simplicial stuff - Boolean network as a gauge field

This is a bad idea. First, causal sets are, to put it politely "not well-respected" in theoretical physics. These kinds of discrete structures tend to break a significant amount of the structure of the theories; enough that they typically can't reduce to anything that looks close to any existing theory in any limit.

In particular, they break Lorentz invariance pretty directly, which is tied up with energy and momentum conservation. The standard "hope" is that that this invariance is really "approximate" somehow, and becomes exact in some limit. But this can't really happen, because, in some sense the "errors" cannot average out, they can only add together.

In particular, you no longer have properties like $E = m$ in a particle's rest frame, because this formula is a consequence of local Lorentz invariance. So, what you end up with, is having to fine-tune because making this approximate means saying something like "$E=m+delta$" for a particle. But what about when you have $10^{23}$ particles? Well, now, that $delta$ term has to be fine-tuned by hand to be $10^{-23}$ to keep the microscopic deviations of order one (which is already way too big). Then you can ask what happens if you boost a collection of particles (e.g., a lead ion) to near the speed of light, etc....

There are many other problems with discreteness, too, but this is an immediate and fatal one.

Incidentally, due to the seriousness of making spacetime discrete, a lot of the people who try to do these things (causal sets, dynamical triangulations, LQG, etc) like to claim that their theories do not actually do this. But this is basically a lie; there's no politer way to put it. If you ever see someone claim they do not make spacetime discrete, but do sometime like that, walk away, because they're probably a crackpot.

Also, the idea that spacetime has to be "discrete" somehow is a huge (but, sadly, common) misunderstanding of what quantum mechanics means. There is nothing "inherently" discrete, or any need to introduce any discreteness into physics other than as a computational tool (e.g., lattice field theory) or as a physical system whose initial conditions make this a convenient approximation (e.g., a regular solid structure, crystals, etc).

The initial idea behind thinking about these kinds of things wasn't so bad, though. It comes from the thinking of, in GR, interpreting the abstract points on a spacetime manifold as corresponding to "events," and the geodesic structure as connecting events together (see, for example, the first few chapters of Misner, Thorne, and Wheeler's GR book). This lead to trying to think of, instead of the geometric structure, the structure of "events", not in the sense of the topological structure of the manifold, but as some other structure inherited by how one can connect things together with geodesics. This naturally partitions things into "all things that could have effected the point $x$" and "all things that the point $x$ can effect," with the hope that by studying the structure of these kinds of causal relations, one can reformulate GR in a new (but equivalent!) way that may fit together nicely with the "observables" formulation of quantum mechanics.

It turns out, this doesn't work so well. The structures that you get are so uncontrolled and pathological and unconstrained, that the only thing you can do to get anything sensible is by just doing GR. AFAIK, almost all physicists (and certainly all of the top people in GR) abandoned this approach by the '80s.

You can find some references for this kind of stuff if you look, but it does not really lead anywhere.

Unfortunately, in the past 10 or 15 years, a handful of people came along and misunderstood what these people were doing, and started building crazy discrete models analogous to these, sometimes in combination with taking "lattice GR" ideas way too seriously, and produced a bunch of nonsense.

And, really, the only thing stopping most physicists from calling these ideas outright crackpot nonsense, is the involvement of actual GR bigshots in the ancestors to these ideas decades ago! Although, there are certainly a few well known physicists out there who are famous for getting very angry when these ideas are brought up ;).

If you're really determined to discover in detail why these ideas are so wrong, other than working out stuff for yourself, or learning why things are the way they are in GR and QFT (which is tough!--but you should!) you may be in some trouble. Physicists aren't really in the habit of writing papers about theories that they think are obviously, fundamentally broken. We just tend to make fun of them when talking to our colleagues and otherwise ignore them! So you can't find much in the way of refutations of specifics in published literature. But if you search carefully you can find the occasionally angry rant published somewhere, and there are a few unpublished papers on the arxiv about why these theories are all broken. The only author I remember offhand doing this is Peeters about LQG's many problems, but there are others if you look.

gt.geometric topology - How well can we localize the "exoticness" in exotic R^4?

My question concerns whether there is a contradiction between two particular papers on exotic smoothness, Exotic Structures on smooth 4-manifolds by Selman Akbulut and Localized Exotic Smoothness by Carl H. Brans. The former asserts:

"Let $M$ be a smooth closed simply connected $4$-manifold, and $M'$ be an exotic copy of $M$ (a smooth manifold homeomorphic but not diffeomorphic to $M$). Then we can find a compact contractible codimension zero submanifold $Wsubset M$ with complement $N$, and an involution $f:partial Wto partial W$ giving a decomposition:
$M=Ncup_{id}W$, $M'=Ncup_{f}W$."

The latter states:

"Gompf's end-sum techniques are used to establish the existence of an infinity of non-diffeomorphic manifolds, all having the same trivial ${bf R^4}$ topology, but for which the exotic differentiable structure is confined to a region which is spatially limited. Thus, the smoothness is standard outside of a region which is topologically (but not smoothly) ${bf B^3}times {bf R^1}$, where ${bf B^3}$ is the compact three ball. The exterior of this region is diffeomorphic to standard ${bf R^1}times {bf S^2}times{bf R^1}$. In a space-time diagram, the
confined exoticness sweeps out a world tube..."

and further:

"The smoothness properties of the ${bf R^4_Theta}$... can be
summarized by saying the global $C^0$ coordinates, $(t,x,y,z)$, are
smooth in the exterior region $[a,infty){bftimes S^2times R^1}$ given
by $x^2+y^2+z^2>a^2$ for some positive constant $a$, while the closure of
the complement of this is clearly an exotic ${bf B^3times_Theta
R^1}$. (Here the 'exotic' can be understood as referring to the
product which is continuous but cannot be smooth...)"

The theorem from the first paper applies to closed manifolds. Is it generalizable to open manifolds (such as $mathbb{R}^4$)? If so, then its confinement of "exoticness" to a compact submanifold seems inconsistent with the world tube construction implied in the statements from the second paper.

computational complexity - How unhelpful is graph minors theorem?

To answer some of your questions:
1. Yes, testing such properties is usually hard. For instance, before the graph minors theorem we had properties for which no algorithm was known at all! (Maybe a recursively enumerable algorithm was known, I don't remember.) After the graph-minors theorem, such properties became testable in polynomial time! Now that's a big jump from no algorithm known to polynomial time. (If I remember correctly, the polynomial is like O(n log n), which is almost linear time.)

As for 2. and 3., I don't know any property which we can easily test, for which we don't know the forbidden minors. I'd like to know such properties, if they are known. It seems to me that if we have an algorithm for easily testing a property, we really understand the property, and therefore should be able to come up with a list of forbidden minors somehow. Of course, this is just a feeling.

EDIT: I've been corrected in the comments. Please read Gil Kalai and David Eppstein's comments.

gn.general topology - Do the empty set AND the entire set really need to be open?

I see a little more in the question now. It seems that the OP is proposing to eliminate the axioms that the empty space and the total space are open but maintain the axioms that arbitrary (nonempty!) unions and finite (nonempty!) intersections of open sets are open.

In this case there is a little content here, because you can try to figure out whether or not $emptyset$ and $X$ will then be open or not.

Note one disturbing fact: with the elimination of the above axiom, there is nothing to imply the existence of any open sets at all! Whether this is good or bad, if it happens there is nothing further to say, so let's assume that there is at least one open set.

Claim: If $X$ is a Hausdorff with more than one point, then the empty set and the total space are open.

Proof: Indeed, the Hausdorff axiom asserts that any two points have disjoint open neighborhooods, so the intersection of these is empty. The same axiom says in particular that every point has at least one open neighborhood (!) which is clearly equivalent to
$X$ being open.

Claim: If $X$ is T1 with more than one point, then the total space is open, but the empty set need not be.

Proof: T1 means that the singleton sets are closed. If there are least two of them, take
their intersection: this makes the empty set closed, hence the total space open. On the other hand, the cofinite topology on an infinite set is T1 and the empty set is not open (if we do not force it to be).

I am coming around to agree with K. Conrad in that having $X$ be open in itself may not be just a formality. A topological space in which some point has no neighborhood sounds like trouble...I guess I was thinking that if you get into real trouble, you can just throw $X$ back in as an open set! If you want to put it that way, this is some kind of completion functor from the OP's generalized topological spaces to honest topological spaces.

ag.algebraic geometry - Weierstrass points on rigid-analytic surfaces

Quick note: I am going to assume you want to talk about complete curves. One can, of course, have a curve with punctures in algebraic geometry, and I'm not sure how you'd want to define a Weierstrass point on it. In rigid geometry, you have even more freedom: you can have the analogue of a Riemann surface with holes of positive area, and I think (not sure) you can also build the analogue of a Riemann surface of infinite genus. I'm going to assume you are not thinking about these issues.

What you want is the rigid GAGA theorem. I'm not sure what the best reference is; I refreshed my memory from Coleman's lectures, numbers 23-25. Rigid GAGA says:

Let $mathcal{X}$ be a projective rigid analytic variety. Then

(1) $mathcal{X}$ is the analytification of an algebraic variety $X$.

(2) The analytificiation functor from coherent sheaves on $X$ to coherent sheaves on $mathcal{X}$ is an equivalence of categories.

(3) The cohomology of a coherent sheaf is naturally isomorphic to that of its analytification.

Thus, if we define Weierstrass points by the condition that the dimension of $H^0(mathcal{O}(kp), X)$ is higher than expected, we will get the same points whether we work algebraically or analytically. It shouldn't be too hard to show that your favorite definition is equivalent to this.

Of course, all I've done is tell you how to translate between analysis and algebra. The algebra itself may be very difficult, as Felipe Voloch points out.

qa.quantum algebra - Quantum channels as categories: question 1.

A quantum channel is a mapping between Hilbert spaces, $Phi : L(mathcal{H}_{A}) to L(mathcal{H}_{B})$, where $L(mathcal{H}_{i})$ is the family of operators on $mathcal{H}_{i}$. In general, we are interested in CPTP maps. The operator spaces can be interpreted as $C^{*}$-algebras and thus we can also view the channel as a mapping between $C^{*}$-algebras, $Phi : mathcal{A} to mathcal{B}$. Since quantum channels can carry classical information as well, we could write such a combination as $Phi : L(mathcal{H}_{A}) otimes C(X) to L(mathcal{H}_{B})$ where $C(X)$ is the space of continuous functions on some set $X$ and is also a $C^{*}$-algebra. In other words, whether or not classical information is processed by the channel, it (the channel) is a mapping between $C^{*}$-algebras. Note, however, that these are not necessarily the same $C^{*}$-algebras. Since the channels are represented by square matrices, the input and output $C^{*}$-algebras must have the same dimension, $d$. Thus we can consider them both subsets of some $d$-dimensional $C^{*}$-algebra, $mathcal{C}$, i.e. $mathcal{A} subset mathcal{C}$ and $mathcal{B} subset mathcal{C}$. Thus a quantum channel is a mapping from $mathcal{C}$ to itself.

Proposition A quantum channel given by $t: L(mathcal{H}) to L(mathcal{H})$, together with the $d$-dimensional $C^{*}$-algebra, $mathcal{C}$, on which it acts, forms a category we call $mathrm{mathbf{Chan}}(d)$ where $mathcal{C}$ is the sole object and $t$ is the sole arrow.

Proof: Consider the quantum channels

$begin{eqnarray*}
r: L(mathcal{H}_{rho}) to L(mathcal{H}_{sigma}) &
qquad textrm{where} qquad &
sigma=sum_{i}A_{i}rho A_{i}^{dagger} \
t: L(mathcal{H}_{sigma}) to L(mathcal{H}_{tau}) &
qquad textrm{where} qquad &
tau=sum_{j}B_{j}sigma B_{j}^{dagger}
end{eqnarray*}$

where the usual properties of such channels are assumed (e.g. trace preserving, etc.). We form the composite $t circ r: L(mathcal{H}_{rho}) to L(mathcal{H}_{tau})$ where

$begin{align}
tau & = sum_{j}B_{j}left(sum_{i}A_{i}rho A_{i}^{dagger}right)B_{j}^{dagger} notag \
& = sum_{i,j}B_{j}A_{i}rho A_{i}^{dagger}B_{j}^{dagger} \
& = sum_{k}C_{k}rho C_{k}^{dagger} notag
end{align}$

and the $A_{i}$, $B_{i}$, and $C_{i}$ are Kraus operators.

Since $A$ and $B$ are summed over separate indices the trace-preserving property is maintained, i.e. $$sum_{k} C_{k}^{dagger}C_{k}=mathbf{1}.$$ For a similar methodology see Nayak and Sen (http://arxiv.org/abs/0605041).

We take the identity arrow, $1_{rho}: L(mathcal{H}_{rho}) to L(mathcal{H}_{rho})$, to be the time evolution of the state $rho$ in the absence of any channel. Since this definition is suitably general we have that $t circ 1_{A}=t=1_{B} circ t quad forall ,, t: A to B$.

Consider the three unital quantum channels $r: L(mathcal{H}_{rho}) to L(mathcal{H}_{sigma})$, $t: L(mathcal{H}_{sigma}) to L(mathcal{H}_{tau})$, and $v: L(mathcal{H}_{tau}) to L(mathcal{H}_{upsilon})$ where $sigma=sum_{i}A_{i}rho A_{i}^{dagger}$, $tau=sum_{j}B_{j}sigma B_{j}^{dagger}$, and $eta=sum_{k}C_{k}tau C_{k}^{dagger}$. We have

$begin{align}
v circ (t circ r) & = v circ left(sum_{i,j}B_{j}A_{i}rho A_{i}^{dagger}B_{j}^{dagger}right) = sum_{k}C_{k} left(sum_{i,j}B_{j}A_{i}rho A_{i}^{dagger}B_{j}^{dagger}right) C_{k}^{dagger} notag \
& = sum_{i,j,k}C_{k}B_{j}A_{i}rho A_{i}^{dagger}B_{j}^{dagger}C_{k}^{dagger} = sum_{i,j,k}C_{k}B_{j}left(A_{i}rho A_{i}^{dagger}right)B_{j}^{dagger}C_{k}^{dagger} notag \
& = left(sum_{i,j,k}C_{k}B_{j}tau B_{j}^{dagger}C_{k}^{dagger}right) circ r = (v circ t) circ r notag
end{align}$

and thus we have associativity. Note that similar arguments may be made for the inverse process of the channel if it exists (it is not necessary for the channel here to be reversible). $square$

Question 1: Am I doing the last line in the associativity argument correct and/or are there any other problems here? Is there a clearer or more concise proof? I have another question I am going to ask as a separate post about a construction I did with categories and groups that assumes the above is correct but I didn't want to post it until I made sure this is correct.

soft question - Most helpful math resources on the web

Sloane's OEIS has already been mentioned.

A similarly useful site is ISC, Simon Plouffe's Inverse Symbolic Calculator.

Here you enter the decimal expansion of a number to as many places as you know, and the search engine makes suggestions of symbolic expressions that the expansion might be derived from. The answer might involve pi, e, sin, cosh, sqrt, ln, and so on.

Sometimes, it becomes difficult to calculate symbolically. Therefore, you can proceed numerically instead, and hope to recover the exact symbolic solution at the end, using ISC: sometimes proving that an answer is correct can be easier than calculating, or discovering, it in the first place.

It can also be useful for discovering simplifications of nested radicals, for example.

ca.analysis and odes - Cone in a metric space

EDIT: As pointed out by Pete below, it seems I misunderstood the question, so what I write below is not relevant. Apologies!

---

This is not the general answer, but in riemannian geometry there is a notion of cone. If $(M,g)$ is a riemannian manifold, then its metric cone is $mathbb{R}^+ times M$, with $mathbb{R}^+$ the positive real half-line parametrised by $r>0$, with metric
$$dr^2 + r^2 g$$
The best example is of course $(M,g)$ the unit sphere in $mathbb{R}^n$ and its cone is then $mathbb{R}^n setminus lbrace 0rbrace$. In this case (and in this case only) the metric extends smoothly to the origin, but in general the apex of the cone is singular.

This is used as a local model for conical (!) singularities and there is a nice interplay between the geometry of $M$ and that of its cone. The most dramatic use of the cone I know is that it turns the problem of determining which riemannian spin manifolds admit real Killing spinors into a holonomy problem, namely the determination of which metric cones admit parallel spinors.

Some of this generalises to the pseudo-riemannian setting; although this is perhaps not as useful as in the riaemannian setting as the holonomy classification in indefinite signatures (except for lorentzian) is still lacking.

ag.algebraic geometry - A reference: the splitting principle for exterior powers of coherent sheaves?

My guess would be that the formula you want does not extend to the case of coherent sheaves. As indicated in Mariano and David answers (which has unfortunately been deleted), the best hope to compute is via a resolution $mathcal F$ of $E$ by vector bundles. In general, for 2 perfect complexes $mathcal F, mathcal G$ of vector bundles, there is a formula for the localized chern classes
$$ch_{Ycap Z}(mathcal F otimes mathcal G) = ch_Y(mathcal F)ch_Z(mathcal G)$$, with $Y,Z$ being the respective support. Unfortunately, this only gives the right formula for the "derived tensor product".

So to mess up the formula, one can pick $E$ such that $Tor^i(E,E)$ are non-trivial. I think an ideal sheaf of codimension at least 2 would be your best bet for computation purpose.

lo.logic - Proof assistants for mathematics

Are you aware of the Archive of Formal Proof for Isabelle? It's a collection of formalized mathematics (and some program verification). Reading the papers there, and browsing the Isabelle theory file sources is a good way to learn.

The Isar tutorial is also a good place to look, if you want to write proofs that look like informal mathematics (as opposed to tactic style). It's quite hard to get the hang of at first (mostly due to lack of documentation), but once you get it, it's a lot easier to work with than plain lists of tactics.

If you're wanting to formalise anything with name binders (lambda-calculus, FOL, programming languages, pi-calculus, etc.) you should also check out the Nominal package for Isabelle which again helps with abstracting the proofs.

gr.group theory - Realizability of irreducible representations of dihedral groups

The name of the concept you are looking for is the Schur index. The Schur index is 1 iff the representation can be realized over the field of values. The Schur index divides the degree of the character.

In your case, the the Schur index is either 1 or 2. You can use a variety of tests to eliminate 2, but for instance:

Fein, Burton; Yamada, Toshihiko. "The Schur index and the order and exponent of a finite group." J. Algebra 28 (1974), 496–498. MR427442 DOI: 10.1016/0021-8693(74)90055-6

shows that if the Schur index was 2, then 4 divides the exponent of G.

In other words, all of your representations are realizable over the field of values.

Isaacs's Character Theory of Finite Groups has most of this in it, and I found the rest of what I needed in Berkovich's Character Theory collections. Let me know if you want more specific textbook references.

Edit: I went ahead and looked up the Isaacs pages, and looks like textbook is enough here: Lemma 10.8 on page 165 handles induced irreducible characters from complemented subgroups, and shows that the Schur index divides the order of the original character. Taking the subgroup to be the rotation subgroup and the original character to be faithful (or whichever one you need for your particular irreducible when n isn't prime), you get that the Schur index divides 1. The basics of the Schur index are collected in Corollary 10.2 on page 161.

At any rate, Schur indices are nice to know about, and if Isaacs's book doesn't have what you want, then Berkovich (or Huppert) has just a silly number of results helping to calculate it.

Edit: Explicit matrices can be found too. If n=4k+2 is not divisible 4, and G is a dihedral group of order n with presentation ⟨a,b:aa=b^n=1, ba=ab^(n-1)⟩, then one can use companion polynomials to give an explicit representation (basically creating an induced representation from a complemented subgroup). Send a to [0,1;1,0], also known as multiplication by x. Send b to [0,-1;1,z+1/z], also known as the companion matrix to the minimum polynomial of z over the field Q(z+1/z), where z is a primitive (2k+1)st root of unity.

Compare this to the more direct choice of a=[0,1;1,0] and b=[z,0;0,1/z]. If you conjugate this by [1,z;z,1], then you get my suggested choice of a representation.

In general, finding pretty, (nearly-)integral representations over a minimal splitting field is hard (and there may not be a unique minimal splitting field), but in some cases you can do it nicely.

Let me know if you continue to find this stuff interesting. I could ramble on quite a bit longer, but I think MO prefers focussed answers.

arithmetic geometry - Families of genus 2 curves with positive rank jacobians

My guess for some examples is the family of (genus 2) hyperelliptic curves y²=degree 6 poly in x passing through n "randomly chosen" rational points (for n=2,3,4,5, or 6). The family of such curves has dimension 7-n, and I would guess that if a hyperelliptic curve has n "random" rational points on it then the Q rank of its Jacobian is usually at least n-1.

An obvious place to look for explicit examples is
MR1406090 Cassels, J. W. S.; Flynn, E. V. Prolegomena to a middlebrow arithmetic of curves of genus 2. London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. ISBN: 0-521-48370-0

matrices - Number of unique determinants for an NxN (0,1)-matrix.

I'm interested in bounds for the number of unique determinants of NxN (0,1)-matrices. Obviously some of these matrices will be singular and therefore won't have a determinant. While it might also be interesting to ask what number of NxN (0,1)-matrices are singular or non-singular, I'd like to ignore singular matrices altogether in this question.

To get a better grasp on the problem I wrote a computer program to search for the values given an input N. The output is below:
1x1: 2 possible determinants
2x2: 3 ...
3x3: 5 ...
4x4: 9 ...
5x5: 19 ...

Because the program is simply designed to just a brute force over every possible matrix the computation time grows with respect to $O(2^{N^2})$. Computing 6x6 looks like it is going to take me close to a month and 7x7 is beyond hope without access to a cluster. I don't feel like this limited amount of output is enough to make a solid conjecture.

I have a practical application in mind, but I'd also like to get the bounds to satiate my curiosity.

definitions - Q-construction and Gabriel-Zisman Localization

It might be a stupid question.

When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but morphism between two objects $M$ and $M'$ are isomorphism class of following diagrams:

$M'leftarrow Nrightarrow M$

where the first morphism is admissible epimorphsim and second is admissble monomorphism.

I found the shape of this diagram is similar the definition of Gabriel-Zisman localization for category. Suppose we have categroy $C$, then objects in $Sigma_{T}^{-1}C$ are the same as those in $C$ but morphism between two objects $M,M'$ is the equivalent class of following diagrams:

$M'leftarrow Nrightarrow M$,where the first morphism $sin Sigma_{T}$.

The composition law is also similar to the above one.

Of course, the equivalence relations of these two constructions are different. But, are there any relationship between these two constructions? Maybe Quillen Q-constructions are inspired by the "localization constructions" ?

Any retags are welcome

ac.commutative algebra - What is the right definition of the Picard group of a commutative ring?

This is a rather technical question with no particular importance in any case of actual interest to me, but I've been writing up some notes on commutative algebra and flailing on this point for some time now, so I might as well ask here and get it cleared up.

I would like to define the Picard group of an arbitrary (i.e., not necessarily Noetherian) commutative ring $R$. Here are two possible definitions:

(1) It is the group of isomorphism classes of rank one projective $R$-modules under the
tensor product.

(2) It is the group of isomorphism classes of invertible $R$-modules under the tensor product, where invertible means any of the following equivalent things [Eisenbud, Thm. 11.6]:

a) The canonical map $T: M otimes_R operatorname{Hom}_R(M,R) rightarrow R$ is an isomorphism.
b) $M$ is locally free of rank $1$ [edit: in the weaker sense: $forall mathfrak{p} in operatorname{Spec}(R), M_{mathfrak{p}} cong R_{mathfrak{p}}$.]
c) $M$ is isomorphic as a module to an invertible fractional ideal.

What's the difference between (1) and (2)? In general, (1) is stronger than (2), because projective modules are locally free, whereas a finitely generated locally free module is projective iff it is finitely presented. (When $R$ is Noetherian, finitely generated and finitely presented are equivalent, so there is no problem in this case. This makes the entire discussion somewhat academic.)

So, a priori, if over a non-Noetherian ring one used (1), one would get a Picard group that was "too small". Does anyone know an actual example where the groups formed in this way are not isomorphic? (That's stronger than one being a proper subgroup of the other, I know.)

Why is definition (2) preferred over definition (1)?

mathematics education - Relating Category Theory to Programming Language Theory

The most immediately obvious relation to category theory is that we have a category consisting of types as objects and functions as arrows. We have identity functions and can compose functions with the usual axioms holding (with various caveats). That's just the starting point.

One place where it starts getting deeper is when you consider polymorphic functions. A polymorphic function is essentially a family of functions, parameterised by types. Or categorically, a family of arrows, parameterised by objects. This is similar to what a natural transformation is. By introducing some reasonable restrictions we find that a large class of polymorphic functions are in fact natural transformations and lots of category theory now applies. The standard examples to give here are the free theorems.

Category theory also meshes nicely with the notion of an 'interface' in programming. Category theory encourages us not to look at what an object is made of but how it interacts with other objects, and itself. By separating an interface from an implementation a programmer doesn't need to know anything about the implementation. Similarly category encourages to think about objects up to isomorphism - it doesn't precisely what sets our groups are made of, it just matters what the operations on our groups are. Category theory precisely captures this notion of interface.

There is also a beautiful relationship between pure typed lambda calculus and cartesian closed categories. Any expression in the calculus can be interpreted as the composition of the standard functions that come with a CCC: like the projection onto the factors of a product, or the evaluation function. So lambda expressions can be interpreted as applying to any CCC. In other words, lambda calculus is an internal language for CCCs. This is explicated in Lambek and Scott. This means that the theory of CCCs is deeply embedded in Haskell, say, because Haskell is essentially pure typed lambda calculus with a bunch of extensions.

Another example is the way structurally recursing over recursive datatypes can be nicely described in terms of initial objects in categories of F-algebras. You can find some details here.

And one last example: dualising (in the categorical sense) definitions turns out to be very useful in the programming languages world. For example, in the previous paragraph I mentioned structural recursion. Dualising this gives the notions of F-coalgebras and guarded recursion and leads to a nice way to work with 'infinite' data types such as streams. Working with streams is tricky because how do you guard against inadvertently trying to walk the entire length of a stream causing an infinite loop? The appropriate dual of structural recursion leads to a powerful way to deal with streams that is guaranteed to be well behaved. Bart Jacobs, for example, has many nice papers in this area.

soft question - How to respond to "I was never much good at maths at school."

It's hard to see this question as something specifically relevant to mathematics. That is, it sounds pretty much the same as asking `What are good ways of carrying on a conversation?' Of course there's no simple answer to that either. To the extent that I'm able to give anyone advice, I would simply call for general sympathy. If you've already established a reasonable dialogue, I don't see that the response you mention would have to carry any problematic connotations. For example, asking plenty of sincere questions about the other person's interests would be one easy way of eventually having a serious discussion about your own.

I would like to remark that the prevalence of a negative reaction to the mathematical profession appears to me greatly exaggerated. I can very honestly say I've never encountered it myself in any social situation. Unfavorable and unrealisic comparison to the arts appears with surprising frequency, as in Lockhart's Lament. But I wonder if such commentators have ever compared the income or employability of an average mathematician with that of an average artist, musician, or writer. My feeling is we are often misled by the fact that the stars in art, music, or literature are so much more prominent than their mathematical counterparts. I'd rather not do it now, but I believe it's rather easy to establish that mathematicians do very well on the average, and this is generally a good thing about the profession. Average status is alright, and we give our stars due respect without overinflating them. The absence of publically recognizable gloss keeps us humble enough to be suitably happy. One could say the same for academic apitutude in general, brought up in one of the answers above, which again is quite well-rewarded on the average. Muddled thinking surrounding 'creativity' has been detrimental to education in many ways, but it's really quite sad if the rhetoric keeps even my fellow mathematicians from recognizing their overall good fortune! (Incidentally, I won't go into them here, but I also have reasons to consider mathematicians especially fortunate even among academics.)

By the way, I really don't like the answers that suggest blaming former teachers. This is an unfortunately common and irresponsible response to a complex question. I don't want to express this in too harsh a manner, but I might ask Qiaochu if he'd heard the teachers' side of the story before reaching his conclusions. Most primary and secondary school teachers I know of are dedicated people working under frequently difficult circumstances, doing what they can to teach basic and necessary skills. Perhaps their own skills are not optimal, but neither are mine.

Maybe I should nevertheless conclude this ramble with a brief answer to the original question: With genuine sympathy, say `Oh, I'm sorry to hear that,' and proceed with good humor.

P.S. I apologize for the moralistic tone of this reply. My impression is many users of this site are quite young, and I'm still a Confucian at heart.

Added, 18 November:

I feel compelled to add a few comments given the seriousness of the topic. That is, I would like people to consider the possibility that many perceived problems arise simply because of the overwhelming importance of mathematics.

Jonah Sinick made a number of thoughtful comments in a separate email, and one point he brought up was that the math he learned prior to the university looks nothing like the kind he knows and loves now. This is probably true. But since the comparison with music or sports is made frequently, consider how much a beginner at the violin practicing the scales resembles a professional performing Beethoven. And then how about running around the field for hours to build up stamina vs. the antics of NBA players? Obviously, there are many things whose mastery requires patient tolerance of the basic skills. Why then so much more perceived difficulty with regard to mathematics? This is because the entire population must learn it to a certain level, much higher, for example, than typical violin skills. Society in general at present considers mathematics that important, and much of the dysfunction flows out of this situation. I leave it to you to imagine a scenario where every child had to perform basic violin pieces as well as basic arithmetic operations, regardless of their inclination or talent. And then imagine we needed as many violin teachers as teachers of mathematics.

I hope none of you wish to argue then that mathematics is in fact not so important as to require such massive social investment. This is not because it runs against our interest, but because that's a really difficult case to make in our modern age, and probably too long to fit into the margins of this webpage. In short, don't blame teachers, don't blame the children, don't blame mathematicians. Blame industrial civilization, if you're so inclined.

The main thing I find surprising in many such discussions is that everyone realizes playing the violin or basketball well requires many hours of boring practice that lead up to the joy that comes with the different levels of mastery. But mathematics, and perhaps some other academic subjects, are supposed to be constantly concerned with being 'fun'. And then, I don't see the violinists themselves viewing the lack of inspiration in scales as a fundamental problem.

Of course I hope as many teachers as possible are able to convey some real sense of joy in regard to the mathematics they teach, and various notions of creativity can be occasionally useful and inspiring. But given the sheer scale of the task at hand, I wouldn't hope for methodological ingenuity to bring about overall improvement at any easily measurable pace.

Yet another addition.

I realize as I reread my own paragraphs that I've been explaining mostly one side of the story. So I'm adding a link to a short article that attempts to present a more balanced perspective.

applications - Robust black box function minimization with extremely expensive cost function

There is an enormous amount of information about the common applied math problem of minimizing a function.. software packages, hundreds of books, research, etc.
But I still have not found a good reference for the case where the function to be sampled is extremely expensive.

My specific problem is an applied one of computer science, where I have a simulation which has databases with a dozen parameters that affect voxel sizes, cache distribution, tree branching, etc. These numeric parameters don't affect the algorithm correctness, just runtime. I want to minimize runtime.

Thus, I can treat the problem like a black box minimization. My cost function is runtime, which I want to minimize. I don't have derivatives, and I can treat it like a black box.
I have a decent starting point and even rough scales of each parameter. There can be interations and correlations between parameters and even noise in time measurements (luckily small.)

So why not just throw this into a standard least-squares minimization tool, using any package out there? Because my timing samples each take 8 hours to run.. so each data point is precious, and the algorithms I find tend to ignore this cost. A classic Levenberg-Marquand procedure freely "spends" samples and doesn't even remember the full history of each sample taken (instead updating some average statistics).

So my question is to ask for a pointer to iterative function minimization methods which use the minimum number of samples of the function. Ideally it would work where I could pass in a set of already-sampled locations and the value at each location, and the algorithm would spit out a single new location to take the next sample (which may be an exploratory sample, not a guess at a best minimum location, if the algorithm thinks it's worthwhile to test.)

I can likely take hundreds of samples, but only hundreds, and most multidimensional minimization methods expect to take millions.

Currently I am doing doing the minimization manually daily, using my own ad-hoc invention. I have say 40 existing timing samples to my 15-parameter model. I fit all my existing samples to a sum of independent quadratics (making the big initial assumption that each parameter is independent) then look at each of the N*(N-1)/2 ~=100 possible correlation coefficients of the full quadratic matrix. I find the few single matrix entries that when allowed to change from 0.0, give the best quadratic fit to my data, and allow those few entries to be their best least-squares fit. I also give locations with small (faster) values higher weight in the fit (a bit ad hoc, but useful to throw out behavior distant from the minimum) Once I have this matrix, I manually look at graphs in each of the major eigenvalue directions and eyeball locations which seem to need better sampling. I recombine all these guesses back into a new sample location. Each day, I tend to generate 4 new points, set up a run to test them over the next day, and repeat the whole thing again after the computation is done. Weekends get 10 point batches!

Thanks for any ideas! This question likely doesn't have a perfect "best" answer but I'm stuck at what strategy would work best when the evaluation cost is so huge.

Testing for Riemannian Isometry

In most physics situations one gets the metric as a positive definite symmetric matrix in some chosen local coordinate system.

Now if on the same space one has two such metrics given as matrices then how does one check whether they are genuinely different metrics or just Riemannian Isometries of each other (and hence some coordinate change can take one to the other).

If in the same coordinate system the two matrices are different then is it proof enough that they are not isometries of each other? (doing this test over say a set of local coordinate patches which cover the manifold)

Asked otherwise, given two ``different" Riemannian Manifolds how does one prove the non-existence of a Riemannian isometry between them?

There have been two similar discussions on mathoverflow at this and this one.
And this article was linked from the later.

In one of the above discussions Kruperberg had alluded to a test for local isometry by checking if the Riemann-Christoffel curvatures are the same locally. If the base manifold of the two riemannian manifolds is the same then one can choose a common coordinate system in which to express both the given metrics and then given softwares like mathtensor by Mathew Headrick this is probably not a very hard test to do.

So can one simply patch up such a test through out a manifold to check if two given riemannian manifolds are globally isometric?

How does this compare to checking if the metrics are the same or not in a set of common coordinate patches covering the manifold?

I somehow couldn't figure out whether my query above is already getting answered by the above two discussions.

lo.logic - On statements provably independent of ZF + V=L

The Incompleteness theorem provides exactly the requested independence. (But did I sense in your question that perhaps you thought otherwise?)

The Goedel Incompleteness theorem says that if T is any consistent theory interpreting elementary arithmetic with a computable list of axioms, then T is incomplete. Goedel provided a statement σ, such as the "I am not provable" statement, which is provably equivalent to Con(T), or if you prefer, the Rosser sentence, "there is no proof of me without a shorter proof of my negation", such that T does not prove σ and T does not prove ¬σ.

This establishes Con(T) implies Con(T + σ) and Con(T + ¬σ), as you requested. [Edit: one really needs to use the Rosser sentence to make the full conclusion here.]

In particular, this applies to the theory T = ZFC+ V=L, since this theory interprets arithmetic and has a computable list of axioms. Thus, this theory, if consistent, is incomplete, using the corresponding sentences σ above. Since it is also known (by another theorem of Goedel) that Con(ZF) is equivalent to Con(ZFC + V=L). This establishes the requrested implication:

• Con(ZF) implies Con(ZFC + V=L + σ) and Con(ZFC + V=L + ¬σ)

The Incompleteness theorem can be proved in a very weak theory, much weaker than ZFC or even PA, and this implication is similarly provable in a very weak theory (PA suffices).

One cannot provably omit the assumption Con(ZF) of the implication, since the conclusion itself implies that assumption. That is, the existence of an independent statement over a theory directly implies the consistency of the theory. So since we cannot prove the consistency of the theory outright, we cannot prove the existence of any independent statements. But in your question, you only asked for relative consistency (as you should to avoid this triviality), and this is precisely the quesstion that the Incompleteness theorem answers.

ct.category theory - Why is a monoid with closed symmetric monoidal module category commutative?

(I have rewritten parts to respond to potential objections, since someone disliked this answer. I didn't change the thrust of my discussion, but I did alter some places where I might have been slightly too glib. If I missed something, please feel free to correct me in the comments!)

The situation is even better than that! Suppose we are given an $E_1$-algebra $A$ of a presentable symmetric monoidal $infty$-category $mathcal{C}$.

Call an $E_n$-monoidal structures on the $infty$-category $mathbf{Mod}(A)$ of left $A$-modules allowable if $A$ is the unit and the right action of $mathcal{C}$ on $mathbf{Mod}(A)$ is compatible with the $E_n$ monoidal structure, so that $mathbf{Mod}(A)$ is an $E_n$-$mathcal{C}$-algebra. Then the space of allowable $E_n$-monoidal structures is equivalent to the space of $E_{n+1}$-algebra structures on $A$ itself, compatible with the extant $E_1$ structure on $A$. (This is even true when $n=0$, if one takes an $E_0$-monoidal category to mean a category with a distinguished object.) The object $A$, regarded as the unit $A$-module, admits an $E_n$-algebra structure that is suitably compatible with the $E_1$ structure an $A$. [Reference: Jacob Lurie, DAG VI, Corollary 2.3.15.]

Let's sketch a proof of this claim in the case Peter mentions. Suppose $A$ is a monoid in a presentable symmetric monoidal category $(mathbf{C},otimes)$. Suppose $mathbf{Mod}(A)$ admits a monoidal structure (not even a priori symmetric!) in which $A$, regarded as a left $A$-module, is the unit. I claim that $A$ is a commutative monoid. Consider the monoid object $mathrm{End}(A)$ of endomorphisms of $A$ as a left $A$-module; the Eckmann-Hilton argument described below applies to the operations of tensoring and composing to give $mathrm{End}(A)$ the structure of a commutative monoid object. The multiplication on $A$ yields an isomorphism of monoids $Asimeqmathrm{End}(A)$.

In the case you mention, the result amounts to the original Eckmann-Hilton, as follows. If $X$ admits magma structures $circ$ and $star$ with the same unit (Below, Tom Leinster points out that I only have to assume that each has a unit, and it will follow that the units are the same. He's right, of course.) with the property that

$$(acirc b)star(ccirc d)=(astar c)circ(bstar d)$$

for any $a,b,c,din X$, then (1) the magma structures $circ$ and $star$ coincide; (2) the product $circ$ is associative; and (3) the product $circ$ is commutative. That is, a unital magma in unital magmas is a commutative monoid.

big list - Examples of common false beliefs in mathematics

Here are two things that I have mistakenly believed at various points in my "adult mathematical life":

For a field $k$, we have an equality of formal Laurent series fields $k((x,y)) = k((x))((y))$.

Note that the first one is the fraction field of the formal power series ring $k[[x,y]]$. For instance, for a sequence ${a_n}$ of elements of $k$, $sum_{n=1}^{infty} a_n x^{-n} y^n$ lies in the second field but not necessarily in the first. [Originally I had $a_n = 1$ for all $n$; quite a while after my original post, AS pointed out that that this actually does lie in the smaller field!]

I think this is a plausible mistaken belief, since e.g. the analogous statements for polynomial rings, fields of rational functions and rings of formal power series are true and very frequently used. No one ever warned me that formal Laurent series behave differently!

[Added later: I just found the following passage on p. 149 of Lam's Introduction to Quadratic Forms over Fields: "...bigger field $mathbb{R}((x))((y))$. (This is an iterated Laurent series field, not to be confused with $mathbb{R}((x,y))$, which is usually taken to mean the quotient field of the power series ring $mathbb{R}[[x,y]]$.)" If only all math books were written by T.-Y. Lam...]

Note that, even more than KConrad's example of $mathbb{Q}_p^{operatorname{unr}}$ versus the fraction field of the Witt vector ring $W(overline{mathbb{F}_p})$, conflating these two fields is very likely to screw you up, since they are in fact very different (and, in particular, not elementarily equivalent). For instance, the field $mathbb{C}((x))((y))$ has absolute Galois group isomorphic to $hat{mathbb{Z}}^2$ -- hence every finite extension is abelian -- whereas the field $mathbb{C}((x,y))$ is Hilbertian so has e.g. finite Galois extensions with Galois group $S_n$ for all $n$ (and conjecturally provably every finite group arises as a Galois group!). In my early work on the period-index problem I actually reached a contradiction via this mistake and remained there for several days until Cathy O'Neil set me straight.

Every finite index subgroup of a profinite group is open.

This I believed as a postdoc, even while explicitly contemplating what is probably the easiest counterexample, the "Bernoulli group" $mathbb{B} = prod_{i=1}^{infty} mathbb{Z}/2mathbb{Z}$. Indeed, note that there are uncountably many index $2$ subgroups -- because they correspond to elements of the dual space of $mathbb{B}$ viewed as a $mathbb{F}_2$-vector space, whereas an open subgroup has to project surjectively onto all but finitely many factors, so there are certainly only countably many such (of any and all indices). Thanks to Hugo Chapdelaine for setting me straight, patiently and persistently. It took me a while to get it.

Again, I blame the standard expositions for not being more explicit about this. If you are a serious student of profinite groups, you will know that the property that every finite index subgroup is open is a very important one, called strongly complete and that recently it was proven that each topologically finitely generated profinite group is strongly complete. (This also comes up as a distinction between the two different kinds of "profinite completion": in the category of groups, or in the category of topological groups.)

Moreover, this point is usually sloughed over in discussions of local class field theory, in which they make a point of the theorem that every finite index open subgroup of $K^{times}$ is the image of the norm of a finite abelian extension, but the obvious question of whether this includes every finite index subgroup is typically not addressed. In fact the answer is "yes" in characteristic zero (indeed $p$-adic fields have topologically finitely generated absolute Galois groups) and "no" in positive characteristic (indeed Laurent series fields do not, not that they usually tell you that either). I want to single out J. Milne's class field theory notes for being very clear and informative on this point. It is certainly the exception here.

ag.algebraic geometry - Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?

So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and Ginzburg. This method has some great successes in geometric representation theory: the most popular is probably Springer theory and character sheaves.

The rough idea of this technique is that just as the Ext algebra of the constant sheaf on a topological space with itself is the cohomology of the space (and the Yoneda product is cup product), Ext's between pushforwards of constant sheaves can be calculated using the Borel-Moore homology of fiber products, and Yoneda product will again have a realization as convolution product.

Now, I'm interested in pushing this method a bit further to work in the microlocal world. Microlocal perverse sheaves/D-modules are a new geometric category, where one forgets about some closed subset of the cotangent bundle, and declares any map which is an isomorphism on vanishing cycles (which are microlocal stalks) away from this locus to be an isomorphism.

My question: If I have a constant sheaf (in D-module language, the D-module of functions) on a smooth variety, or maybe a pushforward of one, is there some way of calculating the Ext's in the microlocal category topologically as well, hopefully using the topology of the characteristic variety?

oa.operator algebras - Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?

Actually, I think the channel is not extremal because I suspect you are misquoting the Landau-Streater result. So I will state it here.

To be precise, for anyone unfamiliar with the field, a quantum channel is a trace-preserving, completely-positive linear map on density matrices (positive semidefinite matrices with unit trace), of potentially different sizes. A basic theorem in quantum information says that every quantum channel from $mtimes m$-dimensional to $ntimes n$-dimensional density matrices can be written in Kraus form:
$$ rho mapsto sum_{i=1}^N A_i rho A_i^dagger, text{ for linear operators } A_k colon mathbb{C}^m to mathbb{C}^n text{ satisfying } sum_k A_k^dagger A_k = I_m. $$

It is easy to show that the set of quantum channels between systems of fixed dimension is convex. It also easy to show that the set of channels that map $frac{1}{m} I_m$ to a fixed density matrix $sigma$ is convex. Now the theorem of Landau-Streater says that if $m = n$, a channel with Kraus form as above is extremal in this latter set if and only if the $N^2$ linear operators $A_i^dagger A_j oplus A_j A^dagger_i$ (of size $2m times 2m$) are linearly independent. It seems you have instead been working with $mtimes m$ matrices. But I think that even if you were to continue and apply the theorem correctly, you would only prove or disprove extremality in the convex subset of unital channels, i.e. those for which $frac 1m I$ is a fixed point. So potentially you could strengthen Ben-Or's conclusion by showing non-extremality in this subset, or otherwise you might conclude extremality there, which would tell you nothing about extremality in the entire set of channels.

at.algebraic topology - Must a Strong deformation retractible 3-manifold be homeomorphic to $mathbb{R}^3$?

JG, maybe a good place to look for background is the paper of Chang, Weinberger, and Yu: Taming 3-manifolds using scalar curvature. They prove that if your M (contractible) is complete and if scal is uniformly positive, then it is homeomorphic to $mathbb{R}^3$...this is weaker than assuming $sec>0$ and using something like the Soul Theorem.

Also, check out Ross Geoghegan's "Topological methods in group theory."

terminology - p-split Hecke characters

Let $K$ be a quadratic imaginary field, $bf n$ an ideal in the ring of integers
${cal O}_K$ and $xi$ an algebraic Hecke character of type $(A_0)$ for the modulus $bf n$. One knows (from Weil) that there exists a number field $E=E_xisupseteq K$ with the property that $xi$ takes values in $E^times$.

Let $p$ be a prime that splits in $K$. Consider the following condition: there exists an unramified place $vmid p$ in $E$ with residue field $k_v={Bbb F}_p$ such that $xi$ takes values in the group of $v$-units in $E$.

The condition implies the existence of a $p$-adic avatar of $xi$ with values in ${Bbb Z}_p^times$.

I would like to know:

1) to the best of your knowledge, has been this condition considered somewhere? does it have a "name"?

2) I'm tempted to say that $xi$ is $p$-split if the condition is satisfied (and that $v$ splits $xi$). Would this name conflict with other situations that I should be aware of?

oc.optimization control - Minimizing a function containing an integral

It looks like a mixed-integer dynamic optimization problem. Your problem can be rewritten as follows: (notice the transformation of the integral into a differential equation? It's a standard trick. Also, note that you need an initial condition for $Y$)

$min_{x(t)} L(T)$

s.t.
$ frac{dL(t)}{dt} = AR(t)-x(t)$, with $L(0) = 0$

$ frac{dR(t)}{dt} = ax(t)R(t)Y(t) - bR(t)$, with $R(0) = R_{0}$

$frac{dY(t)}{dt}=−x(t)R(t)Y(t)$, with $Y(0) = Y_{0}$

$x(t) = delta(t) x_{min} + (1 - delta(t)) x_{max}$ where $delta(t) in {0,1}$

To solve this problem numerically, simply discretize the differential equations using backward Euler (easy), or implicit Runge Kutta (harder, but more accurate). Pose this as a Mixed Integer Nonlinear Program (MINLP) and use one of these solvers to find the solution.

Bonmin
https://projects.coin-or.org/Bonmin

Couenne
https://projects.coin-or.org/Couenne

These solvers will traverse the branch-and-bound tree more intelligently and efficiently than your method of enumerating every single case, which will grow with the no. of discretization grid points you have. (e.g. let's say you discretize over 20 points; the no. of cases you have to search is $2^{20} = 1048576$. Not nice.)

With a branch-and-bound|cut|reduce MINLP solver (and a bit of luck), on average you are unlikely to hit the worst case scenario where every single case is enumerated.

There are other ways of solving this problem -- multiple-shooting, sequential dynamic optimization, etc. In my opinion, optimal control methods (Pontryagin's maximum principle) are typically intractable on problems like this.

soft question - What are some examples of narrowly missed discoveries in the history of mathematics?

In the book The Scientists by John Gribbin, he mentions that, in his search for the theory of general relativity, Einstein apparently wrote down a correct equation that would have led him to correctly discovering the rest of the equations for general relativity very quickly. But, he did not see the equation for what it was and ran down the wrong path for two entire years before coming back to the correct equation. Here's the quote from the book:

"Einstein himself is often presented as the prime example of someone who did great things alone, without the need for a community. This myth was fostered, perhaps even deliberately, by those who have conspired to shape our memory of him. Many of us were told a story of a man who invented general relativity out of his own head, as an act of pure individual creation, serene in his contemplation of the absolute as the First World War raged around him.

It is a wonderful story, and it has inspired generations of us to wander with unkempt hair and no socks around shrines like Princeton and Cambridge, imagining that if we focus our thoughts on the right question we could be the next great scientific icon. But this is far from what happened. Recently my partner and I were lucky enough to be shown pages from the actual notebook in which Einstein invented general relativity, while it was being prepared for publication by a group of historians working in Berlin. As working physicists it was clear to us right away what was happening: the man was confused and lost - very lost. But he was also a very good physicist (though not, of course, in the sense of the mythical saint who could perceive truth directly). In that notebook we could see a very good physicist exercising the same skills and strategies, the mastery of which made Richard Feynman such a great physicist. Einstein knew what to do when he was lost: open his notebook and attempt some calculation that might shed some light on the problem.

So we turned the pages with anticipation. But still he gets nowhere. What does a good physicist do then? He talks with his friends. All of a sudden a name is scrawled on the page: 'Grossman!!!' It seems that his friend has told Einstein about something called the curvature tensor. This is the mathematical structure that Einstein had been seeking, and is now understood to be the key to relativity theory.

Actually I was rather pleased to see that Einstein had not been able to invent the curvature tensor on his own. Some of the books from which I had learned relativity had seemed to imply that any competent student should be able to derive the curvature tensor given the principles Einstein was working with. At the time I had had my doubts, and it was reassuring to see that the only person who had ever actually faced the problem without being able to look up the answer had not been able to solve it. Einstein had to ask a friend who knew the right mathematics.

The textbooks go on to say that once one understand the curvature tensor, one is very close to Einstein's theory of gravity. The questions Einstein is asking should lead him to invent the theory in half a page. There are only two steps to take, and one can see from this notebook that Einstein has all the ingredients. But could he do it? Apparently not. He starts out promisingly, then he makes a mistake. To explain why his mistake is not a mistake he invents a very clever argument. With falling hearts, we, reading the notebook, recognize his argument as one that was held up to us as an example of how not to think about the problem. As good students of the subject we know that the agument being used by Einstein is not only wrong but absurd, but no one told us it was Einstein himself who invented it. By the end of the notebook he has convinced himself of the truth of a theory that we, with more experience of this kind of stuff than he or anyone could have had at the time, can see is not even mathematically consistent. Still, he convinced himself and several others of its promise, and for the next two years they pursued this wrong theory. Actually the right equation was written down, almost accidentally, on one page of the notebook we looked at it. But Einstein failed to recognize it for what it was, and only after following a false trail for two years did he find his way back to it. When he did, it was questions his good friends asked him that finally made him see where he had gone wrong."

at.algebraic topology - Diffeomorphism of 3-manifolds

Surgery theory aims to measure the difference between simple homotopy types and diffeomorphism types. In 3 dimensions, geometrization achieves something much more nuanced than that. Still, I wonder whether the surgeons' key problem has been solved. Is every simple homotopy equivalence between smooth, closed 3-manifolds homotopic to a diffeomorphism?

In related vein, it follows from J.H.C. Whitehead's theorem that a map of closed, connected smooth 3-manifolds is a homotopy equivalence if it has degree $pm 1$ and induces an isomorphism on $pi_1$. Is there a reasonable criterion for such a homotopy equivalence to be simple? One could, for instance, ask about maps that preserve abelian torsion invariants (e.g. Turaev's).

peer review - Criteria for accepting an invitation to become an editor of a scientific journal

I can't say anything about this particular email you received, but I have received invitations to join the editorial board of two journals, which I declined. One was a very reputable journal, but one I did not publish in and to whose community of authors/readers I did not feel I belonged. (One would be right in asking why I was ever approached and I don't have an answer.) The other case is closer to the email you received. It was an Open Access journal and I declined because that is not a publishing model I support.

I believe that setting all practical considerations aside (whether one has time,...) the litmus test is whether one would be willing to publish in that journal. If the answer is negative, I think that you should decline.

ag.algebraic geometry - Is there a sensible notion of abstract constructible space?

In the past by the term "variety" people understood a subset of projective space locally closed for the Zariski topology. Now we have a more natural notion of abstract algebraic variety, i.e. a scheme that is so and so, and we can conceive non quasi-projective varieties.

Now we have the concept of constructible subset of a variety (or of a scheme), i.e. a finite union of locally closed subsets (subschemes).
We know that the image of a morphism of varieties may fail to be a subvariety of the target, nevertheless it's always a constructible subset thereof.

Is there a reasonable notion of "abstract constructible space"? And would it be of any utility in algebraic geometry?

Edit:
A side question. If we have a map $f:Xto Y$ of -say- varieties, we can put a closed subscheme structure on the (Zariski) closure $Z$ of $f(X)$, as described in Hartshorne's book.
On the other hand we can consider the ringed space (that will not be, in genberal, a scheme) $W$ which is the quotient of $X$ by the equivalence relation induced by $f$. Will there be any relation between $Z$ and $W$?

graph theory - Why are Dynkin diagrams characterized by their eigenvalues?

Yes. For example, with quiver representations, we have a formula

$chi(M,N)=dim Hom(M,N)-dim Ext^1(M,N) = sum d_i(M)d_i(N) - sum_{i to j} d_i(M)d_j(N).$

where $d_i(M)$ is the dimension of M at node i.
The proof is to check that it's true for simples, and then note that the category of representations of the path algebra of a quiver has global dimension 1.

So, what you've noted above is that this is positive definite if and only if the graph is Dynkin. Well, what's good about being positive definite? For one thing, if an object has trivial Ext^1 with itself, then it is rigid, it has no deformations. On the other hand, it also must have $chi(M,M)>0$, since Hom always has positive dimension, and $Ext_1(M,M)=0$.

Thus, if our quiver is not Dynkin, it has dimension vectors where no module can be rigid. On the other hand, if you work a bit harder, you can show Gabriel's theorem:

if the graph is Dynkin, every dimension vector has a unique rigid module and this is indecomposible if and only if $chi(M,M)=1$, that is if $M$ is a positive root of the root system.

ac.commutative algebra - An "Elementary" Math Question Generalized (Ring Theory Perhaps)

The following question is posed in the book "The USSR Olympiad Problem Book: Selected Problems and Theorems of Elementary Mathematics"

"Prove that if integers a_1, ..., a_n are all distinct, then the polynomial

(x-a_1)^2(x-a_2)^2...(x-a_n)^2 + 1

cannot be written as the product of two other polynomials with integral coefficients"

I still haven't solved this in the elementary case, but I want to pose it in a more general setting. (I'll post the elementary answer in a bit, I want to try a bit more to figure it out)

EDIT: Now I have solved it, it's MUCH more trivial than I thought, major brain fart on my part

Suppose we have a ring R[x], and the polynomial written above is factorable in this ring. Suppose further that the coefficients are members of a subset of the field that the ring sits in, and that none of the a_i are equal (which means that some finite fields are of course out). Under what conditions IS the polynomial factorable into a product of two polynomials such that their coefficients sit inside the subset? Does the subset have to be algebraically closed?

(Is factorability even remotely easy in a noncommutative ring? I don't see a priori why a factorization would be unique either in a noncommutative ring).

Motivation: I'm doing research in mathematics education, and am interested in the metacognitive faculties of early college and high school pupils, which, for those not versed in metacognition, is the ability to separate oneself from "nitty-gritty" of the problem and think in more general terms. Alan Schoenfeld is the standard reference on this. I'm looking for problems that are good to pose to students to try and understand their thinking skills, and so am looking through particularly hard problems that do not require a strong background in mathematics. In this particular case I'd like to understand the problem in greater depth myself, and hopefully use my more general knowledge of the situation to aid in my study of how students think about such problems.

Hope this is interesting to someone, and that it isn't too specific.

ag.algebraic geometry - Quotient of an affine variety by an infinite discrete group

This is a pretty basic question, so I'd be happy with either standard references or with explanations. Also, there's a good chance I'm confused about some things in the statement of the question, and corrections of these would also be cool.

If G is a finite group acting on a (commutative) $ mathbb{C} $-algebra A, then we can define $Spec(A)/G$ as $Spec(A^G)$ (where $A^G$ is the invariant sublagebra of A), and there is an algebraic map $Spec(A) to Spec(A^G)$ whose "fibers" correspond to orbits of G on $Spec(A)$.

If L is a lattice in $ mathbb{C} $ then $ mathbb{C} / L$ as a topological space can be given the structure of a scheme (it's an elliptic curve). However, this scheme is not $Spec(mathbb{C}[x]^L)$, since the only polynomials invariant under the action of a lattice are the constant polynomials. Is there a construction which replaces $Spec(mathbb{C}[x]^L)$ in this case?

Now let's assume X is a smooth affine scheme and G is a countable group acting freely on X (in my case $G = SL_2(mathbb{Z})$, but I would guess this isn't too important). Since X is smooth we can also view X as a complex manifold $X^{an}$, and then the quotient $X^{an}/G$ is a topological space. Can X/G naturally be given the structure of a scheme? If the answer is no in general, are there conditions on the data X, G that ensure the answer is yes?

complex multiplication - A problem of Shimura and its relation to class field theory

I would like to write some kind of summary of the above two answers. There is nothing new here.

Consider $mathbb{Q} subset mathbb{Q}(zeta_7)$,(and we fix $zeta_7=e^{frac{2pi i}{7}}$) this is an abelian Galois extension, and the Galois group is $(Z/7Z)^{times}$. Frobenius element over $p$ for $p neq 7$ (7 is the ramification) acts on $zeta_7$ by sending it to its $p$-th power.

Now consider $alpha=zeta_7+zeta_7^{-1}$, which is in $mathbb{Q}(zeta_7)$, consider all its Galois conjugates, which are precisely $alpha_2 = zeta^2 +zeta^{-2}$, $alpha_3=zeta^3+zeta^{-3}$. So we have $mathbb{Q} subset mathbb{Q}(alpha)$ is Galois.

Now $Frob_p$ maps $alpha$ to $zeta^p + zeta^{-p}$, which is equal to $alpha$ if and only if $p equiv 1, -1 (mod 7)$. And they are precisely those primes that are totally split in $mathbb{Q}(alpha)$. For those primes, $Z/p = mathbb{O}_{mathbb{Q}(alpha)}/p$, thus we can always find some $n$, s.t., $alpha_i equiv n(mod p)$, which is equivalent to say that $F(x)$ totally split over $Z/p =F_p$.

I am glad to know this problem and answer since I finally found an explicit example of cyclic Galois extension of $mathbb{Q}$...(which I had been wondering for a while...)

We can also do the similar things for p=13. just take $beta=theta +theta^5+theta^8+theta^{12}$,where $theta$ is the 13-th root of unity, then $Q(beta)$ is again a cyclic Galois extension of $Q$.

---

when I said cyclic above, I meant for cyclic of order 3. Thank Peter for pointing out.

pr.probability - probability in number theory

Have you read The Probabilistic Method by Joel Spencer and Noga Alon?

Although originally developed by Erdos, here's an example of the probabilistic method taken from combinatoricist Po-Shen Loh:

$A_1, dots, A_s subseteq { 1, 2, dots, M }$ such that $A_i not subset A_j$ and let $a_i = |A_i|$. Show that $$ sum_{i=1}^s frac{1}{binom{M}{a_i}} leq 1$$
The hint is to consider a random permutation $sigma = (sigma_1, dots, sigma_M)$. Loh defines the event $E_i$ when ${ sigma_1, dots, sigma_{a_i}} = A_i$. Then he observes the events $E_i$ are mutually exclusive and that $mathbb{P}(E_i)$ is relevant to our problem...

There are probably a lot of olympiad combinatorics problems that can be solved this way. Err... you were asking for number theory, but you will find both in Spencer and Alon's book.

sg.symplectic geometry - What is the meaning of symplectic structure?

Here's how I understand it. Classical mechanics is done on a phase space M. If we are trying to describe a mechanical system with n particles, the phase space will be 6*n*-dimensional: 3*n* dimensions to describe the coordinates of particles, and 3*n* dimensions to describe the momenta. The most important property of all this is that given a Hamiltonian function $H:Mto mathbb R$, and a point of the phase space (the initial condition of the system), we get a differential equation that will predict the future behavior of the system. In other words, the function $H$ gives you a flow $gamma:Mtimes mathbb R to M$ that maps a point p and a time t to a point $gamma_t (p)$ which is the state of the system if it started at p after time t passes.

Now, if we do classical mechanics, we are very interested in changes of coordinates of our phase space. In other words, we want to describe M in a coordinate-free fashion (so that, for a particular problem, we can pick whatever coordinates are most convenient at the moment). Now, if $M$ is an abstract manifold, and $H:Mtomathbb R$ is a function, you cannot write down the differential equation you want. In a sense, the problem is that you don't know which directions are "coordinates" and which are "momenta", and the distinction matters.

However, if you have a symplectic form $omega$ on M, then every Hamiltonian will indeed give you the differential equation and a flow $Mtimesmathbb R to M$, and moreover the symplectic form is precisely the necessary and sufficient additional structure.

I must say, that although this may be related to why this subject was invented a hundred years ago, this seems to have little to do to why people are studying it now. It seems that the main reason for current work is, first of all, that new tools appeared which can solve problems in this subject that couldn't be solved before, and secondly that there is a very non-intuitive, but very powerful, connection that allows people to understand 3- and 4-manifolds using symplectic tools.

---

EDIT: I just realized that I'm not quite happy with the above. The issue is that the phase spaces that come up in classical mechanics are always a very specific kind of symplectic manifold: namely, the contangent bundlde of some base space. In fact, in physics it is usually very clear which directions are "coordinates of particles" and which are "momenta": the base space is precisely the space of "coordinates", and momenta naturally correspond to covectors. Moreover, the changes of coordinates we'd be interested in are always just changes of coordinates of the base space (which of course induce a change of coordinates of the cotangent bundle).

So, a symplectic manifold is an attempts to generalize the above to spaces more general than the cotangent bundle, or to changes of coordinates that mess up the cotangent-bundle structure. I have no idea how to motivate this.

UPDATE: I just sat in a talk by Sam Lisi where he gave one good reason to study symplectic manifolds other than the cotangent bundle. Namely, suppose you are studying the physical system of just two particles on a plane. Then, their positions can be described as a point in $P = mathbb R^2 times mathbb R^2$, and the phase space is the cotangent bundle $M=T^* P$.

Notice, however, that this problem has a lot of symmetry. We can translate both points, rotate them, look at them from a moving frame, etc., all without changing the problem. So, it is natural to want to study not the space M itself, but to quotient it out by the action of some Lie group G.

Apparently, $M/G$ (or something closely related; see Ben's comment below) will still be a symplectic manifold, but it is not usually the cotangent bundle of anything. The difference is significant: in particular, the canonical one-form on $M=T^*P$ will not necessarily descend to $M/G$.

fa.functional analysis - Baire category theorem

Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = cup_{n=1}^infty A_n$. Let $bar{A}$ denote the closure of $A$.

Let's call the following statement (2): at least one of the $bar{A}_n$ contains a ball.

Baire category theorem gives:

Fact1:
(1) $Rightarrow$ (2)

Now take any $xin X$ and consider the closed ball $B = B(x,delta)={y: d(x,y)le delta}$. This is a complete metric space itself and it is covered by $ A_n cap B$. Thus these sets satisfy (1). Fact1 gives us an $n$ such that the closure of $A_n cap B$ contains a ball in $B$. Thus, we have strengthened Fact1 to

Fact1': (1) $Rightarrow$ (2')

where (2') is: For every $xin X$, every neighborhood of $x$ contains a ball that is contained in one of the $bar{A}_n$.

Question: Can (2') be strengthened further? Here are some example statements, both of which are too strong:

• For every $x$, there is a $delta$ such that $B(x,delta)$ is contained in one of the $bar{A}_n$


• For every $x$, there is an $A_n$ such that $bar{A}_n$ contains an open set $G$ with $d(x,G)=0$.

---

Many thanks for the responses. The motivation for this question was as follows.

1) What does it mean for a set $A$ to have a closure with empty interior? Take an element
$a in bar{A}$. This means that $a$ is either in $A$ or there is a sequence in $A$ that converges to $a$. This can be rephrased as $a$ is a point that can be approximated with infinite precision by $A$.' If $bar{A}$ has no interior, perturbing $a$ by a small amount will give an $a'$, such that $a'$ is a finite distance away from $bar{A}$. Thus, $a'$ will be a point that can only be approximated by $A$ with finite precision. Then, one can think of $A$ as a multi resolution grid with discontinuous approximation capability: you perturb any point that can be approximated infinitely well by it and you get a point that can only be approximated with finite precision.

2) $X$ itself can be thought of as the perfect multi resolution grid for itself: every point $xin X$ is well approximated by $X$, and perturbing $x$ will not change this. The way I wanted to think of $X= cup_{n=1}^infty A_n$ was this: for every $x in X$, one of the $A_n$ provides a multiresolution grid around $x$ that has continuous approximation capability. I wanted to think, similar to Leonid's response, that the set of $X$ where this is not possible was to be in some sense to be exceptional.

ho.history overview - Newton and Newton polygon

If memory serves correct the history of Newton's polygon and Puiseaux series has some subtleties, so be a bit wary of secondary historical sources. Histories of mathematics are bursting at the seams with romanticized legends, so it is always best to consult primary sources if you wish to know the real history. The following note from Chrystal's Algebra may serve as a helpful entry into the primary literature.

Historical Note. - As has already been remarked, the fundamental idea of
the reversion of series, and of the
expansion of the roots of algebraical
or other equations in power-series
originated with Newton. His famous"
Parallelogram" is first mentioned in
the second letter to Oldenburg; but is
more fully explained in the
Geometria Analytica (see Horsley's edition of Newton's Works, t. i., p.
398). The method was well understood
by Newton's followers, Stirling and
Taylor; but seems to have been lost
sight of in England after their time.
It was much used (in a modified form
of De Gua's) by Cramer in his
well-known Analyse dea Lignes
Courbea Algebriques (1750). Lagrange
gave a complete analytical form to
Newton's method in his "Memoire sur
l'Usage des Fractions Continues,"
Nouv. Mem. d. l'Ac. roy. d. Sciences d. Berlin (1776). (See OEuvres de
Lagrange, t. iv.)

Notwithstanding its great utility, the
method was everywhere all but
forgotten in the early part of this
century, as has been pointed out by De
Morgan in an interesting account of
it given in the Cambridge
Philosophical Transactions, vol.ix.
(1855).

The idea of demonstrating, a priori,
the possibility of expansions such as
the reversion-formulae of S.18
originated with Cauchy; and to him, in
effect, are due the methods employed
in SS.18 and 19. See his memoirs on
the Integration of Partial
Differential Equations, on the
Calculus of Limits, and on the Nature
and Properties of the Roots of an
Equation which contains a Variable
Parameter,
Exercices d'Analyse et de Physique Mathematique, t. i. (1840), p. 327;
t. ii. (1841), pp. 41, 109. The form
of the demonstrations given in SS. 18,
19 has been borrowed partly from
Thomae, El. Theorie der Analytischen
Functionen einer Complexen
Veranderlichen (Halle, 1880), p. 107;
partly from Stolz,
Allgemeine Arithmetik, I. Th. (Leipzig, 1885), p. 296.

The Parallelogram of Newton was used
for the theoretical purpose of
establishing the expansibility of the
branches of an algebraic function by
Puiseaux in his Classical Memoir on
the Algebraic Functions (Liouv. Math.
Jour., 1850). Puiseaux and Briot and
Bouquet (Theorie des Fonctions
Elliptiques (1875), p. 19) use
Cauchy's Theorem regarding the number
of the roots of an algebraic equation
in a given contour; and thus infer the
continuity of the roots. The
demonstration given in S.21 depends
upon the proof, a priori, of the
possibility of an expansion in a
power-series; and in this respect
follows the original idea of Newton.

The reader who desires to pursue the
subject further may consult Durege,
Elemente der Theorie der Functionen einer Complexen Veranderlichen
Grosse, for a good introduction to
this great branch of modern
function-theory.

The applications are very numerous,
for example, to the finding of
curvatures and curves of closest
contact, and to curve-tracing
generally. A number of beautiful
examples will be found in that
much-to.be-recommended text-book,
Frost's Curve Tracing. -- G. Chrystal: Algebra, Part II, p.370

ct.category theory - Terminology: Is there a name for a category with biproducts?

Many people are familiar with the notion of an additive category. This is a category with the following properties:

(1) It contains a zero object (an object which is both initial and terminal).

This implies that the category is enriched in pointed sets. Thus if a product $X times Y$ and a coproduct $X sqcup Y$ exist, then we have a canonical map from the coproduct to the product (given by "the identity matrix").

(2) Finite products and coproducts exist.

(3) The canonical map from the coproduct to the product is an equivalence.

A standard exercise shows this gives us a multiplication on each hom space making the category enriched in commutative monoids (with unit).

(4) An additive category further requires that these commutative monoids are abelian groups.

I want to know what standard terminology is for a category which satisfies the first three axioms but not necessarily the last.

I can't seem to find it using Google or Wikipedia. An obvious guess, "Pre-additive", seems to be standard terminology for a category enriched in abelian groups, which might not have products/coproducts.

Computer program to solve a system of polynomial equations over a finite field

I have a set of polynomial equations for which I want to know the solutions (actually really the number of solutions). It would be great if I could get a computer to do it, but I'm not sure exactly which programs will do things over finite fields. I surmise that Macaulay 2 may be able to do something like this, but I'm not quite sure how.

advice - How to escape the inclination to be a universalist or: How to learn to stop worrying and do some research.

As an undergraduate we are trained as mathematicians to be universalists. We are expected to embrace a wide spectrum of mathematics. Both algebra and analysis are presented on equal footing with geometry/topology coming in later, but given its fair share(save the inherent bias of professors). Number theory, and more applied fields like numerical analysis are often given less emphasis, but it is highly recommended that we at least dabble in these areas.

As a graduate student, we begin by satisfying the breadth requirement, and thus increasing these universalist tendencies. We are expected to have a strong background in all of undergraduate mathematics, and be comfortable working in most areas at a elementary level. For economic reasons, if our inclinations are for the more pure side, we are recommended to familiarize ourselves with the applied fields, in case we fall short of landing an academic position.

However, after passing preliminary exams, this perspective changes. Very suddenly we are expected to focus on research, and abandon these preinclinations of learning first, then doing something. Professors espouse the idea that working graduate student should stop studying theories, stop working through textbooks, and get to work on research.

I am finding it difficult to eschew my habits of long self-study to gain familiarity with a subject before working. Even during my REU and as an undergrad, I was provided with more time and expectation to study the background.

I am a third year graduate student who has picked an area of study and has a general thesis problem. My advisor is a well known mathematician, and I am very interested in this research area. However, my background in some of the related material is weak. My normal mode of behavior, would be to pick up a few textbooks and fix my weak background. Furthermore, to take many more graduate courses on these subjects. However, both of my major professors have made it clear that this is the wrong approach. Their suggestion is to learn the relevant material as I go, and that learning everything I will need up front would be impossible. They suggest begin to work and when I need something, pick up a book and check that particular detail.

So in short my question is:

How can I get over this desire to take
lots of time and learn this material
from the bottom-up approach, and
instead attack from above, learning
the essentials necessary to move more
quickly to making original
contributions? Additionally, for those of you advising students, do you recommend them the same as my advisor is recommending me?

A relevant MO post to cite is How much reading do you do before attacking a problem. I found relevant advice there also.

As a secondary question, in relation to the question of universalist. I find it difficult to restrain myself to working on one problem at a time. My interests are broad, and have difficulty telling people no. So when asked if I am interested in taking part in other projects, I almost always say yes. While enjoyable(and on at least one occasion quite fruitful), this is also not conducive to finishing a Ph.D.(even keeping in mind the advice of Noah Snyder to do one early side project). With E.A. Abbot's claim that Poincaré was the last universalist, with an attempt at modesty I wonder

How to get over this bred desire to work on everything of interest, and instead focus on one area?

I ask this question knowing full well that some mathematicians referred to as modern universalists visit this site. (I withhold names for fear of leaving some out.)

Also, I apologize for the anonymity.

Thank you for your time!

EDIT: CW since I cannot imagine there is one "right answer". At best there is one right answer for me, but even that is not clear.

Moduli stack of principally polarized abelian varieties

The method is really the same as what Deligne-Mumford do to handle the moduli space of curves (creating a smooth cover from a part of a Hilbert scheme), except facts about curves (e.g., Riemann-Roch and cohomological vanishing results) are replaced with analogues for abelian varieties. Below is a guide to relevant literature for why $mathcal{A}_{g,d}$ is a separated DM stack of finite type over $mathbf{Z}$. (Question had $d = 1$.)

Everything needed about abelian varieties is in Mumford's book on abelian varieties and the (self-contained!) Chapter 6 on abelian schemes in his GIT book. The key ingredients are (i) the "Riemann-Roch and Vanishing" theorems from section 16 of Mumford's book on abelian varieties, (ii) Proposition 6.13 in Mumford's GIT book (a relativization of results proved over an algebraically closed field in his book on abelian varieties), (iii) Proposition 6.11 in GIT, and (iv) Proposition 6.14 in GIT.

The references (i) and (ii) ensure that if you consider an abelian scheme $f:A rightarrow S$ of relative dimension $g > 0$ (with dual $A^{rm{t}}$ and Poincar'e bundle $mathcal{P}$) and a polarization $phi:A rightarrow A^{rm{t}}$ on $A$ of degree $d^2$ then for the resulting
$S$-ample line bundle $mathcal{L} = (1, phi)^{ast}(mathcal{P})$ on $A$ the pushforward $mathcal{E} = f_{ast}(mathcal{L}^{otimes 3})$ is a vector bundle on $S$ whose formation commutes with any base change, has rank determined entirely by $g$ and $d$, and defines a closed immersion $A hookrightarrow mathbf{P}(mathcal{E})$. (In these arguments with relative ampleness, EGA IV$_3$, 9.6.4 is very useful for bootstrapping from fields to a general base.) Thus, upon making a universal (smooth) base change to equip $mathcal{E}$ with a global frame, one gets a map from the new base to a certain Hilbert scheme for a projective space (over $mathbf{Z}$) and Hilbert polynomial determined by $g$ and $d$. Since an abelian scheme has a section, this map naturally factors through the universal object $X$ over that Hilbert scheme (which represents the functor of also specifying a section).

Now (iv) comes in: it says that over $X$, the condition on the universal marked object that it admit a (necessarily unique) structure of abelian scheme with the marked section as the identity is represented by an open subscheme. Restricting to this locus (and again using the Hilbert polynomial to track the rank of the pushforward of its $mathcal{O}(1)$) gives a kind of first approximation to a "universal" abelian scheme $mathcal{A}$, but we need to universally equip it with a polarization whose cube recovers the $mathcal{O}(1)$ from the projective embedding as its "associated" ample line bundle (via the construction with the Poincar'e bundle). That's where (iii) comes in: it says that such a polarization is unique if it exists and that its existence is represented by a closed subscheme of the base. Over that subscheme we get a degree-$d^2$ polarization on an abelian scheme of relative dimension $g$, and the way we arrived at it showed that the base of the family (of finite type over $mathbf{Z}$) is a smooth cover of the moduli stack (the "smooth cover" coming about because of the arrangement to get a global frame for $mathcal{E}$).

Since one checks before any of the above that the moduli stack does satisfy effective descent (thanks to descent theory with the ample line bundle associated to the polarization; see 6.1/7 in the book "Neron Models"), we indeed have an Artin stack of finite type over $mathbf{Z}$. The finite 'etale property of automorphism groups of geometric points implies it is actually a DM-stack (using the "unramified diagonal" criterion for an Artin stack to be a DM stack: Theorem 8.1 in the Laumon/Moret-Bailly book). Finally, to check separatedness one uses the valuative criterion, which is the N'eronian property for abelian schemes over a discrete valuation ring (1.2/8 in "Neron Models"). QED

dg.differential geometry - Is Lang's definition of a tensor bundle nonstandard?

This is a slight variation of the standard definition, as far as I can tell.

First of all, let me restrict to the finite dimensional context, since this is more standard.
Then a typical example of $lambda$ is the functor which sends $V_1,ldots,V_p,W_1,ldots,W_q$ to $V_1^{*}otimescdotsotimes V_p^{*}otimes W_1otimes cdots otimes W_q.$ The corresponding tensor bundle will be $(T_X^*)^{otimes p}otimes T_X^{otimes q}$ (here I mean the usual tensor product of bundles), and I can imagine sections of this being referred to as tensors of type $(p,q)$ classically. (It may be that the $p$ and $q$ would be reversed; I would check the conventions carefully of any reference that used terminology of this kind.)

You could check in Spivak, or on Wikipedia, or in any number of other sources to see the various kinds of terminology that are used for this construction, but whatever the terminology, sections of these sorts of bundles are precisely what are referred to as tensors in classical differential geometry.

In more modern treatments, you may see less of this terminology, because people will just write out explicitly the tensor products of bundles as I did above. But this terminology evolved over a long period of time, and tensors in differential geometry were being considered well before the functorial notion of tensor product of vector spaces was introduced.

As for how more general Lang's definition is, I can't think of any other $lambda$s of the top of my head (and it may be that, if you impose some natural axioms on $lambda$, there essentially are no further examples). As far as I can tell, he has simply abstracted the properties you need to have a functor of vector spaces give rise to a corresponding functor of vector bundles.

[Edit: As pointed out in the comment below, and in Tim Perutz's answer, there are indeed
other $lambda$'s: e.g., there are symmetric tensors and exterior tensors (the latter
giving differential forms, of course), which I mysteriously neglected when I wrote the above; one should certainly single them out, and my statement about there being no other $lambda$s is wrong as it stands --- all these
Schur-type functors are certainly candidate $lambda$s. One thing I hadn't realized, which is pointed out by Tim Perutz, is that the case of densities is also included in Lang's definition.]

semigroups - Membership problem in monoids

Every group is a monoid, and if a group has an undecidable subgroup membership problem, then the corresponding submonoid problem will also be undecidable (provided that we can computably produce $s^{-1}$ from $s$, which would be true if the group operation were computable), since if $G$ is a group, then $x$ is in the subgroup generated by $s_1$, ..., $s_n$ if and only if it is in the submonoid generated by $s_1,...,s_n,s_1^{-1},...s_n^{-1}$. Thus, the subgroup membership problem for $G$ reduces to the submonoid membership problem for $G$. If the former is undecidable, then so is the latter.

But I notice that you state your question (in your first paragraph) not in terms of a presentation of the monoid, using words in a presentation, but in terms of the monoid itself, as a raw operation. That is, a literal reading of your question would seem to have us work with a binary operation on a set of natural numbers that makes it a monoid, and we want to decide questions about the nature of that monoid. Thus, your question would seem to belong more to the subject of computable model theory, which is focused entirely on questions of this sort, than to combinatorial group (or monoid) theory. For example, there is a copy of the infinite cyclic group, obtained simply by using a highly non-computable permutation of the integers, where the induced group operation itself is not computable. One can arrange by diagonalization, since there will be only countably many possible queries about submonoid membership, that the submonoid (or subgroup) membership problem for this copy of the monoid is not computable. Since this monoid is just the infinite cyclic group, it is extremely simply as a monoid, but satisfies your undecidability property the way you have stated it. But probably that is not what you mean. If so, you should be more specific about what the input to your decision algorithm is supposed to be.

at.algebraic topology - visualizing what's going on in based homotopy theory, et al.

I'm reading J.P. May's Concise Course in Algebraic Topology, and I'm having a lot of trouble visualizing how things work in Chapter 8, "Based cofiber and fiber sequences". Of course this is pretty basic stuff, but it's really cool to me that there are such clear topological analogues to the usual exact sequences in homological algebra. Still, I can't even get a clear picture of what a smash product looks like for any but the most basic of spaces, and based cones/suspensions/loopspaces make my head hurt.

a) Will I be alright if in my head I just sort think of a smash product as a usual product (for example), with the understanding that I need to tack on an extra condition that I really shouldn't think too hard about?

b) Why all the fuss about based homotopy theory, anyways?

c) While I'm at it, can anyone suggest a book that is less terse? I feel like this one rarely gives the motivation and visual intuition that I'd like...