complex geometry - How can I see the projection $pi:H^1(X,mathcal{O}_X)rightarrow Pic^{0}(X)$ in terms of holomorphic structures on $Xtimesmathbb{C}$?

Hi, as the title says I'm looking for a way to see the projection $pi:H^1(X,mathcal{O}_X)rightarrow operatorname{Pic}^{0}(X)$ in terms of holomorphic structures on $Xtimesmathbb{C}$. ($X$ is a compact complex manifold and $operatorname{Pic}^0(X)$ is the kernel of $c_1:H^1(X,mathcal{O}_X^{ * })rightarrow H^2(X,mathbb{Z})$ in the long exact sequence induced by the exponential sequence). Since
$$H^1(X,mathcal{O}_X) simeq H_{overline{partial}}^{0,1}$$
by the Dolbeault isomorphism, I take
$[c]in H^1(X,mathcal{O}_X)$,
so I take the corresponding $[gamma]in H_{overline{partial}}^{0,1}$ and a representative $gamma$ of the class $[gamma]$. My wrong thought was that $pi([c])=(Xtimesmathbb{C},overline{partial}+gamma)$, i.e. to associate to $[c]$ the trivial bundle with the "holomorphic structure" $overline{partial}+gamma$, but it is not a holomorphic structure unless $gammawedgegamma=0$! So how can I see explicitly (if it is possible) the map $pi$ in terms of holomorphic structures on the trivial line bundle?

Thank you in advance.

Set theory and alternative foundations

This was going to be a comment to Joel David Hamkins's answer on geometry, but it didn't fit.

+1 This is one of the most clear-minded things I have read on MO. It does not make a mockery of Foundations and still says something non-obvious.

I'm very skeptical of all this business with category theory being a foundation for mathematics. First, whenever anyone talks about it, it always seems to be somebody else's work. It's become something of a meme that "Bill Lawvere has proposals to provide foundations for math through category theory", but we don't ever see details provided.

Second, are we really to understand that we are going to add small integers with arrows and diagrams? Draw circles and lines, and say that the latter meet at most once? I think people work in trans-Euclidean hyperschemes of infinite type so much, they forget that math includes these things.

As Wittgenstein remarks in the Investigations, just because you can express A in terms of B, it does not mean that B actually underlies A.

ag.algebraic geometry - Cohomology of Zariski neighborhoods

Do there exist smooth compact (=complete) connected complex algebraic varieties $Xsubset Y$ and a Zariski neighborhood $U$ of $X$ in $Y$ such that the image of $H^{ast}(U,mathbf{Z})$ in $H^{ast}(X,mathbf{Z})$ under the restriction map is different from the image of $H^*(Y,mathbf{Z})$?

Remark: if one considers the rational cohomology instead of the integral one, the answer is no by Hodge theory: if a class on $X$ comes from a class on $U$, it is the restriction of a class of the "right" weight and hence it extends from $U$ to any smooth compactification, in particular to $Y$. In this argument it is important that cohomology maps induced by regular maps of algebraic varieties are strictly compatible with the weight filtrations: if a weight $k$ class in the target is in the image, then it is the image of a weight $k$ class.

The motivation behind the above question is to understand whether or not the same holds for the integral weight filtrations, which can be defined as the Leray filtrations induced by the open embedding in a compactification as the complement of a divisor with normal crossings, see Weight filtration over the integers.

The above is in a sense the simplest possible situation when strictness may fail. I.e., if the answer to the above question is positive, this would imply that the integral cohomology mappings are not necessarily strictly compatible with the weight filtrations. If the answer is negative, then I would be very interested to know the answer to the same question with "algebraic" replaced with "complex analytic" and "Zariski neighborhood" replaced with "an open set whose complement can be blown up to a divisor with normal crossings".

ag.algebraic geometry - Why a subvariety of a variety of general type is of general type

You need to be careful about what you mean by "a general point". Usually, this means "a point in a certain Zariski open set". So in particular, this statement would say that on a surface of general type all rational and elliptic curves lie in a Zariski closed subset. This is Lang's conjecture, still open I believe (proofs were suggested about 15 yrs ago but then withdrawn).

EDIT: OK, so from the comments and the other answer it appears that it should be a "very general point of $X$", and the statement reduces to showing that if you have a morphism $f:Yto T$ with irreducible $T$ and general fiber $Y_t$, and a finite dominant morphism $pi:Yto X$, then $Y_t$ is also of general type.

The basic reason for that is very simple: if $X$ is of general type then it has lots of pluricanonical forms. You can pull them back to $Y$ (they are differential forms, after all) and get lots of pluricanonical forms on $Y$. Then you can restrict them to $Y_t$ and get lots of pluricaninical forms on $Y_t$.

For a more precise answer, I suggest you look at old papers by Kawamata, Viehweg and Kollar, search for "additivity of Kodaira dimension". There is a whole sequence of $C_{n,m}$ conjectures about the Kodaira dimension of a fibration $Y$ in terms of Kodaira dimensions of $T$ and $Y_t$. Some of them are proved, some are still open.

(Note: general type means "maximal Kodaira dimension", i.e. equal to the dimension of the variety.)

gr.group theory - Can all terms of the Johnson filtration be hom-mapped onto the same nontrival group?

The answer seems to be affirmative. We use the idea of Henry Wilton that the image might be taken as an alternating group $A_q$, a simple one (see his comment above). Let $K=mathcal A(1).$ Then

$mathcal A(m) ge [K,K,ldots,K]=[..[K,K],..,K]qquad (mquad times) qquad (*)$

Take a nontrivial $alpha in [K,K]$ and a surjective homomorphism $Delta: mathrm{Aut}(F_n) to A_q$ which doesn't vanish at $alpha$.

Then
$$
A_q =mathrm{NormalClosure}(Delta(alpha))=Delta([K,K])=Delta(K).
$$
It follows that
$$
Delta( [K,K,ldots,K])=A_q
$$
and by $(*)$ $Delta( mathcal A(m))=A_q $ for every $m ge 1.$

gt.geometric topology - Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and only if the links can be related by a sequence of Kirby moves and isotopies. This is pretty similar to Reidemeister's theorem, which says that two link diagrams correspond to isotopic links if and only if they can be related by a sequence of plane isotopies and Reidemeister moves.

Note however that Kirby moves, as opposed to the Reidemeister moves, are not local: the second Kirby move involves changing the diagram in the neighborhood of a whole component of the link. In "On Kirby's calculus", Topology 18, 1-15, 1979 Fenn and Rourke gave an alternative version of Kirby's calculus. In their approach there is a countable family of allowed transformations, each of which looks as follows: replace a $pm 1$ framed circle around $ngeq 0$ parallel strands with the twisted strands (clockwise or counterclockwise, depending on the framing of the circle) and no circle. Note that this time the parts of the diagrams that one is allowed to change look very similar (it's only the number of strands that varies), but still there are countably many of them.

I would like to ask if this is the best one can do. In other words, can there be a finite set of local moves for the Kirby calculus? To be more precise, is there a finite collection $A_1,ldots A_N,B_1,ldots B_N$ of framed tangle diagrams in the 2-disk such that any two framed link diagrams that give homeomorphic manifolds are related by a sequence of isotopies and moves of the form "if the intersection of the diagram with a disk is isotopic to $A_i$, then replace it with $B_i$"?

I vaguely remember having heard that the answer to this question is no, but I do not remember the details.

matrices - Expected determinant of a random NxN matrix.

As everyone above has pointed out, the expected value is $0$.

I expect that the original poster might have wanted to know about how big the determinant is. A good way to approach this is to compute $sqrt{E((det A)^2)}$, so there will be no cancellation.

Now, $(det A)^2$ is the sum over all pairs $v$ and $w$ of permutations in $S_n$ of
$$(-1)^{ell(v) + ell(w)} (1/2)^{2n-# { i : v(i) = w(i) }}$$

Group together pairs $(v,w)$ according to $u := w^{-1} v$. We want to compute
$$(n!) sum_{u in S_n} (-1)^{ell(u)} (1/2)^{2n-# (mbox{Fixed points of }i)}$$

This is $(n!)^2/2^{2n}$ times the coefficient of $x^n$ in
$$e^{2x-x^2/2+x^3/3 - x^4/4 + cdots} = e^x (1+x).$$

So $sqrt{E((det A)^2)}$ is
$$sqrt{(n!)^2/2^{2n} left(1/n! + 1/(n-1)! right)} = sqrt{(n+1)!}/ 2^n$$

ag.algebraic geometry - Reference for cohomology vanishing

To expand on Emerton's answer: Using the excision sequence, Cartan's result in the algebraic case boils down to showing the following: Let $R$ be a regular local ring, and $I$ and ideal of height at least $3$, then $H^i_I(R)=0$ for $ileq 2$.This follows because: $$H^i_I(R) = lim Ext^i(R/I^n,R)$$

And $I^n$, being height $3$, always contains a regular sequence of length $2$, so the $Ext^i$ vanishes for $ileq 2$ by standard result (see Bruns-Herzog Cohen Macaulay book, Proposition 1.2.10 for example). This argument extends to the case of codimension at least $n$ and vanishing of $H^{n-2}$.

Incidentally, a pretty non-trivial question is to find upper bound for the vanishing of local cohomology modules, in other words, the cohomological dimension of a subvariety $Z$. Many strong results have been obtained after SGA, by Hartshorne, Ogus, Faltings, Huneke-Lyubeznik, etc. All those references can be found in
Lyubeznik's paper (they were mentioned in the very first page) which primarily treated the vanishing of etale cohomology.

ac.commutative algebra - Decomposition of modules using computer packages

Suppose A = C[[x,y,z]]/f(x,y,z) is one of the ADE singularities, where there are finitely many indecomposables P_1,...,P_n. In analogy with character theory of finite groups, we want to set up a situation where Hom(P_i,P_j) = delta _ij. That will allow us to decompose a reflexive A-module into a direct sum of indecomposables (in the same way one decomposes a representation into a direct sum of irreducible representations).

The triangulated category underline{CM}(A) = CM(A)/A has a t-structure with heart CM(A), in which the finitely many indecomposables are the simple objects. The simples satisfy Hom^0(S_i, S_j) = delta ij. To compute Hom^0 in this category use the equation underline{Hom}(M,N) simeq Ext^2_A(M,N). (See Burban and Drozd's survey paper, especially page 46.)

So basically you just have to compute Ext^2 in the complete local ring A. Singular can do this directly.

If you don't want to download Singular, Macaulay2 can do it, although it takes some care, because Macaulay naturally works with graded modules over polynomial rings; (one has to be careful with grading shifts.) For more information on the graded case, see the papers by [Kajiura Saito and Takahashi].

soft question - Origins of names of algebraic structures

I'm not sure that I'm historically accurate, but that is how I always thought about algebraic nomenclature.

1) Group actually comes from group of substitutions. I guess that Galois could have introduced any other word, like "set" of substitutions or "flock" of transformations. Set theory was not yet established, so I guess a collection of functions could be called 'group', 'set' and so on according to the taste.

2) For field, I guess it comes from the meaning of field as "sphere", "subject", "area". It makes sense that such a word could come in talking about "solving an equation in the real field" rather than "solving an equation in the complex field". Then the concept of an abstract field could have followed.

3) Ring comes from "Zahlring", ring of numbers. This, as far as I know, is a terminology due to Dedekind. He was actually working with number rings, of the form $mathbb{Z}[alpha]$, where $alpha$ is integral over $mathbb{Z}$. So for some $n$, $alpha^n$ can be expressed in terms of lower powers of $alpha$; in some sense the components of the basis of $mathbb{Z}[alpha]$ over $mathbb{Z}$ cycle, although this is accurate only when $alpha$ is a root of unity. Hence the name ring of numbers.

4) Ideal is easy. When Dedekind realized that in a ring like $mathbb{Z}[sqrt{-5}]$ unique factorization does not hold, he searched for a substitute. He then realized could restore unique factorization allowing something more general than elements, the ideals. These are now called this way since he thought of them as "ideal elements" of the ring. useful to restore unique factorization. It is a fortunate coincidence that indeed for the rings he was working with (which are now called Dedekind rings), unique factorization for ideals actually holds.

5) Idéle has the same origin, being the contraction of the French "idéal élement", although the wording is inverted with respect to French use.

examples - Connection between bi-Hamiltonian systems and complete integrability

As I understand, the lack of indication on how to obtain first integrals in Arnol'd-Liouville theory is a reason why we are interested in bi-Hamiltonian systems.

Two Poisson brackets
${ cdot,cdot } _{1} , { cdot , cdot } _{2}$ on a manifold $M$ are compatible if their arbitrary linear combination
$lambda { cdot , cdot } _1+mu{cdot,cdot} _2$ is also a Poisson bracket. A bi-Hamiltonian system is one which allows Hamiltonian formulations with respect to two compatible Poisson brackets. It automatically posseses a number of integrals in involution.

The definition of a complete integrability (à la Liouville-Arnol'd) is:

Hamiltonian flows and Poisson maps on a $2n$-dimensional symplectic manifold $left(M,{ cdot, cdot }_Mright)$ with $n$ (smooth real valued) functions $F _1,F _2,dots,F _n$ such that: (i) they are functionally independent (i.e. the gradients $nabla F _k$ are linearly independent everywhere on $M$) and (ii) these functions are in involution (i.e. ${F _k,F _j}=0$) are called completely integrable.

Now, I would like to understand the connections between these two notions, and because I haven't studied the theory, any answer would be helpful. I find reading papers on these subjects too technical at the moment. Specific questions I have in mind are:

Does completely integrable system always allow for a bi-Hamiltonian structure? Is every bi-Hamiltonian system completely integrable? If not, what are examples (or places where to find examples) of systems that posses one property but not the other?

I apologize for any stupid mistakes I might have made above. Feel free to edit (tagging included).

dg.differential geometry - Differentiable function germs on differentiable manifolds

Hello everyone, I was wondering if anyone knew how to prove that the map from $C^{infty}(M)$ to $xi (p)$, that is, from the infinitely differentiable functions on a manifold M to the space of (once)-differentiable function germs, where the map is associating to each f in $C^{infty}(M)$ its class in $xi (p)$ is onto.
By the way, since you ask, the reason I'm interested in this is because its a question that WAS on my final for differential topology, I've tried to work it out since then but no luck so far, this is not homework it's just curiosity now, hope its ok ill have to check the post regultaions, sorry, if not just tell me and i'll delete the question...

mirror symmetry - Higher genus closed string B-model

This is a great question I wish I understood the answer to better.
I know two vague answers, one based on derived algebraic geometry and one based on string theory.
The first answer, that Costello explained to me and I most likely misrepeat,
is the following. The B-model on a CY X as an extended TFT can be defined in terms of
DAG: we consider the worldsheet $Sigma$ as merely a topological space or simplicial set (this is a reflection of the lack of instanton corrections in the B-model), and consider the mapping space $X^Sigma$ in the DAG sense. For example for $Sigma=S^1$ this is the derived loop space (odd tangent bundle) of $X$.. In this language it's very easy to say what the theory assigns to 0- and 1-manifolds: to a point we assign coherent sheaves on $X$, to a 1-manifold cobordism we assign the functor given by push-pull of sheaves between obvious maps of mapping spaces (see e.g. the last section here). For example for $S^1$ we recover Hochschild homology of $X$. Now for 2-manifold bordisms we want to define natural operations by push-pull of functions, but for that we need a measure -- and the claim is the Calabi-Yau structure (together with the appropriate DAG version of Grothendieck-Serre duality, which Kevin said Lurie provides) gives exactly this integration...

Anyway that gives a tentative answer to your question: the B-model assigns to a surface $Sigma$ the "volume" of the mapping space $X^Sigma$, defined in terms of the CY form.
More concretely, you chop up $Sigma$ into pieces, and use the natural operations on Hochschild homology, such as trace pairing and identification with Hochschild cohomology (and hence pair-of-pants multiplication).. of course this last sentence is just saying "use the Frobenius algebra structure on what you assigned to the circle" so doesn't really address your question - the key is to interpret the volume of $X^Sigma$ correctly.

The second answer from string theory says that while genus 0 defines a Frobenius manifold you shouldn't consider other genera individually, but as a generating series -- i.e. the genus is paired with the (topological) string coupling constant, and together defines a single object, the topological string partition function, which you should try to interpret rather than term by term. (This is also the topic of Costello's paper on the partition function). BTW for genus one there is a concrete answer in terms of Ray-Singer torsion, but I don't think that extends obviously to higher genus.

As to how to interpret it, that's the topic of the famous BCOV paper - i.e. the Kodaira-Spencer theory of gravity. For one thing, the partition function is determined recursively by the holomorphic anomaly equation, though I don't understand that as "explaining" the higher genus contributions. But in any case there's a Chern-Simons type theory quantizing the deformation theory of the Calabi-Yau, built out of the Kodaira-Spencer dgla in a simple looking way, and that's what the B-model is calculating.
A very inspiring POV on this is due to Witten, who interprets the entire partition function as the wave function in a standard geometric quantization picture for the middle cohomology of the CY (or more suggestively, of the moduli of CYs). This is also behind the Givental quantization formalism for the higher genus A-model, where the issue is not defining the invariants
but finding a way to calculate them.

Anyway I don't know a totally satisfactory mathematical formalism for the meaning of this partition function (and have tried to get it from many people), so would love to hear any thoughts. But the strong message from physics is that we should try to interpret this entire partition function - in particular it is this function which appears in a million different guises under various dualities (eg in gauge theory, as solution to quantum integrable systems, etc etc...)

ag.algebraic geometry - Quotient of abelian variety by an abelian subvariety

Let us work over $mathbb{C}$.

The inclusion $u colon B to A$ induces a surjection $hat{u} colon A^{vee} to B^{vee}$.
By general facts on Abelian varieties, the kernels of $u$ and $hat{u}$ have the same number of connected components. Since $u$ is injective, its kernel is trivial, so it follows $ker hat{u}=(ker hat{u})_0$; in other words $ker hat{u}$ is an Abelian subvariety of $A^{vee}$.

Therefore we have an exact sequence of Abelian varieties
$$0 to ker hat{u} to A^{vee} to B^{vee} to 0.$$ By dualizing it, we obtain $$0 to B to A to (ker hat{u})^{vee} to 0,$$
that is $C = (ker hat{u})^{vee}$.

gr.group theory - Does $mathrm{Aut}(mathrm{Aut}(...mathrm{Aut}(G)...))$ stabilize?

I don't know about non-stabilizing, but rigidity provides many examples that stabilize quickly.

1) Let π be the fundamental group of a finite volume hyperbolic manifold M of dimension ≥ 3 with no symmetries (that is, no nontrivial self-isometries). Negative curvature implies that π is centerless, so the map π -> Aut(π) is injective. Mostow-Prasad rigidity says that Out(π) = Isom(M), so the lack of isometries implies that Out(π) is trivial and Aut(π) = π. [This works verbatim for lattices in higher-rank semi-simple Lie groups subject to appropriate conditions.]

2) Let π=F_d be a free group of rank 2≤d<∞. Then Aut(F_n) is a much larger group; however, Dyer-Formanek showed that Out(Aut(F_n)) is trivial. Thus since Aut(F_n) is clearly centerless, we have Aut(Aut(F_n)) = Aut(F_n).

3) Interpolating between these two examples, if π=π₁(S_g) is the fundamental group of a surface of genus g≥2, then Aut(π) is the so-called "punctured mapping class group" Mod_g,*, which is much bigger than π. Ivanov proved that Out(Mod_g,*) is trivial, and since Mod_g,* is again centerless, we have Aut(Aut(π₁(S_g))) = Aut(π₁(S_g)).

In each of these cases, rigidity in fact gives stronger statements: Let H and H' be finite index subgroups of G = Aut(F_n) or Mod_g,*. (This class of groups can be widened enormously, these are just some examples.) Then any isomorphism from H to H' comes from conjugation by an element of G, by Farb-Handel and Ivanov respectively. In particular, Aut(H) is the normalizer of H in G. Rigidity gives the same conclusion for H = π₁(M) as in the first example and G = Isom(Hⁿ) [which is roughly SO(n,1)]. It seems that by carefully controlling the normalizers, you could use this to construct examples that stabilize only after n steps, for arbitrary large n.

---

Edit: I find the examples of D₈ and D_∞ unsatisfying because even though Inn(D) is a proper subgroup of Aut(D), we still have Aut(D) isomorphic to D. Here is a general recipe for building similarly liminal examples. Let G be an infinite group with no 2-torsion so that Aut(G) = G and H¹(G;Z/2Z) = Z/2Z. (Edited: For example, by rigidity, any hyperbolic knot complement with no isometries has these properties; by Thurston, most knot complements are hyperbolic.) The condition on the 2-torsion implies that for any automorphism G x Z/2Z -> G x Z/2Z, the composition

G -> G x Z/2Z -> G x Z/2Z -> G

is an isomorphism. From this we see that Aut(G x Z/2Z) / G = H¹(G;Z/2Z) = Z/2Z. By examination the extension is trivial, and thus Aut(G x Z/2Z) = G x Z/2Z. However, the image Inn(G x Z/2Z) is the proper subgroup G.

Comments: looking back, this feels very close to your original example of R x Z/2Z. Interesting that it's (seemingly) much harder to find group-theoretic conditions to force the behavior the way you want, while topologically it's easy.

Also, if you instead take G with H¹(G;Z/2Z) having larger dimension, say H¹(G;Z/2Z) = (Z/2Z)², this blows up quickly. You get Aut(G x Z/2Z) = G x (Z/2Z)², but then Aut(Aut(G x Z/2Z)) is the semidirect product of H¹(G;Z/2Z²) = (Z/2Z)⁴ with Aut(G) x Aut(Z/2Z²) = G x GL(2,2). Already the next step seems very hard to figure out. However, if you had enough control over the finite quotients of G, perhaps you could show that the linear parts of these groups don't get "entangled" with the rest, so that the automorphism groups would act like a product of G x (Z/2Z)ⁿ with something else, with n going to infinity. If so, this could yield an example where the isomorphism types of the groups never stabilize.

ag.algebraic geometry - Homology class orthogonal to image of Chern characters?

As Tony Pantev points out, it is easy to make an example by taking non-algebraic classes.

If you impose that $t$ is algebraic, and take $X=Y$, you are very close to stating Grothendieck's Conjecture D. Let $X$ be smooth and projective. The conjecture is that any algebraic class in $H^*(X)$, which is orthogonal to all algebraic classes, is zero torsion.

kt.k theory homology - Any reason why K_23(Z) has order 65520?

More generally, if $F$ is a number field with ring of integers $mathfrak{o}$, and $zeta_F^ast(m)$ is the first nonzero coefficient in the Taylor expansion of $zeta_F$ at $m$, then Lichtenbaum (and Quillen) conjectured that $|zeta_F^ast(1-i)|=frac{# K_{2i-2}(mathfrak{o})_{text{tors}}}{# K_{2i-1}(mathfrak{o})_{text{tors}}}$, times a regulator and some power of 2 (which I believe is not understood in general, although some progress was made on this in Ion Rada's PhD thesis). Hence, odd $K$ groups are related to the denominators of the Bernoulli numbers, and the even ones are related to the numerators. Also, not much cancellation occurs; I think the two $K$-groups can only share factors of 2.

The Voevodsky-Rost theorem might prove the Lichtenbaum conjecture, but I haven't seen anyone come out and say definitely that this is the case.

I don't have much intuition for this, except that the $K$-groups seem to be objects that like to map into étale cohomology groups. In this paper (link to MathSciNet), Soulé constructs Chern class maps from certain $K$-groups to étale cohomology groups. Furthermore, these maps frequently have small (or trivial) kernels and cokernels. I suppose the idea, then, is that $K$-theory is supposed to be a slightly better behaved version of étale cohomology, at least for the purpose of understanding zeta functions.

The rank of $K$-groups of rings of integers was computed by Quillen in the early 70's: it's rank 1 in dimension 0, rank $r_1+r_2-1$ in dimension 1 (Dirichlet's unit theorem), rank 0 in even dimensions $>0$, rank $r_1+r_2$ in dimensions $1pmod 4$ except 1, and rank $r_2$ in dimensions $3pmod 4$.

at.algebraic topology - finite generated group realized as fundamental group of manifolds

Theorem. Every finitely presentable group is the fundamental group of a closed 4-manifold.

Sketch proof. Let $langle a_1,ldots,a_mmid r_1,ldots, r_nrangle$ be a presentation. By van Kampen, the connected sum of $m$ copies of $S^1times S^3$ has fundamental group isomorphic to the free group on $a_1,ldots, a_m$. Now we can quotient by each relation $r_j$ as follows. Realise $r_j$ as a simple loop. A tubular neighbourhood of this looks like $S^1times D^3$. Do surgery and replace this tubular neighbourhood with $S^2times D^2$. This kills $r_j$. QED

There are many restrictions on 3-manifold groups. One of the simplest arises from the existence of Heegaard splittings. It follows easily that if $M$ is a closed 3-manifold then $pi_1(M)$ has a balanced presentation, meaning that $nleq m$.

Other obstructions to being a 3-manifold group were discussed in this MO question.

ct.category theory - Sites which are stacks over themselves

I don't have an answer to your question, but I'm going to post whatever thoughts I had about it. Maybe something here will help someone answer the question, or at least help more people understand what's involved. I'm sorry that it's come out so long.

---

Definitions
(skip this unless you suspect we mean different things by "(pre)stack")

A functor $Fto C$ is a fibered category if for every arrow $f:Uto X$ in $C$ and every object $Y$ in $F$ lying over $X$, there is a cartesian arrow $Vto Y$ in $F$ lying over $f$ (see Definition 3.1 of Vistoli's notes). This arrow is determined up to unique isomorphism (by the cartesian property), so I'll call $V$ "the" pullback of $Y$ along $f$ and maybe denote it $f^*Y$. A fibered category is roughly a "category-valued presheaf (contravariant functor) on $C$".

Given an object $X$ in $C$, let $F(X)$ be the subcategory of objects in $F$ lying over $X$, with morphisms being those morphisms in $F$ which lie over the identity morphism of $X$. I'll call $F(X)$ the "fiber over $X$." Given a morphism $f:Uto X$ in $C$, let $F(Uto X)$ be "the category of descent data along $f$," whose objects consist of an element $Z$ of $F(U)$ and an isomorphism $sigma:p_2^*Zto p_1^*Z$ (where $p_1,p_2:Utimes_XUto U$ are the projections) satisfying the usual cocycle condition over $Utimes_XUtimes_XU$ (see Definition 4.2 of Vistoli's notes). A morphism in $F(Uto X)$ is a morphism $Zto Z'$ in $F(U)$ such that the following square commutes:
$begin{matrix}
p_2^*Z & xrightarrow{sigma} & p_1^*Z \
downarrow & & downarrow\
p_2^*Z' & xrightarrow{sigma'} & p_1^*Z'
end{matrix}$

Suppose $C$ has the structure of a site. Then we say that $F$ is a prestack (resp. stack) over $C$ if for any cover $Uto X$ in $C$, the functor $F(X)to F(Uto X)$ given by pullback is fully faithful (resp. an equivalence). Roughly, a prestack is a "separated presheaf of categories" and a stack is a "sheaf of categories" over $C$.

---

The domain fibration (not your question, but related)

Consider the domain functor $Arr(C)to C$ given by $(Xto Y)mapsto X$. You can check that a cartesian arrow over $f:Uto X$ is a commutative square
$begin{matrix}
U & xrightarrow{f} & X \
downarrow & & downarrow\
Y & = & Y
end{matrix}$
If I haven't made a mistake,

• This fibered category is a prestack iff every cover $Uto X$ is an epimorphism.


• It is a stack if furthermore every cover $Uto X$ is the coequalizer of the projection maps $p_1,p_2:Utimes_XUto U$. This last condition is equivalent to saying that every object $Y$ of $C$ satisfies the sheaf axiom with respect to the morphism $Uto X$. In particular, the domain fibration is a stack if and only if the topology is subcanonical.

---

The codomain fibration (your question)

Consider the codomain functor $Arr(C)to C$ given by $(Uto X)mapsto X$. You can check that a cartesian arrow over a morphism $f:Uto X$ is a cartesian square
$begin{matrix}
V & to & U \
downarrow & & downarrow\
Y & xrightarrow{f} & X
end{matrix}$
There is a general result that says that the fibered category of sheaves on a site is itself a stack (I usually call this result "descent for sheaves on a site"). If you're working with the canonical topology on a topos (where every sheaf is representable), it follows that the codomain fibration is a stack. If the topology is subcanonical, then objects are sheaves, so descent for sheaves tells you that the pullback functor is fully faithful (i.e. the codomain fibration is a prestack), but when you "descend" a representable sheaf, it may no longer be representable, so the codomain fibration may not be a stack. In your question you say that being a prestack is actually equivalent to the topology being subcanonical, but I can't see the other implication (prestack⇒subcanonical).

Supposing the codomain fibration is a prestack, saying that it is a stack roughly says that when you glue representable sheaves along a "cover relation," you get a representable sheaf, but with the strange condition that the "cover relation" you started with came from a relation where you could glue to get a representable sheaf. That is, given this diagram, where the squares on the left are cartesian ($Rightarrow$ is meant to be two right arrows), can you fill in the "?" so that the square on the right is cartesian?
$begin{matrix}
Z' & Rightarrow & Z & to & ?\
downarrow & & downarrow & & downarrow\
Utimes_XU & Rightarrow & U & to & X
end{matrix}$

A more natural (to me) condition is to ask that the only sheaves you can glue together from representable sheaves are already representable. That is, if $RRightarrow U$ is a "covering relation" (i.e. each of the maps $Rto U$ is a covering and $Rto Utimes U$ is an equivalence relation), then the quotient sheaf $U/R$ is representable. I would call such a site "closed under gluing."

For example, the category of schemes with the Zariski topology is closed under gluing (it's the "Zariski gluing closure" of the category affine schemes). The category of algebraic spaces with the etale topology is closed under gluing (it's the "etale gluing closure" of the category of affine schemes). In fact, I think that a standard structure theorem for smooth morphisms and a theorem of Artin (∃ fppf cover ⇒ ∃ smooth cover) imply that the category of algebraic spaces with the fppf topology is closed under gluing.

soft question - The work of Thurston

There are several sources for Thurston's hyperbolization theorem, some published, some not.

Off the top of my head:

1) M.Kapovich, Hyperbolic manifolds and discrete groups.

2) J. Hubbard's Teichmuller theory volume II (not yet published)

3) J. Morgan, H. Bass (eds). The Smith conjecture. (English)
Papers presented at the symposium held at Columbia University, New
York, 1979.
Pure and Applied Mathematics, 112.
Academic Press, Inc., Orlando, Fla., 1984. xv+243 pp.

For only the case of manifolds that fibre over S^1

1) J-P. Otal, The hyperbolization theorem for fibred 3-manifolds.

Of course there's also the new non-Thurston proofs using Ricci flow.

Oh, and regarding that anecdote about repelling people from a field -- I've only heard that comment attributed to one mathematician and it was in reference to Thurston's early work on foliations. I don't think that's a widely held belief, but I wasn't alive then so I'm just going on 2nd hand comments.

reference request - Manifolds of continuous mappings.

I'm planning a short course on few topics and applications of nonlinear functional analysis, and I'd like a reference for a quick and possibly self-contained construction of a structure of a Banach differentiable manifold for the space of continuous mappings $C^0(K,M)$, where $K$ is a compact topological space (even metric if it helps) and $M$ is a (finite dimensional) differentiable manifold.

A construction of a differentiable structure of Banach manifold for this space can be found e.g. in Lang's book Fundamentals of differential geometry (1999). The main tools are the exponential map and tubular nbds (having fixed a Riemannian structure on $M$. This is OK but I believe there should be something even more basic.

Does anybody have a reference for alternative constructions (not necessarily elementary) ?

special functions - modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder

$R_{k,m} equiv H_{k} ~ mod H_m$

for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial (orthogonal under the weight $w(x) = e^{-x^2}$) and $deg R_{k,m} leq m-1$. I haven't been able to find anything online, neither could compute it through the recurrence relation of Hermite polynomials...

Update:

The motivation for my question is as follows. The $m$-point Gauss-quadrature is obtained by placing the nodes at the roots of $H_m$ and choosing the weights accordingly such that integrating any polynomial (with respect to weight $w$) of order $leq 2m-1$ is exact. Now I want to know the error formula for polynomials of degree $k geq m$, especially $H_k$. By computing $H_k$ modulo $H_m$, the integration error is given by the integration of the remainder $R_{k,m}$.

real analysis - existence of antiderivatives of nasty but elementary functions

This shows that "elementary function" needs a good definition. We do NOT want to allow, for example $f(x) = 1$ when $x$ rational and $f(x) = -1$ when $x$ irrational. Even though $f^2 = 1$, this $f$ is not an algebraic function.

So, correctly defined, an elementary function is an analytic function on a domain in the complex plane, such that ...... [fill in the usual conditions]

Added later. My advice: For "elementary function" do not use the popularized form of the
definition as in Wikipedia. Instead, use a definition from
the actual mathematics papers. (Papers with proofs, not
just quickie approximate definitions for the masses.)

For example

"Integration in Finite Terms", Maxwell Rosenlicht,
The American Mathematical Monthly 79 (1972), 963--972.
Stable URL: http://www.jstor.org/stable/2318066

Everything is carried out in differential fields ... In particular, every function involved is infinitely differentiable ... None of those
"discontinuous elementary functions" mentioned in the question.
Not even $|x| = sqrt{x^2}$ is elementary.

===========

"Algebraic Properties of the Elementary Functions of Analysis",
Robert H. Risch,
American Journal of Mathematics 101 (1979) 743--759.
Stable URL: http://www.jstor.org/stable/2373917

He also works in differential fields. Some quotes:

The elementary functions of a complex variable $z$ are those analytic functions that are built up from the rational functions of $z$ by successively applying algebraic operations, exponentiating, and taking logarithms. As is well known, this class includes the trigonometric and basic inverse trigonometric functions.

[Part II]
Suppose $mathbb{C}(z, theta_1, dots, theta_m) = mathcal{D}_m$ is the abstract field, isomorphic to a field of meromorphic functions on some
region $R$ of the complex plane, ...

==========

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.

graph theory - How to estimate the growth of the probability that $G(n, M)$ contains a $k$-clique

You might take a look at Chapter VII of Bollobas. In particular,
Theorem VII.1.7 -- which is simple enough that he doesn't bother providing a proof -- states that the expected number of $k$-cliques in $G(n,M)$ is, setting $N={n choose 2}$ and $K={k choose 2}$,
$$
{n choose k} {n-K choose M- K} {N choose M}^{-1}.
$$
Also, Theorem VII.3.7 states that if $M=o(n^{-2/(k-1)})$ then with probability tending to one, $G_{n,M}$ contains no $k$-clique, whereas if $M/n^{-2/(k-1)} to infty$ then with probability tending to one $G_{n,M}$ does contain a $k$-clique. I know this doesn't fully answer your question but it may help.

Incidentally, (you probably already realize that) it is a priori possible (though I don't think it is the case) that, for example, $t_k(M+1)-t_k(M) geq frac{1}{mathrm{poly}(n)}$ for all ${k choose 2}leq M leq lceil frac{(k-1)N}{k}rceil$, since all we really know by Turán is that
$$
sum_{M=K}^{lceil(k-1)N/krceil} (t_k(M+1)-t_k(M)) = 1.
$$

triangulated categories - Why do people "forget" Verdier abelianization functor?(Looking for application)

I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal cohomological functor) is really powerful and useful formalism. The professor assigned many problems concerning the property of localization functor in triangulated category.He strongly suggested us using abelianization functor to do these problems

If we do these problems in triangulated category, we have to work with various axioms TRI to TRIV which are not very easy to deal with. But if we use Verdier abelianization functor, we can turn the whole story to the abelian settings. Triangulated category can be embedded to Frobenius abelian category(projectives and injectives coincide). Triangulated functors become exact functor between abelian categories. Then we can work in abelian category. Then we can easily go back(because objects in triangulated category are just projectives in Frobenius abelian category, we can use restriction functor). In this way, it is much easier to prove something than Verider did in his book.

My question is:

1. What makes me surprised is that Verdier himself even did not use Abelianization in his book to prove something. I do not know why?(Maybe I miss something)




2. I wonder whether there are any non-trivial application of Verdier abelianization functor in algebraic geometry or other fields?

Thank you

homological algebra - Sums of injective modules, products of projective modules?

1. Under what assumptions on a noncommutative ring R does a countable direct sum of injective left R-modules necessarily have a finite injective dimension?




2. Analogously, under what assumptions on R does a countable product of projective left R-modules necessarily have a finite projective dimension?

These questions arise in the study of the coderived and contraderived categories of (CDG-)modules, or, if one wishes, the homotopy categories of unbounded complexes of injective or projective modules.

There are some obvious sufficient conditions and some less-so-obvious ones. For both #1 and #2, it clearly suffices that R have a finite left homological dimension.

More interestingly, in both cases it suffices that R be left Gorenstein, i.e., such that the classes of left R-modules of finite projective dimension and left R-modules of finite injective dimension coincide.

For #1, it also suffices that R be left Noetherian. For #2, it suffices that R be right coherent and such that any flat left module has a finite projective dimension.

Any other sufficient conditions?

The category of finite locally-free commutative group schemes

Voici tout ce qu'il y a à savoir sur cette catégorie FL/S.

En toutes généralités, c'est une catégorie exacte au sens de Quillen. Plus précisément, le plongement de FL/S dans la catégorie des faisceaux de groupes abéliens fppf sur $S$ font de FL/S une sous-catégorie stable par extensions dans cette catégorie abélienne de faisceaux fppf. Dans une catégorie exacte on dispose d'une notion de monomorphismes et épimorphismes stricts: ce sont ceux qui peuvent s'insérer dans une suite exacte. Alors, si $f:Grightarrow H$ est un morphisme de schémas en groupes, $f$ est un monomorphisme strict si et seulement si c'est une immersion fermée. De plus, $f$ est un épimorphisme strict si et seulement si c'est un morphisme fidèlement plat.

En général cette catégorie exacte n'est pas abélienne. Comme rappelé précédemment c'est cepdendant le cas si $S$ est le spectre d'un corps.

Maintenant supposons que $S$ soit le spectre d'un anneau de valuation d'inégales caractéristiques que je note $mathcal{O}_K$.

Lorsque $e_{K/mathbb{Q}_p} < p-1$ Raynaud a montré que c'est une catégorie abélienne. De plus, le foncteur fibre générique $Gmapsto Gotimes K$ est pleinement fidèle et identifie FL/S à une sous-catégorie abélienne de FL/$spec(K)$ (i.e., après un choix d'une clôture algébrique $overline{K}$ de $K$, une sous-catégorie abélienne de la catégorie des $Gal(overline{K}|K)$-modules discrets finis en tant que groupe abélien).

Lorsque $e geq p-1$ le résultat précédent est faux. Néanmoins on a le résultat suivant: dans FL/S tout morphisme possède un noyau et un conoyau. Plus précisément, si $f$ est un morphisme dans FL/S alors le platifié de $ker f$ ($ker f =$ noyau usuel dans la catégorie des schémas en groupes non-nécessairement plats) (platifié= on tue la $p$-torsion) est un noyau dans $FL/S$ du morphisme $f$. On construit de même l'image de $f$ comme adhérence schématique de l'image en fibre générique. Cependant, la catégorie précédente n'est pas abélienne. Soit en effet $K=mathbb{Q}_p (zeta_p)$ et
$$
f: mathbb{Z}/pmathbb{Z} longrightarrow mu_p
$$
le morphisme qui à $bar{1}in mathbb{Z}/pmathbb{Z}$ associe $zeta_pin mu_p$. Alors, $f$ est un isomorphisme en fibres génériques. On en déduit que dans FL/S les noyaux et conoyaux de $f$ sont nuls. Ce n'est cependant pas un isomorphisme !

nt.number theory - The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt correct, but he doesn't give references, and the thought of ploughing through Artin's collected works seems a bit daunting to me, so I thought I'd ask here first.

Background.

If $V$ is a smooth (affine or projective) curve over a finite field $k$ of size $q$, then $k$ has (up to isomorphism) a unique extension $k_n$ of degree $n$ over $k$ (so $k_n$ has size $q^n$) and one can define $N_n$ to be the size of $V(k_n)$. Completely concretely, one can perhaps imagine the case where $V$ is defined by one equation in affine or projective 2-space, for example $y^2=x^3+1$ (note that this equation will give a smooth curve in affine 2-space for $p$, the characteristic of $k$, sufficiently large), and simply count the number of solutions to this equation with $x,yin k_n $to get $N_n$. Let $F_V(u)=sum_{ngeq1}N_nu^n$ denote the formal power series associated to this counting function.

Now it turns out from the "formalism of zeta functions" that this isn't the most ideal way to package the information of the $N_n$, one really wants to be doing a product over closed points of your variety. If $C_d$ is the number of closed points of $V$ of degree $d$, that is, the number of closed points $v$ of (the topological space underlying the scheme) $V$ such that $k(v)$ is isomorphic to $k_d$, then one really wants to define
$$Z_V(u)=prod_{dgeq1}(1-u^d)^{-C_d}.$$
If one sets $u=q^{-s}$ then this is an analogue of the Riemann zeta function, which is a product over closed points of $Spec(mathbf{Z})$ of an analogous thing.

Now the (easy to check) relation between the $C$s and the $N$s is that $N_n=sum_{d|n}dC_d$, and this translates into a relation between $F_V$ and $Z_V$ of the form
$$uZ_V'(u)/Z_V(u)=F_V(u).$$

This relation also means one can compute $Z$ given $F$: one divides $F$ by $u$, integrates formally, and then exponentiates formally; this works because $f'/f=(log(f))'$.

The reason I'm saying all of this is just to stress that this part of the theory is completely elementary.

The Weil conjectures in this setting.

The Weil conjectures imply that for $V$ as above, the power series $Z_V(u)$ is actually a rational function of $u$, and make various concrete statements about its explicit form (and in particular the location of zeros and poles). Note that they are usually stated for smooth projective varieties, but in the affine curve case one can take the smooth projective model for $V$ and then just throw away the finitely many extra points showing up to see that $Z_V(u)$ is a rational function in this case too.

How to prove special cases in 1923?

OK so here's the question. It's 1923, we are considering completely explicit affine or projective curves over explicit finite fields, and we want to check that this power series $Z_V(u)$ is a rational function. Dieudonne states that Artin manages to do this for curves of the form $y^2=P(x)$ for "many polynomials $P$ of low degree". How might we do this? For $P$ of degree 1 or 2, the curve is birational to projective 1-space and the story is easy. For $V$ equals projective 1-space, we have
$$F_V(u)=(1+q)u+(1+q^2)u^2+(1+q^3)u^3+ldots=u/(1-u)+qu/(1-qu)$$
from which it follows easily from the above discussion that
$$Z_V(u)=1/[(1-u)(1-qu)].$$
For polynomials $P$ of degree 3 or 4, the curve has genus 1 and again I can envisage how Artin could have approached the problem. The curve will be birational to an elliptic curve, and it will lift to a characteristic zero curve with complex multiplication. The traces of Frobenius will be controlled by the corresponding Hecke character, a fact which surely will not have escaped Artin, and I can believe that he was now smart enough to put everything together.

For polynomials of degree 5 or more, given that it's 1923, the problem looks formidable.

Q1) When Dieudonne says that Artin verified (some piece of) the Weil conjectures for "many polynomials of low degree", does he mean "of degree at most 4", or did Artin really move into genus 2?

How much further can we get in 1931?

Now this one really surprised me. Dieudonne claims that in 1931 F. K. Schmidt proved rationality of $Z_V(u)$, plus the functional equation, plus the fact that $Z_V(u)=P(u)/(1-u)(1-qu)$, for $V$ an arbitrary smooth projective curve, and that he showed $P(u)$ was a polynomial of degree $2g$, with $g$ the genus of $V$. This is already a huge chunk of the Weil conjectures. We're missing the statement that $P(u)$ has all its rots of size $q^{-1/2}$ (the "Riemann hypothesis") but this is understandable: one needs a fair amount of machinery to prove this. What startled me (in my naivity) was that I had assumed that all this was due to Weil in the 1940s and I am obviously wrong: "all Weil did" was to prove RH. So I have a very basic history question:

Q2) However did Schmidt do this?

---

EDIT: brief summary of answers below (and what I learned from chasing up the references):

A1) Artin didn't do anything like what I suggested. He could explicitly compute the zeta function of an arbitrary given hyperelliptic curve over a given finite field by an elegant application of quadratic reciprocity. See e.g. the first of Roquette's three papers below. The method in theory works for all genera although the computations quickly get tiresome.

A2) Riemann-Roch. Express the product defining $Z$ as an infinite sum and then use your head.

cv.complex variables - Power series for meromorphic differentials on compact Riemann surfaces

Suppose I have a compact Riemann surface of $g>1$ given by the quotient $H/Gamma$ where I do know $Gamma$ explicit. Is there a way to write down the power series of meromorphic functions, differentials, quadratic differentials, and so on explicitly if one does know these sections explicitly (for example in a hyperelliptic picture of the surface).
I know that there is the subject of automorphic forms, but the literature I have seen about this concentrates on modular groups. Moreover it is typically written for number theorists.
Is there literature for compact surfaces (for example $Y^2=Z^6-1$), and maybe readable for geometers.

Thank you.

nt.number theory - When is $Pn^2-2an+frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, a square?

It is easy to show that the following problems are equivalent.

a. When is $Pn^2-2an+frac{a^2-k}{P}$ , with $P$ Prime, $k=a^2 mod P$, and $n$ any integer, a square?

and

b. When is $X^2-PY^2=k$ solvable in integers?

So, any suggestions on problem a ? How fast would an algorithm used to compute this run ?

nt.number theory - Heuristically false conjectures

Just run across this question, and am surprised that the first example
that came to mind was not mentioned:

Fermat's "Last Theorem" is heuristically true for $n > 3$,
but heuristically false for $n=3$ which is one of the easier
cases to prove.

if $0 < x leq y < z in (M/2,M]$ then $|x^n + y^n - z^n| < M^n$.
There are about $cM^3$ candidates $(x,y,z)$ in this range
for some $c>0$ (as it happens $c=7/48$), producing values of
$Delta := x^n+y^n-z^n$ spread out on
the interval $(-M^n,M^n)$ according to some fixed distribution
$w_n(r) dr$ on $(-1,1)$ scaled by a factor $M^n$ (i.e.,
for any $r_1,r_2$ with $-1 leq r_1 leq r_2 leq 1$
the fraction of $Delta$ values in $(r_1 M^n, r_2 M^n)$
approaches $int_{r_1}^{r_2} w_n(r) dr$ as $M rightarrow infty$).

This suggests that any given value of $Delta$, such as $0$,
will arise about $c w_n(0) M^{3-n}$ times. Taking $M=2^k=2,4,8,16,ldots$
and summing over positive integers $k$ yields a rapidly divergent sum
for $n<3$, a barely divergent one for $n=3$, and a rapidly convergent
sum for $n>3$.

Specifically, we expect the number of solutions of $x^n+y^n=z^n$
with $z leq M$ to grow as $M^{3-n}$ for $n<3$ (which is true and easy),
to grow as $log M$ for $n=3$ (which is false), and to be finite for $n>3$
(which is true for relatively prime $x,y,z$ and very hard to prove [Faltings]).

More generally, this kind of analysis suggests that for $m geq 3$
the equation $x_1^n + x_2^n + cdots + x_{m-1}^n = x_m^n$
should have lots of solutions for $n<m$,
infinitely but only logarithmically many for $n=m$,
and finitely many for $n>m$. In particular, Euler's conjecture
that there are no solutions for $m=n$ is heuristically false for all $m$.
So far it is known to be false only for $m=4$ and $m=5$.

Generalization in a different direction suggests that any cubic
plane curve $C: P(x,y,z)=0$ should have infinitely many rational points.
This is known to be true for some $C$ and false for others;
and when true the number of points of height up to $M$ grows as
$log^{r/2} M$ for some integer $r>0$ (the rank of the elliptic curve),
which may equal $2$ as the heuristic predicts but doesn't have to.
The rank is predicted by the celebrated conjecture of Birch and
Swinnerton-Dyer, which in effect refines the heuristic by accounting
for the distribution of values of $P(x,y,z)$ not just
"at the archimedean place" (how big is it?) but also "at finite places"
(is $P$ a multiple of $p^e$?).

The same refinement is available for equations in more variables,
such as Euler's generalization of the Fermat equation;
but this does not change the conclusion (except for equations such as
$x_1^4 + 3 x_2^4 + 9 x_3^4 = 27 x_4^4$,
which have no solutions at all for congruence reasons),
though in the borderline case $m=n$ the expected power of $log M$ might rise.

Warning: there are subtler obstructions that may prevent a surface from
having rational points even when the heuristic leads us to expect
plentiful solutions and there are no congruence conditions that
contradict this guess. An example is the Cassels-Guy cubic
$5x^3 + 9y^3 + 10z^3 + 12w^3 = 0$, with no nonzero rational solutions
$(x,y,z,w)$:

Cassels, J.W.S, and Guy, M.J.T.:
On the Hasse principle for cubic surfaces,
Mathematika 13 (1966), 111--120.

at.algebraic topology - Group Completions and Infinite-Loop Spaces

A well-written discussion of the group completion can be found on pp. 89--95 of
J.F. Adam: Infinite loop spaces, Ann. of Math. studies 90 (even though he only
discusses a particular group completion of a monoid). In particular you
assumption of commutativity comes in under the assumption that $pi_0(M)$ is
commutative which makes localisation with respect to it well-behaved
(commutativity is not the most general condition what is needed is some kind of
Øre condition).

In any case if you really want conclusions on the homotopy equivalence level I
think you need to put yourself in some nice situation for instance requiring
that all spaces be homotopy equivalent to CW-spaces. If you don't want that you
should replace homotopy equivalences by weak equivalences, if not you will
probably find yourself in a lot of trouble. In any case I will assume that we
are dealing with spaces homotopy equivalent to CW-complexes.

Starting with 1) a first note is that your conditions does not have to involve
an arbitrary ring $R$. It is enough to have $R=mathbb Z$ and one should
interpret the localisation in the way (for instance) Adams does:
$H_ast(X,mathbb Z)=bigoplus_alpha H_ast(X_alpha,mathbb Z)$, where
$alpha$ runs over $pi_0(X)$, and a $beta$ maps $H_ast(X_alpha,mathbb Z)$
to $H_ast(X_{alphabeta},mathbb Z)$. Then your group completion condition is
that the natural map $mathbb Z[pi_0(Y)]bigotimes_{mathbb Z[pi_0(X)]}
H_ast(X,mathbb Z)rightarrow H_ast(Y,mathbb Z)$ should be an isomorphism.
This then implies the same for any coefficient group (and when the coefficient
group is a ring $R$ you get your condition). (Note that for this formula to even make sense we need at least associativity for the action of $pi_0(X)$ on the homology. This is implied by the associativity of the Pontryagin product of $H_*(X,mathbb Z)$ which in turn is implied by the homotopy associativity of the H-space structure.)

Turning now to 1) it follows from standard obstruction theory. In fact maps into
simple (hope I got this terminology right!) homotopy types, i.e., spaces for
which the action of the fundamental groups on the homotopy groups is trivial (in
particular the fundamental group itself is commutative). The reason is that the
Postnikov tower of such a space consists of principal fibrations and the lifting
problem for maps into principal fibrations is controlled by cohomology groups
with ordinary coefficients. Hence no local systems are needed (they would be if
non-simple spaces were involved). The point now is that H-spaces are simple so
we get a homotopy equivalence between any two group completions and as
everything behaves well with respect to products these equivalences are H-maps.

Addendum:
As for 2) it seems to me that this question for homotopy limits can only be solved under supplementary conditions. The reason is that under some conditions we have the Bousfield-Kan spectral sequence (see Bousfield, Kan: Homotopy limits, completions and localizations, SLN 304) which shows that $varprojlim^s(pi_s X_i)$ for all $s$ will in general contribute to $pi_0$ of the homotopy limit. As the higher homotopy groups can change rather drastically on group completion it seems difficult to say anything in general (the restriction to cosimplicial spaces which the OP makes in comments doesn't help as all homotopy limits can be given as homotopy limits over $Delta$. Incidentally, for homotopy colimits you should be in better
shape. There is however an initial problem (which also exists in the homotopy limit case): If you do not assume that the
particular group completions you choose have any functorial properties it is not
clear that a diagram over a category will give you a diagram when you group
complete. This can be solved by either assuming that in your particular
situation you have enough functoriality to get that (which seems to be the case
for for instance May's setup) or accepting "homotopy everything" commutative
diagrams which you should get by the obstruction theory above. If this problem
is somehow solved you should be able to conclude by the Bousfield-Kan spectral
sequence $injlim^ast H_*(X_i,mathbb Z)implies
H_*(mathrm{hocolim}X_i,mathbb Z)$. We have that localisation is exact and
commutes with the higher derived colimits so that we get upon localisation a
spectral sequence that maps to the Bousfield-Kan spectral sequence for ${Y_i}$
and is an isomorphism on the $E_2$-term and hence is so also at the convergent.

As for 3) I don't altogether understand it. Possibly the following gives some
kind of answer. For the H-space $coprod_nmathrm{B}Sigma_n$ which is the
disjoing union of classifying spaces of the symmetric groups its group
completion has homotopy groups equal to the stable homotopy groups of spheres
which shows that quite dramatic things can happen to the homotopy groups upon
group completion (all homotopy groups from degree $2$ on of the original space
are trivial).

ag.algebraic geometry - Complex torus, C^n/Λ versus (C*)^n

I'm having trouble distinguishing the various sorts of tori.

One definition of torus is the algebraic torus. Groups like SU(2,ℂ/ℝ) and SU(3,ℂ/ℝ) have important subgroups that are topologically a circle and a torus, and I guess those were some of the most important Lie groups so the name torus stuck. Groups like SL(2,ℂ) and SL(n+1,ℂ) have a similar important subgroup isomorphic to ℂ* and (ℂ*)ⁿ, so the name torus gets applied to them too. In general, one calls the multiplicative group of an arbitrary field a torus in many situations, sometimes denoting the entire lot of them as Gm.

Another definition of a topological torus is a direct product of circles. A standard way to construct various flat geometries on a torus is to take ℝⁿ and quotient out by a discrete rank n lattice Λ, for instance ℝ/ℤ or ℂ/ℤ[i]. A complex torus is defined analogously as ℂⁿ/Λ where Λ is a rank 2n lattice (since ℂⁿ has real rank 2n).

One reads in various places that every abelian variety is a complex torus, but not every complex torus is an abelian variety. The notation ℂⁿ/Λ is usually nearby.

Is the multiplicative group of the field, Gm or ℂ*, an abelian variety?

In other words, is an algebraic torus over the complexes a complex torus?

Is an abelian variety isomorphic as a group to ℂⁿ/Λ, or just topologically?

My dim memory of elliptic curves was that they were finitely generated abelian groups, but since they are uncountable that doesn't make any sense. Presumably I am thinking of their rational points. However, ℂⁿ/Λ is always an abelian group, so I don't see what the fuss is about deciding when it is an abelian variety. It seems likely to me the group operations are different.

ag.algebraic geometry - Quasi-coherent sheaves of O_X-algebras

Under some slightly stronger hypothesis (Noetherian is certainly enough) we may write
$mathcal A$ as the union of its coherent subsheaves. If $mathcal E$ is a coherent subsheaf, then the subalgebra of $mathcal A$ that it generates will also be coherent,
because this can be tested locally, where it then follows from your assumptions. Thus in this case, $mathcal A$ is the union of coherent $mathcal O_X$-algebras.

I'm not sure how good a notion coherent is outside of the Noetherian context. If no-one
gives an answer in the non-Noetherian context, then you might want to look at the stacks project, which discusses this kind of "coherent approximation to quasi-coherent sheaves" in some generality, if I remember correctly.

What do mathematicians currently do in conformal field theory (or more general field theory)

CFT/QFT/TFT/etc. is a huge subject...

Here are some random references off the top of my head...

Segal, "The definition of conformal field theory".

Costello, "Topological conformal field theories and Calabi-Yau categories" -- This is (essentially) the 2d version of the (Hopkins-)Lurie/Baez-Dolan cobordism hypothesis that Lennart mentions. See also Kontsevich-Soibelman, "Notes on A-infinity...". This stuff is closely related to mirror symmetry, which is - in physics terms - a duality between certain field theories (or sigma models). Mirror symmetry by itself is already a huge enterprise...

See papers by Yi-Zhi Huang for stuff about vertex operator algebras and CFTs.

One can consider string topology from a field theory viewpoint... see for example Sullivan, "String Topology: Background and Present State" and Blumberg-Cohen-Teleman, "Open-closed field theories, string topology, and Hochschild homology". This is actually related to the work of Costello, Lurie, Kontsevich mentioned above -- see e.g. section 2.1 of Costello's paper.

An important problem is that of making rigorous some of the things that physicists do in QFT, such as path integrals. See Costello, "Renormalization and effective field theory" and also Borcherds, "Renormalization and quantum field theory".

There's also Chern-Simons theory... Gromov-Witten theory... Kapustin-Witten theory... Rozansky-Witten theory...

Related MO questions:

A reading list for topological quantum field theory?

Mathematics of path integral: state of the art

Doing geometry using Feynman Path Integral?

soft question - Open source mathematical software

I'll second the votes for Sage, Macsyma as Maxima and Wxmaxima, Scilab, Octave, R, and GAP.

For kids to play with are KGeometry KiG (K-interactive-Geometry), letting you draw out geometric relationships and actively move points around, letting all defined subcomponents change with it: e.g. draw three points, define+draw the line segments between the points, define+draw the perpendicular bisectors of these line segments, define+draw a circle that touches the three points of the triangle. Now drag any of the three points of the triangle around and watch all of the defined components move along to remain the bisectors / intersections / circles consistently. It's a great way to play around with geometric constructions.

Also, you can't go wrong with using awk, sed, and bash on the command line.

dg.differential geometry - Are submersions of differentiable manifolds flat morphisms?

I can show that this is true for your "simple" case.

---

If g(x,y) ∈ C^∞(ℝ²) vanishes on x ≤ 0 then it decomposes as g(x,y) = a(x)G(x,y) where a(x) ∈ C^∞(ℝ) vanishes on x ≤ 0 and G(x,y) ∈ C^∞(ℝ²).

---

This can be shown by proving the statements below. They could possibly be standard results, but I've never seen them before.
First, I'll refer to the following sets of functions.

• Let U be thet set of functions f(x) ∈ C^∞(ℝ) which vanish on x ≤ 0 and are positive on x > 0.


• Let V be the set of functions f: ℝ⁺→ ℝ such that x^-n f(x) → 0 as x → 0, for each positive integer n.

The statements I need to show the main result are as follows.

Lemma 1: For any f ∈ V, there is a g ∈ U such that f(x)/g(x) → 0 as x → 0.

Proof: Choose any smooth function r: ℝ⁺→ ℝ⁺ with r(0) = 1 and r(x) = 0 for x ≥ 1. For example, we can use r(x) = exp(1-1/(1-x)) for x < 1. Then, the idea is to choose a sequence of positive reals α_k → 0 satisfying ∑_k α_k < ∞, and set

$$g(x) = x^{theta(x)}, theta(x)=sum_{k=1}^infty r(x/alpha_k)$$

for x > 0 and g(x) = 0 for x ≤ 0. Only finitely many terms in the summation will be nonzero outside any neighborhood of 0, so it is a well defined expression, and smooth on x > 0. Clearly, θ(x) → ∞ and, therefore, x^-n g(x) → 0 as x → 0. It needs to be shown that all the derivatives of g vanish at 0 so that g ∈ U. As r and all its derivatives are bounded with compact support, r⁽ⁿ⁾(x) ≤ K_nx^-n-1 for some constants K_n. The n^th derivative of θ is

$$theta^{(n)}(x)=sum_kalpha_k^{-n}r^{(n)}(x/alpha_k)le K_nx^{-n-1}sum_kalpha_k$$

which has polynomially bounded growth in 1/x. The derivatives of log(g) satisfy

$$frac{d^n}{dx^n}log(g(x))=frac{d^n}{dx^n}left(log(x)theta(x)right)$$

which also has polynomially bounded growth in 1/x. However, the derivative on the left hand side is g⁽ⁿ⁾(x)/g(x) plus a polynomial in g⁽ⁱ⁾(x)/g(x) for i < n. So, induction gives that g⁽ⁿ⁾(x)/g(x) has polynomially bounded growth in 1/x and, multiplying by g(x), g⁽ⁿ⁾(x) → 0 as x → 0.

By definition of f ∈ V, there is a decreasing sequence of positive reals ε_k such that f(x) ≤ xⁿ for x ≤ ε_n. We just need to make sure that α_k ≤ ε_n+1 for k ≥ n to ensure that g(x) ≥ x^n-1 for ε_n+1 ≤ x ≤ min(ε_n,1). Then f(x)/g(x) goes to zero at rate x as x → 0.
█

Lemma 2: For any sequence f₁,f₂,... ∈ V there is a g ∈ U such that f_k(x)/g(x) → 0 as x → 0 for all k.

Proof: The idea is to apply Lemma 1 to f(x) = Σ_k λ_k|f_k(x)| for positive reals λ_k. This works as long as f ∈ V, which is the case if Σ_k λ_ksup_x≤kmin(x,1)^-k|f_k(x)| is finite, and this condition is easy to ensure.
█

Lemma 3: For any sequence f₁,f₂,... ∈ V there is a g ∈ U such that f_k(x)/g(x)ⁿ → 0 as x → 0 for all positive integers k,n.

Proof: Apply Lemma 2 to the doubly indexed sequence f_k,n = |f_k|^1/n.

The result follows from applying lemma 3 to the triply indexed sequence f_i,j,k(x) = max{|(d^i+j/dxⁱdy^j)g(x,y)|: |y| ≤ k} ∈ V. Then, there is an a ∈ U such that f_ijk(x)/a(x)ⁿ → 0 as x → 0. Set G(x,y) = f(x,y)/a(x) for x > 0 and G(x,y) = 0 for x ≤ 0. On any bounded region for x > 0, the derivatives of G(x,y) to any order are bounded by a sum of terms, each of which is a product of f_ijk(x,y)/a(x)ⁿ with derivatives of a(x), so this vanishes as x → 0. Therefore, G ∈ C^∞(ℝ²).
█

In fact, using a similar method, the simple case can be generalized to arbitrary submersions.

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Let p: M →N be a submersion. If h ∈ C^∞(N) and g ∈ C^∞(M) satisfy hg = 0 then, g = aG for some G ∈ C^∞(M) and a ∈ C^∞(N) satisfying ha = 0.

---

Very Rough Sketch:
If S ⊂ N is the open set {h≠0} then g and all its derivatives vanish on p^-1(S).
The idea is to choose a smooth parameter u:N-S →ℝ⁺ which vanishes linearly with the distance to S. This can be done locally and then extended to the whole of N (I'm assuming manifolds satisfy the second countability property). As all the derivatives of g vanish on p^-1(S), u^-ng tends to zero at the boundary of S. This uses the fact that p is a submersion, so that u also goes to zero linearly with the distance from p^-1(S) in M.

Then, following a similar argument as above, a can be expressed a function of u so that g/a and all its derivatives tend to zero at the boundary of S. Finally, G=0 on the closure of p^-1(S) and G=g/a elsewhere.
█

I suppose the next question is: does proving the special case above of a single g and h reduce the proof of flatness to algebraic manipulation?

ac.commutative algebra - Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?

This can be done in a few steps in probably any computer algebra package. You take the generators of your original ideal $I$, and bi-homogenize them, as described in the question. Then saturate with respect to the two hyperplanes at infinity, which are defined by the equation $x_0 y_0$.

For example, the diagonal in $mathbb A^3 times mathbb A^3$ is defined by $x_1 - y_1$, $x_2 - y_2$, and $x_3 - y_3$. If I wanted to use this to compute the ideal of the diagonal in $mathbb P^3 times mathbb P^3$, I would use the following commands in Macaulay2:

 r = QQ[x0,x1,x2,x3,y0,y1,y2,y3]
 i = ideal(x1*y0-y1*x0, x2*y0-y2*x0, x3*y0-y3*x0)
 saturate(i, x0*y0)

The code in Singular would be:

 ring r = 0, (x0,x1,x2,x3,y0,y1,y2,y3), dp;
 ideal i = x1*y0-y1*x0, x2*y0-y2*x0, x3*y0-y3*x0;
 LIB "elim.lib";
 sat(i, x0*y0);

pr.probability - A probabilistic inequality

I think it's true.
My reasoning goes like this (and I kept making mistakes with the algebra/arithmetic so check carefully)

$P(S_6 lt 3 delta) le P(hbox{exactly 4 }X_ihbox{s less than }delta)+P(hbox{exactly 5 }X_ihbox{s less than }delta)+P(hbox{all 6 }X_ihbox{s less than }delta)$. So if we let $p=P(X_1 lt delta)$, then the right hand part of this inequality is

$(p^4)(10p^2-24p+15)$. I then plugged $2p-(p^4)(10p^2-24p+15)$ into R and got a function that looks always non-negetive between 0 and 1 (shouldn't be too bad to show this by derivatives but I'm lazy and have computer power).

groupoids - How difficult is Morse theory on stacks?

The title is a little tongue-in-cheek, since I have a very particular question, but I don't know how to condense it into a pithy title. If you have suggestions, let me know.

Suppose I have a Lie groupoid $G rightrightarrows G_0$, by which I mean the following data:

• two finite-dimensional (everything is smooth) manifolds $G,G_0$,


• two surjective submersions $l,r: G to G_0$,


• an embedding $e: G_0 hookrightarrow G$ that is a section of both the maps $l,r$,


• a composition law $m: G times_{G_0} G to G$, where the fiber product is the pull back of $G overset{r}to G_0 overset{l}leftarrow G$, intertwining the projections $l,r$ to $G_0$.


• Such that $m$ is associative, by which I mean the two obvious maps $G times_{G_0} G times_{G_0} G to G$ agree,


• $m(e(l(g)),g) = g = m(g,e(r(g)))$ for all $gin G$,


• and there is a map $i: G to G$, with $icirc l = r$ and $icirc i = text{id}$ and $m(i(g),g) = e(r(g))$ and $m(g,i(g)) = e(l(g))$.

Then it makes sense to talk about smooth functors of Lie groupoids, smooth natural transformations of functors, etc. In particular, we can talk about whether two Lie groupoids are "equivalent", and I believe that a warm-up notion for "smooth stack" is "Lie groupoid up to equivalence". Actually, I believe that the experts prefer some generalizations of this — (certain) bibundles rather than functors, for example. But I digress.

Other than that we know what equivalences of Lie groupoids are, I'd like to point out that we can work also in small neighborhoods. Indeed, if $U_0$ is an open neighborhood in $G_0$, then I think I can let $U = l^{-1}(U_0) cap r^{-1}(U_0)$, and then $U rightrightarrows U_0$ is another Lie groupoid.

Oh, let me also recall the notion of tangent Lie algebroid $A to G_0$ to a Lie groupoid. The definition I'll write down doesn't look very symmetric in $lleftrightarrow r$, but the final object is. The fibers of the vector bundle $A to G_0$ are $A_y = {rm T}_{e(y)}(r^{-1}(y))$, the tangent space along $e(G_0)$ to the $r$-fibers, and $l: r^{-1}(y) to G_0$ determines a God-given anchor map $alpha = dl: A to {rm T}G_0$, and because $e$ is a section of both $l,r$, this map intertwines the projections, and so is a vector bundle map. In fact, the composition $m$ determines a Lie bracket on sections of $A$, and $alpha$ is a Lie algebra homomorphism to vector fields on $G_0$.

Suppose that I have a smooth function $f: G_0 to mathbb R$ that is constant on $G$-orbits of $G_0$, i.e. $f(l(g)) = f(r(g))$ for all $gin G$. I'd like to think of $f$ as a Morse function on "the stack $G_0 // G$". So, suppose $[y] subseteq G_0$ is a critical orbit, by which I mean: it is an orbit of the $G$ action on $G_0$, and each $y in [y]$ is a critical point of $f$. (Since $f$ is $G$-invariant, critical points necessarily come in orbits.) If $y$ is a critical point of $f$, then it makes sense to talk about the Hessian, which is a symmetric pairing $({rm T}_yG_0)^{otimes 2} to mathbb R$, but I'll think of it as a map $f^{(2)}_y : {rm T}_yG_0 to ({rm T}_yG_0)^*$. In general, this map will not be injective, but rather the kernel will include $alpha_y(A_y) subseteq {rm T}_yG_0$. Let's say that the critical orbit $[y]$ is nondegenerate if $ker f^{(2)}_y = alpha_y(A_y)$, i.e. if the Hessian is nondegenerate as a pairing on ${rm T}_yG_0 / alpha_y(A_y)$. I'm pretty sure that this is a condition of the orbit, not of the individual point.

Nondegeneracy rules out some singular behavior of $[y]$, like the irrational line in the torus.

Anyway, my question is as follows:

Suppose I have a Lie groupoid $G rightrightarrows G_0$ and a $G$-invariant smooth function $f: G_0 to mathbb R$ and a nondegenerate critical orbit $[y]$ of $f$. Can I find a $G$-invariant neighborhood $U_0 supseteq [y]$ so that the corresponding Lie groupoid $U rightrightarrows U_0$ is equivalent to a groupoid $V rightrightarrows V_0$ in which $[y]$ corresponds to a single point $bar y in V_0$? I.e. push/pull the function $f$ over to $V_0$ along the equivalence; then can I make $[y]$ into an honestly-nondegenerate critical point $bar y in V_0$?

I'm assuming, in the second phrasing of the question, that $f$ push/pulls along the equivalence to a $V$-invariant function $bar f$ on $V_0$. I'm also assuming, so if I'm wrong I hope I'm set right, that ${rm T}_{bar y}V_0 cong {rm T}_yG_0 / alpha_y(A_y)$ canonically, so that e.g. $bar f^{(2)}_{bar y} = f^{(2)}_y$.

big list - nonstandard analysis book recommendation

Hey, I just peeked at your MathOverflow page and saw that you are interested in "spatial and visual arguments". So I tell you something else (which you didn't ask for):

There also is another version of analysis with nilpotent infinitesimals, i.e. elements which are not zero, but some power of which is zero. In classical logic this contradicts the field axioms, but in intuitionistic logic it can be done. J.L. Bell's Primer of Infinitesimal Anlysis develops basic analysis on these grounds, by assuming (axiomatically) that you have something like the real numbers with nilpotents. Proofs become much easier even than in Nonstandard Analysis. Only in an appendix he addresses the existence of models for his axioms - they live in toposes.

As is very nicely laid out in the preface of Moerdijk/Reyes' "Models for Smooth Infinitesimal Analysis", it is these infinitesimals which were (implicitly) used by classical geometers like Cartan, and are (implicitly) used by physicists until today. They illustrate their point with a visual proof of Stokes' theorem using nilpotents.

In the settings of Moerdijk/Reyes (which are certain toposes) there also exist real numbers which combine the two kinds of infinitesimals, nilpotents and invertibles.

arithmetic geometry - bibl. q.s on Dwork's "p-adic cycles", Mazur's "p-adic variations":

You could read Mazur's article in the $p$-adic monodromy volume. And also Katz's Travaux de Dwork, as well as his two articles on Serre--Tate theory (LNM 828?), and the accompanying article of Deligne and Illusie on K3 surfaces. You could also read Gross's Tameness Criterion paper in Duke from the late 80s, which uses Dwork's ideas and related $p$-adic techniques. And there is Nygaard's article on the Tate conjecture for K3's over finite fields.

Dwork is difficult, and I don't recommend reading him in a vacuum or for casual entertainment. But his ideas and insights are very deep, and very original. (His actual techniques are very involved, and I am not sure that I would recommend learning them before you learned some more standard ideas from $p$-adic geometry, such as are explained in the above references.)

soft question - How many different representations of pi can we come up with?

Let me explain: a friend of a friend is opening a new pizza restaurant called "Pi", and he's looking to decorate his walls with pi-related material: formulas, equations, theorems w/ proof, diagrams, etc. Any suggestion is welcome, so long as it meets these two criteria:

1. It has to be mathematically correct.


2. It has to be either a representation of pi itself or lead directly to a representation of pi.

So for example, this is okay: $sum_{n=1}^{infty} frac{1}{n^2}$ (because it equals $frac{pi^2}{6}$)
But this is not: $frac{22}{7}$.

How many can we come up with?

ag.algebraic geometry - Recover a morphism from its pullback

EDIT: The original question has been answered, but another difficulty in the proof has appeared. See below.

Let $f,g : X to Y$ be two morphisms of schemes such that the induced pullback functors $f^* , g^* : Qcoh(Y) to Qcoh(X)$ are isomorphic. Can we conclude $f=g$?

If $X$ and $Y$ are affine, then this is quite easy; simply use naturality to conclude that the isomorphism $f^* M cong g^* M$ is multiplication with a global unit in $mathcal{O}_Y$, which is independent from $M$, and deduce $f^# = g^#$. Ok then the claim is also true if $X$ is not affine. But what happens when $Y$ is not affine? The problem is basically, that you cannot lift sections to global sections. Also, the reduction to the affine case works only if we already know that

There is an open affine covering ${U_i}$ of $Y$, such that $f^{-1}(U_i) = g^{-1}(U_i)$ and the pushforward with $U_i to Y$ maps quasicoherent modules to quasicoherent modules.

Laurent Moret-Bailly has proven below that $f$ and $g$ coindice as topological maps. Thus, only the latter concerning the pushforwards is unclear (to me). Everything is ok when $Y$ is noetherian or quasiseparated. What about the general case?

PS:
I'm also interested in questions like this one; is it possible to recover scheme theoretic properties from the categories of quasi-coherent modules? If anyone knows literature about this going beyond Gabriel's and Rosenberg's, please let me know.

ag.algebraic geometry - Is a sub-stack of a scheme a scheme?

Let $f:mathcal{X}rightarrow Y$ be a morphism from an Artin stack to a scheme such that $f$ is an immersion. Then $mathcal{X}$ is automatically an algebraic space, so we're done by Knutson, Algebraic spaces, II.6.16.

Additions prompted by Brian's comment

Assume that $f:mathcal{X}rightarrow Y$ is a schematic map, and that $Y$ is a scheme; then $f$ is the pullback of $f$ over the map of schemes $mathrm{id}_Y$, so $mathcal{X}$ must be a scheme. Knutson needs lemma II.6.16 because he doesn't use the now-standard definition of schematic, but atlases instead.

When using immersion, I always mean $jcirc i$, where $i$ is a closed immersion and $j$ an open one, following EGA I. But I understand that this is not a better choice than the other way round, and that they are only equivalent when the morphism is quasicompact.

nt.number theory - Twin Prime Conjecture Reference

Euclid never made a conjecture about the infinitude of twin primes.

It is possible to guess that he was making a conjecture on the basis of his text but it requires wishful thinking.

Here is the paper where de Polignac makes his general conjecture (which if true also implies the twin prime conjecture).

Regarding the NOVA show, Goldston makes a comment to those behind the NOVA segment (with a response) here:

http://discussions.pbs.org/viewtopic.pbs?t=45116

No one really knows if Euclid made the twin prime conjecture. He does have a proof that there are infinitely many primes, and he or other Greeks could easily have thought of this problem, but the first published statement seems to be due to de Polignac in 1849. Strangely enough, the Goldbach conjecture that every even number is a sum of two primes seems less natural but was conjectured about 100 years before this.

pr.probability - Polish spaces in probability

One simple thing that can go wrong is purely related to the size of the space (polish spaces are all size $leq 2^{aleph_0}$). When spaces are large enough product measures become surprisingly badly behaved. Consider Nedoma's pathology: Let $X$ be a measure space with $|X| > 2^{aleph_0}$. The diagonal in $X^2$ is not measurable.

We'll prove this by way of a theorem:

Let $U subseteq X^2$ be measurable. $U$ can be written as a union of at most $2^{aleph_0}$ spaces of the form $A times B$.

Proof: First note that we can find some countable collection $A_i$ such that $U subseteq sigma(A_i times A_j)$ (proof: The set of $V$ such that we can find such $A_i$ is a sigma algebra containing the basis sets).

For $x in {0, 1}^mathbb{N}$ define $B_x = bigcap { A_i : x_i = 1 } cap bigcap { A_i^c : x_i = 0 }$.

Consider all sets which can be written as a (possibly uncountable) union of $B_x times B_y$ for some $y$. This is a sigma algebra and obviously contains all the $A_i times A_j$, so contains $A$.

But now we're done. There are at most $2^{aleph_0}$ of the $B_x$, and each is certainly measurable in $X$, so $U$ can be written as a union of $2^{aleph_0}$ sets of the form $A times B$.

QED

Corollary: The diagonal is not measurable.

Evidently the diagonal cannot be written as a union of at most $2^{aleph_0}$ rectangles, as they would all have to be single points, and the diagonal has size $|X| > 2^{aleph_0}$.

tag removed - Strong Bezout's Identity?

Let ${ a_i }_{i=1}^N $ be a set of elements of the ring of integers, $mathbb{Z}_D$ and define $g = text{gcd}(a_1, a_2,ldots, a_N, D)$. Then Bezout's Identity states that there exists another set ${ x_i }_{i=1}^N $ such that

$sum_{i=1}^N a_i x_i equiv g bmod D$

For my work, I needed to show that such a solution set ${ x_i }_{i=1}^N $ exists with an ADDITIONAL requirement that $x_1$ must be coprime to $D$. I managed to prove this stronger version of Bezout's Identity using Chinese Remainder Representation (correctly I hope).

My question : Is this result well-known under another name? Do you know of any references discussing this result? Or is this a special case of an even stronger form of Bezout's Identity?

rt.representation theory - Signed and unsigned Hecke algebra canonical basis

You probably know all of this already, but here goes...

Write $C'_w = T_w + sum_{x < w} p_{x,w} T_x$ where $p_{x,w} in umathbb{Z}[u]$. Now, the other basis can be defined by applying the involutive automorphism $b: mathcal{H}_n to mathcal{H}_n$, given by $b(T_w)=T_w$ and $b(u)=-u^{-1}$.

Define $C_w := b(C'_w)$.

Since, $b$ commutes with the bar involution, this basis is bar invariant as well.

Explicitly, $C_w = T_w + sum_{x < w} (-1)^{ell(w)+ell(x)} bar p_{x,w} T_x$.

So $C_w = bar{P}^{-1} P C'_w$ which seems hard to compute in general.

rt.representation theory - Is there analogue of Peter-Weyl theorem for non-compact or quantum group

Marty's answer discusses the Plancherel formula, and in a comment on his answer, I mentioned Harish-Chandra's work on the Plancherel formula in the case of reductive Lie groups. Yemon Choi's answer also mentions the case of semisimple Lie groups as being easier than the general case. The point of this answer is to elaborate slightly on my comment, and to point out that, while the semi-simple case might be easier, it is a very substantial piece of mathematics; indeed, it is essentially Harish-Chandra's life's work.

When Harish-Chandra began his work, it was known (thanks to Mautner?) that semisimple Lie groups were type I, and hence that for such a group $G$, the space $L^2(G)$ admits a
well-defined direct integral decomposition into irreducibles (typically infinite
dimensional, if $G$ is not compact). However, this is a far cry from knowing the precise
form of the decomposition.

The most fundamental, and difficult, question, turns out to
be whether there are any atoms in the Plancherel measure, i.e. whether $L^2(G)$ contains
any non-zero irreducible subspaces, i.e. whether the group $G$ admits discrete series.
This was solved by Harish-Chandra, who established his famous criterion: $G$ admits discrete series if and only if it contains a compact Cartan subgroup. He also gave a complete enumeration of the discrete series representations up to isomorphism, and described
their characters via formulas analogous to the Weyl character formula.

Harish-Chandra then want on to describe
the Plancherel measure on $L^2(G)$ inductively in terms of direct integrals
of parabolic inductions of discrete series represenations of Levi subgroups of $G$.
(It is the appearance of Levi subgroups, which are always reductive but typically
never semisimple, that also forces one to generalize from the semisimple to the reductive case.)

After completing the theory of the Plancherel measure for reductive Lie groups, he then went on to develop the analogous theory for $p$-adic reductive groups. However, in this case, one still doesn't have a complete enumeration of the discrete series representations in general: there are certain "atomic" discrete series representations, called "supercuspidal", which have no analog for Lie groups, and which aren't yet classified in general (i.e. for all $p$-adic reductive groups).

Harish-Chandra's work, as well as standing on its own as an amazing edifice, was a central
inspiration for Langlands in his development of the Langlands program, and remains
at the core of the Langlands proram today.

For a very nice introduction to Harish-Chandra's work, and the surrounding circle of ideas,
one can read this article by Rebecca Herb.

ho.history overview - What is Shimura referring to by "an incorrect formula given by Minkowski... known to most experts."

I do not know where in the original sources, but the topic under discussion is the mass formula for integral quadratic forms. The accepted source with correct information is Conway and Sloane,

Low Dimensional Lattices. IV. The Mass Formula

Proceedings of the Royal Society of London, A 419, 259-286 (1988).

In the actual publication, the tables are sprinkled throughout, and I found it difficult to read the text. I have some sort of preprint around here where the tables are all at the end, easier to find what you want. But it still takes some real patience, and in fact some imagination, to use properly.

I can recommend the book by the same Conway and Sloane, called Sphere Packings, Lattices, and Groups.

terminology - Do you need to say what left-unique and right-unique means?

I am talking about a relation that is what Wikipedia describes as left-unique and right-unique. I never heard these terms before, but I have heard of the alternatives (injective and functional). The question is, which terminology do you recommend? Should I include short definitions? (The context is a text in the area of formal methods. I'm not sure if this helps.)

These are some trade-offs that I see:

• I think that left-unique and right-unique are not widely known, but I'm not sure at all.


• functional is overloaded


• injective sounds too fancy (subjective, of course)


• left-unique and right-unique are symmetric (good, of course)

Edit: It seems the question is unclear. Here are more details. I describe sets X and Y and then say:

1. now we must find an injective and functional relation between sets X and Y such that...


2. now we must find a left-unique and right unique relation between sets X and Y...

Which one do you recommend? What other information would you add? The relation does not have to be total. For example, various different ranges correspond to different 'feasible' relations. Technically I should not need to say that the relation does not have to be total, but will many people assume that it has to be total if I don't say it?

ra.rings and algebras - characterization of a submodule

First, Let me talk about the "correct definition" of module over non-unital ring(not necessarily commutative) and how this definition coincide with usual definition of module over unital ring in particular case

First we study $R-mod_{1}$={category of associative action of $R$ on $k$-mod}= {($M$,$Rbigotimes _{k}Mrightarrow M$).

$r_{1}(r_{2}z)=(r_{1}r_{2})z$}

Let $R_{1}=Rbigoplus k$ be an untial $k$-algebra with usual multiplication. And we have the categorical equivalence as: $R-mod_{1}approx R_{1}-mod$

Now,we define module over non-unital algebra $R$ as $R-mod=R_{1}-mod/(Tors_{R_{1}})^{-}$, where $(Tors_{R_{1}})^{-}$ is Serre subcategory of $R_{1}-mod$

$R_{1}-modoverset{q_{R}^{*}}{rightarrow}R-mod$ is a localization functor having right adjoint functor.

Trivial Example:

if $R$ has is an unital $k$-algebra. Then $R_{1}-mod$ is equivalent to $R-mod$

Less Trivial example in commutative case:

Consider affine line $k[x]$. Let $R=xk[x]$(maximai ideal of $k[x]$). Then $R-mod$=$Qcoh(mathbb{A}^{1}-{0}$). It is a cone.

Toy general case:

Let $m$ is a two-sided proper ideal of associative commutative unital ring $A$. Then: we have

$m-mod$=$A-mod/({Mepsilon A-mod|mcdot M=0})^{-}$, where$T^{-}$ is smallest Serre category containing $T$. It is clear that is equivalent to Qcoh(Complement of $mathbb{V}(m)$),where
$mathbb{V}(m)$ is closed subvariety determined by $m$.

Now, I should stop here and write another(maybe)post on definition of sub-module. There are several reference:

Gabriel, Pierre Des catégories abéliennes. (French) Bull. Soc. Math. France 90 1962 323--448
Kontsevich-Rosenberg Noncommutative spaces and flat descent

Gabber-RameroAlmost Ring Theory

pr.probability - Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?

For some reason my thinking is very fuzzy today, so I apologize for the following rather silly question below...

Let $T$ be an ergodic transformation of $(X,Omega, mathbb{P})$ and let $X$ be partitioned into $n < infty$ disjoint sets $R_j$ of positive measure. For $x in R_k$ define $tau(x) := inf {ell>0:T^ell x in R_k}$. The Kac lemma (see, e.g. http://arxiv.org/abs/math/0505625) gives that $int_{R_k} tau(x) dmathbb{P}(x) = 1$.
Now $int_X tau(x) dmathbb{P}(x) = sum_k int_{R_k} tau(x) dmathbb{P}(x) = n$, or equivalently $mathbb{E}tau = n$.

Can anyone provide a sanity check on the above assertion that the expected return time is just the size of the partition? I've never seen this explicitly stated as a corollary of the Kac lemma, which seems odd.

ac.commutative algebra - Commutative Noetherian Domains of Krull Dimension One

This follows from a direct generalization of the Noether normalization lemma. It is covered in these notes from Mel Hochster. These notes prove it in a pretty general form (when the base ring is only an integral domain rather than a field).

Edit: A sufficient condition is that the algebra is finitely generated, but it is clearly not necessary.

Edit 2: I misread the question. I thought he was asking if A is finitely generated over some polynomial algebra (including infinitely generated polynomial algebras).