations of this quiver are not tractable, even working over a field.) In this case, I'm not asking for a full classification, only in a restricted algebraic theory that captures what is taught as diagram chasing.

Maybe the properties of a diagram that I listed in the second paragraph already yield a wild theory. It's fine to ditch some of them as necessary to have a tractable answer. Or to restrict to the category $textbf{Vect}(k)$ if necessary, although I am interested in greater generality than that.

To make an analogy, there is a theory of Lie bracket words. There is an algorithm related to Lyndon words that tells you when two sums of Lie bracket words are formally equal via the Jacobi identity. This is a satisfactory answer, even though it is not a classification of actual Lie algebras. In the case of commutative diagrams, I don't know a reasonable set of axioms — maybe they are related to triangulated categories — much less an algorithm to characterize their formal implications.

(This question was inspired by a mathoverflow question about George Bergman's salamander lemma.)

---

David's reference is interesting and it could be a part of what I had in mind with my question, but it is not the main part. My thinking is that diagram chasing is boring, and that ideally there would be an algorithm to obtain all finite diagram chasing arguments, at least in the acyclic case. Here is a simplification of the question that is entirely rigorous.

Suppose that the diagram $D$ is finite and acyclic and that all pairs of paths commute, so that it is equivalent to a functor from a finite poset category $mathcal{P}$ to the abelian category $mathcal{A}$. Suppose that the only other decorations of $D$ are that: (1) certain arrows are the zero morphism, (2) certain vertices are the zero object, and (3) certain composable pairs of arrows are exact. (Actually condition 2 can be forced by conditions 1 and 3.) Then is there an algorithm to determine all pairs of arrows that are forced to be exact? Can it be done in polynomial time?

This rigorous simplification does not consider many of the possible features of lemmas in homological algebra. Nothing is said about projective or injective objects, taking kernels and cokernels, taking direct sums of objects and morphisms (or more generally finite limits and colimits), or making connecting morphisms. For example, it does not include the snake lemma. It also does not include diagrams in which only some pairs of paths commute. But it is enough to express the monomorphism and epimorphism conditions, so it includes for instance the five lemma.

Canonical basis for the extended quantum enveloping algebras

I am trying to understand some construction done by Lusztig in his book on quantum groups. Given some Cartan datum, let $U=U_q(mathfrak{g})$ the standard quantized enveloping algebra of the Kac-Moody algebra $mathfrak{g}$. Its negative part $U^{-}$ (the subalgebra of $U$ generated by the $f_i$'s) has a canonical basis $B$. In his book Introduction to Quantum Groups (don't be fooled by the title!), Lustzig constructs the non-unital extension $dot{U}$ and proves that it has a canonical basis $dot{B}$.

As a set, $dot{B}$ is in bijection with $Btimes X times B$ (where $X$ is the lattice of weights, normally called $P$ in any other books on quantum groups), and its elements are described with the rather cryptic notation $bdiamond_zeta b''$. The definition of those elements is however very obscure and non-explicit. I would appreciate finding an easier, more explicit, description, like the one given in section 25.3 in Lustzig's book for $U_q(mathfrak{sl}_2)$, also described in great detail by Lauda in the first part of his paper A categorification of quantum sl(2). It is not clear to me how this description can be extended to more complicated quantum groups.

Does anyone know any similar simple description of the canonical basis $dot{B}$, even in some other particular cases? I am also very interested of knowing if is there any relation with crystals, akin to the equivalence between $B$ and Kashiwara's crystal basis $B(infty)$.

dg.differential geometry - Change of coordinates introduced through dx

Ok here are the partial derivatives $k_M$:

they are dual to ${b^M} = {dy^mu,dtheta^alpha}$ ie:

$b^M(k_N)=delta_N^M$

Writing $k_N=(k_N)^{mu}frac{partial}{partial x^mu}+(k_N)^{alpha}frac{partial}{partial theta^alpha}$ and solving $b^nu(k_mu)=delta^nu_mu,;b^mu(k_alpha)=0; ; b^alpha(k_nu)=0;;b^alpha(k_beta)=delta^alpha_beta$ one finds:

$(k_mu)^nu=delta^nu_mu,;(k_mu)^alpha=0;;(k_alpha)^beta=delta^beta_alpha;;(k_alpha)^nu=-eta_alpha^{;;mu}$

So:

$frac{partial}{partial y^mu} = frac{partial}{partial x^mu}$

$frac{partial}{partial y^alpha} = -eta^{;;mu}_alpha frac{partial}{partial x^mu} + frac{partial}{partial theta^alpha}$

rt.representation theory - Which is the correct universal enveloping algebra in positive characteristic?

This is an extension of this question about symmetric algebras in positive characteristic. The title is also a bit tongue-in-cheek, as I am sure that there are multiple "correct" answers.

Let $mathfrak g$ be a Lie algebra over $k$. One can define the universal enveloping algebra $Umathfrak g$ in terms of the adjunction: $text{Hom}_{rm LieAlg}(mathfrak g, A) = text{Hom}_{rm AsAlg}(Umathfrak g, A)$ for any associative algebra $A$. Then it's easy enough to check that $Umathfrak g$ is the quotient of the free tensor algebra generated by $mathfrak g$ by the ideal generated by elements of the form $xy - yx - [x,y]$. (At least, I'm sure of this when the characteristic is not $2$. I don't have a good grasp in characteristic $2$, though, because I've heard that the correct notion of "Lie algebra" is different.)

But there's another good algebra, which agrees with $Umathfrak g$ in characteristic $0$. Namely, if $mathfrak g$ is the Lie algebra of some algebraic group $G$, then I think that the algebra of left-invariant differential operators is some sort of "divided-power" version of $Umathfrak g$.

So, am I correct that this notions diverge in positive characteristic? If so, does the divided-power algebra have a nice generators-and-relations description? More importantly, which rings are used for what?

gn.general topology - The proper name for a kind of ordered space

I'm trying to find the correct term for a specific kind of totally ordered space:

Let $S$ be a totally ordered space with strict total order $<$.

Property: For any two $s_{1}$ and $s_{2}$ in $S$ where $s_1 < s_2$, there must exist some $s_{3}$ such that $s_{1} < s_{3}$ and $s_{3} < s_{2}$.

What is the name of this property? Thank you!

ct.category theory - Why is the concept of topos a "metamorphosis" of the concept of space?

The answers already provided are very good and informative, so I just wish to add something concerning the "metamorphosis" of the very notion of space of which Grothendieck speaks in Semailles.

Every space has its associated topos, but there are topoi which are NOT spatial. You can define categorically the notion of point of a topos, and this definition corresponds to the usual notion of points when one restricts to spatial topoi.

Now, the fact that there are plenty of topoi with no points basically means that one can do topology in a pointless world: you can still formally define notions of compactness, coverings, and well as most of the standard topological (and even homotopical) machinery, directly in a given topos, regardless of its having points or not.

As it turns out, the passage from point-set to pointless topology is not just an idle game: for instance in physics at the Planck level you may still want to talk of topological and geometric properties of space-time, and yet you have no well-defined points.

ra.rings and algebras - Which commutative groups are the group of units of some field?

Another characterization is theorem 2.1 in this paper on the field with one element:

http://arxiv.org/pdf/0911.3537

If H is a commutative group, let H+ be H together with a new element 0. To give a field structure on H+ is equivalent to giving a bijection s:H+ --> H+ that commutes with all of its conjugates-by-H.

Maybe this is similar to Dicker's characterization? Dicker mentions the operation x --> 1-x, while the s in Connes-Consani is meant to be x --> x + 1.

soft question - How do you approach your child's math education?

While pushing each of my three sons on a swing when they were wee lads, I would count-by: 1,2,3,4,5,6,7,8,9,10; then 2,4,6,8,10,12,14,16,18,20; then 3,6,9 etc. The swing push would be over when we got to 100. My wife hung count-by sheets in the kitchen. We spoke about counting by eggs as if they were a fraction of a dozen: 1/12, 1/6, 1/4, 1/3, etc. I taught the youngest how to compute squares in his head when he was in 2nd grade. First, he learned one squared through 10 squared, then 10,20,30, etc squared. Then we played a game: what is 20 squared? what is two times 20? what is twenty squared plus two times twenty? what is twenty squared plus two times 20 plus 1? what is twenty-one squared? These exercises were in the car on a ten minute ride to school. We started to work through computing products as differences of squares.

Certainly I taught the boys some modular arithmetic, and they all attended the math circle --- even started them a bit too young.

Also, they were taught how to count to 1023 on 10 fingers. Lots of cute tricks.
In terms of the mental calculations, even if you can't do the arithmetic quickly, you can teach the child to do so. When the child sees that you struggle with it, then (s)he has someone with whom (s)he can compete.

In addition, I would stress units and developing answers as complete sentences and guiding writing.

Read "Alice in Wonderland" and "A Wrinkle in Time" to the child at about 1st or 2nd grade. Emphasize the connections between math and human development.

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

With the stuff I've seen in the literature of sequence transformations, I've started to love the formulae for Aitken's Δ² process:

$S_n^{prime}=S_{n+1}-frac{(Delta S_n)^2}{Delta^2 S_n}$

and its generalization the Wynn ε algorithm:

$varepsilon_{k+1}^{(n)}=varepsilon_{k-1}^{(n+1)}+frac1{varepsilon_{k}^{(n+1)}-varepsilon_{k}^{(n)}}$

for the latter one especially because it is nicely represented as a lozenge diagram:

finite groups - Reference for this theorem in representation theory?

I am not quite sure about the reference :( I always thought of this fact as follows.

Matrix elements of tensor powers of a representation U are all possible monomials in matrix elements of U, so the space of all their linear combinations are values of all possible polynomials in the matrix elements of U. Now, by definition of H, values of matrix elements of U separate elements of G/H, so every function on G/H (including all irreducible characters) can be written as a polynomial in the matrix elements of U in the case of finite groups, or can be approximated by polynomials with arbitrary precision in the case of compact infinite groups and the ground field being R or C (Stone-Weierstrass).

Now, to complete the proof, we may use orthogonality of matrix elements: if E_{ij} are matrix elements of an irrep V, and F_{ij} --- matrix elements of an irrep W (all thought of as functions on the group), then for the standard bilinear form on the ring of functions C(G) we have (E_{ij},F_{kl})=0 unless V is isomorphic to W and, in the latter case, i=l, j=k (in which case the value is 1) - here I probably want the order of the group to not be divisible by char(k) in the finite case, or the group to be compact, and the field be real/complex in the infinite case. Since irreducible characters can be approximated by polynomials in matrix elements, such a character cannot be orthogonal to all matrix elements of tensor powers and is, therefor, contained in one of them.

reference request - spectral sections - explicit construction

You probably already know this, but the standard method is to assume there is a spectral gap, i.e. a real number not in the spectrum of $D_beta$ for all $beta$. So although, as you point out, zero might not work, another real number might, or more generally, if there is a function $g:Bto R$ so that $g(beta)$ does not lie in the spectrum of $D_beta$ then you can use it to construct a spectral section.

Such functions can always be found locally, and so you can try to patch these local solutions; perhaps this is what underlies Melrose-Piazza.

Another general method, when the space $B$ is an interval and the dependence of $D_beta$ on $beta$ is analytic, is to use the Kato selection lemma, which allows you to analytically continue eigenvectors, and in particular if you take the positive eigenspan at one parameter, you can extend this to a spectral section over the interval, even if there is no spectral gap. Presumably if you are careful you can extend this idea for more general analytic families. Typically families which arise in geometric contexts vary analytically, although this is not always easy to check.

differential topology - Triangulations of exotic 4-spheres

Here is my comment expanded to answer form: The question of existence of exotic 4-spheres (i.e., the smooth Poincaré conjecture) is still open, and (according to Wikipedia) the existence of exotic PL structures is equivalent to it. Therefore, the answer is that no such explicit triangulations are known.

In general, explicit triangulations of higher dimensional manifolds seem to be difficult to write down. I've heard from computer algebra specialists that no one has even written an explicit triangulation of $mathbb{CP}^3$. The chaos surrounding this earlier question might suggest that the problem is subtle.

ra.rings and algebras - Is there a version of the Archimedean property which does not presuppose the Naturals?

It is not surprising that some versions of the Archimedean property concern subsets of the order rather than merely elements. The reason is that the Archimedean property is provably not expressible in a first order manner.

This is because the structure of the reals R, as an ordered field, say, (but one can add any structure at all), has elementary extensions to nonstandard models R^* which are non-Archimedean. This means that any statement in the language of ordered fields that is true in the reals R will also be true in the nonstandard reals. To prove that such models exist is an elementary application of the Compactness theorem, and one can also construct them directly via the ultrapower construction. One can also control the cofinality of the nonstandard order. For example, one can arrange that every countable subset of R^* is bounded. Since all these various nonstandard models R^* satisfy exactly the same first order truths as the standard reals R, but are non-Archimedean, it follows that being Archimedean is not first-order expressible.

Being Archimedean is, of course, second-order expressible, and the usual definition is a second order definition. As Neel mentions in the comments, the natural numbers are identifiable as the smallest subset of the ordered field containing 0 and closed under successor n+1.

If one adds the natural numbers N as a predicate to the original model, so that one is looking at R as an ordered field with a unary predicate holding of the natural numbers, then the nonstandard model R^* will include a nonstandard version N^* of the natural numbers. This new field R^*, which is not Archimedean, will nevertheless appear to be Archimedean relative to the nonstandard natural numbers N^*. For example, for any x and y in R^*, there will be a number n in N^* such that nx > y.

Indeed, one can do amazing things along this line. Suppose that V is the entire set-theoretic universe, and let V^* be a nonstandard version of it (such as an ultrapower by a nonprincipal ultrafilter on the natural numbers). Inside V^*, the structure R^* is thought to be the actual real numbers and so V^* thinks R^* is Archimedean, even though back in V we can see that it is mistaken, precisely because V^* is using the wrong set of natural numbers for its conclusion. The model V^* simply cannot see the true set of natural numbers sitting inside R^*, because it does not have that set.

More generally, one can similarly describe what it should mean for any ordered field F to be Archimedean relative to a subring R. Perhaps this simple idea is the generalization for which you are looking? It is mainly amounting to the question of whether the subring is cofinal in the original order.

Thus, it is very natural to look at the possible cofinalities of the orders that arise in ordered fields (or the other types of structures that you consider). For any infinite regular cardinal κ, one may find an elementary extension of the reals R to a nonstandard ordered field R^*, where the order of R^* has a cofinal κ sequence. To do this, just perform a series of κ many extensions, each with new elements on top of the previous model. In κ many steps, the union of the structures you built will have an order with cofinality exactly κ.

If one only uses the ultrapower construction to construct the nonstandard models, however, then there are limits on the resulting cofinality of the order. Understanding these limits is a large part of Shelah's deep work on PCF (= possible cofinality) theory.

Rado graph containing infinitely many isomorphic subgraphs

I realise that the question is almost three years old, but maybe that's not that long in Maths.

I am not sure who the following is originally due to, but as far as I am aware, it's the standard to show that the Rado graph contains every countable graph. The idea is to start with whatever graph you want and construct the Rado graph around it.

Let $G=G_0$ be any countable graph. For $i>0$ define a new Graph $G_i$ as follows:

• The vertices $V(G_{i})$ of $G_i$ are all of $V(G_{i-1})$ plus an extra vertex $v_A$ for every finite subset $A$ of $V(G_{i-1})$.




• All edges of $V(G_{i-1})$ are also edges of $V(G_{i})$




• Add an edge between $ain V(G_{i-1})$ and $v_{A}in V(G_{i})$ whenever $ain A$.

Let $R$ be the union of the graphs $G_i$ over all $i>0$. Then show that $R$ is the Rado graph. Clearly, $R$ contains $G=G_0$.

One of the nice things about this construction is that you can show without much difficulty that any automorphism of $G$ extends to an automorphism of $R$. So not only does the Rado graph contain every countable graph $G$, but it contains "special" copies of $G$ with the above extension property.

rt.representation theory - When are all characteristic l representations liftable

Another sufficient condition is that if $G$ is solvable, then for every prime $l$, every absolutely irreducible characteristic $l$ representation can be lifted to the complex numbers. In fact, solvability is not really necessary; $l$-solvability suffices.
This is the Fong-Swan theorem.

---

Added later: Since groups with order not divisible by $l$ are trivially $l$-solvable,
this sufficient condition includes, and is more general than the the condition stated by Pete L. Clark.

at.algebraic topology - Verb form of 'homotopy'? 'Homotope'?

Is there a transitive verb, in common use, which means 'deform via a homotopy'? I used to think 'homotope' was the answer, but it produces surprisingly few relevant matches on Google, so now I have my doubts. I want to be able to say things like "By Lemma 17 we can homotope the k-cells (rel boundary) into the subspace Y". I am well aware that I could rearrange the sentence to say something like "...there is a homotopy such that...", but that's not what I want to do. I want a simple, one-word verb with this meaning. I suppose 'deform' might work, but it's not as specific as I would like.

Another form of this question: Does the use of 'homotope' in the sample sentence of the previous paragraph sound strange? Sound standard and idiomatic?

convexity - Determination of a symmetric convex region by parallel sections

Yes.

Assume we have two distinct functions $f$ and $g$ such that $f^{-1}-fequiv g^{-1}-g$.
Take a sequence $x_n=f(x_{n-1})$.
Clearly $f(x_n)-g(x_n)=0$ or $(-1)^n[f(x_n)-g(x_n)]$ has the same sign for all $n$.

Sinse $int_0^1f=int_0^1g$, there are two sequences $x_n$ and $y_n$ as above such that $f(x_n)=g(x_n)$, $f(y_n)=g(y_n)$ and say $(-1)^n[f(x)-g(x)]>0$ for any $xin(x_n,y_n)$.
Note that $x_n,y_nto 0$ and $int_{x_n}^{y_n}|f-g|=const>0$.
It follows that $limsup_{xto0} |f(x)-g(x)|toinfty$, a contradiction.

qa.quantum algebra - Which is the correct version of a quantum group at a root of unity?

There are (at least) five interesting versions of the quantum group at a root of unity.

The Kac-De Concini form:
This is what you get if you just take the obvious integral form and specialize q to a root of unity (you may want to clear the denominators first, but that only affects a few small roots of unity). This is best thought of as a quantized version of jets of functions on the Poisson dual group. It's most important characteristic is that it has a very large central Hopf subalgebre (generated by the lth powers of the standard generators). In particular, its representation theory is sits over Spec of the large center, which is necessarily a group and turns out to be the Poisson dual group. It also has a small quotient Hopf algebra when you kill the large center.

The main sources for the structure of the finite dimensional representations are papers by subsets of Kac-DeConcini-Procesi (the structure of the representations depends on the symplectic leaf in G*, in particular there are "generic" ones coming from the big cell) as well as some more recent work by Kremnitzer (proving some stronger results about the dimensions of the non-generic representations) and by DeConcini-Procesi-Reshetikhin-Rosso (giving the tensor product rules for generic reps). The main application that I know of this integral form is to invariants of knots together with a hyperbolic structure on the compliment and to invariants of hyperbolic 3-manifolds due to Kashaev, Baseilhac-Bennedetti, and Kashaev-Reshetikhin. The hope is that these invariants will shed some light on the volume conjecture.

The Lusztig form:
Here you start with the integral form that has divided powers. Structurally this has a small subalgebra generated by the usual generators (E_i, F_i, K_i) since E^l = 0. The quotient by this subalgebra gives the usual universal enveloping algebra via something called the quantum Frobenius map. The main representation that people look at are the "tilting modules." Tilting modules have a technical description, but the important point is that the indecomposable tilting modules are exactly the summands of the tensor products of the fundamental representations. Indecomposable tilting modules are indexed by weights in the Weyl chamber. The "linkage principle" tells you that inside the decomposition series of a given indecomposable tilting module you only need to look at the Weyl modules with highest weights given by smaller elements in a certain affine Weyl group orbit.

It is the Lusztig integral form (not specialized) that is important for categorification. The Lusztig form at a root of unity is important for relationships between quantum groups and representations of algebraic groups and for relationships to affine lie algebras. The main sources are Lusztig and HH Andersen (and his colaborators). I'm also fond of a paper of Sawin's that does a very nice job cleaning up the literature.

The Lusztig integral form is also the natural one from a quantum topology point of view. For example, if you start with the Temperley-Lieb algebra (or equivalently, tangles modulo the Kauffman bracket relations) and specialize q to a root of unity what you end up with is the planar algebra for the tilting modules for the Lusztig form at that root of unity.

The small quantum group:

This is a finite dimensional Hopf algebra, it appears as a quotient of the K-DC form (quotienting by the large central subalgebra) and as a subalgebra of the Lusztig form (generated by the standard generators). I gather that the representation theory is not very well understood. But there has been some work recently by Roman Bezrukavnikov and others. I also wrote a blog post on what the representation theory looks like here for one of the smallest examples.

The semisimplified category:

Unlike the other examples, this is not the category of representations of a Hopf algebra! (Although like all fusion categories it is the category of representations of a weak Hopf algebra.) You start with either the category of tilting modules for the Lusztig form or the category of finite dimensional representations of the small quantum group and then you "semisimplify" by killing all "negligible morphisms." A morphism is negligible if it gives you 0 no matter how you "close it off." Alternately the negligible morphisms are the kernel of a certain inner product on the Hom spaces. The resulting category is semisimple, its representation theory is a "truncated" version of the usual representation theory. In particular the only surviving representations are those in the "Weyl alcove" which is like the Weyl chamber except its been cut off by a line perpendicular to l times a certain fundamental weight (see Sawin's paper for the correct line which depends subtly on the kind of root of unity).

This example is the main source of modular categories and of interesting fusion categories. Its main application is the 3-manifold invariants of Reshetikhin-Turaev (where this quotient first appears, I think) and Turaev-Viro. For those invariants its very important that your braided tensor category only have finitely many different simple objects.

The half-divided powers integral form:

This appears in the work of Habiro on universal versions of the Reshetikhin-Turaev invariants and on integrality results concerning these invariants. This integral form looks like the Lusztig form on the upper Borel and like the K-DC form on the lower Borel. The key advantage is that in the construction of the R-matrix via the Drinfeld double you should be looking at something like U_q(B+) otimes U_q(B+)* and it turns out that the dual of the Borel without divided powers is the Borel with divided powers and vice-versa. There's been very little work done on this case beyond the work of Habiro.

ac.commutative algebra - Rings of integers of function fields

This might be a somewhat silly and inconsequential question, but it's aroused my curiosity. One has the theorem in commutative algebra that the integral closure of a domain $A$ in its field of fractions $Q(A)$ is the intersection of all the valuation subrings of $Q(A)$ containing $A$. This naturally leads to the impulse to define the ring of integers of an arbitrary field $k$, independent of any particular subring, to be the intersection of all valuation subrings of $k$. Really what one is doing here is taking the integral closure of the prime ring, which is to say the minimal integrally closed subring. This gives the answer one would expect for $mathbb{Q}$ and for number fields. I believe that the ring of integers under this definition of $mathbb{C}$ is $overline{mathbb{Z}}$, and the ring of integers of $mathbb{R}$ is $overline{mathbb{Z}} cap mathbb{R}$. So far, so good.

Now here's the question: what happens in algebraic geometry? We could ask about the ring of integers $mathcal{O}$ of the function field $Q(A)$ of an irreducible affine variety $textbf{Spec
}A$ over an algebraically closed field $k$, and naïvely hope that it coincides with $A$; but there may be many subalgebras of $Q(A)$ whose field of fractions is $Q(A)$, and at most one of these will be its ring of integers. Can we characterise geometrically those affine varieties whose coordinate ring is the ring of integers of its function field?

And what happens for function fields of more general varieties? For projective varieties, whose global ring of functions isn't much to write home about, we might hope for a more interesting answer.

Edit: It occurs to me that as it stands, the (boring) answer to my question is that the ring of integers is always inside the ground field. Let me instead ask the following, related question: for affine or general function fields $k(X)$, is it always true that there is a minimal (but not necessarily unique minimal) subring $R$ such that

$R$ is integrally closed in $k(X)$, and

$k(X)$ is algebraic over the field of fractions $Q(R)$?

If so, what is the geometric meaning of $R$, and when must we have $Q(R) = k(X)$?

ag.algebraic geometry - Finding n points that are equidistant around the circumference of an ellipse

I'll attempt to answer your question by misinterpreting it.

As pointed out in the comments, computing elliptic integrals is not going to be easy. But what if you wanted to find $n$ points arranged around an ellipse which form the vertices of an equilateral polygon? Now the answer to the question is given by a real algebraic variety. It's possible that this question may be computationally more tractable.

There are $2n$ variables $(x_i,y_i)$, $i=1,ldots,n$, and $2n-1$ equations:
$$ frac{x_i^2}{a^2}+frac{y_i^2}{b^2}=1, i=1,ldots, n,$$
$$(x_i-x_{i+1})^2+(y_i-y_{i+1})^2=(x_{i+1}-x_{i+2})^2+(y_{i+1}-y_{i+2})^2, i=1,ldots, n-1,$$
indices taken $(mod n)$.

Also, for geometric reasons, one expects $n-1$ components to this variety, each of which is a circle. For example, if $n=5$, one would expect two solutions which are oriented in different directions, and two solutions which are star shaped.
As one moves around the circle, the solution should move around.

Thus, one expects this variety to be a complete intersection defined by quadratic equations. There are methods from algebraic geometry to find solutions to such equations. There are versions of Newton's recursion which may be effective for finding a numerical solution. The dihedral symmetry might further constrain the solutions. Maybe someone could point you to some references if this sort of solution would suffice for your application?

sg.symplectic geometry - Fukaya categories of hyperkahler reductions: general request for information

At the risk of writing things that are obvious to those listening in: this is Nadler-land, no?

If $X$ is a smooth complex variety with reductive group $G$ acting, and $mu_{mathbb C}: T^*Xrightarrow {mathfrak g}^*$ is the complex moment map, then $mu_{mathbb C}^{-1}(0)/G = T^*(X/G)$ provided one interprets all quotients as stacks.

If $T^*X$ is hyperkahler and we do the hyperkahler quotient for the maximal compact of $G$, picking a nontrivial real moment value $mu_{mathbb R}^{-1}(zeta)$ at which to reduce amounts (by Kirwan) to imposing a GIT stability on $mu_{mathbb C}^{-1}(0)$---i.e. to picking a nice open subset of the cotangent stack $T^*(X/G)$ that is actually a variety. A stack version of Nadler's "microlocal branes" theorem would describe the (suitable, undoubtedly homotopical/derived) exact Fukaya category as the constructible derived category of $X/G$.

Since I'm completely ignorant of how the Nadler-Zaslow/Nadler story actually works, I'd like to then imagine that such an equivalence microlocalizes properly to give an equivalence over the hyperkahler reduction (i.e. the nice open set) as well? Admittedly, by microlocalizing to the stable locus one should avoid all the derived unpleasantness (this should be analogous to what happens in Bezrukavnikov-Braverman's proof of "generic" geometric Langlands for $GL_n$ in characteristic $p$, where by localizing to the generic locus, ${mathcal D}$-module really means ${mathcal D}$-module, not "module over the enveloping algebroid of the tangent complex" or something like that).

Admittedly, I don't have a clue how to deal with the issue that the base $X$ in the important examples is typically affine...maybe if one forces some kind of boundary conditions also in the $X$-direction one could make the Fukaya category nontrivial in Tim's example of the Hilbert scheme??

4 manifolds - PD3 groups and PD4 complexes

I am interested at the moment in what groups can occur as the fundamental group of a 4-manifold (or more generally, a 4-dimensional CW complex) with prescribed conditions on the intersection form. I have what I am hoping is a basic homotopy theory question:

A (orientable) PD-$n$ group is a group $G$ such that the Eilenberg-Maclane space $K(G,1)$ admits "Poincare duality", i.e. there is an $n$-dimensional integer homology class in $K(G,1)$ (thought of as the "fundamental class") such that cap product with it yields an isomorphism between the corresponding cohomology and homology groups (like for closed oriented manifolds). This is more general than saying that $K(G,1)$ admits the structure of an orientable closed manifold of dimension $n$.

Let $G$ be a PD-3 group. Is there any reason why $G$ cannot be the fundamental group of an orientable PD4 complex $X$ with vanishing second homotopy group, $pi_2(X)=0$?

ag.algebraic geometry - What is the relationship between integrable systems and toric degenerations?

I very nearly wrote my PhD thesis on this topic.
Here's as much as I was able to figure out, though it's hardly a direct answer to your question.

1) Say your total space is K"ahler, and your fibers are compact. Then you can define a Levi-Civita connection on any open set consisting of smooth fibers. It turns out that this connection generates symplectomorphisms between the fibers.

2) In toric degenerations, the torus acts on the total space of the family, mostly moving them around, but preserving the zero fiber (which is why it's toric).

1+2?) Now imagine you use (1) to give a map from your general fiber $F_1$ to your special fiber $F_0$. Map further, to ${mathfrak t}^*$, using the moment map on the toric variety.

Now you have an integrable system on $F_1$, stolen from $F_0$!

There's a problem: since $F_0$ isn't smooth, we can't actually use (1) to make the map. The hope is to take limits along the horizontal vector field to define a continuous function $F_1 to F_0$.

3) It turns out that this is the same as following the gradient flow for the norm square of the moment map. And limits of real-analytic gradient flows on smooth varieties are well-understood, by Lojasiewicz. So if your total space is smooth, you can use this to show that the map $F_1 to F_0$ is well-defined, continuous, and smooth away from the singularities in $F_0$.

I never got around to investigating how things change if the total space is singular (as in the Gel'fand-Cetlin-Sturmfels-Gonciulea-Lakshmibai degeneration motivating the questioner, and me too). Of course you can pick a resolution of singularities, and I guess you can
ask that the metric on the exceptional fibers be very very small, and use that to generalize Lojasiewicz' results. But I never worked on this seriously.

Example:

Let the family be $det : C^{2times 2} to C$. Then the $0$ fiber is the cone over $P^1 times P^1$, so a toric variety, but the fiber over $1$ is $SL(2)$. That has a $T^2$ action, by left and right multiplication by its maximal torus, but doesn't have the rescaling action that the $0$ fiber enjoys. One can actually solve the ODE defined by the Levi-Civita/gradient flow and write down the map $SL(2) to det^{-1}(0)$. It collapses $SU(2)$ to the singular point $0$.

What is the integrable system? Regard $SL(2)$ as $T^* S^3$, and the action variable as $(p,vec v) mapsto |vec v|$. This generates unit-speed gradient flow on $T^* S^3$, which breaks down at zero vectors (the $SU(2) = S^3$) because they don't know which direction to go.

ag.algebraic geometry - About a non-obvious (?) link between the jacobians of curves and differentials

To explain my problem, I must give a lemma:

Let $X$, $Y$, $Z$ be curves over $k$ (of characteristic 0) such that the genus of $Z$ is greater than 2, and $pi : X to Y$, $phi : X to Z$ two non-constant morphisms.
If $phi^star(H^0(Z,Omega))subseteqpi^star(H^0(Y,Omega))$, where $Omega$ denotes the sheaf of regular 1-forms in each case, then there exists a non-constant morphism $u: Y to Z$ such that $phi = u circ pi$.

Now, in a proof, I saw the use of this lemma, except that the hypothesis was the inclusion $mathrm{Image}(mathrm{Jac}(Z) to mathrm{Jac}(X)) subseteq mathrm{Image}(mathrm{Jac}(Y) to mathrm{Jac}(X))$, instead of $phi^star(H^0(Z,Omega))subseteqpi^star(H^0(Y,Omega))$. I can guess it is equivalent, but why? Is it related to Grothendieck's duality? Did I miss something obvious?

linear algebra - Sparse approximate representation of a collection of vectors

By triangle inequality, preserving the property you wish for means that you can find "representatives" for each $v$ so that the $ell_1$ distances between any $v, v'$ are preserved to within 2$epsilon$ additive error.

There is a general result by Brinkman and Charikar that says that in general, for a collection of $n$ vectors in an $ell_1$ space, there's no way to construct a set of $n$ vectors in a smaller (e.g $log n$) dimensional space) that preserves distances approximately even multiplicatively (let alone additively). This distinction is important if you have vectors in the original space that are $O(epsilon)$ apart, but otherwise doesn't matter greatly.

Brinkman, B. and Charikar, M. 2005. On the impossibility of dimension reduction in l1. J. ACM 52, 5 (Sep. 2005), 766-788. DOI= http://doi.acm.org/10.1145/1089023.1089026

So I'm guessing that the answer to your first question should be no.

ag.algebraic geometry - Lower bound for characteristic variety

Let K be an algebraically closed field of char. 0, let A_n(K) be the Weyl algebra. Let I in A_n(K) be a left ideal generated by p elements. Set M := A_n / I.

Does the following then hold?

dim Ch(M) geq 2n - p

Here Ch(M) is the characteristic variety of M. (I know that the answer is yes if n = 1, and also if I is generated by homogeneous elements.)

Thank you.

nt.number theory - applications of Artin's holomorphy conjecture

I am waking up an old and already well-answered question, to offer another point of view,

The Artin's conjecture appears very naturally in the context of Chebotarev's density theorem.
In fact, we can see Cheobtarev's contribution as a clever trick to circumvent the Artin conjecture by reducing the proof to cases where it is known (by works of Dirichlet and Hecke).
But the proof of Chebotarev will be much simpler and more natural if we had the Artin's conjecture, which moreover would give better results as far as the error term is concerned.
This is, I think, a good justification of the importance of Artin's conjecture.

To explain the role of Artin's conjecture, let us also assume for simplicity GRH.
Then, for $G=Gal(K/mathbb Q)$ a finite Galois group, and $rho$ an irreducible Artin representation of $G$, the $L$-function $L(rho,s)$ has no zero on Re $s>1/2$ (by GRH)
and no pole either (by Artin), except for a simple pole at $s=1$ if $rho$ is trivial.
Thus the logarithmic derivative, $L'/L(rho,s)$ has no pole on Re $s>1/2$ (except perhaps...): this illustrates clearly the symmetric and complementary role played by Artin and Riemann's conjectures; both poles and zeros of $L$ contribute to simple poles of $L'/L$, and Artin eliminates some of them, Riemann the others. Now standard techniques of analytic number theory allows us, by integrating $L'/L$ on a vertical line $2+i mathbb R$ and moving it near the critical line, to $1/2+epsilon + i mathbb R$, to get an estimate of the quantity:
$$pi(rho,x) = sum_{p^n < x} log(p) tr rho(frob_p)^n$$
where the sum is on prime power less than $x$.
This estimate is $O(x^{1/2+epsilon})$ if $rho$ is non trivial, and $x + O(x^{1/2+epsilon})$ if $rho$ is trivial, because of the pole at $s=1$.

Now let $C$ be subset of $G$ stable by conjugaison, $1_C$ its characteristic function.
Since $1_C$ is a central function, it is a linear combinaison of character of irreducible representation of $G$, say
$$1_C = sum_rho a_rho tr rho.$$
Hence
$pi(C,x) := sum_{p^n< x} log p 1_C(frob_p) = sum_rho a_rho pi_rho(x)$.
Since $a_1$ is easily computed as $|C|/|G|$, we get $pi(C,x)=|C|x/|G| + O(x^{1/2+epsilon})$, which is up to standard manipulation Chebotarev's density theorem.

Hence we can say that Artin's conjecture play to Chebotarev's density theorem a role analog to the role played by the standard conjectures for the Weil's conjecture proved by Deligne. In both cases, a clever and beautiful trick was used (by Chebotarev and Deligne, respectively) to prove a theorem
(Chebotarev's theorem, aka Frobenius' conjecture, and Deligne's theorem, aka the last Weil's conjecture) without proving the conjectural statement that makes the theorem limpid (Artin's conjecture, resp. standard conjectures). This is great, but doesn't make the conjectures any less interesting.

Important addendum A friend of mine made me notice something that kind of weakens significantly the point I was trying to make above. Indeed, for the argument outlined above, one doesn't need GRH + Artin's conjecture: GRH is enough. Or, more precisely,
the part of Artin's conjecture that is needed is already known under GRH. Indeed, it is clear that in the argument the absence of poles of $L(rho,s)$ is only used in the region Re $s>1/2$. But by Brauer's theorem, we know that $L(rho,s)$ is meromorphic on $mathbb C$,
and by elementary reasoning that $prod_rho L(rho,s) = zeta_K(s)$, where $rho$ runs amongst Artin's representations of Gal$(K/mathbb Q)$. Moreover, by Hecke $zeta_K$ has no poles (except a simple one at $s=1$). Hence if all the $L(rho,s)$ have no zero on any given region, then they don't have a polo either on that region -- expected for $L(1,s)$ with its simple pole at $s=1$.

I leave the argument above, because it shows that, the absence of poles for $L(rho,s)$ is important for questions on the distribution of primes, even if it is a theorem rather than a conjecture. Moreover, this argument has an historical interest, as Artin's conjecture was made in the 1920's, and Brauer's theorem proved in 1946.

I ignore if the question of absence of poles on the critical line for the $L(rho,s)$ (the only part of Artin's conjecture that is still open) has any direct application on the
distribution of primes.

computability theory - Is the solution bounded Diophantine problem NP-complete?

A particular quadratic Diophantine equation is NP-complete.

$R(a,b,c) Leftrightarrow exists X exists Y :aX^2 + bY - c = 0$

is NP-complete. ($a$, $b$, and $c$ are given in their binary representations. $a$, $b$, $c$, $X$, and $Y$ are positive integers).

Note that there are trivial bounds on the sizes of $X$ and $Y$ in terms of $a$, $b$, and $c$.

Kenneth L. Manders, Leonard M. Adleman: NP-Complete Decision Problems for Quadratic Polynomials. STOC 1976: 23-29

nt.number theory - Analog to the Chinese Remainder Theorem in groups other than Z_n.

I did a course titled something similar in my undergraduate, and while it didn't teach the following applications, H. Cohen's A Course in Computational Algebraic Number Theory (which I read right after the course, and you should too) does.

As you mention, one can use the Pollard rho algorithm to find a factor $p$ of $N$, in time $O(sqrt p)$. There are two other basic algorithms that use CRT implicitly, both in Cohen's book:

1) Pollard's $p-1$ (and its generalizations, such as Williams' $p+1$): Compute $gcd(a^{n!}-1, N)$. If $p-1 | n!$, then the gcd will be divisible by $p$, and one can factor. This uses CRT implicitly in the following way: we can compute the $gcd(a^{n!}-1,N)$ using only mod-$N$ operations - but we find $p$ because of the existence of CRT. If we accidently get $N$ as the gcd, we can still factor using another application of CRT. Read the above referenced book for details.

2) The much faster Elliptic Curve Method: Initialize an "elliptic curve" $E$ mod $N$ and a point $P$ on it. Compute $(n!)P$. I write "elliptic curve" because we aren't really defining an elliptic curve - $mathbb{Z}/(N)$ is not a field! But, using CRT, we treat it as the combination fields. We hope that the order of $P$ on $E/mathbb{F}_p$ divides $n!$, and mod any other prime dividing $N$, the order does not divide $n!$. In this case $(n!)P$ will "have" $p$ in its denominator, but not the other primes, allowing us to recover $p$. This, again, is using CRT much in the same way that Pollard's Rho does. We compute things only mod $N$ - but we get things that are structurely inherit, such as $p$.

3) A bit of a different kind of computational application of CRT is D. J. Bernstein's "Doubly focused enumeration of locally square polynomial values." (Pages 69--76 in High primes and misdemeanours: lectures in honour of the 60th birthday of Hugh Cowie Williams, edited by Alf van der Poorten, Andreas Stein. Fields Institute Communications 41, American Mathematical Society, 2004. ISBN 0-8218-3353-7).

The author uses CRT explicitly in order to enumerate over numbers satisfying certain congruence properties. It is not cryptographic, but computationally interesting and simple to understand, not to mention record braking.

fourier analysis - Basics of Fast Discrete Sine Transform

I just got started with DCT/DST but I still fail to understand how the fast DST is supposed to work. The idea behind the FFT is rather apparent and there is very intuitive pseudo-code of it on the Wikipedia article on the Cooley-Tukey algorithm.

The DST-I is defined as
$$a_N(k) = sum_{n=0}^{N-1}x_n sinfrac{pi(n+1)(k+1)}{N+1}$$
which can be split into
$a_N(k) = b_{Nover 2}(k)+a_{Nover 2}(k)$ and $a_N(N-k-1)=b_{Nover 2}(k)-a_{Nover 2}(k)$, thus exploiting the symmetry in a similar fashion as for the DFT in case of Cooley-Tukey, with
$$a_{Nover 2}(k)=sum_{n=0}^{{Nover 2}-1} x_{2n+1} sinfrac{pi(n+1)(k+1)}{N+1},$$
$$b_{Nover 2}(k)=sum_{n=0}^{{Nover 2}-1} x_{2n} sinfrac{2pi(2n+1)(k+1)}{N+1}$$
as explained in e.g. in Britanak, Yip, Rao. Discrete Cosine and Sine Transforms: General Properties, Fast Algorithms and Integer Approximations.

However, I don't get it to describe this as a simple, recursive algorithm as my twiddle factors appear to be wrong. My code (prototyping in python...) looks basically like below, however it yields to the wrong result for any N>2 (yes, N=1 is trivial but apparently it is correct for N=2, so I can't be totally off course?). What am I missing? Or is the problem using this approach that $b_{Nover 2}(k)$ actually is the DST-II, thus this cannot be computed this way at all (i.e. need to compute DST-I of odd parts and DST-II of even parts, recursively?). While it is kind of fun figuring out the solution on my own, any hints are greatly appreciated - this has to be described somewhere this simple, or hasn't it?

def We(N,k):
    return math.sin(math.pi*2*(k+1)/(N+1))

def Wo(N,k):
    return math.sin(math.pi*(k+1)/(N+1))

def dst_fast(x_in,N):
    x = x_in[:]
    y = [0]*N
    if N == 1:
        return x
    even = dst_fast(x[::2],N//2)
    odd = dst_fast(x[1::2],N//2)
    for k in range(N//2):
        e = We(N,k) * even[k]
        o = Wo(N,k) * odd[k]
        y[k] = e + o
        y[N-1-k] = e - o 
    return y

gt.geometric topology - Can you explain a step in a proof about 2-sided surfaces in 3-manifolds?

Following is an argument given by Hempel where I am unable to understand his comment about choosing a loop close enough to a surface. Can somebody please elucidate this:

Lemma: If $F$ is a compact connected surface properly embedded in a $3$-manifold $M$ and if $image(i_*:H_1(F;Z/2Z)rightarrow{H_1(M;Z/2Z)})=0$, then $F$ is 2-sided in $M$.

Proof: By regular neighborhood theory, it suffices to show that $F$ separates some connected neighborhood of $F$. If this is not the case then there is a loop $Jsubset{M}$ such that $Jcap{F}$ is a single point, with transverse intersection. We may choose $J$ close enough to $F$ so that $J$ is homologous to zero (mod 2) in $M$. This contradicts homological invariance (mod 2) of intersection numbers. QED.

Doubt: Everything else is clear to me except the bold part in the proof. I don't think we will be able to bring $J$ close to $F$ unless it already bounds a disc. So I can see the proof only in the simply connected $M$ case where after suitably perturbing the loop and using the loop theorem to bound, I can apply an ambient isotopy to bring the disc itself closer to $F$. But how do we see this for the non-simply connected $M$? A complicated loop in $M$ might go all around it.

Reference: Lemma 2.1, Chapter 2 (Heegard Splittings), 3-manifolds(book) by Hempel.

ag.algebraic geometry - Deformations of Tame Coverings

To say that I am a novice at deformation theory is to grossly overestimate my abilities in this area. I've come across the following theorem in a paper, and I'd like to know how far one is able to generalize it:

Theorem

Let $R$ be a complete DVR, $X$ a proper smooth curve over $R$, and $D$ a simple divisor on $X$. Let $bar f:bar Yrightarrow bar X$ be a tame covering of $bar X$ with branch locus contained in Supp($bar D$). Then there exists a unique lifting of $bar f$ to a D-tame covering $f:Yrightarrow X$. In addition, $f$ is a tame covering of $X$, and the branch locus of $f$ is contained in Supp($D$).

Notation

I'm not really sure how much of it is standard, so here's what I mean by the terms in the theorem:

A simple divisor on $X$ - a divisor that has no multiple components when base-changed to any geometric point of the base scheme.

Tame covering of integral varieties over a field ($f:bar Yrightarrow bar X$ in our theorem) - what you think: the ramification indices in codim 1 are coprime to the characteristic.

D-tame covering of schemes over a DVR ($f:Yrightarrow X$ in the theorem) - if for every (natural number) $k$, base change to $R/m^k$ is $Dtimes_{R}R/m^k$ -tame in the following sense:

D-tame covering of schemes over an Artin local ring, $Lambda$ ($R/m^k$ in the previous notation) - a finite, flat morphism which is etale away from Supp($D$)., and such that if $x$ is in Supp($D$), and t is a local equation for D at $x$, then $f_ast(O_Y)_x$ is a t-tame extension of $O_x(X)$. What's that? $A'$ in the following, plays the role of $O_x(X)$ here.

t-tame extensions of a ring $A'$ s.t. $A'/Nil(A')$ is a DVR with parameter $check t$ (and assume (0) is primary in $Spec(A')$, meaning $Spec(A')$ has no imbedded components)- tame when base changed to $A'/tA'$ (where $t$ is some element going to $check t$). $A'/tA'$ is an Artin ring. So what do I mean by tameness of extensions over Artin rings? Well:

A tame extension of an Artin ring $Lambda$ - it is a (finite) product of $Gamma_i$'s, such that for each $Gamma_i$ it is free over $Lambda$ of rank $e_if_i$; the extension of residue fields is $f_i$; $e_i$ is prime to the characterstic of the residue field of $Lambda$; and $Gamma_i$ contains a principal ideal $J$ such that $J^{e_i}$=0, and such that $Gamma_i/J$ is free over $Lambda$ of rank $f$.

Question

Phew, that was a lot of work... So - other than finding the whole thing very confusing, here is a concrete set of questions:

1. Can this be generalized to $X$ a higher dimensional scheme?

2. What would be a good reference to read for this type of deformation theory?

3. Well - just about any insight and context you can give would be great (whether deformation theoretic or about the various generalizations of tameness here.)

This is a bit long winded, but if nothing else - it was good exercise for me to write this.

cv.complex variables - what is the formal definition of multi-valued holomorphic function?

It seems this notion causes some confusion not only on me.
Let me give my own definition, by taking into account with yours.

Let $X$ be an analytic manifold, and let $pi:tilde{X}mapsto X$ be some universal covering of $X$.　A multivalued holomorphic function on $X$ is a holomorphic function on $U$, where $U$ is some contractible open subset of $X$, if the function $picirc f$ on some component $mathcal{C}$ of $pi^{-1}(U)$ can be analytically extended to $tilde{X}$. Denote this function by $tilde{f} _mathcal{C}$

Two multivalued holomorphic functions $(f,U)$ and $(g,V)$ are equivalent, if there exists some component $mathcal{C}$ of $pi^{-1}(U)$ and $mathcal{D}$ of $pi^{-1}(V)$, such that $tilde{f}_mathcal{C} = tilde{g}_mathcal{D}$.

This definition doesn't depend on the choice of universal covering.

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Edit:

Here is a more rigorous definition：

Definition. A multivalued holomorphic function on $X$ based at $x$ is a function $f$ in $mathcal{O}_x$, such that one component of $pi^* f$ can be analytically extended to $tilde{X}$, where $pi: tilde{X}mapsto X$ is some universal covering.

Denote the ring of multivalued holomorphic functions on $X$ based at $x$ as $tilde{mathcal{O}} _x$, which is a subring of $mathcal{O}_x$.
Any loop based at $x$ gives an action on $tilde{mathcal{O}} _x$, which is essentially the monodromy.

Claim. $tilde{mathcal{O}} _x$ is locally free $mathcal{O}(X)$-module with rank the order of fundemental group.

It is essentially equivalent to the defintion that a mutilvalued fundtion is defined as function on covering space. But i feel it is more intuitive and doesen't depend on the choice of covering space.

Under this definition, I can make sense of the solution of mulivalued function of some analytic differential equation, which is actually my original motivation to ask this question.

nt.number theory - Everywhere locally isomorphic abelian varieties

Here's a slight variant of Felipe Voloch's answer, for those who don't have a favorite group cohomology class. Let $C$ be an abelian variety over $mathbb{Q}$. Suppose that all the $overline{mathbb{Q}}$ automorphisms of $C$ are defined over $mathbb{Q}$ and let $P$ be this automorphism group.

Take two classes in $H^1(mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q}), P)$ which are distinct, but become equal in $H^1(mathrm{Gal}(overline{mathbb{Q}_v}/mathbb{Q}_v), P)$ for every $v$. The corresponding twists of $C$ should give you the examples you want.

How have I made things easier? Because I made the action of $mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q})$ on $P$ trivial, I can describe the group cohmology explicitly as
$$H^1(mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q}), P) cong mathrm{Hom}(mathrm{Gal}(overline{mathbb{Q}}/mathbb{Q}), P)/P.$$
Here $P$ acts by conjugation on the target.

Since $P$ is finite, any of these Hom's factor through $mathrm{Gal}(K/mathbb{Q})$ for some finite extension $K/mathbb{Q}$.

So we are now reduced to the following: We must find finite groups $G$ and $P$, an extension $K/mathbb{Q}$ with Galois group $G$, an abelian variety with automorphism group $P$ and two maps $alpha$, $beta: G to P$ such that

• $alpha$ and $beta$ are not conjugated to each other by any element of $P$ but


• when we restrict to any decomposition subgroup group, $alpha$ and $beta$ become conjugate.

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Take $G=(mathbb{Z}/2)^2$ and $P=S_6 times (mathbb{Z}/2)$. We will not use the $(mathbb{Z}/2)$ factor at all in the following; the reason it is there is that the automorphism group of an abeliabn variety always contains a central involution, namely $-1$. Feel free to think of $P$ as $S_6$.

Take $K/mathbb{Q}$ to be any biquadratic extension in which no prime is completely ramified. This condition assures that no decomposition group is the whole of $G$. Let $alpha$ send the generators of $G$ to the elements $(12)(56)$ and $(34)(56)$ of $S_6$. Let $beta$ send the generators of $G$ to $(12)(34)$ and $(13)(24)$. Then $alpha$ and $beta$ are not conjugate in $S_6$, but they become conjugate when restricted to any of the three cyclic subgroups.

The one missing step is to construct an abelian variety with automorphism group $S_6 times (mathbb{Z}/2)$, and all automorphisms defined over $mathbb{Q}$. Dror Spieser, in the comments, points out that we can just take the restriction of scalars of an elliptic curve (without CM) defined over an $S_6$ extension of $mathbb{Q}$. I still don't have a good construction of this but, thanks to Bjorn's answer, I don't need one.

pr.probability - method of moments and Laplace transform from Shepp and Lloyd

Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful:

In equation (15), they claimed it is straightforward that if there is an $F_r$ such that

$$int_0^1 exp(-y/xi) dF_r(xi) = int_y^{infty} frac{E(x)^{r-1}}{(r-1)!} frac{exp(-E(x) -x)}{x} dx $$

then $F_r$ will have moments $G_{r,m}$.

Here

$$G_{r,m} = int_0^{infty} frac{x^{m-1}}{m!} frac{E(x)^{r-1}}{(r-1)!} exp(-E(x)-x) dx$$

and

$$E(x) = int_x^{infty} frac{e^{-y}}{y} dy$$

which is related to the thread Reference request for a "well-known identity" in a paper of Shepp and Lloyd

It looks to me like some sort of Laplace transform, but I can't manage to get the algebra to work, because of the inverse exponent $y/xi$ with respect to $xi$.

I will be happy enough if one can tell me why we are looking at the transform $int_0^1 exp(-y/xi) dF_r(xi)$ instead of the usual moment generating function $int_0^1 exp(-y xi) dF_r(xi)$, or maybe it's a typo?

ag.algebraic geometry - An optimization problem for points on the sphere (master's dissertation)

First, by means of a disclaimer, some background. I am entering the fourth and final year of an undergraduate master's degree in maths, and a quarter of the maximum credit for this year will be for a project on the subject of my choice, which I will describe below. In this project, I am allowed to cite and quote the work of others to any extent so long as it is correctly referenced. My current understanding is that final-year projects in pure maths tend to include little or none of the author's original research, but rather take the form of a summary with perhaps some relatively minor original investigation. I have checked that asking about my chosen subject on this website is within regulations, so long as I cite the thread. Assuming it's alright, please bear in mind that I might quote you unless you ask not to be quoted.

Moving on to the subject of the project, I am interested in "sphere-like" polyhedra. More specifically: given a polyhedron with a fixed number n of sides, I would like to minimize its surface area relative to its volume. Some preliminary research turned up Michael Goldberg's paper "The Isoperimetric Problem For Polyhedra", from Tohoku Mathematical Journal vol. 40 (1934), which can be found at http://staff.aist.go.jp/d.g.fedorov/Tohoku_Math_J_1934_40_226-236.pdf

This is the only paper in English that I was able to find directly addressing the problem. It cites a number of previous non-English papers, which it summarizes as not having established very much, the main result being that a solution polyhedron can be considered as a set of n points on the unit sphere, representing the centres of mass of the polyhedron's faces which are tangent to the sphere at this point. Then, if we define sphericity as inverse to the surface area of such a polyhedron circumscribed about a unit sphere, we are looking to maximize the sphericity given n.

Goldberg also makes some further progress on the problem, and conjectures that the solutions fall into a particular class of polyhedra, which he terms "medial". He published a later paper on the subject of medial polyhedra, which can be found at http://staff.aist.go.jp/d.g.fedorov/Tohoku_Math_J_1936_43_104-108.pdf

As an undergraduate making his first foray into mathematical research, I have little intuition for the directions I should be going in (and hence, apologies if any of the tags I've added are inappropriate); but I have considered looking at what happens if I add the additional constraint that the polyhedron should have a nontrivial symmetry group, as the few known results do (tetrahedron, triangular prism, cube, pentagonal prism, dodecahedron). To this end I thought using the stereographic projection, fixing one of the n points to be at infinity, might make it easier to study the symmetry and perhaps even view the points as roots of a polynomial. Then I may be able to provide new proofs for the known solutions, which might generalize so that I can investigate unknown solutions for small values of n, at least those which have nontrivial symmetry.

In the other direction, I am interested in the asymptotic behaviour of the solution set. Goldberg places some bounds on its behaviour in his first paper; I've been wondering about possible bounds on the minimum or maximum area of a single face in terms of n, or on the maximum number of edges it can have (I've got a gut feeling that most of the faces are going to be hexagons for large enough n). An important question is of course how sphericity behaves asymptotically with respect to n. While I expect the solution set will eventually become quite irregular, is it possible to find a well-behaved family of polyhedra for which sphericity behaves similarly, if not quite as well?

Finally, how if at all does this problem relate to other optimization problems for points on the sphere? The known solutions at least coincide, for example, with those for minimizing the energy of n point charges on a sphere (the Thomson problem), though I suspect the coincidence only occurs because n is small.

In summary, my question is this: are there any English-language publications, or translations, on this subject of which I am currently unaware? Has any further progress been made? If not, what avenues sound like they may be worth exploring? If anyone is interested in pursuing the problem themselves and would be willing to give me permission to cite their work I would of course be delighted.

Many thanks,

Robin

gn.general topology - A question about totally disconnected point sets.

Let $E_c$ be the complete Erdös space (Erdös, Annals of Math vol 41 1940), defined as the subspace of $ell^2(mathbb{N})$ where all coordinates are irrationals. It is polish (separable and completely metrizable) and totally disconnected, but admits a "connectification" namely a (still polish) topology on $E_ccup{p}$ that makes it connected (and of course induces the one on $E_c$). The crucial point is the fact that any nonempty closed and open subset of $E_c$ is unbounded. Then, as in Bill Johnson's answer $E_c$ is the union of the closed and totally disconnected subspaces $overline{B}(0,n)cup{p}$, $ngeq 1$. It remains to remark that, like any polish space, $E_ccup{p}$ embeds as a closed subset of $H$ (as remarked in Gerald Edgar's answer).

co.combinatorics - What are the Applications of Hypergraphs

I've done some work which made me appreciate the view that labelled hypergraphs are one of the most widely appropriate, general ways to represent data on stateful machines. In computer science, we commonly want to divide up state into a number of possibly overlapping data structures, which will contain and be referenced by pointers.

This lends itself to the following representation: data structures are hyperedges. Non-pointer data within data structures are labels of the associated hyperedge. And pointers are represented by vertices, possibly (not always!) needing an attribute to indicate which hyperedge is the source and which is the target of the pointer. Computation, then, is graph rewriting.

As Qiaochu says, hyperedges are absurdly general. Likewise, the notion of data. To make this useful, one needs to constrain the form the hyperedges take. What is nice is that the need to constrain the way that state is represented is perfectly matched with the need in useful programming langauges to contrain the representation of data, and furthermore, one can often cleanly map the programming-driven constraints into reasonable constraints on the hypergraphs.

The idea crops up again and again the literature on graph transformations. A good stepping off point is Drewes, Habel, & Kreowski, 1997, Hyperedge Replacement Graph Grammars, In Rozenberg, Handbook of Graph Grammars and Computing by Graph Transformations.

examples - Applications of the Chinese remainder theorem

The Mayan calendar system uses a number of different periodic processes, and provides a simple but very important example of a practical use of the CRT.

The Tzolkin, or Day Count, has twenty weekdays (Ik, Akbal,... Auau) and thirteen numbers, 1-13. Each day, the day name advances, and so does the number. For example, 7 Ik is followed by 8 Akbal. These name/number pairs repeat in a 260 day cycle, which has been in continuous uniterrupted use since at least 600BC.

The Haab, or Vague Year, is a 365 day year consisting of 19 months (Pop, Uo, ..., Cumku, Uayeb). The first 18 months have 20 days and Uayeb has five days. The Haab runs 0 Pop, 1 Pop, ..., 19 Pop, 0 Uo, 1 Uo,..., 4 Uayeb and then repeats to 0 Pop.

Together, the Tzolkin and Haab form the calendar round, with dates given by Tzolkin then Haab, for example, 7 Ik 0 Cumku. This cycle repeats every 18980 days, about 52 years, which means that a calendar round date is good for most practical purposes (such as birth dates).

The earlier Mayan period, from around the 1st century BC to the 13th century AD, also featured a system known as the long count, recently made somewhat famous by the fact that it finished a 5126 year cycle on December 21, 2012.
Long count dates have Kin (days) which run 0-19. 20 Kin make one Uinal, 18 Uinal make one Tun, 20 Tun make one Katun, and 20 Katun make one Baktun. Dates are written with Baktun first, as, for example, 9.7.17.12.14. After 13 Baktun, the date goes back to zero, so that 12.19.19.17.19 was followed by 0.0.0.0.0.

One major problem with studying the Mayan calendar is that the long count dates fell out of use hundreds of years before the Spanish arrived, and it is nontrivial to decide which Mayan long count dates correspond to which dates in the modern western calendar system - the 'correlation problem'.

The key document, the Chronicle of Oxcutzcab, says that a tun ended 13 Ahua 8 Xul in AD 1539, thus tying together the long count (tun ending), the calendar round, and the Julian calendar. From ancient records, the long count is known to have begun on 4 Ahua, 8 Cumku. So, given 0.0.0.0.0 = 4 Ahua 8 Cumku, one needs to solve $x$.0.0 = 13 Ahua 8 Xul. The day number gives the equation $360 x equiv 9 pmod{13}$. Since 8 Xul is 125 days after 8 Cumku, the Haab gives the second equation $360 x equiv 125 pmod{365}$.

So, there's a simple little use of CRT: Solve for $x$, and find $x equiv 924 pmod{949}$.

To finish the story, the year AD 1539 contains the long count tun 924 = 2.6.4.0.0 plus some multiple of 949 tun = 2.7.9.0.0. There is enough historical evidence to guess the date to within 949 tun (about 935 years), and so one learns that 11.16.0.0.0 is in AD 1539. Finally, the calendar round is still in use and so one can determine that 11.16.0.0.0 is November 12, 1539. I'll leave it as an exercise to determine that December 21, 2012 really was 0.0.0.0.0.

rt.representation theory - Why should we consider D-module on flag variety of Lie algebra?

Why don't we stay at D-module on base affine space but go to study flag variety of Lie algebra?

I remembered there are nice papers of Bernstein-Gelfand-Gelfand and Gelfand-Kirillov discussing the relationship of representation of Lie algebra and correspondence differential operators on base affine space.

What about Grassmannian? Can we consider the D-module on Grassmannian?

books - Text for an introductory Real Analysis course.

I'm not a fan of the Pfaffenberger text. For example, look at the proof of the chain rule. The proof sticks to the "derivative as slope" idea, and so has to consider the special case where one derivative is zero. This isn't very elegant, and causes confusion in what should be a straightforward proof -- IMO when students are first being exposed to something as elementary as analysis, simplicity should be an overriding concern.

Apostol, Buck and Bartle, those are texts that I like pretty well. Or the lecture notes used at the University of Alberta for their honours calculus sequence Math 117, 118, 217, 317 (available on-line) -- pretty well based on Apostol.

There's a few subtle issues going on here. Some departments view analysis as something people learn after they go through a service-level calculus sequence. Some departments treat calculus as part of an analysis sequence -- ie students only see calculus through the eyes of analysis. What book you choose is largely determined by what path your department is comfortable with.

lo.logic - intuitionistic interpretation of classical logic

There exists a translation, T, applicable to both propositions and proofs with the property that:

If $P$ is a classical proof of $phi$ then $P^T$ is an intuitionistic proof of $phi^T$.

(Trivially, any intuitionistic proof is also a perfectly good classical one.)

The details of some versions of the translation of the proof can be found in computer science texts under the name "CPS translation", though with different notation. Surprisingly this translation has an alter-ego as one of the stages when compiling certain programming languages.

Update: Adding (1) more detail and (2) something on the CPS translation.

(1) These translations don't allow you to completely remove double-negation elimination from classical proofs. But they do allow you to pull all of the double negation eliminations all the way out of the body of the proof to the very last line. So starting with a classical proof of $phi$ we can translate it into an intuitionistic proof of $negnegphi$ and then we can tack one double negation elimination step onto the end to turn this back into a classical proof of $phi$.

(2) According to the Curry-Howard isomorphism, a proof $P$ of a theorem $phi$ can be interpreted as a program $P$ that produces a result of type $phi$. When we convert a program to "continuation passing style", instead of just accepting a result of type $phi$ back from our program, we instead write a program that accepts an extra argument (known as a continuation) that tells the program what it should do with its result. Ie. instead of writing a program of type $phi$ we write a program of type $(phirightarrow k)rightarrow k$. The continuation is the extra argument of type $phirightarrow k$ and we now get a final result of type $k$. The CPS translation takes an already existing program of type $phi$ and converts it to one of type $(phirightarrow k)rightarrow k$. It's a bit fiddly but you can get a translation by following your nose, so to speak. Every function that your program calls also has to be modified so that it too uses the continuation and you thread the continuation all the way through the code.

But by the Curry-Howard isomorphism, this means you're translating a proof of the proposition $phi$ into a proof of $(phirightarrow k)rightarrow k$. This is basically a version of the Godel-Gentzen translation as described in Constructivism in Mathematics: an Introduction Studies in Logic and the Foundations of Mathematics (by A. S. Troelstra and D. van Dalen.) A nice result of this is that you can now interpret classical proofs as computer programs.

I think it is a pretty amazing fact that you can take some esoteric mathematics comparing two systems of logic and turn it directly into code that can be used in a compiler.

linear algebra - Sum of two unitary matrix is equal to every matrix?

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sg.symplectic geometry - Classification of symplectic surfaces

This is not a complete answer to the question, and I don't know if a complete answer is written down anywhere in the literature. In the first revison of the answer I tried to adress all the 10 comments that my previous answer recieved. The second revison contains a conjecture (that I am 99% sure of) describing the complete answer to this question.

The first point is that the classification of symplectic surfaces can not be simpler than the classification of surfaces up to a diffeo. And the classification up to a diffeo of non-compact surfaces is quite a delicate subject. It is desribed for example in the paper http://www.jstor.org/pss/1993768, 1963, ON THE CLASSIFICATION OF NONCOMPACT SURFACES, IAN RICHARDS. In particular one phenomena apears here -- a certain ideal boundry of the surface. This idel boundary is a totally disconnected, compact separable space. I guess a good illustration will be a disk from which we throw away a Cantor set on the x axis, Cantor set been the ideal boundary.

Let us give now some examples that illustrate the additional phenomena that happen for non-compact surfaces, if we take in account the symplectic form. First of all there is the simplest case when the surface has finite topological type. In this case we have two topological invariants, the fundamental group, a free group on $n$ generators plus the number of punctures (or boundary components) $m$. In this case a complete classification of symplectic forms can be given. Either the surface has a bouned are $A$, in this case this area is the only invariant. Or it has an infinite area. In this case there are $m$ types of surfaces. Namely for every boundary (or puncture) we can check if it has on open neighborhood that his finite area, or not. The number of components near which the area is unbounded can be any between $1$ and $m$.

Example. Consider $S^2$ with two delited points. Then either it has finite area, or it is symplectomorphic to an infinite cylinder $S^1times R^1$ with the form $ds wedge dt$, or it is symplectomorphic to $R^2setminus 0$, with the form $dxwedge dy$.

If the number of punctures is countable, and every puncture has a neighborhood that is diffeomorphic to a punctured disk, then the situation should be very similar to what I have described above. Namely 1) the area can be bounded. 2) The number of UNBOUNDED punctures for wich every neighborhood has infinite area is bounded, in such surfaces are enumerated by natural numbers. 3) The number of unbounded punctures is infinite, in this case we just need to cound the number of bounded punctures, that can be finite of inifinite.

CONJECTURE. Here is the conjecture telling what should be the complete answer to the question.

Take a non-compact surface. Then the set of symplectic structures of infinite area on it is in one to one correspondence with closed non-empty subsets of its ideal boundary.
For every bounded A>0 there is a unique symplectic strucutre on the surface with given area.

In the case of a surface with puncutres, the ideal boundary is just the union of punctures. Below the statement of this conjecture is proven for some simplest examples of surfaces. I think, that the general case should not be much different.

For the simple examples that I have descirbed the proof should be ALMOST identical to Moser's argument. Indeed the simplest examples have the following property:
these surfaces can be decomposed in a countable union of compact pieces, all of which apart from one piece are annuli, and one piece is a compact surface with a boundary. Now, on a compact surface with a boundary (as well as on the cylinder) the symplectic form is uniquely defined by its area -- this can be done by Moser argument (we need that in a neighbrohood of each boundary the symplectic form is strandard, which is automatic in our case). Now the symplectomorphism can be constructed inductively.

Consider for example the case of $R^2$ of infinite area. We can take any exostion of $R^2$ by cylinders. For example we indroduce some coordinates on $R^2$, and conisder cylinders of radiuce $n,n+1$. We don't care what is the exact expression of w. The crucial thing for us is that the sum of areas tends to infinity. Now we take the standard $R^2$ and take a decompositon in concetnric cylinders of needed area. Then define the symplectomorphism from the standrd $R^2$ cylinder by cylinder. For the class of surfaces, that I dealed with same thing should work. Though I am not sure that this is the best "proof" of the statement.

I think it is not hard to prove this conjecture as well in the case of the complement to a Cantor set in the unit disk.