Planar graph drawing

this is not a well defined question. Perhaps you mean that you want to know if a face of a planar embedding of a planar graph is convex ? Note that it's always possible to draw a planar graph so all faces (with the exception of the outer face) are convex

nt.number theory - Why do zeta functions contain so much information?

This question is a major open problem. One theoretical framework has been proposed that would explain why all the various zeta and L-functions (the ones attached to number fields, function fields, modular forms, algebraic varieties, Hecke characters, ...) have the same nice properties (analytic continuation, special values that "contain so much information," ...). It is known as the theory of motives. It is somewhat stymied by the fact that no one can (with proof) define the right category of motives, nor has anyone been able to for some 40 years. Milne's lecture notes "What is a motive?" are a great introduction to the theory; http://www.jmilne.org/math/xnotes/mot.html

For a particular answer to your question, the so-called Equivariant Tamagawa Number Conjecture in the theory of motives encompasses every one of the "information-packed" formulas we have about special values of L-functions. Of course, no one expects a proof any time soon (since such a proof would at once prove long outstanding questions about the category of motives, Birch and Swinnerton-Dyer's conjecture and the Stark conjectures).

Here's one more concrete idea. One important gadget in algebraic geometry is etale cohomology. Most L-functions that we know of can be defined in terms of the eigenvalues of various Frobenius maps (global and local) acting on etale cohomology groups. In many cases, by clever arguments using analogues of classical Betti cohomology theorems (Poincare duality, Lefschetz fixed point theorems, ...), we can express the results on special values of L-functions in terms of etale cohomology. In other words, the conceptual reason that L-functions defined from different objects behave so similarly to one another is that all these objects are algebro-geometric, and etale cohomology ought to be an awful lot like Betti cohomology. Of course, this is more believable for a variety over the complexes than it is for Spec Z, but it's a start.

gt.geometric topology - Is a simple loop in a spine of a strongly irreducible Heegaard splitting primitive in the fundamental group?

New Answer: Take a 2-bridge knot, and perform hyperbolic
Dehn filling (so that the core of the Dehn filling is geodesic),
and so that the filling slope has intersection number $>1$ with
the meridian. Then the meridian will not be primitive, since it
will be a multiple of the core of the Dehn filling. 2-bridge
knots have a genus two Heegaard splitting, which has a spine
for the handlebody which is a wedge of two meridians
at the bottom. This remains a spine in the
Dehn filling, so the loops represented by the meridians are not
primitive. This also works for the (2,n) torus knots (which are
2-bridge), so I think Charlie's answer is right (at least for
many small Seifert fibered-spaces).

Old (non)Answer:
Here's almost an example. All punctured torus bundles have
Heegaard genus $leq 3$, and many have Heegaard genus 3 (I discussed
this once in my defunct blog). One may find a genus 3 Heegaard splitting
of any once punctured torus bundle by taking two copies of a
fiber, tubing them together along the boundary on one side,
and adding a handle to the other side (drill out discs from
both fibers, and glue an annulus in). By a theorem of
Moriah-Rubinstein, most Dehn fillings will also have
Heegaard genus 3 if the punctured torus bundle does.

The Heegaard splitting of the punctured torus bundle
has one side which is a handlebody,
and the other side a compression body. We may think of the
handlebody as a product neighborhood of the fiber (which is
a punctured torus) with a 1-handle attached. We may find a
spine for the handlebody which consists of a wedge of two
loops which is a spine for the punctured torus, together with
another loop going through the 1-handle.

Now, the peripheral curve of the punctured torus is not primitive
in Dehn fillings along curves which intersect the longitude
multiple times. This curve is represented in the spine not
as an embedded curve, but has multiplicity two (since it is
a commutator of the generators). If we choose a small punctured torus bundle
of genus three, most Dehn fillings will be small (non-Haken) of
Heegaard genus 3, and so this Heegaard surface will be strongly
irreducible. But the peripheral curve will not be primitive,
since it will be a multiple of the core of the Dehn filling. However,
it is not embedded in the spine.

Even though this doesn't answer the question, it gives a strategy
for trying to find an example. Namely, if one can find a 1-cusped
hyperbolic 3-manifold which is small, and contains an incompressible
surface with boundary, such that the boundary slope is a generator in
the surface,
and such that a tubular neighborhood of the surface (or a slight modification by
drilling a hole in a tubular neighborhood)
is a minimal genus Heegaard splitting, then many Dehn fillings will
have the desired property.

ag.algebraic geometry - Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my memory it appears this was posed by Grothendieck as part of the big program of motives.

Consider classes of complex algebraic varieties X modulo relations

    [X] - [Y] = [XY], 
    [X x Y] = [X] x [Y], 

Also, if you're familiar with taking inverse of an affine line, let's do that too:
$$ exists mathbb A^{-1}quad text{such that}quad [mathbb A] cdot [mathbb A^{-1}] = [mathbb A^0].$$

(+ if you want, you can also take idempotent completion and formal completion by A^-1).

It's not hard to see that you can add (formally) and multiply (geometric product as above) those things, so they form a ring. Let's denote this ring  Mot (It's actually very close to what Grothendieck called baby motives.)

And for things that form a ring you can study their Spec. For example, you can talk about points of the ring — each point is by definition a homomorphism to complex numbers.

Question: what are the properties of Spec Mot? How to describe its points?

For example, one point is Euler characteristics $chi in text{Spec},mathbf{Mot}$, since it's additive and multiplicative (it's even integral!) Any other homomorphism to complex numbers is thus sometimes called generalized Euler characteristics.

There's also a plane there given by mixed Hodge polynomials (that is, polynomials whose coefficients are weighted Hodge numbers $h^{p,q}_k$), since Hodge polynomial at a given point satisfies those relations too (see the references below).

As Ben says below, things would become even more interesting if we considered this ring for schemes over $mathbb Z$, because then each $q$ would give a generalized Euler characteristic $chi_q$ that counts points of $X(mathbb F_q).$

Are there any other points? Any more information?

fourier analysis - How does one use the Poisson summation formula?

The existing answers list some important situations where Poisson Summation plays a role, the application to proving the functional equation of $theta$ and hence of $zeta$ being my personal favourite. My best answer to Tim's question as he actually asked it might be: why not have it in mind to try using it whenever you have a discrete sum that you are having trouble estimating, especially if you fancy your chances of understanding the Fourier transform of the summands. You'll end up with a different sum and it might be a lot easier to understand, and you might even be able to approximate your first sum by an integral (the term $hat{f}(0)$ in the Poisson summation formula).

To explain a little more with an example, there's a whole theory concerned with the estimation of exponential sums $sum_{n leq N} e^{2pi i phi(n)}$. There are two processes called A and B that can be used to turn a sum like this into something you might be better positioned to understand. Process A is basically Weyl/van der Corput differencing (Cauchy-Schwarz) and process B is essentially Poisson summation. It's not a very straightforward task to put together a theory of how these processes interact, and how they may best be combined to estimate your sum, and in fact this is in general something of an art. The 10 lectures book by Montgomery contains a nice exposition, and there's a whole LMS lecture note volume by Graham and Kolesnik if you want more details.

I want to share a perhaps slightly obscure paper of Roberts and Sargos (Three-dimensional exponential sums with monomials, Journal fur die reine und angewandte Mathematik (Crelle) 591), in which they use Poisson Summation in the form of Process B mentioned above arbitrarily many times to establish the following rather simple-to-state result: the number of quadruples $x_1,x_2,x_3,x_4$ in $[X, 2X)$ with

$$|1/x_1 + 1/x_2 - 1/x_3 - 1/x_4| leq 1/X^3$$

is $X^{2 + o(1)}$. In other words, the quantities $1/a + 1/b$ tend to avoid one another to pretty much the same extent as random numbers of the same size. Very very roughly speaking (I don't really understand the argument in depth) the proof involves looking at exponential sums $sum_x e^{2pi i m/x}$, and it is these that are transformed repeatedly using Poisson summation followed by other modifications (it being reasonably pointless to try and apply Poisson sum twice in succession).

graph theory - Combining DAGs into an acyclic tournament

Since your question is somewhat open ended, here's an observation, although it doesn't go anywhere yet.

A 2SAT instance is a decision problem in which given a set of variables $V$ and a formula comprising a conjunction of clauses over them, each clause being distinct and containing exactly two distinct literals, one wishes to know whether there is a truth assignment to the variables that makes the formula true.

Each 2SAT instance induces a digraph on $V cup overline{V}$: an arc from $u$ to $v$ exists if there is a clause $overline{u} lor v$.

Conjecture: If this digraph is a DAG, then the 2SAT instance must be satisfiable.
If the 2SAT instance is satisfiable then the digraph must have exactly one variable in each of its strongly connected components.

Moreover, such ``2SAT digraphs'' are transposable: reversing their arcs gives a digraph isomorphic to the original.

Your question could be interpreted as being about a collection of 2SAT instances where one is allowed to negate the literals in all the clauses of any instance.

ct.category theory - Image of composite morphisms

This cannot happen in a regular category. Below I give a proof using the sequent calculus of subobjects in a regular category. It can be deciphered using the book 'Sketches of an Elephant Volume 2' by Peter T. Johnstone, in particular chapter D1.

I write $beta:=I$ and $gamma:=J$. I hope the definition of image given in your book is the same as mine, namely the image of a subobject (~mono) $S$ under a morphism $phi$ is the least subobject of the codomain of $phi$ through which $phicircoverline{S}$ factors, where $overline{S}in S$.

Assume we know
$exists x(alpha(x)wedge f(x)=y) dashvvdash_{y:Y}quad beta(y)$ and $exists y(beta(y)wedge g(y)=z) dashvvdash_{z:Z}quad gamma(z)$. We then want to prove two things. The first is that $exists x(alpha(x)wedge g(f(x))=z)vdash_{z:Z}quad gamma(z)$, the second that $gamma(z)vdash_{z:Z} quad exists x(alpha(x)wedge g(f(x))=z)$.

For the first we have the following.
$alpha(x)wedge g(f(x))=z$
$vdash_{x:X,z:Z} quadalpha(x)wedge g(f(x))=z wedge f(x)=f(x)$
$vdash_{x:X,z:Z}quad alpha(x)wedge g(f(x))=z wedge beta(f(x))$
$vdash_{x:X,z:Z}quad gamma(g(f(x)))$. Therefore $alpha(x)wedge g(f(x))=zvdash_{x:X,z:Z}quad gamma(z)$ and hence $exists x(alpha(x)wedge g(f(x))=z)vdash_{z:Z}quad gamma(z)$.

The second also holds. First note that $beta wedge g(y)=z$
$vdash_{y:Y,z:Z}quad exists x(alpha(x)wedge f(x)=y)wedge g(y)=z$
$vdash_{y:Y,z:Z}quad exists x(alpha(x)wedge f(x)=ywedge g(y)=z)$
$vdash_{y:Y,z:Z}quad exists x(alpha(x)wedge g(f(x))=z)$ from which we may conclude that $gamma(z)vdash_{z:Z} quad exists y(beta(y)wedge g(y)=z)vdash_{z:Z} quad exists x(alpha(x)wedge g(f(x))=z)$.

rt.representation theory - What is a representation?

Let G be a compact Lie group with a maximal torus T, then what does the complex representation of G $alpha:Grightarrow U(n)$ mean? Does it mean that regarding
$alpha(g)$ as an isometry on $C^n$ for $gin G$?

Also, it will be good someone let me know the representations $alpha:Grightarrow U(n)$ for $G=G_2,F_4,E_6,E_7,E_8$? I don't have J.F.Adams's book "Lectures on exceptional lie groups"at hand.

ag.algebraic geometry - Is an algebraic bijection from a projective variety to itself necessarily an isomorphism?

Let $X$ be a projective variety. Assume there is an algebraic map $f: X rightarrow X$ that is a bijection. I am thinking of $X$ as a variety, not a scheme, so by a bijection I mean a bijection on closed points. Most likely I am working over the complex numbers, so if you like I mean a bijection on complex points. Can you conclude that $f$ has an algebraic inverse?

I think this is not immediately obvious, since it is not true that any algebraic bijection between two projective varieties is an isomorphism. For instance, there is an algebraic bijection from ${Bbb P}^1$ to a cuspidal cubic in ${Bbb P}^2$ given by $[x,y] rightarrow [x^3, x^2y, y^3]$. So if this is true one must use the fact that the map is from $X$ to itself.

I am interested in cases where $X$ is both singular and reducible (although is of pure dimension, if that helps), so a complete answer would cover any such case. Alternatively, if it is not true that such a map has an algebraic inverse, I would like an explicit counter example.

nt.number theory - Minimal basis of set of positive integers

The exponent can be arbitrarily close to $frac{1}{2}$: If $n=r^2$ then $A+B=lbrace 0,1,2,cdots n-1 rbrace$ where $A=lbrace 0,1,2,cdots r-1rbrace$ and $B=lbrace 0,r,2r,cdots,(r-1)rrbrace$. This means that $S=A cup B$ is a set of size $2r=2sqrt{n} $ with $S+S supset lbrace 0,1,2,cdots n-1 rbrace$. So by iteration (as described in the old answer) we can have exponent $frac{1}{2}+log_n(2)$.

Modifying this a bit gives a construction I probably have seen before: Let $A=lbrace 0,1,4,5,16,17,20,21cdotsrbrace$ be the (infinite) set of non-negative integers whose binary expansion is $0$ in all the odd positions and $B=2A$ the set of those with $0$ in all the even positions. Here both have exponent $frac{1}{2}$ and $A+B=mathbb{N}_0$. Then $S=A cup B$ has about $2sqrt{N}$ members less than $N$ and $S+S=mathbb{N}_0.$

---

Old Answer Here is a nice ad hoc start: The set $S=lbrace 0, 1, 3, 5, 6, 13, 15, 16, 18, 25, 26, 28, 30, 31 rbrace$ has $S+S=lbrace 0,1,2,cdots,62 rbrace$ (It is the start of a sequence in the OEIS with next term 63, but I'm not sure if that helps). Then $T=S+63S$ has 196 members the largest being $1984$ and $T+T=lbrace 0,1,2,cdots,3968 rbrace$. Now $14=31^{0.63944}$ and $196=3968^{0.63699}$ and further iterating will continue to give examples sparser than $k^{2/3}$. The same trick can be done with any finite example. The densities decrease but only very very slightly.
LATER the set $S'=lbrace 0, 1, 3, 5, 6, 13, 14,17, 18, 25, 26, 28, 30, 31 rbrace$ also has $S'+S'=lbrace 0,1,2,cdots,62 rbrace.$

Observe that the places where $S$ has a jump from $j$ to $2j+1$ are at $0,1,6,31$ (numbers of the form $frac{5^j-1}{4}$) and that there is a central symmetry for $0,1$ and for $0,1,3,5,6$ and for $S$. This suggests a treasure hunt for an symmetric extension with largest member $1+5+25+125=156$.

UPDATE For any one of the four choices $(a,b)=(68,69),(68,72),(69,70),(70,71)$ the set $V=S cup lbrace 63, 64, a,b,156-b,156-a, 92, 93 rbrace cup (S+125)$ has $36$ members, the largest being $156$. and $V+V=lbrace 0,1,2,cdots,312rbrace$ There is only one way to similarly extend $S'$ to a set of size $36$ with the same properties, it has lower half $S' cup lbrace 63, 65, 67, 69 rbrace$. These are better than the examples above since $36=312^{0.62398}$. Also, these can be iterated as above.

Maybe an even lower density is possible for $W=V cup lbrace ?? rbrace cup(V+625)$ where the set in the middle is made of several pairs $q,781-q$. I'd guess 8 pairs leading to $88=1562^{0.60885}$ but that is pure speculation. Of course there might be better examples which are not symmetric (or even ones which are). There turns out to be no advantage at this stage to putting the exact middle of $78$ in $V$ (for any of the 5 choices). I don't think it would help get a better $W$ but I am not sure.

What do gerbes and complex powers of line bundles have to do with each other?

If L is any line bundle on a space (scheme, whatever) X, A is any (additive) abelian group, and a an element of A, there is a natural construction of an A-gerbe $L^a$ as follows. By definition, $L^a$ should be a "sheaf of categories", or stack (not algebraic) on X, and here are its categories of sections. Identify L with its total space, which is a $mathbb{G}_m$-bundle on X, and for any open set U in X, let $L^a(U)$ be the category of all A-torsors on $L|_U$ whose monodromy about each fiber of $L|_U to U$ is a.

One can check that this really is a gerbe: it is locally nonempty, since if L is trivial over U, you can write $L|_U = mathbb{G}_m times U$ and then pull back the unique A-torsor on $mathbb{G}_m$ with monodromy a. It has a natural action of A-torsors on X, given by pulling up along the bundle map $L to X$ and tensoring. And this action is free and transitive, since the difference of two a-monodromic torsors on $L|_U$ has trivial monodromy on each fiber and therefore descends to X.

Why do I call this $L^a$? Suppose that $L = mathcal{O}_X(D)$ for a divisor D, where for simplicity let's say that D is irreducible of degree n; then L gets a natural trivialization on $U = X setminus D$ having a pole of order n along D. As shown above, this induces a trivialization $phi$ of $L^a$ on U, and if we pick a small open set V intersecting D and such that D is actually defined by an equation f of degree n, then we get a second (noncanonical) trivialization $psi$ of $L^a$ on V. You can check that the difference $psi^{-1} phi$, which is an automorphism of the trivial gerbe on $U cap V$, is in fact described by the A-torsor $mathcal{T} = f^{-1}(mathcal{L}_a)$, where $f colon U cap V to mathbb{G}_m$ and $mathcal{L}_a$ is the A-torsor of monodromy a. Since f has degree n, $mathcal{T}$ has monodromy na about D. Thus, it is only reasonable to say that the natural trivialization $phi$ has a pole of order na, which is consistent with the behavior of the trivialization of L itself on U, when raising to integer powers.

What does this have to do with twisting of differential operators? Suppose we have some kind of sheaves (D-modules, locally constant sheaves, perverse sheaves; technically, they should form a stack admitting an action of A-torsors). On the one hand, one could mimic the above construction of $L^a$ to describe a-monodromic sheaves on L, and this is what is often called twisting. On the other hand, there is a natural way to directly twist sheaves by the gerbe $L^a$ without mentioning L at all (that is, you can twist by any A-gerbe). The procedure is as follows: a twisted sheaf is the assignment, to every open set U in X, of a collection of sheaves on U parameterized by the sections of $L^a(U)$, and compatible with tensoring by A-torsors. Of course, since if $L^a(U)$ is nonempty this is the same as giving just one sheaf, this is sort of overkill, but the choice of just one such sheaf is noncanonical whereas this description is canonical. These collections should be compatible with the restriction functors $L^a(U) to L^a(V)$ when $V subset U$. It is an exercise to reader to check that this is the same as the other definition of twisting :)

Man, you asked the right question at the right time. My thesis is all about this stuff.

ho.history overview - Genealogy of the Lagrange inversion theorem

A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, is the Lagrange inversion theorem for power series (even in the formal context).

Starting from the exact sequence

$0 rightarrow mathbb{C} rightarrow mathbb{C}((z)) xrightarrow{D} mathbb{C}((z)) ;xrightarrow{ mathrm{Res} }; mathbb{C} rightarrow 0,$

and using the simple rules of the formal derivative D, and of the formal residue Res (that by definition is the linear form taking the formal Laurent series $f(z)=sum_{k=m}^{infty}f_k z^k in mathbb{C}((z))$ to the coefficient $f_{-1}$ of $z^{-1}$) one easily proves:

(Lagrange inversion formula): If $f(z):=sum_{k=1}^{infty}f_k z^kin mathbb{C}[[z]]$ and $g(z):=sum_{k=1}^{infty}g_k z^kinmathbb{C}[[z]]$ are composition inverse of each other, the
coefficients of the (multiplicative) powers of $f$ and $g$ are linked
by the formula $$n[z^n]g^k=k[z^{-k}]f^{-n},$$ and in particular (for
$k=1$), $$[z^n]g=frac{1}{n} mathrm{Res}( f^{-n} ).$$

(to whom didn't know it: enjoy computing the power series expansion of the composition inverse of $f(z):=z+z^m$,
or of $f(z):=zexp(z),$ and of their powers).

My question: what are the generalization of this theorem in wider context. I mean, in the same way that, just to make one example, the archetypal ODE $u'=lambda u$ procreates the theory of semigroups of evolution in Banach spaces.

Also, I'd be grateful to learn some new nice application of this classic theorem.

(notation: for $f=sum_{k=m}^{infty}f_k z^k in mathbb{C}((z))$ the symbol $[z^k]f$ stands, of course, for the coefficient $f_{k}$)

career - Teaching statements for math jobs?

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

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

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

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

books - Erratum for Cassels-Froehlich

Hendrik Lenstra says:

Below my 51 errata that I didn't see on your list or in William Stein's mail
yet. Most are of a typographical nature, but some have mathematical
substance. I did at the present occasion not verify the correctness of those.

And: I did not do any proofreading of my list either!! I trust you will apply
your own sound judgment.

Good luck!!

And best regards,

Hendrik

Errata for Cassels-Fr"ohlich
copied by Hendrik Lenstra from his own copy
Jan 13, 2010

Page 3, Proposition 1. This Proposition is misstated, and the proof has the
wrong reference: Chapter II, section 10 has no such result, but Chapter II,
section 5 does. The Theorem in the latter section is the correct formulation:
it is not the extension of the valuation, but the completion that one wants to
be unique. More or less coincidentally, the Proposition is correct as stated
(exercise!), but that statement is neither used (by anybody) nor proved (in the
book).

Page 45, line 5: for "=n" read "=n+1".

Page 52, part (3) of the first definition: for "K" read "V".

Page 54, line -5: replace roman "A" by italic "A" (twice).

Page 73, line 6: replace "vica" by "vice".

Page 75, line 1: replace "(19.9)" by "(19.10)".

Page 78, first line of display (A.19): replace "b_{ij}" by "b_{1j}".

Page 78, line -8 (display (A.24)): replace the third subscript "P" by "R".

Page 78, line -6: replace the third subscript "P" by "R".

Page 79, line -8: remove the three commas within the parentheses.

Page 98, the lower "delta" in the diagram should have a "hat" (the upper
one has one, though it is barely visible in my copy).

Page 123, last line before section 2.5: replace roman "C" by italic "C".

Page 129, line 10: replace "]" by "])".

Page 130, line 1: replace "2.7" by "2.8".

Page 130, line 14: replace "2.5" by "1.5".

Page 131, last line before Corollary 1: replace "2.7" by "2.8".

Page 131, line -10: replace the second "K_{nr}" by "K_{nr}^*".

Page 135, line 6 of Lemma 4: replace the second "M)" by "M))".

Page 139, line 14 (the first display): put ")" before the final ".".

Page 140, line 3 of section 2.3: replace "H^2(G,Z)" by "H^1(G,Q/Z)" (with
Q, Z boldface), since the isomorphism delta hasn't been applied yet!

Page 140, line -8: replace "s" by "s_alpha".

Page 140, line -2: replace "Prop. 2" by "Prop. 1". Also, the proof is
confused. One defines s'alpha to be (alpha,L'/K), and the fact that
salpha maps to s'_alpha under the natural map G^{ab} -> (G/H)^{ab}
FOLLOWS from the equality of character values rather than playing a role
in the proof of that equality.

Page 141, first line after the first diagram: replace the last "K" by "K'".

Page 143, line -3: replace "Lubin" by "Lubin-".

Page 147, first line after Definition: put ")" at the end.

Page 150, proof of Proposition 1: (c) is not proved that way.

Page 150, line -10: for "left-and" read "left- and".

Page 151, line 13: replace the last "[a]" by "[b]".

Page 154, line 18: replace "r_pi" by "r_pi(omega)".

Page 154, line 19: in my copy, there is the scrawled complaint "why is K_pi
from sec. 3.6 equal to K_pi from section 2.8?", and a three line additional
argument, which reads as follows: "Adopt the definition of K_pi as in sec.3.6
(or Theorem 3(b)). Then r(pi) is trivial on K_pi (def. of r), and so is
vartheta(pi) by Cor. to Prop. 6. Also r(pi) and vartheta(pi) are F on
K_{nr}. Hence r and vartheta agree on pi, hence on all of K^*. (Hence also
K_pi=Kpi !)

Page 154, line 2 of section 3.8: replace "2.3" by "2.7".

Page 154, line -8: replace "I_K" by "I'_K".

Page 155, line -11: replace roman "G" by italic "G".

Page 156, line 3: replace "3.3" by "3.4".

Page 156, line 10: replace "beta_j" by "beta^j".

Page 157, line 9: replace "intertia" by "inertia".

Page 158, line -4: replace "|" by "/".

Page 168, line 5: replace roman "F" by italic "F".

Page 168, line -16: replace roman "C" by italic "C".

Page 170, line -18: replace "U^S an arbitrarily small" by "U^S contained in
an arbitrarily small" (because U^S is generally not open).

Page 175, line 2 after the diagram: for "N_{M/K}" read "N_{M/K}J_M".

Page 179, line 12: put ")" before the second "=".

Page 183, line 1: there is no "Proposition 2". Probably "Proposition 2.3" is
meant.

Page 183, display (7): replace the second "K" by "K^*".

Page 192, line -11: replace "infiinte" by "infinite".

Page 211, line -12 (counting the footnote as -1): for "does or does not"
read "does not or does". (This is what I scrawled, I did not verify it at the
present occasion!)

Page 211, line -7: for "seq" read "seq.".

Page 236, line 5: for "2.5", read "1.2, Prop. 1".

Page 353, line 4: for "(lambda,b)_v", read "(b,lambda)_v".

Page 360, last line of Exercise 5.1: for "4.3", read "4.4".

Page 366, under "Tchebotarev, N.,", also list "165," and "227,".

----------------------THE---END-------------------------------------------------

toric varieties - Relationship between topological cohomology and $ell$-adic cohomology

The way to study the topology of the situation was introduced by Khovanski in
"Newton polyhedra, and toroidal varieties" Funkcional. Anal. i Priložen. 11
(1977), no. 4, 56--64, 96. His result (if I have interpreted it correctly) is
that $X$ may be compactified as a hypersurface in a projective toric variety to
a smooth variety with normal crossings such that each stratum is of the same
form as $X$. As far as I can see this construction works uniformly so that we
would get the same construction over a (suitable) mixed characteristic discrete
valuation ring. Then desired isomorphism then follows from the smooth and proper
base change theorem. (I have some vague recollection that this comment is also to be
found somewhere in SGA but I am not going to do any wading looking for it...)

Addendum: Let me first note that the right setup to even formulate the
question is a scheme $S$ with functions on it giving the coefficients of $f$.
The latter polynomial should then be non-degenerate in the sense that all its
fibres over (geometric) points of $S$ should be non-degenerate. The statement is
then that if $picolon Xto S$ is the scheme of zeroes of $f$ in the constant torus
over $S$, then $R^ipi_astmathbb Z_ell$ is locally constant commuting with base
change where $ell$ is invertible in $mathcal O_S$ (together with comparison
theorem of $ell$-adic cohomology and classical for a complex point of $S$). As
this statement is only dealing with the $ell$-adic sheaves $R^ipi_astmathbb Z_{ell}$
we may use the definition of $ell$-adic sheaf introduced by Jouanolou in SGA V.
Kohvanski's method then should give a compactification of $X$ by a smooth
$S$-scheme with complement of relative normal crossings. The two theorems used,
proper base change and vanishing of vanishing cycles, then follows directly from
the case of finite coefficients (by Jouanolou's very definition).

dg.differential geometry - Geometric imagination of differential forms

The $k$-forms that are easiest to describe are those with $k in {0,1,n-1,n}$. A 0-form on an $n$-manifold is a function. A 1-form on an $n$-manifold, if you imagine it in $n+1$ dimensions, is like an arrangement of shingles on a roof: At each point of the manifold, it defines a directional slope, which as other people have said, is the same as a dual vector on tangent vectors. An $n$-form is a density, i.e., an entity that you can integrate over the manifold. And an $(n-1)$-form is a flux (like, say, describing oil coming out of a well): At each point it has a null tangent direction, and it assigns a non-zero volume to each cross section.

Of course you can think of any $k$-form as a $k$-dimensional flux, and for general values of $k$ you might as well. But when $k$ is 1 or $n-1$, it is somewhat easier to visualize the condition that the form is closed. A 1-form is closed when the shingles locally mesh as the slope of a smooth roof, i.e., the form is locally integrable. An $(n-1)$-form is closed when the flux is locally conservative, which for instance is the case with fluid flow. In fact, theorem: A closed, non-zero $(n-1)$-form is equivalent to a 1-dimensional foliation with a transverse volume structure.

The reason that other values of $k$ are harder is that while you do get an entirely analogous algebraic integrability condition when the form is closed, you might not get the same kind of geometric integrability. A non-zero 1-form has an $(n-1)$-dimensional kernel at each point. (Although the visualization that I suggested is in $n+1$ dimensions, it is also true in $n$ dimensions that these tangent hyperplanes mesh when the 1-form is closed.) A non-zero $(n-1)$-form has a 1-dimensional kernel at each point. But a $k$-form for other values of $k$ doesn't usually have a kernel. (Okay, a maximum rank 2-form in odd dimensions also has a 1-dimensional kernel, and it is equivalent to a 1-foliation with a transverse symplectic structure.)

I have heard the statement that only 1-forms and 2-forms are any good. (Well, that's an overstatement, but they are more important than the others except for maybe $0$ and $n$.) In particular, symplectic forms show up a lot, so it is important to try to imagine them even though by definition they have no kernels. I think of a symplectic form as a calibration for a local complex structure. (Or an almost complex structure, which might be all that exists globally.) I.e., among the different tangent 2-planes of a symplectic $2n$-manifold, the ones that are complex lines have the greatest pairing with the symplectic form, while the ones that are real planes have vanishing pairing, and the pairing minimum is achieved by complex lines with the wrong orientation.

---

One more remark: The geometric picture of a foliation with a transverse volume structure holds for closed $k$-forms that are also non-zero simple forms (i.e., wedge products of linearly independent 1-forms). I think it's a theorem that any closed $k$-form is locally a sum of closed, simple $k$-forms. If that's correct, then that's also a way to visualize a closed $k$-form, as an algebraic superposition of volumed foliations. $k=1$ and $k=n-1$ are special cases in which every non-zero form is simple.

mg.metric geometry - $C^1$ isometric embedding of flat torus into $mathbb{R}^3$

A group of french mathematicians and computer scientists are currently working on this. The project is named Hévéa, and has already produced a few images. Edit: a few images and the PNAS paper have been released, see http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html

Just a few word to explain what I understood of their method (which is by using h-principle) from the few image I saw in preview. Start with a revolution torus. The meridians are cool, because they all have the same length, as expected from those of a flat torus. But the parallels are totally uncool, because their lengths differ greatly: they witness the non-flatness of the revolution torus.

Now perturb your torus by adding waves in the direction of the meridians (like an accordion), with large amplitude on the inside and small amplitude on the outside. If you design this perturbation well, you can manage so that the parallels now all have the same length. Of course, the perturbed meridian have now varying lengths! So you do the same thing by adding small waves in another direction, getting all meridians to have the same length again. You can iterate this procedure in a way so that the embedding converges in the $C^1$ topology to a flat embedded torus. But to prove that the precise perturbation you chose in order to get a nice image does converge, and that your maps are embeddings needs work (getting an immersion is easier if I remember well).

Also, the Hévéa project plans to draw images of Nash spheres, that is $C^1$ isometric embeddings of spheres of radius $>1$ inside a ball of unit radius.

nt.number theory - Heuristic justification for Goldbach's conjecture

On the Wikipedia page of Goldbach's conjecture, a heuristic justification is given, which did not completely satisfy me. It roughly goes as follows:

• randomly define a subset integers in accordance with the prime number
  theorem


• Let $K_n$ be the random variable, counting the number of ways the
  natural number $2n$, can be written as
  a sum of two members of this set.

Then $E[K_n]rightarrow infty$ .

The problem is that, although the mean goes to infinity, it still might be true that the probability that $K_n>0$ for all $n$ is zero.

So I thought of a different heuristic, and I am curious about whether anything is known about it:

Let $mathcal P$ be the collection of
all subsets of odd numbers whose
density agrees with the prime number
theorem, and let $mathcal G$ be the
collection of subsets for which
Goldbach's property holds (i.e. every
even number can be written in at least
one way with two members of the set).
Let $mu$ be the uniform product
measure of the space ${0,1}^{mathbb
> N}$. Then the quantity $$
> frac{mu(mathcal P cap mathcal
> G)}{mu(mathcal P)} $$
is (significantly) greater than zero.

Edit: As pointed out in the comments,
$mu(mathcal P) = 0$, so this
quantity is meaningless as it is, but
I think it can be formalized in some
way.

I do not know if this is easy or almost as difficult as the original problem. But it would be a very convincing heuristic for me in that, it would tell me how much of Goldbach's conjecture is already explained by the prime number theorem.

I would appreciate answers, or references to any known results, or reasons if this kind of heuristic is not relevant, if that is the case.

knot theory - complexity of counting homomorphisms

This is a question I have thought about and asked a number of people, but have never got an answer beyond "It should be true that..."

Given a finitely generated group $G$ (eg. a link group $G_L:=pi_1(S^3-L)$ for a link $L$) and a finite group $H$ we want to count homomorphisms from $G$ to $H$. For link groups as above, this is an invariant of $L$.

My question: (for which $H$) is there a polynomial-time algorithm (in the number of generators and relations for $G$) for computing $N(G,H):=|Hom(G,H)|$ (particularly for $G_L$)?

Some things I know:
1) If $L$ is a knot and $H$ is nilpotent then $N(G_L,H)$ is constant (M. Eisermann)
2) D. Matei; A. I. Suciu, have an algorithm for solvable $H$, but the complexity is not clear.
3) The abelianization of $G_L$ is just $Z^c$, $c$ the number of components, so for $H$ abelian it is easy.

A wild conjecture is that it should always be "FPRASable" i.e. there exists a fully polynomial randomized approximation scheme for the computation.

lattices - Proof of glb and lub of Lexicographic Product of poset

Surely this information is in Ralph Mackenzie's book on universal algebra.

But let me just sketch some quick proofs. First, note that you were right to consider complete lattices, since in general the lexicographic order on lattices may not be a lattice order, as I explain in this MO answer. So we suppose that $L_1$ and $L_2$ are complete lattices.

My suggestion is that you focus on the order, rather than on the algebraic operations as you defined them. That is, we can define the lexical order on the product $L_1times L_2$, and then prove that this order is a lattice order. The lexical order is just the dictionary order, placing (a,b) below (a',b') just in case $aleq a'$ or $a=a'$ and $bleq b'$. It is not difficult to prove that this order is reflexive, transitive and antisymmetric, so it is indeed a partial order. Furthermore, when the lattices are complete, it is a lattice order (that is, has lub's and glb's).
To see this, consider two points $(a,b)$ and $(a',b')$.
The lub of $(a,b)$ and $(a',b')$ is easily seen to be $(lub(a,a'),0)$, if $a$ and $a'$ are incomparable, since this is an upper bound, and any other upper bound must be at least as big as this. The lub of $(a,b)$ and $(a',b')$ is clearly $(a',b')$, if $alt a'$; and it is $(a,lub(b,b'))$, if $a = a'$. Similarly, the glb of $(a,b)$ and $(a',b')$ will be $(glb(a,a'),1)$, if $a$ and $a'$ are incomparable; it is $(a,b)$, if $a lt a'$; and it is $(a,glb(b,b'))$, if $a = a'$. These answers amount to the definition of $vee$ and $wedge$ you gave in your question.

The essence of this suggestion is that you should derive the operations you mentioned as expressing the right lub and glb properties for the lexical order, as a consequence of the order, rather than defining those operations first and then trying to prove it has the lattice algebraic properties.

Now, knowing that this is a lattice order, we may easily derive that lub and glb are commutative and associative, etc., since these properties hold in any lattice order. That is, once you get the order correct, the algebraic properties follow for free:

• Note that $glb(a,b)$ depends only on the set {a,b} and not on the pair $langle a,brangle$, so we clearly get commutativity $glb(a,b)=glb(b,a)$.


• The same is true for $lub(a,b)$, for the same reason.


• These operations are associative, since if d=glb(a,glb(b,c)), then d is an lower bound of a and glb(b,c), and therefore a lower bound of all three a,b,c. Thus, it is an lower bound of glb(a,b) and c and so is at most glb(glb(a,b),c). The converse is similar.


• One can show lub is associative in a very similar manner.


• I'm not sure what you mean by idempotent, but clearly glb(a,a)=a, since obviously a is the least upper bound of a and a.


• A stronger result would be to show that the lexical order on $L_1times L_2$ is also a complete lattice order, which it is, since any subset $Xsubset L_1times L_2$ has a lub in the product simply by taking the lub's in the first coordinate, and then glb among elements of the second coordinate that work with that first element. This argument does not work when the lattices are not complete, unless $L_1$ is linear.

ca.analysis and odes - Why is $ frac{pi^2}{12}=ln(2)$ not true?

This question may sound ridiculous at first sight, but let me please show you all how I arrived at the aforementioned 'identity'.

Let us begin with (one of the many) equalities established by Euler:

$$ f(x) = frac{sin(x)}{x} = prod_{n=1}^{infty} Big(1-frac{x^2}{n^2pi^2}Big) $$

as $(a^2-b^2)=(a+b)(a-b)$, we can also write: (EDIT: We can not write this...)

$$ f(x) = prod_{n=1}^{infty} Big(1+frac{x}{npi}Big) cdot prod_{n=1}^{infty} Big(1-frac{x}{npi}Big) $$

We now we arrange the terms with $ (n = 1 land n=-2)$, $ (n = -1 land n=2$), $( n=3 land -4)$ , $ (n=-3 land n=4)$ , ..., $ (n = 2n land n=-2n-1) $ and $(n=-2n land n=2n+1)$ together.
After doing so, we multiply the terms accordingly to the arrangement. If we write out the products, we get:

$$ f(x)=big((1-x/2pi + x/pi -x^2/2pi^2)(1+x/2pi-x/pi - x^2/2pi^2)big) cdots $$
$$
cdots big((1-frac{x}{(2n)pi} + frac{x}{(2n-1)pi} -frac{x^2}{(2n(n-1))^2pi^2})(1+frac{x}{2npi} -frac{x}{(2n-1)pi} -frac{x^2}{(2n(2n-1))^2pi^2)})big) $$

Now we equate the $x^2$-term of this infinite product, using
Newton's identities (notice that the '$x$'-terms are eliminated) to the $x^2$-term of the Taylor-expansion series of $frac{sin(x)}{x}$. So,

$$ -frac{2}{pi^2}Big(frac{1}{1cdot2} + frac{1}{3cdot4} + frac{1}{5cdot6} + cdots + frac{1}{2n(2n-1)}Big) = -frac{1}{6} $$

Multiplying both sides by $-pi^2$ and dividing by 2 yields

$$sum_{n=1}^{infty} frac{1}{2n(2n-1)} = pi^2/12 $$

That (infinite) sum 'also' equates $ln(2)$, however (According to the last section of this paper).

So we find $$ frac{pi^2}{12} = ln(2) . $$

Of course we all know that this is not true (you can verify it by checking the first couple of digits). I'd like to know how much of this method, which I used to arrive at this absurd conclusion, is true, where it goes wrong and how it can be improved to make it work in this and perhaps other cases (series).

Thanks in advance,

Max Muller

(note I: 'ln' means 'natural logarithm)
(note II: with 'to make it work' means: 'to find the exact value of)

lo.logic - Are there any good nonconstructive "existential metatheorems"?

A basic principle of first order logic is that, if a theorem follows from a list of axioms, then it follows from some finite subset of that list. This can often give nontrivial consequences. For example:

(1) If $T$ is any true first order statement about characteristic zero fields, then there is a constant $N$ so that $T$ is true for fields of characteristic $>N$.

Proof: take your list of axioms to be the standard field axioms plus, in case (1), the axiom $1+1+1+cdots+1 neq 0$ where the sum is over $p$ ones for each prime $p$.

A similar example is:

(2) If $T$ is any true first order statement about algebraically closed fields, there is a constant $N$ such that $T$ is true for any field $K$ in which every polynomial of degree $leq N$ has a root.

In both of these cases, it can be easier to (informally) prove a result about characteristic zero/algebraically closed fields than it is to extract the constant $N$.

Here is a different example. Recall that Ramsey's theorem (in a special case) says:

(3) For any positive integer $N$, there is an integer $M$ such that, for any bicoloring of the complete graph on $M$ vertices, there is a monochromatic complete subgraph on $N$ vertices.

As I understand it, the original proof was to show

(3') For any bicoloring of the complete graph on infinitely many vertices, there is an monochromatic infinite complete subgraph.

This clearly implies

(3'') For any positive integer $N$, and any bicoloring of the complete graph on infinitely many vertices, there is an monochromatic complete subgraph on $N$ vertices.

Writing (3'') formally, you wind up with an infinite sequence of axioms, saying that the graph has more than $k$ vertices for every $k$. Since only finitely many of those axioms are used in the proof of (3''), we see that (3) must hold. Of course, there are now direct proofs of (3), but my understanding is that the logic theory proof came first.

ag.algebraic geometry - Stalks of sheaf-hom

By the way, here's a counterexample for the most sweeping generalization ("it's always an isomorphism"), which I found online in a book called "Topological Invariants of Stratified Spaces". Let X = [0,1] and F be the skyscraper sheaf Z at 0. Let G be the constant sheaf Z. If U contains 0 then Hom(F|U,G|U)=0, so Hom(F,G)_0 = 0, but of course Hom(Z,Z)=Z.

What about for coherent O_X modules on a ringed space X that need not be a scheme? Say, a complex manifold? Just idle curiosity...

ag.algebraic geometry - If $Omega_{X/Y}$ is locally free of rank $mathrm{dim}left(Xright)-mathrm{dim}left(Yright)$, is $Xrightarrow Y$ smooth?

A variation on Ishai's example is a closed embedding: its sheaf of relative differentials is 0, hence free of finite rank, even though it needn't be smooth.

However, k[e] / e^2 over k is not actually a counterexample (except in characteristic 2). The module of relative differentials of Spec k[e] / e^2 over Spec k is not free if the characteristic of k is not 2. Let A = k[e] and B = k[e] / e^2. Then

Omega_B = Omega_A (x) B / d(e^2) = k[e] / (e^2, 2e)

via the isomorphism Omega_A --> A : dt --> 1. This is not isomorphic to B unless 2 = 0.

On the other hand, you can conclude that B is smooth if its cotangent complex is a vector bundle in degree 0. In the case of k[e] / e^2, the cotangent complex is

[ I_{B/A} / I_{B/A}^2 ---> Omega_A (x) B ] = [ e^2 A / e^4 A ---> B de ]

in degrees [-1,0] and the differential is the universal derivation. (I write I_{B/A} for the ideal of B in A.) Even in characteristic 2, the differential has a kernel, so the cotangent complex is not concentrated in degree 0.

qa.quantum algebra - What is Out(G-mod) for a finite group G?

Following the notation of Etingof-Nikshych-Ostrik what is Out(G-mod) for a finite group G?

That is what are all bimodule cateogries over the fusion category G-mod of complex G-modules which have the property that they're just G-mod as a left (resp. right) G-module up to equivalence of bimodules? I think this is the same as the group of tensor autoequivalences of G-mod which do not come from conjugating by 1-dimensional objects, but ENO only prove that in the case where there are no 1-dimensional objects.

If you have an automorphism of the group G then you get an automorphism of G-mod. I'm not totally sure though if outer automorphisms of G necessarily give outer automorphisms (for example could you have an outer automorphism that fixed all conjugacy classes?) and I also don't know whether all outer automorphisms come up this way.

The reason that I'm asking is that I want to understand the relationship between Out(C) and Out(D) where C and D are Morita equivalent fusion categories. However, I realized I don't have enough examples where I understand what Out(C) actually is.

oa.operator algebras - Ideals in Factors

One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I_n$, type $II_1$, and type $III$ factors are algebraically simple (any 2-sided ideal must contain a projection. All projections are comparable in a factor, so you can show 1 is in the ideal). Ideals in $B(H)$ ($dim(H)=infty$, $H$ separable) have been studied extensively. What about ideals in $II_infty$ factors?

One might think, since every $II_infty$ factor $M$ can be written as $Noverline{otimes} B(H)$ for $N$ a $II_1$ factor, if $Isubset B(H)$ is an ideal, then $Notimes I$ is a 2-sided ideal. This is false. One needs to take the ideal generated by $Notimes I$. What does that mean from a von Neumann algebra viewpoint? Is it the same as taking the norm closure?

We can also describe some ideals in terms of the trace. One has the equivalent of the Hilbert-Schmidt operators: $$I_2={xin M | tr(x^ast x)<infty}$$ and the trace class operators:
$$I_1={xin M | tr(|x|)<infty}=I_2^ast I_2 =left{sum^n_{i=1} x_i^ast y_i | x_i, y_iin I_2right}.$$
What is the relation of $I_j$ to $Notimes L^j(H)$ for $j=1,2$ (where $L^2(H)$ is the Hilbert-Schmidt operators and $L^1(H)$ is the trace class operators in $B(H)$)? Is $I_j$ the norm closure of $Notimes L^j(H)$?

lo.logic - A question about definability in first order theories based upon classical logic

Not weakened at all!

In PA and ZFC (and a wide class of other f-o theories), every defining formula is equivalent to one with no closed subformula. Let's start with ZFC. Suppose $varphi(x)$ is a defining formula, with $x$ free; now let $z$ be any variable not appearing anywhere in $varphi$, and for each subformula $psi$ of $varphi$, define a new formula $psi^z(z)$ (with all the same free variables as $psi$, plus $z$) by:

• if $psi$ is an atomic formula, take $psi^z := psi land (z=z)$;


• if $psi = top$, take $psi^z := (z=z)$


• if $psi = bot$, take $psi^z := forall z'. z in z'$


• if $psi = psi_1 land psi_2$, then take $psi^z := psi_1^z land psi_2^z$

…and so on: all the remaining cases (non-nullary connectives and quantifiers) just commute with $(-)^z$, same as $land$. Anyway, we've defined these new versions of all subformulas; by induction, they all have $z$ free, have no closed subformulas, and are equivalent to the original versions; so up at the top we apply it to our original formula, and have $varphi^z(z,x)$; now $forall z. varphi^z (z,x)$ is equivalent to $varphi(x)$ and has no closed subformula.

Now, this relied on the fact that ZFC has (in most presentations) no closed terms (indeed, no terms except variables), for the atomic formula case to work: in PA, for instance, $0=0$ gets bumped up to $0=0 land z=z$, which still has a closed subformula. So for eg PA, we have to work a bit harder at defining that case:

• for $psi$ an atomic formula $R(t_1,ldots,t_n)$, take $psi^z := exists w. [w = t_1 land R(w,ldots,t_n) land z=z]$.

This now makes it all work again! But, this relied on all basic relations $R$ taking at least one argument. In any language with this property, we're good. The one thing we can't generally deal with is theories with nullary relation symbols, aka propositional constants — although we can sometimes still handle them like we handled $top$ and $bot$ in $ZFC$.

(Actually, I'm not quite sure what you mean by “how would ZFC and PA be weakened if we changed the definition of a defining formula” – since I don't know axiomatisations of those theories that involve this term. But the standard axiomatisation of ZFC does involve functions (in the replacement axiom), which are just defining formulas $f(x,y)$ with an extra free variable $y$, so I guess what you have in mind might be something like strengthening the definition of function allowed in there? But I think the above construction should answer the question, in any case!)

gr.group theory - The finite subgroups of SU(n)

This is really a comment on the top answer above, but since new users can't comment, I'll let someone else manually transfer the information to the right place.

There is a further mistake in the list of Fairbairn, Fulton and Klink (repeated in the list of Hanahy and He), which appears to be a misunderstanding of the classification by Blichfeldt et al. Two of the cases in that classification consists of semidirect products of abelian groups by $A_3$ and $S_3$. However, it is not specified which abelian groups can occur in this fashion!

Fairbairn, Fulton and Klink mistakenly assume that the abelian group in question has to be $mathbb{Z}/nmathbb{Z} times mathbb{Z}/n mathbb{Z}$ for some positive integer $n$, thus giving rise to the groups they denote $Delta(3n^2)$ and $Delta(6n^2)$. However, this is not the case.

Example 1: $A_3$ acts on the copy of $mathbb{Z}/7mathbb{Z}$ generated by the diagonal matrix with entries $e^{2pi i/7}, e^{4 pi i/7}, e^{8 pi i/7}$; this example occurs inside the exceptional subgroup of order 168. More generally, if $m,n$ are positive integers and $m^2+m+1 equiv 0 pmod{n}$, then $A_3$ acts on the copy of $mathbb{Z}/nmathbb{Z}$ generated by the diagonal matrix with entries $e^{2pi i/n}, e^{2m pi i/n}, e^{2m^2 pi i/n}$.

Example 2: $S_3$ acts on the copy of $mathbb{Z}/3mathbb{Z} times mathbb{Z}/9mathbb{Z}$ generated by the diagonal matrices with entries $e^{2pi i/9}, e^{2pi i/9}, e^{14 pi i/9}$
and $1, e^{2pi i/3}, e^{4pi i/3}$; this example occurs inside the exceptional subgroup of order 648.

I don't know a reference for the complete classification of the abelian groups that can occur inside the semidirect product. Yau and Yu don't say any more than Blichfeldt et al, though they do at least provide a helpful rewrite of the classification in modern language.

ag.algebraic geometry - When does Tannakian theory work over affine schemes besides fields?

If I understand your question correctly you are asking whether or not there is a characterization of those A-linear functors C-->Proj(A) which are equivalent to the forgetful functor Rep(G)-->Proj(A), where G is an affine group scheme over A and Rep(G) is the category of representations of G whose underlying A-module is finitely generated projective.

In this case the categories Rep(G) need not be abelian because Proj(A) is in general not abelian, so the classical Tannakian formalism probably won't help you. However, as in the classical case we can rephrase the problem in terms of comodules and Hopf algebras: An affine group scheme G over A is of the form Spec(H) for some Hopf algebra H, and a representation of G is the same as an H-comodule.

The classical characterization of Deligne (found in "Categories Tannakiennes", Grothendieck Festschrift Vol. II) is split up into the following parts:

1) Every faithful exact functor w:C-->Vect(k) is equivalent to a forgetful functor Comod(L(w))-->Vect(k) for some k-coalgebra L(w).

2) If the category C has a symmetric monoidal structure and if w is a strong monoidal functor, then L(w) is a bialgebra.

3) If C is rigid, then L(w) is a Hopf algebra.

For some time now I've been working on a generalization of step 1) to the case of arbitrary rings A, where we replace Vect(k) by Proj(A). I have recently uploaded a paper (here) which contains the following result (see Corollary 9.8):

Let C be an A-linear category (with finite direct sums) and w:C-->A-Mod an A-linear functor whose image is contained in Proj(A). We say that a diagram F:D-->C is w-rigid if the colimit of wF:D-->A-Mod is finitely generated and projective. If

a) w reflects isomorphisms,

b) the category el(w) of elements of w is cofiltered (equivalently, w is flat),

c) C has colimits of w-rigid diagrams and w preserves them,

then there is a flat coalgebra L(w) such that w:C-->Proj(A) is equivalent to the forgetful functor Comod(L(w))-->A-Mod, where Comod(L(w)) denotes the category of L(w)-comodules whose underlying A-module is finitely generated and projective. Condition a) and c) are necessary conditions. I don't know if b) is a necessary condition.

I have convinced myself that result 2) from above should work at this level of generality, and I think that 3) should not cause any trouble either. In other words, if your category C is a symmetric monoidal category where every object has a dual and if w is a strong monoidal functor satisfying a)-c), then the coalgebra L(w) should be a Hopf algebra, and C is in fact a category of representations of an affine group scheme.

analytic number theory - PNT for general zeta functions, Applications of.

Hi Anweshi,

Since Emerton answered your third grey-boxed question very nicely, let me try at the first two. Suppose $L(s,f)$ is one of the L-functions that you listed (including the first two, which we might as well call L-functions too). (For simplicity we always normalize so the functional equation is induced by $sto 1-s$.) This guy has an expansion $L(s,f)=sum_{n}a_f(n)n^{-s}$ as a Dirichlet series, and the most general prime number theorem reads

$sum_{pleq X}a_f(p)=r_f mathrm{Li}(x)+O(x exp(-(log{x})^{frac{1}{2}-varepsilon})$.

Here $mathrm{Li}(x)$ is the logarithmic integral, $r_f$ is the order of the pole of $L(s,f)$ at the point $s=1$, and the implied constant depends on $f$ and $varepsilon$.

Let's unwind this for your examples.

1) The Riemann zeta function has a simple pole at $s=1$ and $a_f(p)=1$ for all $p$, so this is the classical prime number theorem.

2) The Dedekind zeta function (say of a degree d extension $K/mathbb{Q}$) is a little different. It also has a simple pole at $s=1$, but the coefficients are determined by the rule: $a(p)=d$ if $p$ splits completely in $mathcal{O}_K$, and $a(p)=0$ otherwise. Hence the prime number theorem in this case reads

$|pleq X ; mathrm{with};p;mathrm{totally;split;in};mathcal{O}_K|=d^{-1}mathrm{Li}(x)+O(x exp(-(log{x})^{frac{1}{2}-varepsilon})$.

This already has very interesting applications: the fact that the proportion of primes splitting totally is $1/d$ was very important in the first proofs of the main general results of class field theory.

3) If $rho:mathcal{G}_{mathbb{Q}}to mathrm{GL}_n(mathbb{C})$ is an Artin representation then $a(p)=mathrm{tr}rho(mathrm{Fr}_p)$. If $rho$ does not contain the trivial representation, then $L(s,rho)$ has no pole in neighborhood of the line $mathrm{Re}(s)geq 1$, so we get

$sum_{pleq X}mathrm{tr}rho(mathrm{Fr}_p)=O(x exp(-(log{x})^{frac{1}{2}-varepsilon})$.

The absence of a pole is not a problem: it just means there's no main term! In this particular case, you could interpret the above equation as saying that "$mathrm{tr}rho(mathrm{Fr}_p)$ has mean value zero.

4) For an elliptic curve, the same phenomenon occurs. Here again there is no pole, and $a(p)=frac{p+1-|E(mathbb{F}_p)|}{sqrt{p}}$. By a theorem of Hasse these numbers satisfy $|a(p)|leq 2$, so you could think of them as the (scaled) deviation of $|E(mathbb{F}_p)|$ from its
"expected value" of $p+1$. In this case the prime number theorem reads

$sum_{pleq X}a(p)=O(x exp(-(log{x})^{frac{1}{2}-varepsilon})$

so you could say that "the average deviation of $|E(mathbb{F}_p)|$ from $p+1$ is zero."

Now, how do you prove generalizations of the prime number theorem? There are two main steps in this, one of which is easily lifted from the case of the Riemann zeta function.

1. Prove that the prime number theorem for $L(s,f)$ is a consequence of the nonvanishing of $L(s,f)$ in a region of the form $s=sigma+it,;sigma geq 1-psi(t)$ with $psi(t)$ positive and tending to zero as $tto infty$. So this is some region which is a very slight widening of $mathrm{Re}(s)>1$. The proof of this step is essentially contour integration and goes exactly as in the case of the $zeta$-function.




2. Actually produce a zero-free region of the type I just described. The key to this is the existence of an auxiliary L-function (or product thereof) which has positive coefficients in its Dirichlet series. In the case of the Riemann zeta function, Hadamard worked with the auxiliary function $ A(s)=zeta(s)^3zeta(s+it)^2 zeta(s-it)^2 zeta(s+2it) zeta(s-2it)$. Note the pole of order $3$ at $s=1$; on the other hand, if $zeta(sigma+it)$ vanished then $A(s)$ would vanish at $s=sigma$ to order $4$. The inequality $3<4$ of order-of-polarity/nearby-order-of-vanishing leads via some analysis to the absence of any zero in the range $s=sigma+it,;sigma geq 1-frac{c}{log(|t|+3)}.$ In the general case the construction of the relevant auxiliary functions is more complicated. For the case of an Artin representation, for example, you can take $B(s)=zeta(s)^3 L(s+it,rho)^2 L(s-it,widetilde{rho})^2 L(s,rho otimes widetilde {rho})^2 L(s+2it,rho times rho) L(s-2it,widetilde{rho} times widetilde{rho})$. The general key is the Rankin-Selberg L-functions, or more complicated L-functions whose analytic properties can be controlled by known instances of Langlands functoriality.

If you'd like to see everything I just said carried out elegantly and in crystalline detail, I can do no better than to recommend Chapter 5 of Iwaniec and Kowalski's book "Analytic Number Theory."

Does there exist a Riemann surface corresponding to every field extension? Any other hypothesis needed?

Zariski introduced an abstract notion of Riemann surface associated to, for example, a finitely generated field extension $K/k$. It's a topological space whose points are equivalence classes of valuations of $K$ that are trivial on $k$, or equivalently valuation rings satisfying $ksubset R_vsubset K$. If $A$ is a finitely generated $k$-algebra inside $K$ then those $R_v$ which contain $A$ form an open set.

In the case of a (finitely generated and) transcendence degree 1 extension all of these valuation rings are the familiar DVRs -- local Dedekind domains -- and they serve to identify the points in the unique complete nonsingular curve with this function field. (There is also the trivial valuation with $R_v=K$, which corresponds to the generic point of that curve.)

In higher dimensions there are lots of complete varieties to contend with -- you can keep blowing up. Also there are more possibilities for valuations. Most of the valuation rings are not Noetherian. A curve in a surface gives you a discrete valuation ring, consisting of those rational functions which can meaningfully be restricted to rational functions on the curve: those which do not have a pole there. A point on a curve on a surface gives you a valuation whose ring consists of those functions which do not have a pole all along the curve, and which when restricted to the curve do not have a pole at the given point. The value group is $mathbb Ztimes mathbb Z$ lexicographically ordered. A point on a transcendental curve in a complex surface, or more generally a formal (power series) curve in a surface gives you a valuation by looking at the order of vanishing; the value group is a subgroup of $mathbb R$.

This space of valuations has something of the flavor of Zariski's space of prime ideals in a ring: it is compact but not Hausdorff, for example. It can be thought of as the inverse limit, over all complete surfaces $S$ with this function field, of the space (Zariski topology) of points $S$.

big picture - Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation:

What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?

I believe this to be a serious question because it actually is oftentimes a good idea casually to identify isomorphism classes. To bring up an intermediate-level example I've alluded to often, consider the classification of topological surfaces. When I explain it to students, I do somewhat consciously write equalities as I manipulate one shape into another homeomorphic one. I even do it rather quickly to encourage intuitive associations that are likely to be useful. In any case, for arguments of that sort, it would be really tedious, and probably pointless, to write down isomorphisms with any precision.

Meanwhile, at other times, I've also joined in the chorus of criticism that greets the conflation of equality and isomorphism.

The problem is it's quite challenging to come up with really striking examples where this care is rewarded. Let me start off with a somewhat specialized class of examples. These come from descent theory. The setting is a map $$Xrightarrow Y,$$ which is usually submersive, in some sense suitable to the situation. You would like criteria for an object $V$ lying over $X$, say a fiber bundle, to arise as a pull-back of an object on $Y$. There is a range of formalism to deal with this problem, but I'll just mention two cases. One is when $Y=X/G$, the orbit space of a group action on $X$. For $V$ to be pulled-back from $Y$, we should have $g^*(V)simeq V$ for each $gin G$. But that's not enough. What is actually required is that there be a collection of isomorphisms $$f_g: g^*(V)simeq V$$ that are compatible with the group structure. This means something like $$f_{gh}=f_gcirc f_h,$$ except you have to twist in an obvious way to take into account the correct domain. So you see, I have at least to introduce notation for the isomorphisms involved to formulate the right condition. In practice, when you want to construct something on $Y$ starting from something on $X$, you have to specify the $f_g$ rather precisely.

Another elementary case is when $X$ is an open covering $(U_i)$ of $Y$. Then an object on $Y$ is typically equivalent to a collection $V_i$ of objects, one on each $U_i$, but with additional data. Here as well, $V_i$ and $V_j$ obviously have to agree on the intersections. But that's again not enough. Rather there should be a collection of isomorphisms $$phi_{ji}: V_i|U_icap U_jsimeq V_j|U_icap U_j$$
that are compatible on the triple overlaps:
$$phi_{kj}circ phi_{ji}=phi_{ki}.$$ Incidentally, for something like a vector bundle, since any two of the same rank are locally 'the same,' it's clear that keeping track of isomorphisms will be the key to the transition from collections of local objects to a global object. The formalism is concretely applied in situations where you can define some objects only locally, but would like to glue them together to get a global object. For a really definite example that comes immediately to mind, there is the determinant of cohomology for vector bundles on a family of varieties over a parameter space $Y$. Because a choice of resolution is involved in defining this determinant, which might exist only locally on $Y$, Knudsen and Mumford struggled quite a bit to show that the local constructions glue together. Then Grothendieck suggested the remedy of defining the determinant provisionally as a signed line bundle, which then allowed them to nail down the correct $phi_{ji}$. These days, this determinant is a very widely useful tool, for example, in generating line bundles on moduli spaces.

I apologize if this last paragraph is a bit too convoluted for non-specialists. Part of my reason for writing it down is to illustrate that my main examples for bolstering the 'keep track of isomorphisms' paradigm are a bit too advanced for most undergraduates.

So, to conclude, I'd be quite happy to hear of better examples. As already suggested above, it would be nice to have them be accessible but substantively illuminating. If you would like to discuss, say, different bases for vector spaces, it would be good if the language of isomorphism etc. clarifies matters in a really obvious way, as opposed to a sets-and-elements exposition.

Added: Oh, if you have advanced examples, I would certainly like to hear about them as well.

Added: I see now there are three levels at least to distinguish:

Regarding objects as equal vs. regarding them as isomorphic vs. paying attention to specific isomorphisms.

I somehow conflated the two transitions in the course of asking the question. Of course I'm happy to see good examples illustrating the nature of either, but I'm especially interested in the second refinement.

Added yet again:: I'm grateful to everyone for contributing nice examples, and to Urs Schreiber who put in some effort to instruct me over at the n-category cafe. As I mentioned to Urs there, it would be especially nice to see examples of the following sort.

1. One usually thinks $X=Y$;




2. A careful analysis encourages the view $Xsimeq Y$;




3. This perspective leads to genuinely new insight and benefit.

Even better would be if some specific knowledge of the isomorphism in 2. is important. Of course, more than two objects might be involved. I was initially hoping for some input from combinatorics, with the emphasis on 'bijective proofs' and all that. Anything?

Added, 14 May:

OK, I hope this will be the last addition. Because this question flowed over to the n-category cafe, I ended up having a small discussion there as well. I thought I'd copy here my last response, in case anyone else is interested.

n-cafe post:

I suppose it's obvious by now that I'm using a specific request to drive home the need for 'small but striking examples' in favor of category theory.

Last fall, Eugenia Cheng told me of a visit to some university to give a colloquium talk. The host greeted her with the observation that he doesn't regard category theory as a field of research. OK, he was probably a bit extreme, but milder versions of that view are quite common. Now, one possible response is to regard all such people as unreasonable and talk just to friends (who of course are the reasonable people!). This is not entirely bad, because that might be a way to buy time and gain enough stability to eventually prove the earth-shattering result that will show everyone! Another way is to take up the skepticism as a constructive everyday challenge. This I suppose is what everyone here is doing at some level, anyways.

Other than the derived loop space, which is not exactly small, Urs' examples are all of the simple subtle sort that can, over time, contribute to a really important change in scientific outlook and maybe even the infrastructure of a truly glorious theory. For example, I agree wholeheartedly about the horrors of the old tensor formalism. But it's not unreasonable to ask for more striking accessible evidence of utility when it comes to the current state of category theory.

The importance of small insights and language that gradually accumulate into the edifice of a coherent and powerful theory is the usual interpretation of Grothendieck's 'rising sea' philosophy. However, the process is hardly ever smooth along the way, especially the question of acceptance by the community. I'm not a historian, but I've studied arithmetic geometry long enough to have some sense of the changing climate surrounding etale cohomology theory, for example, over the last several decades. The full proof of the Weil conjectures took a while to come about, as you know. Acceptance came slowly with many bits and pieces sporadically giving people the sense that all those subtleties and abstractions are really worthwhile. Fortunately, the rationality of the zeta function was proved early on. However, there was a pretty concrete earlier proof of that as well using $p$-adic analysis, so I doubt it would have been the big theorem that convinced everyone. One real breakthrough came in the late sixties when Deligne used etale cohomology to show that Ramanujan's conjecture on his tau function could be reduced to the Weil conjectures. There was no way to do this without etale cohomology and the conjecture in question concerned something very precise, the growth rate of natural arithmetic functions. This could even be checked numerically, so impressed people in the same way that experimental verification of a theoretical prediction does in physics. Clearly something deep was going on. Of course there were many other indications. The construction of entirely new representations of the Galois group of $mathbb{Q}$ with very rich properties, the unification of Galois cohomology and topological cohomology, a clean interpretation of arithmetic duality theorems that gave a re-interpretation of class field theory, and so on.

For myself, being a fan of you folks here, I believe this kind of process is going on in category theory. But I don't think you have to be too unreasonable to doubt it. In a similar vein, I don't agree with Andrew Wiles' view that physics will be irrelevant for number theory, but also think his pessimism is perfectly sensible.

I think I'm trying to make the obvious point that the presence of pessimists can be very helpful to the development of a theory, in so far as the optimists interact with them in constructive ways. I haven't been coming to this site much lately. But I did catch David's recent post on Frank Quinn's article, which ended up as a catalyst for my MO question.

At the Boston conference following the proof of Fermat's last theorem, I've been told Hendrik Lenstra said something like this: 'When I was young, I knew I wanted to solve Diophantine equations. I also knew I didn't want to represent functors. Now I have to represent functors to solve Diophantine equations!' So should we conclude that he was foolish to avoid representable functors for so long? I wouldn't.

This response to the MO question brings up the importance of knowing the specific isomorphism between some Hilbert spaces given by the Fourier transform. This is an excellent example, especially when we consider how it relates to the different realizations of the representations of the Heisenberg group and the attendant global issues, say as you vary over a family of polarizations. But I couldn't resist recalling Irving Segal's insistence that 'There's only one Hilbert space!' Obviously, he knew, among many other things, the different realizations of the Stone-Von-Neumann representation as well as anyone, so you can take your own guess as to the reasoning behind that proclamation. He certainly may have lost something through that kind of philosophical intransigence. But I suspect that he, and many around him, gained something as well.

co.combinatorics - Intersection homology for toric varieties

There is a simple and beautiful description in terms of commutative algebra (repeatedly calculating global sections and taking a projective cover). The work of Braden-MacPherson cited by Alexander is relevant, but only for certain toric varieties (those admitting affine pavings). Also, the Braden-MacPherson paper is really aimed at handling the case of flag varieties etc., which is more complicated than toric varieties.

I think the first combinatorial description was given by Bernstein and Lunts at the end of their book on equivariant sheaves:

Bernstein, Joseph; Lunts, Valery
Equivariant sheaves and functors. LNM 1578. Berlin: Springer-Verlag.

This was then abstracted to arbitrary (perhaps non-rational) polytopes here:

Bressler, Paul and Lunts, Valery, Intersection Cohomology on Nonrational Polytopes,
Compositio Mathematica, Volume 135, Issue 3, pp 245-278.
http://arxiv.org/abs/math/0002006

There is parallel work by BBFK:

Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, and Ludger Kaup
Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) Volume 54, Number 1 (2002), 1-41.

All of this is summarized quite nicely in Kirwan and Wolf, An introduction to Intersection Cohomology Theory, Second Edition, Chapman and Hall, 2006.

ca.analysis and odes - roots of sum of two polynomials

Though in general you won't have a closed-form expression for the roots of your polynomials, it's possible to write down perturbation series for roots of a polynomial in a single variable in terms of the coefficients. These are basically the Puiseux series mentioned in this question.

This paper by Bernd Sturmfels (MR) sketches out the "global picture" of such series, though it's fairly complicated and I personally am not clear on whether there's a simple algorithm to decide which is the proper choice of series that will converge. See also the article "Algebraic equations and hypergeometric series" by M Passare, A Tsikh in the book: The Legacy of Niels Henrik Abel (MR).

What I've just written is probably a little unclear so I'll describe the simplest example. Suppose you'd like to write down a series for the roots of $a_2x^2+a_1x+a_0=0$. There are a pair of series which converges when $left|frac{a_1^2}{4a_0a_2}right|<1$ and a pair which converges when $left|frac{a_1^2}{4a_0a_2}right|>1$, and you can derive the first pair of series by treating $a_1x$ as a perturbation to the equation $a_2x^2+a_0=0$ and you can derive one of the second pair of series by treating $a_2x^2$ as a perturbation to $a_1x+a_0=0$ and the other by treating $a_0$ as a perturbation to the equation $a_2x^2+a_1x=0$.

By plugging in the coefficients of $P_n(x)+AQ_n(x)$ into the appropriate series I just described and looking at the leading order terms as functions of $A$, you will be able to derive the scaling of the corrections to the roots of $P_n(x)$.

Apologies for the rather unexplicit answer, but this is just at the limit of what I understand.

motivic cohomology - Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My understanding is that for smooth projective schemes $X$ over $mathbf{Q}$, the following hold or are conjectured to:

1. Certain Ext groups of powers of the Tate motive (base-changed to $X$) are isomorphic to certain motivic cohomology groups of $X$ and to certain pieces of algebraic K-groups of $X$. (I should tensor all these groups with $mathbf{Q}$. I believe there are further subtleties if we don't do this.)




2. If instead of extensions of abstract motives as in #1, we look at extensions in categories of realizations (Hodge structures, $p$-adic Galois representations, ...), the map from the algebraic K-groups to the Ext groups in #1 can be described concretely in terms of polylogarithms, or rather the appropriate version of the polylogarithm for the given realization. In particular, for $K_1$, we get extensions of $mathbf{Q}$ by $mathbf{Q}(1)$ described by logarithms of nonzero numbers. This is more or less Kummer theory.




3. The order of vanishing of the L-functions of $X$ at integers are determined by the ranks of these groups.

Now, everything above was about motives over $mathbf{Q}$ (or over schemes over $mathbf{Q}$) and with coefficients in $mathbf{Q}$. My question is then this:

*Is there an analogous picture if we consider motives over a number field $F$ and with coefficients in $F$?

The motive part is easy: instead of Tate motives, we should look at motives associated to algebraic Hecke characters over $F$ with values in $F$. ("Hecke motives"?) These are surely the same as motives of $F$-linear rank $1$. And then we can consider the Ext groups (in the $F$-linear category). Are these isomorphic to $F$-linear analogues of motivic cohomology or algebraic K-groups? Are these maps given by analogues of polylogarithms, and are there relations to analogues of L-functions?

The only new case where I have a clue is where $F$ is an imaginary quadratic field of class number one. Let $E$ be the elliptic curve over $F$ with complex multiplication by $F$. Then there is a "Kummer map" from $E(F)$ to the Galois cohomology group $H^1(G_F,T_l(E))$ (for any prime $l$), which can presumably be viewed as an Ext group of two Hecke motives (though I suppose there is a Weil-Châtelet obstruction to the map being an isomorphism?). So this suggests that in the $F$-linear world, the role of the K-group $K_1(L)=L^*$, for $L$ an extension of $mathbf{Q}$, would be played by $E(L)$, for $L$ an extension of $F$. I know there is a paper by Beilinson-Levin on elliptic polylogarithms, but I haven't invested the energy to penetrate it. I didn't notice anything in it about complex multiplication, though.

That's all I got. Any ideas?

(All this actually came up rather naturally in some daydreaming about $F$-linear analogues of de Rham-Witt cohomology and topological Hochschild homology, so I'd like to hope there's some connection to reality.)

at.algebraic topology - Geometric model for classifying spaces of alternating groups

Probably the right thing to do is to express the classifying space of $A_n$ as the non-trivial double cover of the classifying space of $S_n$. A point in the classifying space is then a set of $n$ points in $mathbb{R}^infty$ with a "sign ordering". A sign ordering is an equivalence class of orderings of the points, i.e., ways to number them from 1 to $n$, up to even permutations. I coined the term "sign ordering" by analogy with a cyclic ordering. But that name aside, the idea comes up all the time in various guises. For instance an orientation of a simplex is by definition a sign ordering of its vertices.

This is in the same vein as your other examples and you can of course do something similar with any subgroup $G subseteq S_n$. You can always choose an ordering of the points up to relabeling by an element of $G$.

---

A bit more whimsically, you could call the configuration space of $n$ sign-ordered points in a manifold "the configuration space of $n$ fermions". Although a stricter model of the $n$ fermions is the local system or flat line bundle on $n$ unordered points, in which the holonomy negates the fiber when it induces an odd permutation of the points. This local system is similar to the sign-ordered space in the sense that the sign-ordered space is the associated principal bundle with structure group $C_2$.

nt.number theory - Primes P such that ((P-1)/2)!=1 mod P

This is an attempt to justify the answer $1/2$ based on the Cohen-Lenstra heuristics. There will be a lot of nonsensical steps, and I am not an expert, so this should be viewed with caution.

As is observed above, this is equivalent to determining $h(p) mod 4$, where $h(p)$ is the class number of $mathbb{Q}(sqrt{-p})$. Since $p$ is odd and $3 mod 4$, the only ramified prime in $mathbb{Q}(sqrt{-p})$ is the principal ideal $(sqrt{-p})$. Thus, there is no $2$-torsion in the class group and $h(p)$ is odd.

For any odd prime $q$, let $a(q,p)$ be the power of $q$ which divides $h(p)$. We want to compute the average value of
$$prod_{q equiv 3 mod 4} (-1)^{a(q,p)}.$$

First nonsensical step: Let's pretend that the CL-heuristics work the same way for the odd part of the class group of $mathbb{Q}(sqrt{-p})$, that they do for the odd part of the class group of $mathbb{Q}(sqrt{-D})$. We just saw above that the fact that $p$ is prime constrains the $2$-part of the class group; this claim says that it does not effect the distribution of anything else.

Then we are supposed to have:
$$P(a(q,p)=0) = prod_{i=1}^{infty} (1-q^{-i}) = 1-1/q +O(1/q^2),$$
$$P(a(q,p)=1) = frac{1}{q-1} prod_{i=1}^{infty} (1-q^{-i}) = 1/q +O(1/q^2),$$
and
$$P(a(q,p) geq 2) = O(1/q^2).$$

If you believe all of the above, then the average value of $(-1)^{a(p,q)}$ is $ 1-2/q+O(1/q^2)$.

Second nonsensical step: Let's pretend that $a(q,p)$ and $a(q',p)$ are uncorrelated. Furthermore, let's pretend that everything converges to its average value really fast, to justify the exchange of limits I'm about to do.

Then
$$E left( prod_{q equiv 3 mod 4} (-1)^{a(q,p)} right) = prod_{q equiv 3 mod 4} left( 1- 2/q + O(1/q^2) right)$$.`

The right hand side is zero, just as if $h(p)$ were equally like to be $1$ or $3 mod 4$.

formalization - reducing a theorem to set theory using first order logic

Deducing a non-trivial theorem directly from ZFC is a tedious business. First you will need to define the integers in terms of sets. The natural numbers are most commonly encoded as von Neumann ordinals. Then you have to define addition and multiplication. These are functions, which are typically encoded as sets of ordered pairs from $mathbb{N}timesmathbb{N}$ to $mathbb N$. An ordered pair is typically encoded as $(x,y) := lbracelbrace xrbrace,lbrace x,yrbracerbrace$. Then you will have to define primes, etc.

If you really want to go through this exercise, then I would recommend learning Mizar, which is a system for formal proofs. Mizar is based on Tarski-Grothendieck set theory, which is a slight extension of ZFC. Most of the groundwork that I've described above has already been done by previous users of Mizar, so that you just need to "drill down" through the existing definitions in order to figure out how to do things, and don't have to encode it all from scratch yourself.

ct.category theory - When are all split monomorphisms complemented?

This is true in boolean categories (extensive + terminal object + (T : 1 → 1 + 1) is subobject classifier), but for any monic, not just split monics. There's quite a nice and easy read on extensive categories from 93:

Carboni et al. Introduction to extensive and distributive categories. Journal of Pure and Applied Algebra (1993) vol. 84 pp. 145-158

However, modulo axiom of choice, all monics in Set split, and I suspect this could be true for any boolean category.

at.algebraic topology - map of manifolds inducing iso on top cohomology, but not surjective on one other cohomology group

I think I have one: consider a torus in $mathbb{R}^{3}$ embedded as a surface of rotation, e.g. rotate a circle of radius $1$ in the zy plane with center $(0,2,0)$ around the $z$ axis. Now put a small sphere with center $(0,2,0)$ and radius $epsilon$. Then the map $T rightarrow S^{2}$ given by projection towards the center of the sphere should give an isomorphism in $H^{2}$ (since the degree is 1) but it is obviously not surjective in $H^{1}$

lo.logic - Conservation of Hyperarithmetic Sentences over AC and CH.

First, let me remark that in your question, you can combine AC and CH together, rather than having two separate conservation results as you did. In fact, you can ramp CH up to GCH and more, including such principles as $Diamond$ or others, without any problem. That is, the conservation result is that ZFC + GCH proves $varphi$ if and only if ZF proves $varphi$, for a large class of statements $varphi$, including the arithmetic statements, as you mentioned, but much more.

The phenomenon extends completely up the hyperarithmetic hierarchy and beyond, beyond even the analytic sentences up into the projective hiearchary at the level of $Sigma^1_2$. (In this hiearchy, the hyperarithmetic statements are $Delta^1_1$ and the analytic statements $Sigma^1_1$.)

This absoluteness result is the content of the Shoenfield Absoluteness Theorem, which asserts that any $Sigma^1_2$ statement is absolute between between any two transitive models of set theory $Vsubset W$ having the same ordinals. In particular, a $Sigma^1_2$ statement holds in the universe if and only if it holds in the constructible universe $L$, where both AC and GCH hold.

Thus, the $Sigma^1_2$ statements provable in ZFC+GCH are the same as those provable in ZF.

ag.algebraic geometry - Dualizing sheaf on singular curves

I am trying to understand the stabilization map, which takes a prestable curve (a curve with some marked points, and at worst nodal singularities) and returns a stable curve (a curve with some marked points, at worst nodal singularities, and finite automorphisms). Essentially what the stabilization map does is it contracts all of the unstable components of the curve, that is, those with infinite automorphisms.

There is furthermore supposed to be a map from the moduli of prestable curves to the moduli of stable curves, and therefore we need a nice description of the stabilization map which works well in flat families.

One possible such description is supposed to be: Take the dualizing sheaf omega_C of the prestable curve C; then take L := omega_C(x₁ + ... + x_n), where the x_i are the marked points; then some large power of L will be generated by global sections, and the stabilization of C is the image of C under the corresponding map to projective space.

In particular, if C' is an unstable component of C, then L restricted to C' should be O_C', as the image of C' should be a point.

I have several dumb questions:

1. Why is omega_C an invertible sheaf? Does it follow from, e.g., Hartshorne III.7.11?




2. Why is some large power of L generated by global sections?




3. How do we know that this power can be chosen to be constant in families?




4. How do we see that L restricted to unstable C' is O_C'?

My more general question is: Given a singular curve, or at least a prestable curve, what explicit information can we deduce about the dualizing sheaf? For example, is there a way to figure out what the dualizing sheaf looks like when restricted to an irreducible component, or at least a smooth irreducible component?

lo.logic - Proof of Gödel incompleteness

OK, if there's a model M of ZF which can be embedded in another model N via m, we cannot conclude that N thinks m is a model.

However, it seems to me the same proof can be carried without it, only using the following reverse implication for the infinite set of axioms ZF:

(*) if N is a model of ZF and m is any inside model of ZF, it can be lifted to an 'outside' model M, which moreover satisfies A iff N thinks m does

A is the diagonal sentence, saying there's a model which does not satisfy A

Now we can prove:
(1) if ZF is consistent, then A.

Proof: by the completeness Theorem ZF has a model M. if M is negative we're done. otherwise M has a negative sub-model which can be lifted by (*) to a negative model in the 'real world'. so again A is true.

(2) ( the inside version of (1) - for any theorem we can prove it has a valid proof - just write it down explicitly)
ZF proves that : if ZF is consistent, then A.

Now, if ZF is consistent, and it proves its consistency, then using modus ponens on (1) and 'inside' modus ponens of (2) we get:
(1') A
(2') ZF proves A. (Thus, any model satisfies A)

which is a contradiction: A is true iff there's a model that does not satisfy A.

Am I still missing something?

tag removed - Can a limit to zero of a limit to zero assume they're both going at the same rate?

My title can be a bit confusing, so here's a bit of background.
The corollary to the Fundamental Theorem of Calculus says that $int_a^bf(x)dx = F(b)-F(a)$, assuming that F'(x) = f (x), or that the area under the curve f (x) from x = a to x = b is equal to the difference of values of the antiderivative of f (x) at a and b.

The following is my attempt to prove it.
1: The area under the curve of f (x) from x = a to x = b is equal to the area of the rectangles under the curve as you take more and more rectangles. See this image:

Mathematically speaking, it's $int_a^bf(x)dx = lim_{hto 0} sum_{n=1}^{(b-a)/h}hcdot f(a+hn)$
2: Let us replace our measly f (x) with its definition, in terms of the derivative of F (x), namely $f(x) = lim_{jto 0}frac{F(x+j)-F(x)}{j}$. Thus, our first equation becomes

$lim_{hto 0} sum_{n=1}^{(b-a)/h}hcdot lim_{jto 0}frac{F(a+hn+j)-F(a+hn)}{j}$

Now, my question is, since both h and j are going to zero via a limit, can I assume that they are effectively the same? Can I simply replace all instances of j with an h and rid myself of an unnecessary second limit? If I could, my proof would continue as follows:

3: Replacing all j's with h's yields:
$lim_{hto 0} sum_{n=1}^{(b-a)/h}hcdot frac{F(a+h(n+1))-F(a+hn)}{h}$, and the *h*s can cancel out: $lim_{hto 0} sum_{n=1}^{(b-a)/h}F(a+h(n+1))-F(a+hn)$.
4: Thankfully, this becomes a telescoping series, as seen here:
$F(a+h(1))-F(a+0h) + F(a+h(2))-F(a+1h) + F(a+h(3))-F(a+2h) + ... = -F(a-h) + F(b-h)$
$ + F(a+h(frac{b-a}{h}))-F(a+h(frac{b-a}{h}-1) = F(b) - F(b-h)$
which, together, yields -F (a - h) + F (b) as the sum.
Putting this back in, we get $ lim_{h to 0} -F(a - h) + F(b) = F(b) - F(a) = int_a^bf(x)dx = F(b)-F(a) $

However, steps 3 and 4 require the ability for me to assume that h and j are the same thing. My teacher (who admittedly doesn't deal with this too often), whom I asked first, said that perhaps h and j are going to 0 at different rates. However, I do not think that the concept of a limit to 0 at a rate actually means anything.
So the question I bring to you is: Is the operation that I performed to go from step 2 to step 3 a valid operation? If so, why? If not, why not?

Thanks for your help!
-Gabriel Benamy