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.

---

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).

differential topology - Understanding iterated integrals

As mentioned by Emerton, iterated integrals only work well for unipotent representations of $pi_1(X,x)$.

The reason for this is that differential forms are abelian objects: for paths $gamma_i$, and a closed 1-form
$alpha in Omega^1(X)$,
$$
int_{gamma_1gamma_2} alpha = int_{gamma_1} alpha + int_{gamma_2} alpha = int_{gamma_2gamma_1} alpha
$$
That and homotopy invariance implies that integration induces a pairing
$$
int: H^1(Omega(X)) otimes mathbb{Q} [pi_1(X,x)]^{ab} to mathbb{C}
$$
where $mathbb{Q}[pi_1(X,x)]^{ab} = H_1(X;mathbb{Q})$.

By considering iterated integrals, we can go one step further. The above pairing has a generalization as
$$
int: H^0(Ch^{leq n}(X)) otimes mathbb{Q}[pi_1(X,x)]/J_x^{n+1} to mathbb{C}
$$
where $Ch^{leq n}(X)$ is the lenght $leq n$ part of Chen's complex and $J$ is the augmention ideal generated by the $(gamma-1)$.
So iterated integrals describe the pro-unipotent (Malcev) completion $pi_1^{uni}(X,x)$ of $pi_1(X,x)$.
And $varinjlim_n H^0(Ch^{leq n}(X))$ can be thought of as the Hopf algebra of functions on the (pro-unipotent) de Rham fundamental group. One can also define a Hodge and weight filtration and get a pro-mixed Hodge structure on $varprojlim mathbb{Q}[pi_1(X,x)]/J^n$.

This allows to extend the correspondance between unipotent local systems and unipotent representations the fundamental group to the de Rham and even the Hodge or motivic setting. Of course there are technical conditions for things to go smoothly. Basically one needs $X$ to be a unipotent $K(pi,1)$ in the sense that $H^i (pi_1^{uni}(X,x),mathbb{Q}) to H^i(X,mathbb{Q})$ is an isomorphism. In the language of rational homotopy theory this corresponds to 1-minimality.

PS: It is not clear how to proceed to go beyond the unipotent setting. I think Hain, Matsumoto and Terasoma have a generalization of the bar construction that works for more general "relative completions" but nothing has been published yet.

homology - Poincaré quasi-isomorphism

Both the cell complex, $C$, and the dual cell complex $C'$ are refined by the first barycentric subdivision $BC$. There are maps $C to BC$ and $C' to BC$, sending a cell $sigma$ to the sum of all cells of the same dimension contained in $sigma$; these maps are both quasi-isomorphisms.

So, if you allow me to formally invert quasi-isomorphisms, I'm done.

Is the question whether there is an honest map of chain complexes between $C$ and $C'$, without subdividing?

---

UPDATE Here is something you can do, and something you can't do.

With $C$ and $BC$ as above, and $r : C to BC$ the refinement map, there is a homotopy inverse $s: BC to C$. (More precisely, $C to BC to C$ is the identity, and $BC to C to BC$ is homotopic to the identity.) Working the same trick with $r' : C' to BC$, we get quasi-isomorphisms between $C$ and $C'$ which are homotopy inverse to each other. As you will see, however, this construction is very nongeometric and inelegant.

Construction: Let $q:BC to Q$ be the cokernel of $C to BC$. An easy computation checks that each $Q_i$ is free. Since $C to BC$ is a quasi-isomorphism, $Q$ is exact. An exact complex of free $mathbb{Z}$ modules must be isomorphic to a direct sum of complexes of the form $cdots to 0 to mathbb{Z} to mathbb{Z} to 0 to cdots$. Choose such a decomposition of $Q$, so $Q_i = A_{i+1} oplus A_{i}$ and the map $Q_i to Q_{i-1}$ is the projection onto $A_{i}$.

Now, consider the map $q_i^{-1}(A_i) to A_i$ in degree $i$. This is surjective, and $A_i$ is free, so choose a section $p^1_i$. We also define a map $p^2_i$ from the $A_{i+1}$ summand of $Q_{i}$ to $BC_i$ by $p^2_i = d p^1_{i+1} d^{-1}$. In this way, we get maps $p_i = p^1_i oplus p^2_i: Q_{i} to BC_i$ which give a map of chain complexes.

We note that $qp: Q to Q$ is the identity. Therefore, $1-pq$, a map from $BC to BC$, lands in the subcomplex $C$ and gives a section $s:BC to C$. Proof of the claim about homotopies will be provided on request.

---

On the other hand, here is something you can't do: Get the quasi-isomorphism to respect the symmetries of your original space. For example, let $C$ be the chain complex of the cube, and $C'$ the chain complex of the octahedron. I claim that there is no quasi-isomorphism $C to C'$ which commutes with the group $S_4$ of orientation preserving symmetries.

Consider what would happen in degree $0$. A vertex of the cube must be sent to some linear combination of the vertices of the octahedron. By symmetry, it must be set to
$$a (mbox{sum of the "near" vertices}) + b (mbox{sum of the "far" vertices})$$
for some integers $a$ and $b$. But then the map on $H_0$ is multiplication by $3(a+b)$, and cannot be $1$.

---

I imagine you want something stronger then my first answer, but weaker than my second. I am not sure what it it, though.

gn.general topology - existence of a connected set with given connected projections.

Edit: This answer is wrong. But, since Anton based his answer on this idea, I am leaving it up.

No. Let $A$, resp. $B$, be the graph $x=f(y)$, resp. $x=g(z)$, for some continuous functions $f$ and $g$. Pick numbers $0<d<b<a<c<1$, and arrange things so that $f(0)=g(0)=0$, $f(1)=g(1)=1$, and $f(y)$ grows from 0 to $a$, then decreases to $b$, then finally increases to 1, whereas $g(z)$ grows from $0$ to $c$, decreases to $d$, and increases to 1. (Here, $a$, $b$, $c$, $d$ are function values, i.e., values of $x$, not of the arguments $y$ or $z$.)

Now if $C$ projects to $A$ and $B$, respectively, it is not hard to see that $(0,0,0)in C$ and $(1,1,1)in C$. Trying to move $(x,y,z)$ continuously from the former to the latter, a contemplation of how $x$ must grow and shrink in order to maintain $x=f(y)$ and $x=g(z)$ leads to a contradiction. (Hard to put in words, but a pair of graphs reveals it I think.)

nt.number theory - Finite set of (perfect power) polynomial values?

As Qiaochu said in the comments, you must include Pell type equations as a special case, because they are the only counter example. At least for $k=2$, Siegel's theorem on integral points on algebraic curves implies that if your polynomial $P(x)$ has at least three distinct roots then $P(n)=m^2$ has only finitely many solutions. So your conjecture is, in particular, true for irreducible polynomials of degree higher than 2.

Anyway, for the general question for any exponent it's better that you read the full story in
"The diophantine equation f(x)=g(y)" by Y. Bilu and R.F. Tichy.

ag.algebraic geometry - When is the base change morphism an isomorphism?

This is a rewrite of a previous question, which was in turn a follow up question to Leray-Hirsch principle for étale cohomology The motivation is to clarify some points in Torsten Ekedahl's answer there. (Some time ago I also posted a brief version as a comment there.)

Let $Xto Y$ be a surjective morphism of connected algebraic varieties over an algebraically closed field $k$ and let $F$ be a constructible sheaf on $X$ whose stalks are of order prime to $char (k)$. For any closed geometric point $yin Y$ we have the natural base change map $(R^i f_*F)_yto H^i(X_y,F)$ where $X_y$ is the fiber over $y$.

Can one deduce that all $R^i f_{ast} F,i>0$ are zero assuming that for each closed $y$ we have $H^i(X_y,F)=0,i>0$? If the answer is yes, I would be interested in knowing whether there is an analog of this for the $l$-adic cohomology with $mathbf{Q}_l$ coefficients.

[upd: nope, as per Dustin's comment below.]

Note that in the 'etale case Proposition 4.12, Chapter VI in Milne, 'Etale cohomology gives a stronger conclusion ($H^{ast}(X,F')to H^{ast}(Y,f^{-1}F')$ is an iso for any $F'$) under a stronger hypothesis that the $H^{ast}(X_y,F'')$ vanishes in positive degrees for any geometric point $y$, which may or may not be closed, and for all $F''$.

So here is a side-question: does it suffice to check the vanishing for closed points?

At first I thought the answer would be yes, which is why I accepted Torsten's answer to the above question, but then I realized I don't know how to prove this.

[upd: .. and for a good reason since it's false.]

Remark: if $f$ is proper, the statement is true by the proper base change theorem, but as pointed out by Dustin Clausen in the above thread, assuming $f$ smooth does not help. In fact, if $f:mathbf{A}^2setminus 0 to mathbf{A}^1$ is the projection to one of the coordinate axes, then (unless I'm mistaken) $R^3 f_*underline{A}$ is non-zero at the origin (where $A$ is a finite group of order prime to $char(k)$), but the cohomology of the preimage of the origin vanishes in degrees $>1$.

Here is some more motivation and one more question. If $G$ is a Lie group that acts on a sufficiently nice topological space $X$, then the stack cohomology of the quotient is simply the cohomology of the Borel construction $Xtimes EG/G$ where $EG$ is the universal $G$-bundle and the action is diagonal. The Borel construction is mapped to $X/G$ (assuming the topological quotient is reasonable) and the fiber over $[x]in X/G,xin X$ is the classifying space of the stabilizer of $x$. So if the action is with finite stabilizers, $H^{ast}(X/G,mathbf{Q})$ is isomorphic to the cohomology of the Borel quotient. One can try to mimick this in the 'etale setting, but there are some technical difficulties: in the topological case there are some tools (maximal compact subgroup, the existence of slices for compact group actions, ...) which don't seem to have easy analogs.

However, I would still guess that if $G$ is an algebraic group that acts with finite stabilizers on an algebraic variety $X$ (I'd be willing to assume it smooth but not affine), the cohomology of the quotient with $mathbf{Q}_l$ coefficients should be the same as the cohomology of the Borel construction and it seems plausible that someone has looked into this before.

So I would like to ask: is there a reference for that?

(Note that although $EG$ does not exist as an algebraic variety, it can be approximated by algebraic varieties: e.g. if $G=GL_n(k)$, one can take the spaces of $n$-frame bundles in $k^N,Ntoinfty$.)

at.algebraic topology - Mumford conjecture: Heuristic reasons? Generalizations? ... Algebraic geometry approaches?

Here is a belated and perhaps too naive answer for your first question, i.e. a heuristic reason for why to believe in the Mumford conjecture.

One can think of cohomology classes on $M_g$ as characteristic classes for families of curves. That is, a class is the same thing as a rule such that whenever you are given a smooth family of curves $X to B$ of genus $g$, this rule associates a cohomology class in $H^bullet(B)$ in a functorial manner. (Topologically, you can also think of it as a characteristic class of oriented surface bundles of genus $g$, since $mathrm{MCG}(Sigma_g) simeq mathrm{Diff}^+(Sigma_g)$.)

In a similar way, one might then think of a cohomology class on $M_infty$ as a rule that assigns a cohomology class to a family of curves of arbitrary genus, or as a characteristic class of arbitrary oriented surface bundles. Of course this should be made precise since "functoriality" of such a characteristic class seems meaningless without any way of comparing surface bundles of different genus, and so one needs to define comparison maps between different moduli spaces by working with boundary components, gluing on tori/pants, etc. Nevertheless the intuitive picture is clear: a class on $M_infty$ should be a rule that assigns in a uniform manner, to any family of curves $X to B$ whatsoever, a cohomology class in $H^bullet(B)$.

Now the $kappa$ classes seem to fit the bill for being classes on $M_infty$: given any $pi colon X to B$ whatsoever, we may form the vertical tangent bundle, take its Euler class, multiply, push forward. This seems as canonical as one could hope for. Are there any others? Well, there are the $lambda$-classes, i.e. the Chern classes of $pi_ast Omega_{X/B}^1$, but Mumford showed via Grothendieck-Riemann-Roch that these are polynomials in the $kappa$'s. It's hard to think of anything else.

Now surjectivity of $mathbf Q[kappa_1,kappa_2,ldots] to H^bullet(M_infty)$ asserts that these obvious classes are really the only ones that you can write down in a uniform way, and injectivity says that there are no uniform relations between the $kappa$'s (i.e. all relations are "low-genus accidents"). Now one might believe in the Mumford conjecture simply because people thought hard about these things and could not find any other genus-invariant characteristic classes, nor any genus-invariant relations between the $kappa$'s. Mumford's conjecture is the simplest possible explanation for this failure.

co.combinatorics - Asymptotics of a hypergeometric series/Taylor series coefficient.

I was planning on figuring this problem out for myself, but I also wanted to try out mathoverflow. Here goes:

I wanted to know the asymptotics of the sum of the absolute values of the Fourier-Walsh coefficients of the "Majority" function on 2n+1 binary inputs. Long story short, that boiled down to finding the asymptotics of the following quantity:

L[n] := sum{k=0..n} F[n,k],

where

F[n,k] := (2n+1)!! / [(2k+1) k! (n-k)!].

Here is the set of steps that a naive person such as me always follows in such a scenario:

Step 1. Type it into Maple. In this case, Maple reports back

L[n] = ( (2n+1) (2n choose n) / 2^n ) hypergeom([1/2, -n], [3/2], -1).

I don't know exactly what this hypergeom is, nor how to evaluate its asymptotics. I can't seem to get Maple to tell me (with asympt()) either.

Step 2. Evaluate the first few terms (1, 4, 14, 48, 166, 584, 2092, etc.) and type it into OEIS. It's clearly sequence A082590. There are many definitions given for this sequence; One definition is (basically) the above hypergeometric formula. Another is that it is

2^n sum{k=0..n} 2^(-k) (2k choose k).

Given this, it's pretty easy to deduce from Stirling that the rough asymptotics is Theta(2^(2n) / sqrt(n)). Actually, I originally didn't notice this simple definition on the OEIS page. Instead, the simplest thing I noticed was the title definition,

the [x^n] coefficient of 1/((1-2x) sqrt(1-4x)).

Since that was the simplest thing I initially noticed, I wondered how to find the asymptotics of that. I was sure many people would know that, but mathoverflow didn't exist at the time. So...

Step 3. Email Doron Zeilberger out of the blue, asking for help. He very kindly responded: "You could use the contour integral... or my favorite way would be to use my Maple packages:

read AsyRec:
read EKHAD:
Asy(AZd( 1/(1-2*x)/(1-4*x)^(1/2)/(x^(n+1)),x,n,N)[1],n,N,2);

which produces

2^(2*n)(1/n)^(1/2)(1+3/8/n+121/128/n^2)."

Great! There's the precise answer! But I don't know much about how Zeilberger's packages work, don't know what the contour integral is, and anyway, it's all roundabout story.

If you were writing this in a paper, what would be the shortest way to go from the definition of L_n to the fact

L[n] = (2^(2n) / sqrt(n)) (1+o(1))?

soft question - Which math paper maximizes the ratio (importance)/(length)?

Any of three papers dealing with primality and factoring that are between 7 and 13 pages:

First place: Rivest, R.; A. Shamir; L. Adleman (1978). "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems". Communications of the ACM 21 (2): 120–126.

Runner-up: P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, Proc. 35nd Annual Symposium on Foundations of Computer Science (Shafi Goldwasser, ed.), IEEE Computer Society Press (1994), 124-134.

Honorable mention: Manindra Agrawal, Neeraj Kayal, Nitin Saxena, "PRIMES is in P", Annals of Mathematics 160 (2004), no. 2, pp. 781–793.

pr.probability - How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

After Bill's latest clarifications in the commentary on Critch's answer, I think the question is interesting again. My take:

One thing that always seemed to fall through the cracks when I learned about probability theory is that probability is intricately tied to information, and probabilities are only defined in the context of information. Probabilities aren't absolute; two people who have different information about an event may well disagree on its probability, even if both are perfectly rational. Similarly, if you get new information relevant to a certain event, then you should probably reevaluate what you think is the probability that it will occur. Your particular problem is interesting because the new information you get isn't enough for you to revise that probability by purely mathematical considerations, but I'll get to that in good time.

With the previous paragraph in mind, let's compare two games:

G1. You are given two closed doors, A and B, with two numbers behind them, and your goal is to choose the door with the higher number. You are given no information about the doors or numbers.

G2. You are given two closed doors, A and B, with two numbers behind them, and your goal is to choose the door with the higher number. You are allowed to look behind one of the doors and then make your choice.

For the first game, by symmetry, you clearly can't do better than choosing a door randomly, which gives you a success probability of exactly 1/2. However, the second game has a chance of being better. You are playing for the same goal with strictly more information, so you might expect to be able to do somewhat better. [I had originally said that it was obviously better, but now I'm not so sure that it's obvious.] The tricky thing is quantifying how much better, since it's not clear how to reason about the relationship between two numbers if you know one of the numbers and have no information about the other one. Indeed, it isn't even possible to quantify it mathematically.

"But how can that be?" you may ask. "This is a mathematical problem, so how can the solution not be mathematically definable?" There's the rub: part of the issue is that the problem isn't formulated in a mathematically rigorous way. That can be fixed in multiple ways, and any way we choose will make the paradox evaporate. The problem is that we're asked to reason about "the probability of answering the question correctly," but it's not clear what context that probability should be computed in. (Remember: probabilities aren't absolute.) In common probability theory problems and puzzles, this isn't an issue because there is usually an unambiguous "most general applicable context": we should obviously assume exactly what's given in the problem and nothing else. We can't do that here because the most general context, in which we assume nothing about how the numbers $x$ and $y$ are chosen, does not define a probability space at all and thus the "probability of answering the question correctly" is not a meaningful concept.

Here's a simpler ostensible probability question that exhibits the same fallacy: "what's the probability that a positive integer is greater than 1,000,000?" In order to answer that, we have to pick a probability distribution on the positive integers; the question is meaningless without specifying that.

As I said, there are multiple ways to fix this. Here are a couple:

I1. (Tyler's interpretation.) We really want the probability of answering the question correctly given a particular $x$ and $y$ to be greater than 1/2. (The exact probability will of course depend on the two numbers.)

I2. (Critch's interpretation.) More generally, we want the probability of answering correctly given a particular probability distribution for $(x,y)$ to be greater than 1/2. (The exact probability will of course depend on the distribution.)

(Those two are actually equivalent mathematically.) Clearly, if we knew what that distribution was, we could cook up strategies to get a success probability strictly above 1/2. That's pretty much obvious. It is not nearly as obvious that a single strategy (such as the one in the statement of the question) can work for all distributions of $(x,y)$, but it's true, as Bill's proof shows. It's an interesting fact, but hardly paradoxical now.

Let me summarize by giving proper mathematical interpretations of the informal statement "there is a strategy that answers the question correctly with probability strictly greater than 1/2," with quantifiers in place:

(1a) $exists text{ strategy } S: forall x, y: exists delta > 0$: $S$ answers correctly on $x$, $y$ with probability at least $1/2 + delta$.

(1b) $exists text{ strategy } S: forall text{ probability distributions } P text{ on } mathbb{N}^2: exists delta > 0$: $S$ answers correctly, when $x$, $y$ are chosen according to $P$, with probability at least $1/2 + delta$.

I think with the proper quantifiers and the dependence on $x$ and $y$ explicit, it becomes a cool mathematical result rather than a paradox. Actually, based on my arguments at the beginning, it's not even that surprising: we should expect to do better than random guessing, since we are given information. However, simply knowing one number doesn't seem very useful in determining whether the other number is bigger, and that's reflected in the fact that we can't improve our probability by any fixed positive amount without more context.

Edit: It occurred to me that the last part of my discussion above has a nonstandard-analytical flavor. In fact, using the first version of the formula for simplicity (the two being equivalent), and the principle of idealisation, I think we immediately obtain:

(2) $exists text{ strategy } S: exists delta > 0: forall text{ standard }x, y:$ $S$ answers correctly on $x$, $y$ with probability at least $1/2 + delta$.

(Please correct me if I'm wrong.)
The number $delta$ is not necessarily standard, and a basic argument shows that it must actually be smaller than all standard positive reals, i. e., infinitesimal. Thus, we can say that being able to look behind one door gives us an unquantifiably small, infinitesimal edge over random guessing. That actually meshes pretty well with my intuition! (It might still a nontrivial observation that the strategy $S$ can be taken standard; I'm not sure about that...)

ag.algebraic geometry - Complex vector bundles that are not holomorphic

Here is the answer to the question, kindly explained to me by Burt Totaro.

EDITED. This is an OPEN PROBLEM.

0) Apperently in the case of CP^n exitence of a complex bundle without holomorphic structure is still an OPEN PROBLEM. Though it is belived that there should be plenty of examples starting from $nge 5$, coming from topologically indecomposable rank two bundles, apperently no such bundle was proven to be non-holomorphic as for today.

1) A toplologically non-trivial rank 2 complex bundle with $c_1=0$, $c_2=0$ was constructed in

Rees, Elmer, Some rank two bundles on $P_{n}C$, whose Chern classes vanish. Variétés analytiques compactes (Colloq., Nice, 1977).

It was also claimed in this article that this bundle does not admit a holomorphic structure. But this claim was deduced from an article that contained a gap. So it is yet unknown if this particular bundle has holomorphic strucutre or not. This is discussed in
M. Schneider. Holomorphic vector bundles on P^n. Seminaire Bourbaki
1978/79, expose 530.

This is why Okonek and Schneider write in their book p. 137 that this is an open problem.

2) On the positive side it is proven that every complex vector bundle on a smooth projective rational 3-fold has an holomorphic structure.

C. Banica and M. Putinar. On complex vector bundles on projective
threefolds. Invent. Math. 88 (1987), 427-438.

3) If one wants to construct examples of bundles on projective manifolds that are not necesserely Fanos it is possible to use the fact that the integral Hodge conjecture fails. Namelly there are elements in $H^{2p}(X,Z)$ which are in $H^{p,p}$
but which are not represented by an algebraic cycle. Kollar gave
such examples with $dim(X)=3$. A recent reference, which refers back to earlier results, is:

C. Soule and C. Voisin. Torsion cohomology classes and algebraic cycles
on complex projective manifolds. Adv. Math. 198 (2005), 107-127

4) One reason to expect examples of such bundles in higher dimensions is
Schwarzenberger's conjecture that every rank-2 algebraic vector bundle E,
on $P^n$ with $nge 5$ is a direct sum of two line bundles. So, for example,
if $c_1(E)=0$ then $c_2(E)=-d^2$ for some integer d, according to the
conjecture.

homological algebra - The sharp 3x3 lemma: a proof by universal properties?

I was reading this paper a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category.

I'll reproduce the diagram:

$$ matrix{
&&0&&0&&0
cr&&downarrow&&downarrow&&downarrow
cr &&A_1 & & B_1& &C_1
cr &&downarrow & &downarrow&&downarrow
cr &&A_2 & to & B_2 & to & C_2 &
cr &&downarrow &&downarrow&&downarrow
cr 0&to&A_3 & to & B_3 & to & C_3
} $$

With all rows and columns exact. (This diagram lives in an abelian category).

Show that there exists an exact sequence $A_1to B_1to C_1$ making the diagram commute.

Sure, it's not too hard with elements, I mean, it's just part of the snake lemma. However, proving it with universal properties is another story. By the universal property of the kernel, there are natural maps $A_1 to B_1$, and $B_1 to C_1$. Proving that this is exact is another story entirely. I believe I was able to show (I tried this a few months ago) that the top left corner (not counting zeros) is cartesian (a pullback square), but I still couldn't prove exactness.

I repeat, this is for a proof without elements. It should rely only on the definition of an abelian category and universal properties. If you happened not to click the link to the paper, the whole point is a proof without elements. I'd really like to see at least one proof in homological algebra actually done from the definition, just because it would be extremely instructive.

qa.quantum algebra - Are the “identity object axioms” in the definition of a braided monoidal category needed? (Answered: No)

I am asking because the literature seems to contain some inconsistencies as to the definition of a braided monoidal category, and I'd like to get it straight. According Chari and Pressley's book ``A guide to quantum groups," a braided monoidal category is a monoidal category $mathcal{C}$ along with a natural system of isomorphisms $sigma_{U,V}: U otimes V rightarrow V otimes U$ for all pairs of objects $U$ and $V$, such that

(i) The ``Hexagon" axioms (two commutative diagrams) hold.

(ii) The ``identity object" axioms: $rho_V= lambda_V circ sigma_{{bf 1},V}: {bf 1} otimes V rightarrow V$
and
$lambda_V= rho_V circ sigma_{V, {bf 1}}: {V} otimes {bf 1} rightarrow V$,
where $lambda_V$ and $rho_V$ are the isomorphisms of $V otimes {bf 1}$ and ${bf 1} otimes V$ with $V$ that are part of the definition of monoidal category. See Chari-Pressley Definitions 5.2.1 and 5.2.4. They use the term "quasitensor category," but note on p153 that the term "braided monoidal category" is equivalent.

However, in some references (ii) seems to have been dropped. I am thinking in particular of Definition 3.1 is this
expository paper,
and the
wikipedia article.
The wikipedia article goes further, and suggests that (ii) somehow follows from (i) and the axioms of a monoidal category. So, my questions are.

1) Is (ii) needed? That is if we do not impose (ii), does it follow from (i) and the axioms of a monoidal category?

2) If (ii) is needed, can someone provide an example demonstrating why? That is, provide an example of a monoidal category $mathcal{C}$ along with maps $sigma_{U,V}$ such that (i) holds but (ii) fails. Alternatively, if (ii) is not needed, I'd like a proof (or reference to a proof) that it follows from other axioms.

cv.complex variables - Restriction of a complex polynomial to the unit circle

I am pretty sure that the following statement is true. I would appreciate any references (or a proof if you know one).

Let $f(z)$ be a polynomial in one variable with complex coefficients. Then there is the following dichotomy. Either we can write $f(z)=g(z^k)$ for some other polynomial $g$ and some integer $k>1$, or the restriction of $f(z)$ to the unit circle is a loop with only finitely many self-intersections. (Which means, more concretely, that there are only finitely many pairs $(z,w)$ such that $|z|=1=|w|$, $zneq w$ and $f(z)=f(w)$.)

EDIT. Here are a couple reasons why I believe the statement is correct.

1) The statement is equivalent to the following assertion. Consider the set of all ratios $z/w$, where $|z|=1=|w|$ and $f(z)=f(w)$ (here we allow $z=w$). If $f$ is a nonconstant polynomial, then this set is finite.

[[ Here is a proof that the latter assertion implies the original statement. Suppose that there are infinitely many pairs $(z,w)$ such that $|z|=1=|w|$, $zneq w$ and $f(z)=f(w)$. Then some number $cneq 1$ must occur infinitely often as the corresponding ratio $z/w$. However, this would imply that $f(cz)=f(z)$ (as polynomials). It is easy to check that this forces $c$ to be a root of unity, and if $k$ is the order of $c$, then $f(z)=g(z^k)$ for some polynomial $g(z)$. ]]

Going back to the latter assertion, note that the set of all such ratios is a compact subset of the unit circle, and it is not hard to see that 1 must be an isolated point of this set. So it is plausible that the whole set is discrete (which would mean that it is finite).

2) If I am not mistaken, experiments with polynomials that involve a small number of nonzero monomials (such as 2 or 3) also confirm the original conjecture.

at.algebraic topology - H-space structure on infinite projective spaces

There's also a different way of writing down the $H$-space structure, that I like for its algebro-geometric flavor. (I'll talk about $mathbb{C}P^infty$ here, and $mathbb{R}P^infty$ should be analogous.)

Regarding $mathbb{C}P^infty$ as a classifying space for complex line bundles, we know that this $H$-space structure is supposed to implement "tensor product of line bundles". In a (not very explicit) sense this tells us the homotopy class of $mathbb{C}P^infty times mathbb{C}P^infty to mathbb{C}P^infty$: It represents the line bundle $mathcal{O}(1,1) = p_1^* mathcal{O}(1) otimes p_2^* mathcal{O}(1)$. We can use this description to write down a much more explicit (and classical) explicit representative.

First, let's recall what the analogous picture looks like for finite projective spaces. The line bundle $mathcal{O}(1,1)$ determines (upon picking generating sections) the Segre map
$mathbb{C}P^n times mathbb{C}P^m to mathbb{C}P^{nm+n+m}$ which takes (in homogeneous coordinates)

$([X_0:ldots:X_n] , [Y_0:ldots:Y_m]) mapsto[X_0 Y_0: ldots : X_i Y_j: ldots: X_n Y_m]$

where I'm choosing to be vague on the precise ordering of the coordinates.
(In the end this won't matter up to homotopy, as the maps will become homotopic upon composing with $mathbb{C}P^{nm+n+m} hookrightarrow mathbb{C}P^infty$.)

The analogous formula with infinitely many homogeneous coordinate makes just as much sense, one just has to a good ordering of pairs of non-negative integers. Such an infinite Segre map gives another realization of the $H$-space structure.

computer science - Programming Languages based on Category Theory

CAML by definition is Categorical Abstract Machine Language,

however I am not cetain that you can say that an language explicitly uses category theory. Perhaps you are asking "Are there languages that allow Category Theory Concepts to be easily represented?" or perhaps you are asking if the compilation or interpretation of a particular programming language uses Category Theory in its implementation?

While technically, all Turing-complete capable languages should be equivalently able to express the same set of computations, some languages do so more elegantly than others, allowing the programmer or mathematician to be more eloquent.

I would say LISP and SCHEME, even though based on lambda-calculus, are more connected to the spirit of category theory in concept. While the numbers and integers are conceptually defined as atomic and can be built up from primitives in concept and in theory; in practice, the implementations of SCHEME and LISP and (CLU) tend to take shortcuts to speed up implementation.

The hierarchical ability to pass functions and functions of functions (etc.) as first-class parameters to functions in LISP and SCHEME let you be able to emulate the actions or morphisms of category theory better in that language than others. You just have to start from the ground up, as I have not yet seen a library or package in LISP or SCHEME for category theory.

ag.algebraic geometry - "Every scheme as a sheaf" references?

It is a very nice question. Functor view point algebraic geometry was proposed by Gabriel and later developed by Grothendieck.

Actually, Kontsevich and Rosenberg developed noncommutaive algebraic geometry completely based on this point of view explicitly. They take the presheaf $text{Alg}^{op} rightarrow text{Set}$ as a noncommutative space and developed flat descent theory, the theory of noncommutative smooth space, theory of noncommutative stack. As an interesting example, they defined noncommutative grassmannian, group scheme, general flag vairety as a presheaves and using descent theory of quasi coherent sheaves to glue affine presheaves together according to the so called "smooth topology".

I attended a lecture course last semester, he proved a theorem which "shows" that "One can do algebraic geometry only using presheaves rather than sheaves, if one need sheaves, just take sheafification and all the properties will hold".

There are the following references:

All these papers are available in Max Plank preprint series (search Kontsevich or Rosenberg in "author" and leave other blank empty).

If you take a look at the first paper, just disregard the notion of "Q-category" which is a technique tool to generalize grothendieck topologies because in noncommutative case, flat morphism does not respect to base change in general.

algebraic k theory - Maps between K-groups induced by rings homomorphism

Let $f: Rto S$ be a map between two commutative Noetherian rings. Let $G_0(R)=K_0(mod R)$ be the Grothendieck group of finite generated modules over $R$. It means $G_0(R)$ is the quotient of the free abelian group on all isomorphism class of finitely generated modules over $R$ by the subgroup generated by relations coming from short exact sequences.

If $fd_RS<infty$, one can define a map $f^*:G_0(R)to G_0(S)$ by:
$$f^*([M]) = sum_{igeq0} (-1)^i [Tor_R^i(M,S)] $$

(for reference, see Section 7, Chapter 2
of Weibel's book on K-theory.

Now, if $R$ is not regular or $S$ is not a complete intersection in $R$, then having finite flat dimension is somewhat a miraculous condition. So my question is: Can a map $f^*$ be defined in a more general situation than for finite flat dimension maps?

EDIT: Let me elaborate a little bit because of some interesting answers and comments below (especially Clark's answer). The main motivation I have in mind is the case of $R$ being a hypersurface. Then most $R$- modules have infinite resolutions, but it is well known that their resolution is eventually periodic. So, even though they are not homologically finite, the modules can be homologically described with finite data (i.e. finite number of matrices).

The fact above is crucial in many results I know about hypersurfaces. For a random recent example, see here. In particular, in this situation one can define( at least when $S$ is finite $R$-module):

$$f^*([M])= [Tor^{2n}(M,S)] - [Tor^{2n-1}(M,S)]$$

for sufficiently big $n$.

So that's one of the reasons I wonder if there is more systematic map one can define.