lo.logic - nontrivial theorems with trivial proofs

A while back I saw posted on someone's office door a statement attributed to some famous person, saying that it is an instance of the callousness of youth to think that a theorem is trivial because its proof is trivial.

I don't remember who said that, and the person whose door it was posted on didn't remember either.

This leads to two questions:

(1) Who was it? And where do I find it in print---something citable? (Let's call that one question.)

(2) What are examples of nontrivial theorems whose proofs are trivial? Here's a wild guess: let's say for example a theorem of Euclidean geometry has a trivial proof but doesn't hold in non-Euclidean spaces and its holding or not in a particular space has far-reaching consequences not all of which will be understood within the next 200 years. Could that be an example of what this was about? Or am I just missing the point?

linear algebra - Submultiplicative matrix norm: Max Norm

The inequality $|AB|_{max} leq |A|_{a}|B|_{b}$ for all $A$, $B$ can be achieved or destroyed just by rescaling the norms $|cdot|_a$ and $|cdot|_b$. Let's suppose that we're considering $d times d$ matrices. If we just make sure that the two norms $|cdot|_a$ and $|cdot|_b$ are scaled so that both of them have the property $|C|_i geq sqrt{d}|C|_{max}$ for all $d times d$ matrices $C$, then the desired inequality follows from the elementary inequality $|AB|_{max} leq d.|A|_{max}|B|_{max}$. Conversely, if the norms are rescaled so that both of them give norm $frac{1}{2}$ to the identity matrix, then the inequality clearly cannot hold since $|Id|_{max}=1$. The fact that such rescalings exist follows from the fact that norms on a finite-dimensional space are pairwise equivalent.

The point of this is that there are a lot of norms on the space of matrices if we don't make any additional requirements on them. Is this the kind of answer you were looking for? Or do you want the two norms to have additional properties?

edited: there was a typo on the main inequality

dg.differential geometry - Do symmetric spaces admit isometric embeddings as intersections of quadrics?

While preparing a seminar I gave today, the following question arose. I asked the seminar participants, but nobody knew the answer. Hence I'm asking it here in MO.

Background

Recall that a complete, connected and simply connected pseudoriemannian manifold $(M,g)$ is a symmetric space if the Riemann curvature tensor is parallel with respect to the Levi-Civita connection: $nabla R = 0$.

Typical examples are the (simply-connected) spaces of constant curvature: sphere, hyperbolic space, (anti) de Sitter spacetimes,... all of which admit local isometric embeddings as quadrics in some flat pseudoriemannian manifold $mathbb{R}^{p,q}$. Recall that the flat metric on $mathbb{R}^{p,q}$ is given in flat coordinates by
$$sum_{i=1}^p (dx_i)^2 - sum_{i=1}^q (dx_{p+i})^2.$$

For example, the sphere with unit radius of curvature embeds in $mathbb{R}^{n+1}$ as the quadric
$$x_0^2 + x_1^2 + cdots + x_n^2 = 1,$$
whereas the hyperbolic space embeds in $mathbb{R}^{1,n}$ as one sheet of the quadric
$$-x_0^2 + x_1^2 + cdots + x_n^2 = -1,$$
again for unit radius of curvature.

Similarly, and again for unit radii of curvature, $n$-dimensional de Sitter spacetime is the universal covering space of the quadric
$$-x_0^2 + x_1^2 + cdots + x_n^2 = 1$$
in $mathbb{R}^{1,n}$, whereas $n$-dimensional anti de Sitter spacetime is the universal covering space of the quadric
$$-x_0^2 + x_1^2 + cdots + x_{n-1}^2 - x_n^2 = -1.$$

This continues to be the case for other spaces of constant curvature in other signatures.

Other riemannian symmetric spaces, such as the grassmannians, can also admit isometric embeddings, this time in projective spaces, whose image is the intersection of a number of quadrics. This is the celebrated Plücker embedding. Notice that grassmannians do not (generally) have constant sectional curvature.

The remaining nontrivial lorentzian symmetric spaces -- the $n$-dimensional Cahen-Wallach spaces -- can also be locally embedded isometrically in $mathbb{R}^{2,n}$ as the intersection of two quadrics.
In particular this shows that all the indecomposable lorentzian symmetric spaces (in dimension $>1$, at least), which are the (anti) de Sitter and Cahen--Wallach spacetimes, can be locally embedded isometrically as the intersection of quadrics in some pseudoeuclidean space.

Question

Is this also the case for the other simply-connected (pseudo)riemannian symmetric spaces?

Perhaps asking about quadrics is too strong, so perhaps a weaker question is

Are simply-connected symmetric spaces always (locally) algebraic?

Here by locally algebraic I mean that they are the universal covering space of an algebraic space.

geometry - What are the possible images of a square under an area-preserving map?

Let S be the open unit square in R^2: the set of points (x,y) with 0 < x < 1 and 0 < y < 1. Consider an area-preserving smooth map S --> R^2, that is, a map whose Jacobian has determinant 1 at every point. What can the image of S look like?

Can the image of the square have a smooth boundary? I think you can smooth out the two corners on top with a transformation of the form (x,y) --> (x,y+f(x)), but this makes the other two corners sharper.

(I had added, and now removed, some nonsense about Gromov's nonsqueezing theorem, which does not hold in dimension 2, and doesn't say what I thought it said besides.)

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."

career - Choosing postdocs

In this period I have been applying for some postdoc positions. One of the main difficulties is that there is no central organization, so the times and practices for applying vary considerably even in the same country. Even being able to start a postdoc exactly when the previous one ends is not always simple.

Here I have a question about etiquette. What is people expected to do when moving from one postdoc to another one? A typical example would be a position which starts a few months before the previous one is finished. Or it is conceivable that one may have to accept a position in a short time, only to find later that he has won another position (for which he had already applied) which is "better" for various reasons, for instance it lasts longer or is in a place which is more favourable for personal reasons. An extreme case would be that on getting a permanent position while still doing a postdoc.

What is the expected etiquette in such cases? I'm interested mainly in what happens in Europe, if the opinion changes worldwide, but it is probably useful to hear from anyone about this matter.

lo.logic - Is there a name for a family of finite sequences that block all infinite sequences?

Let ${bf N}^omega = bigcup_{m=1}^infty {bf N}^m$ denote the space of all finite sequences $(N_1,ldots,N_m)$ of natural numbers. For want of a better name, let me call a family ${mathcal T} subset {bf N}^omega$ a blocking set if every infinite sequence $N_1,N_2,N_3,N_4,ldots$ of natural numbers must necessarily contain a blocking set $(N_1,ldots,N_m)$ as an initial segment. (For the application I have in mind, one might also require that no element of a blocking set is an initial segment of any other element, but this is not the most essential property of these sets.)

One can think of a blocking set as describing a machine that takes a sequence of natural number inputs, but always halts in finite time; one can also think of a blocking set as defining a subtree of the rooted tree ${bf N}^omega$ in which there are no infinite paths. Examples of blocking sets include

1. All sequences $N_1,ldots,N_m$ of length $m=10$.


2. All sequences $N_1,ldots,N_m$ in which $m = N_1 + 1$.


3. All sequences $N_1,ldots,N_m$ in which $m = N_{N_1+1}+1$.

The reason I happened across this concept is that such sets can be used to pseudo-finitise a certain class of infinitary statements. Indeed, given any sequence $P_m(N_1,ldots,N_m)$ of $m$-ary properties, it is easy to see that the assertion

There exists an infinite sequence $N_1, N_2, ldots$ of natural numbers such that $P_m(N_1,ldots,N_m)$ is true for all $m$.

is equivalent to

For every blocking set ${mathcal T}$, there exists a finite sequence $(N_1,ldots,N_m)$ in ${mathcal T}$ such that $P_m(N_1,ldots,N_m)$ holds.

(Indeed, the former statement trivially implies the latter, and if the former statement fails, then a counterexample to the latter can be constructed by setting the blocking set ${mathcal T}$ to be those finite sequences $(N_1,ldots,N_m)$ for which $P_m(N_1,ldots,N_m)$ fails.)

Anyway, this concept seems like one that must have been studied before, and with a standard name. (I only used "blocking set" because I didn't know the existing name in the literature.) So my question is: what is the correct name for this concept, and are there some references regarding the structure of such families of finite sequences? (For instance, if we replace the natural numbers ${bf N}$ here by a finite set, then by Konig's lemma, a family is blocking if and only if there are only finitely many finite sequences that don't contain a blocking initial segment; but I was unable to find a similar characterisation in the countable case.)

finite fields - A quadratic form

For $d=q-1$, let $f$ be a polynomial vanishing at all points of $mathbb F_q$ except $0$. Let $f_1$ $f_2$ such that $f_1+f_2=f$, and define $A$ such that $A(f_1)=1$ and $A(f_2)=-1$ and $A(g)=0$ for all other $g$. Then, the only terms surviving are

$$A(f_1)A(f_2)phi(f_1+f_2)+A(f_2)A(f_1)phi(f_2+f_1)+A(f_1)^2phi(2f_1)+A(f_2)^2phi(2f_2).$$

Then, we can compute the value to be

$$-phi(f)+-phi(f)+phi(0)+phi(0)=-(q-1)+-(q-1)+q+q=2.$$

Now, I can't prove that this is minimal, but this does show that $q$ isn't minimal for all $d$.

I'll edit this answer if I come up with anything for the $frac{q}{2}-1$ case.

EDIT: Whenever 2 can be achieved, it is minimal, because the value must be an even number. The summation breaks up to $sum_{fneq g} A(f)A(g)phi(f+g)+sum_f A(f)^2phi(2f)$. The latter is a multiple of $q$, which is even, so it suffices to show that the first sum is even. For all $f,g$, we get two terms, $A(f)A(g)phi(f+g)$ and $A(g)A(f)phi(f+g)$ that are equal, and, being equal integers, their sum is even. Every term in the first sum is of this form, and so the first sum must be even. So we have a sum of even integers, so the minimal number must be even, so the example above shows that 2 is minimal for $d=q-1$. In fact, it shows that 2 is minimal for $dgeq q-1$, by just throwing in extra copies of the fact $x-1$.

Still haven't worked it out for $d$ less than $q-1$, but there's a proof for large $d$.

Edit 2: In fact, the same line of argument shows that for $d$ less than $q$, we can bound above by $2(q-d)$, so for $d=q/2-1$, we can bound above by $2q-2(q/2-1)=2q-q+2=q+2$, don't yet know if this is sharp.

at.algebraic topology - Simplicial homotopy book suggestion for HTT computations

Echoing what Urs said, "But you should all be using the nLab more: if you want literature on quasi-categories, look up the references section on quasi-categories! :-) " and in regard to your question in particular, "See the nLab page on simplicial sets for links."

By looking at this nLab page on simplicial sets, in the references section you'll have found (as of this date, July 21, 2011 10:19 pm, EST):

Greg Friedman: An elementary illustrated introduction to simplicial sets

which is chock full of pictures illustrating the geometric ideas underlying the combinatorics of simplicial sets. As Friedman discusses in the introduction on the second and third pages of this paper,

"Here, for the most part, you won't
find many complete proofs of theorems,
and so these notes will not be
completely self-contained. Rather, I
try primarily to show by example how
the very basic combinatorics,
including the definitions, arise out
of geometric ideas and to show the
geometric ideas underlying the most
elementary proofs and properties."

Suffice it to say, those notes may be the end of your search in finding a concise introduction to simplicial sets that also helps develop your geometric intuition and the computation of products. You also might find the references I gave in answer to the question here helpful.

rt.representation theory - Generalization of Schur's lemma (Update)

I am not a mathematician nor physicist. I just know the basics of the representation theory. In my research, I realized that there is an orthogonality relation between the unitary group matrix elements as follows:

$$I_1 = int mathrm{D}mathbf{U} ; U_{i j}^{(mathbf{r})} U_{k l }^{*(mathbf{r}^{prime})} = frac{1}{ d_{ mathbf{r} } } delta_{mathbf{r} mathbf{r}^{prime} } delta_{i k} delta_{j l}
$$

where $mathbf{U} in mathcal{U}(N)$, $mathrm{D}mathbf{U}$ is the standard Haar measure, $U_{ij}^{(mathbf{r})}$ denotes the $(i,j)$-th element of the representation matrix of $mathbf{U}$, and $d_{ mathbf{r} }$ is the dimension of the irreducible representation $mathbf{r}$.

Now, I need to know the answer for this integral:

$$I_2 = int mathrm{D} mathbf{U} ; U_{i_1 j_1}^{(mathbf{r})} U_{ k_1 l_1 }^{ * ( mathbf{r} ) } U_{ i_2 j_2 }^{(mathbf{r}^{prime})} U_{ k_2 l_2 }^{* ( mathbf{r}^{ prime prime } ) }
$$

I appreciate any help.

p.s. Here is my conjecture for the answer:

$$
I_2 = delta_{ mathbf{r}^{prime} mathbf{r}^{prime prime} } times left{
eqalign{
frac{1}{ d_{ mathbf{r} } d_{ mathbf{r}^{ prime } } -1 } delta_{ i_1 k_1 } delta_{ j_1 l_1 } delta_{ i_2 k_2 } delta_{ j_2 l_2 } ( 1- delta_{ mathbf{r} mathbf{r}^{prime} } ) \
+ delta_{ mathbf{r} mathbf{r}^{prime} } left[
eqalign{
frac{ 1 }{ d_{ mathbf{r} }^2 -1 }
( delta_{ i_1 k_1 } delta_{ j_1 l_1 } delta_{ i_2 k_2 } delta_{ j_2 l_2 }
+ delta_{ i_1 k_2 } delta_{ j_1 l_2 } delta_{ i_2 k_1 } delta_{ j_2 l_1 } ) \
- frac{ 1 }{ d_{ mathbf{r} } ( d_{ mathbf{r} }^2 -1 ) } ( delta_{ i_1 k_1 } delta_{ j_1 l_2 } delta_{ i_2 k_2 } delta_{ j_2 l_1 } + delta_{ i_1 k_2 } delta_{ j_1 l_1 } delta_{ i_2 k_1 } delta_{ j_2 l_2 } ) }
right]
} right} $$

UPDATE:

I have been advised that it might be helpful if I can find the tensor product of two irreducible representations, $ mathbf{s} = mathbf{r} otimes mathbf{r}^{prime}$, which most likely leads to a reducible representation, and then I need to decompose $mathbf{s}$ into its irreducible components (by using the Clebsch–Gordan coefficients, according to wikipedia), to be able to use the Schur's lemma to get the answer!!!

However, it is hard for me to do this, and needs awful background.

fa.functional analysis - Criteria for boundedness of power series

Consider a power series $sum_{n=0}^{infty} a_n x^n$ that is convergent for all real
x, thus defining a function $f: mathbb{R} to mathbb{R}$.
Can one give necessary and sufficient criteria the sequence of the coefficients $(a_n)$ has to meet in order for $f$ to be bounded on $mathbb{R}$? (Let's disregard the trivial case that $a_0$ is the only non-zero coefficient and let's call a sequence "function-bounded" if the power series is bounded.) Criteria for boundedness seem to be far more difficult to obtain than the usual criteria for convergence of a power series, here some remarks:

a) A necessary condition for $sum_n a_n x^n$ to be bounded is that there are infinitely many non-zero coefficients which change sign infinitely many times.

b) The boundedness of $f$ is an "unstable" property of the sequence of coefficients: any non-zero change in any finite subset (except $a_0$) will destroy boundedness. Thus the linear subspace of all function-bounded sequences is rather "sparse" in the vector space of all sequences representing convergent power series.

c) On the other hand, the linear subspace of all function-bounded sequences contains at least all power series of functions that can be written as $cos circ h$ with $h$ an entire, real-analytic function, and the algebraic span of these functions. One could conjecture that this space is already the space of all bounded functions that can be represented as power series[EDIT: seems to be refuted, cf. comment below]. And perhaps this could be a starting point for deducing the criteria.

EDIT (conjecture added):
Is is true, that every power series $sum_{n=0}^{infty} a_n x^n$ that is convergent for all real $x$ can be modified only by changing the signs of the terms to a convergent power series $sum_{n=0}^{infty} epsilon_n a_n x^n, quad epsilon_n in {pm1}$ that is bounded for all real $x$?
Example: One can modifify the signs of the power series of the exponential function $sum_{n=0}^{infty} x^n/n!$ pretty easily to a bounded power series by $epsilon_n = +1$ for $n = 0 or 1 mod 4$ and $epsilon_n = -1$ for $n = 2 or 3 mod 4$, yielding the function $sin(x) + cos(x)$. (One can modify the signs pretty easily a bit further such that the power series is not only bounded on the real axis, but also on the imaginary axis - but this is not the question here).
I have neither succeeded in finding a counterexample nor in prooving this conjecture.

EDIT2:
Thanks for the nice counterexample. I would like to improve the conjecture as follows: Define a power series $sum_{n=0}^{infty} a_n x^n$ as nondominant, if for all $x in mathbb{R}$ the absolute value of every term $a_kx^k$ is smaller or equal than the sum of the absolute values of all the other terms: $|a_kx^k| le sum_{n neq k} |a_n x^n|$. The improved conjecture is: Is is true, that every nondominant power series $sum_{n=0}^{infty} a_n x^n$ that is convergent for all real $x$ can be modified only by changing the signs of the terms to a convergent power series $sum_{n=0}^{infty} epsilon_n a_n x^n, quad epsilon_n in {pm1}$ that is bounded for all real $x$?

enumerative combinatorics - Which came first: the Fibonacci Numbers or the Golden Ratio?

The answer for either of these is "hundreds of millions of years" due to their emergence/use in biological development programs, the self-assembly of symmetrical viral capsids (the adenovirus for example), and maybe even protein structure. Because of their close relationship I'd be hard pressed to say which 'came first'.

If you google for it, you'll find plenty of books and papers. However, be extremely careful about examples without a well-explained functional role... there are an arbitrarily large number of coincidences out there if you're looking for them, and humans excel at numerology.

nt.number theory - References and applications involving the Krull Toplogy

I recommend Chapter 8 of Jacobson's Basic Algebra II as a good general reference for the Krull topology and its applications in Galois theory. As is usual for BAI and BAII, if you read other books first you will get very excited at the depth of coverage of this topic.

A few remarks:

1) The Krull topology can actually be defined on $operatorname{Aut}(E/F)$, for any field extension $E/F$. Indeed it is just the subspace topology it inherits from the compact open topology on the set of all maps from $E$ to $E$, where $E$ is given the discrete topology. (Strangely, this one topology -- the preferred function space topology in all my mathematical travels -- gets many names in this case. Jacobson calls it the finite topology. I have also heard it referred to as the "hull-kernel" topology -- ugh!.) If I am not mistaken, it is always totally disconnected and Hausdorff but need not be compact if $E/F$ is transcendental.

2) There have been some efforts (including by me!) to extend Galois theory to transcendental field extensions. The Krull topology comes up in the general case, e.g. in some papers of T. Soundararajan.

3) The Krull topology is also the topology associated to the Galois connection on $operatorname{Aut}(E/F)$, hence it comes up in universal algebra, order theory, mathematical logic, etc. I don't know enough about these fields to point you to any particularly interesting application there, but someone else here certainly might.

Addendum: For instance, here is one of the papers I had in mind in 2) above:

Soundararajan, T.
Galois theory for general extension fields.
J. Reine Angew. Math. 241 1970 49--63.

The general aim of this paper is to investigate exhaustively general Galois correspondences on the level of fields. Specifically, the author considers correspondences when a topology is involved on the group of automorphisms.

Let $E$ be an extension of a field $K$ and $G_0$ the full group of $K$-automorphisms of $E$. We say $(E/K;G_0)$ is a Krull Galois system if there is a 1--1 Galois correspondence between all the intermediate fields of $E/K$ and all the ``Krull-closed'' sub-groups of $G_0$. It is classical that every separable algebraic normal extension allows a Krull Galois theory. The first theorem in this paper states the converse to the above. The author calls the triple $(E/K,G,tau)$ a topological Galois system if there exists a 1--1 correspondence between all intermediate fields of $E/K$ and all $tau$-closed sub-groups of $G$, where $tau$ is some topology on $G$. Theorem 4 catalogues the conditions on the topology $tau$ in order that $E$ may be separable algebraic normal over $K$, where $(E/K,G,tau)$ is a topological Galois system. The next section deals with generalized topological Galois systems where there is a 1--1 Galois correspondence between all intermediate fields [all $tau$-closed sub-groups of $G$] and some sub-groups of $G$ [some intermediate subfields of $E/K$]. The last section deals with a characterization of Krull topology and concludes with the following theorem: Let $(E/K,G,tau_1)$ be a topological Galois system such that $(G,tau_1)$ is a compact topological group. Then $E$ is separable algebraic normal over $K$, $G$ is the Galois group of $E/K$ and $tau_1$ is the Krull topology.

This paper is lucidly presented and is highlighted by illustrative examples and useful remarks. (MathSciNet review by N. Sankaran)

Do DG-algebras have any sensible notion of integral closure?

I like this question a lot. It deserves an answer, and I really wish I had a good one. Instead, I offer the following idea. Maybe it has some merit?

Background

Let me fix some terminology. Suppose $f:Rto S$ a homomorphism of (classical, commutative) rings.

1. An element $sin S$ is said to be integral over $R$ if there is a monic polynomial $pin (R/ker f)[x]$ such that $s$ is a root of $p$; this is equivalent to saying that the subring $(R/ker f)[s]subset S$ is finite over $(R/ker f)$.




2. We say that $f$ is integrally closed if it is a monomorphism and if every element of $S$ that is integral over $R$ is in $R$.




3. At the opposite extreme, we say that $f$ is integrally surjective if every element of $S$ is integral over $R$. (This turns out to be equivalent to being a colimit of proper homomorphisms of finite presentation.)




4. Among the integrally surjective homomorphisms are the elementary integrally surjective homomorphisms, i.e., homomorphisms of the form $Rto (R/mathfrak{a})[x]/(p)$, where $R$ is of finite presentation, $mathfrak{a}subset R$ is a finitely generated ideal, and $p$ is any monic polynomial.

The classical integral closure construction can now be described as a unique factorization of every homomorphism $f:Rto S$ into an integrally surjective homomorphism $Rtooverline{R}$ followed by an integrally closed monomorphism $overline{R}to S$.

In §3.6 of Mathieu Anel's (really cool!) paper, he describes this factorization system and the "proper topology" constructed from it. In particular, he observes that integrally closed monomorphisms are precisely those morphisms satisfying the unique right lifting property with respect to all elementary integrally surjective homomorphisms.

Making this work for $E_{infty}$ ring spectra

Since you expressed interest in getting integral closure off the ground for $E_{infty}$ ring spectra, I'll work in that context.

We can use Andre Joyal's theory of factorization systems in ∞-categories (see §5.2.8 of Jacob Lurie's Higher Topos Theory) to try to play this same game in the ∞-category $mathcal{C}$ of connective $E_{infty}$ ring spectra. (For reasons that will become clear, I'm worried about making this fly for nonconnective $E_{infty}$ ring spectra.) Once a set $I$ of elementary integrally surjective morphisms is selected, integrally closed monomorphisms are determined as the class of maps that are right orthogonal to $I$, and the integral closure construction is a factorization system on $mathcal{C}$.

So what should $I$ be? I think there might be some flexibility here, depending on your aims, but here's a proposal:

1. Start with the coherent connective $E_{infty}$ ring spectra that are of finite presentation (over the sphere spectrum). (Concretely, these are the connective $E_{infty}$ ring spectra $A$ with the following properties: (1) $pi_0A$ is of finite presentation, (2) for every integer $n$, $pi_nA$ is a finitely presented module over $pi_0A$, and (3) the absolute cotangent complex $L_A$ is a perfect $A$-module.) We'll only need these kinds of $E_{infty}$ ring spectra in our construction of $I$.




2. Among these $E_{infty}$ ring spectra, consider the set $I'$ of all morphisms $Ato B$ of finite presentation that induce a surjection on $pi_0$. Let's call the maps of $I'$ quotients.




3. Now we need to enlarge $I'$ to allow ourselves morphisms that act as though they are of the form $Ato A[x]/(p)$ for $p$ monic. For any of our connective $E_{infty}$ ring spectra $A$, we can consider any finitely generated and free $E_{infty}$-$A$-algebra $A[X]$ (i.e., the symmetric algebra on some free and finitely generated $A$-module), and we can consider quotients (in the sense above) $A[X]to B$ where $B$ is almost perfect as an $A$-module (equivalently, $pi_nB$ is finitely presented as an $pi_0A$-module for every integer $n$); let us add the resulting composites $Ato A[X]to B$ to our set $I'$ to obtain the set $I$.

Now the morphisms that are right orthogonal to $I$ can be called integrally closed monomorphisms of $E_{infty}$ ring spectra; call the set of them $S_R$. The morphisms that are left orthogonal to that can be called the integrally surjective morphisms of $E_{infty}$ ring spectra. The integral closure would be a factorization system $(S_L, S_R)$. One shows the existence of a factorization (via a presentation argument), and it follows from general nonsense (more precisely, 5.2.8.17 of HTT) that it is unique.

Three observations

1. It's not clear that $I$ is big enough for all purposes. One might want to allow shifts of free modules to generate our finitely generated and free $E_{infty}$-$A$-algebras in the description above. I haven't thought carefully about this.




2. It's not so obvious how to talk about quotients $Ato B$ when dealing with nonconnective guys. One wants to say that the fiber (in the category of $A$-modules) is "not any more nonconnective than $A$." I'm not quite sure how to formulate this. In any case, that's why I restricted attention to the connective guys above.




3. Predictably, this is definitely not compatible with the usual integral closure: if I take two classical rings $R$ and $S$ and a ring homomorphism $Rto S$, the integral closure of $HR$ in $HS$ is not in general an Eilenberg-Mac Lane spectrum. If $R$ is a $mathbf{Q}$-algebra, the two notions are compatible, however.

Making any computation

... seems really hard. But maybe that's not such a big surprise. After all, integral closures are hard to compute classically as well.

lo.logic - Do you know any good introductory resource on sequent calculus?

Gentzen, 1934, 'Investigations into Logical Deduction' — This is very readable, and introduces so many ideas that later synthetic works invariably miss some. If you're serious, this, and some other papers of Gentzen's, are indispensable.

Stan Wainer has written some excellent introductory texts. I don't think any are freely available for download, although PDFs are washing about here and there. Wainer, 1997, 'Basic Proof Theory with Applications to Computation', in Schwichtenberg, Logic of Computation, Springer Verlag, I strongly recommend.

But the best starting point is probably Proofs and Types, as recommended by Neel, reading at least up to the proof of cut elimination. A warning: Girard's style is a little slippery, and it is common for students to say they have read it, who turn out to have absorbed the opinions but little of the results.

Postscript — If you care about the fine technicalities of matching up normal proofs in natural deductions with cut-free proofs in sequent calculus, Ungar, 1992 Normalization, cut-elimination, and the theory of proofs is a good text, generously made freely available as part of the Stanford Medieval and Modern Thought Digitization Project. This literature is a bit tricky, because the two proof calculi are formulated, and their metatheory have come about in a somewhat different manner. The literature doesn't date back to Gentzen, except to the trivial extent that the two calculi are shown to have equivalent expressive strength, because the theory of normalisation for natural deduction was not fixed untilo Prawitz, 1965, Natural Deduction.

nt.number theory - Is there a nice proof of the fact that there are (p-1)/24 supersingular elliptic curves in characteristic p?

One argument (maybe not of the kind you want) is to use the fact that the wt. 2 Eisenstein series on $Gamma_0(p)$ has constant term (p-1)/24.

More precisely: if ${E_i}$ are the s.s. curves, then for each $i,j$,
the Hom space $L_{i,j} := Hom(E_i,E_j)$ is a lattice with
a quadratic form (the degree of an isogeny), and we can form the corresponding
theta series $$Theta_{i,j} := sum_{n = 0}^{infty} r_n(L_{i,j})q^n,$$
where as usual $r_n(L_{i,j})$ denotes the number of elements of degree $n$.
These are wt. 2 forms on $Gamma_0(p)$.

There is a pairing on the $mathbb Q$-span $X$ of the $E_i$ given by $langle E_i,E_jrangle
= $ # $Iso(E_i,E_j),$ i.e. $$langle E_i,E_jrangle = 0 text{ if } i neq jtext{ and equals # }Aut(E_i) text{ if }i = j,$$
and another formula for $Theta_{i,j}$ is
$$Theta_{i,j} := 1 + sum_{n = 1}^{infty} langle T_n E_i, E_jrangle q^n,$$
where $T_n$ is the $n$th Hecke correspondence.

Now write $x := sum_{j} frac{1}{text{#}Aut(E_j)} E_j in X$. It's easy to see
that for any fixed $i$, the value of the pairing $langle T_n E_i,xrangle$
is equal to $sum_{d |n , (p,d) = 1} d$. (This is just the number of $n$-isogenies
with source $E_i,$ where the target is counted up to isomorphism.)
Now
$$sum_{j}
frac{1}{text{#}Aut(E_j)} Theta_{i,j} =
bigg{(}sum_{j} frac{1}{text{#}Aut(E_j)}bigg{)} + sum_{n =1}^{infty} langle T_n E_i, xrangle
q^n
= bigg{(}sum_{j}frac{1}{text{#}Aut(E_j)}bigg{)} + sum_{n = 1}^{infty} bigg{(}sum_{d | n, (p,d) = 1} dbigg{)}q^n.$$

Now the LHS is modular of wt. 2 on $Gamma_0(p)$, thus so is the RHS. Since we know
all its Fourier coefficients besides the constant term, and they coincide with those of the Eisenstein series, it must be the Eisenstein series.
Thus we know its constant term as well, and that gives the mass formula.

(One can replace the geometric aspects of this argument, involving s.s. curves and Hecke
correspondences, with pure group theory/automorphic forms: namely the set ${E_i}$ is
precisely the idele class set of the multiplicative
group $D^{times}$, where $D$ is the quat. alg. over $mathbb Q$ ramified at $p$ and $infty$. This formula, writing the Eisenstein series as a sum of theta series, is then
a special case of the Seigel--Weil formula, I believe, which in general, when you pass to constant
terms, gives mass formulas of the type you asked about.)

fa.functional analysis - Topological dual and the notions of "smaller" and "larger" than...

To answer your final question: Let $Phi supset Psi$. Consider $Phi' subset Phi^*$, the former is the continuous linear functionals on $Phi$, and the latter is the set of all linear functionals on $Phi$. Then the restriction of $Phi'$ on $Psi$ is obviously continuous, so $Phi' subset Psi'subset Psi^*$.

Therefore if you make a space smaller, you makes its dual bigger.

Intuitively speaking, elements of $Psi'$ need to be continuous on fewer objects, and hence has fewer constraints; thus $Psi'$ contains more objects.

For your original question: your interpretation is sort-of okay. The point is that infinite dimensional Hilbert spaces admit dense proper subspaces (hope I am getting the notation correct). And in particular you can have two dense subspaces of a Hilbert space with one strictly contained in the other. You may want to review volume 2 of Reed and Simon.

ct.category theory - When is the Yoneda product graded commutative?

Another starting point is to think of ${rm Ext}(A,A)$ as the derived endomorphism ring of the object $A$ and recall Schur's lemma. If $A$ is a finitely-generated simple module over a ring $R$, then ${rm Hom}_R(A,A)$ is a division algebra. For example, if $R$ is a $k$-algebra over an algebraically closed field $k$, then ${rm Hom}_R(A,A)$ is isomorphic to $k$ (so, in particular, it is commutative.) Via Freyd-Mitchell embedding, this should give some idea what to expect in degree $0$.

Going back the question, then, the examples one might have in mind are categories of modules over a group ring or enveloping algebra of a graded Lie algebra: in these cases, ${rm Ext}(k,k)$ is group- or Lie algebra cohomology, respectively, and has a graded-commutative cup product, where $k$ is the trivial module.

Perhaps there is a suitable "semisimplicity" hypothesis one could impose on the category so that ${rm Ext}(A,A)$ is graded-commutative for all simple objects $A$?

geometry - Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections

My general question concerns what we can learn about an arbitrary, three-dimensional convex polytope (or convex hull of an arbitrary polytope) strictly from the surface areas of its two-dimensional projections on a plane as it 'tumbles' in 3-space (i.e. as it rotates along an arbitrary, shifting axis).

If it's helpful, please imagine the following physical set-up:

We take an arbitrary three-dimensional convex polytope, and fix the center of mass to a coordinate in 3-space, $(x_0, y_0, z_0)$, located at some distance, $D$, above a flat surface. While this prohibits translation of the center of mass, the polytope is still allowed to tumble freely (i.e. it is allowed rotation around an arbitrary axis centered at the fixed coordinate).

There is no 'gravity' or other force to stabilize the tumbling polytope in a particular orientation. Over time it will continue to tumble randomly. (The 'physical' set-up is only meant for descriptive reasons.)

We shine a beam of coherent light on the tumbling polytope, larger than the polytope's dimensions, and continually record the area of the resulting shadow, or two-dimensional projection on the surface. To be clear, the area of the two-dimensional projection is the only information we are allowed to observe or record, and we are allowed to do so over an arbitrary length of time.

---

My question is - From observing the area of the tumbling polytope's shadow, or two-dimensional projection over time, what can we learn about it's geometry? To what extent can we characterize and/or reconstruct the polytope from its changing shadow (extracting the surface area for example - hat tip to Nurdin Takenov)?

Do we gain anything by watching the evolution of the convex polytopes shadow as it tumbles (part of the point for the physical example), as opposed to an unordered collection of two-dimensional projections?

Update - Nurdin Takenov (and Sergei Ivanov in later comments) nicely points out that we can use the average surface area of the two-dimensional projection to find the surface area of the tumbling convex polygon. Might we be able to find it's volume?

(Addendum - I would be really neat if somebody could point me to any algorithms in the literature... or available software.... that let's me calculate and characterize two-dimensional surface projections of convex polytopes!)

gr.group theory - Are all connected solvable affine algebraic groups supersolvable?

The basic question is whether there is a notion of chief factor of a connected solvable algebraic group that matches my intuition. A few smaller assertions are sprinkled through the explanation, and the implicit question is if these are correct (are unipotent groups nilpotent, are their chief factors all isomorphic to subgroups of Ga, etc.).

In J.S. Milne's course notes on the basic theory of algebraic groups, theorem 14.30:

A connected solvable smooth group over a perfect field has a connected unipotent normal subgroup whose quotient is of multiplicative type.

In other words, the derived subgroup is contained in the unipotent radical.

Now, a connected group acts on a group of multiplicative type trivially, by 13.21.

I did not see it mentioned, but I believe unipotent groups are nilpotent in both the group theoretic sense and whatever fancy definition one might cook up for these functors. I think that the lower central factors should be direct products of subgroups of the additive group Ga.

By the Jordan decomposition or 13.13 or 13.15, I think any action of Gm on (Ga)^n is diagonal.

It looks like a connected solvable affine algebraic group over an algebraically closed field has a chief series consisting of subgroups of Ga and Gm, all of which I would describe as being one-dimensional.

The analogy with finite groups takes the unipotent radical to be O_p(G), the p-core, and so it appears that a finite group of solvable algebraic type always has [G,G] <= O_p(G), so that not only is G supersolvable nilpotent-by-abelian, it is p-closed and its eccentric chief factors are all for the same prime p. In the algebraic case, the central chief factors would be the subgroups of Gm, and the eccentric chief factors would be the subgroups of Ga.

In other words, connected solvable affine algebraic groups over algebraically closed fields are very dissimilar from finite solvable groups in that the representations they define on their own chief factors are all one-dimensional. More briefly, connected solvable affine algebraic groups over algebraically closed fields are supersolvable.

Edit: I think the answer to my question is relatively simple: "supersolvable" is a little tricky to directly generalize, but "nilpotent-by-abelian" is quite easy and true, and still implies that any chief factors will be one-dimensional. In Jim's answer, it appears Borel-Serre-Mostow (at least by the time they are translated into Russian) also considered these groups to be "supersolvable", so the name is reasonable, even if the correct definition is just "nilpotent-by-abelian".

nt.number theory - Sequence of Diophantine Equations

Is there some (huge) positive integer $M$ with the following property:
for any $z>M$, there exist positive integers $x, y_{1}, y_{2},..., y_{z}$
such that $x^x$ $=$ $y_{1}^{y_{1}}$+ $y_{2}^{y_{2}}$+ ... +$y_{z}^{y_{z}}$
?

[Please remark that the $y$'s are $geq$ $1$ and need not
to be necessarily distinct.]

As a [rather naive] way to attack this problem [which may (perhaps) be
related to some works of Robinson, Matiasevich, M. Davis, and Chao-Ko],
I'm thinking about lots of $1$'s, lots of $2$'s, and lots of
$(x-1)$'s.
Also, let us observe that, if $z$ has this property and $y_{i}$ $=$ $2$
for some $i$, then $z+3$ has the same property, too...

lo.logic - How are the two natural ways to define ''the category of models of a first-order theory T'' related?

Background/Motivation: Inspired by an interesting question by Joel, I've been wondering about the relationship between two very natural ways to define the category of ''all models of T'' where T is a first-order theory.

Let us assume that T is a complete theory with infinite models. Then on the one hand we can define the category Mod(T) whose objects are all the models of T and whose morphisms are all homomorphisms in the sense of model theory -- i.e. functions $varphi: M rightarrow N$ such that

For any $n$-ary relation $R$ in the language of $T$, if $M models R^M(a_1, ldots, a_n)$, then $N models R^N(varphi(a_1), ldots, varphi(a_n))$;

and

$varphi(f^M(a_1, ldots, a_n)) = f^N(varphi(a_1), ldots, varphi(a_n))$ for any $n$-ary function symbol $f$ in the language of $T$.

Also, one can define another category Elem(T) whose objects are also all the models of T, but whose morphisms are only the elementary embeddings, that is, functions which preserve the truth of all first-order formulas. As a model theorist, I'm more used to thinking about the category Elem(T), and this latter category arises naturally if one cares about which sets are definable but one does not particularly care about which sets are definable by positive quantifier-free formulas.

Question: What sorts of category-theoretic properties automatically transfer from Mod(T) to Elem(T), or from Elem(T) to Mod(T)?

To be clear, by a ``category-theoretic property'' I mean something that is preserved by an equivalence of categories.

Another related question is:

Question: Suppose we have a set of category-theoretic properties which we know characterize all the categories C which are equivalent to Mod(T) for some T [or to Elem(T) for some T]. Can we use this to characterize the categories which are equivalent to Elem(T) [respectively, Mod(T)] for some T?

Here are a couple of basic facts I know, which may or may not be useful here. First of all, every category Elem(T) is equivalent to a category Mod(T') for some other theory T' -- namely, the "Morleyization'' T' of T, where we expand the language by adding new predicates for every definable set (and iterating $omega$ times), thereby forcing T' to have quantifier elimination. However, it is certainly not true that every category Mod(T) is equivalent to a category of the form Elem(T') -- for instance, Mod(T) might not have colimits of $omega$-directed chains, but Elem(T) always will (by Tarski's elementary chain theorem).

Addendum: As Joel pointed out, there is a third possible notion of ''morphism'' for this category: the ``strong homomorphisms'' $varphi: M rightarrow N$ which commute with the interpretations of function symbols and have the property that for any $n$-ary relation $R$ in the language of $T$, $$M models R^M(a_1, ldots, a_n) Leftrightarrow N models R^N(varphi(a_1), ldots, varphi(a_n)).$$

I'd also be interested to learn about any relationships between the category of models with strong homomorphisms and the other two categories above.

nt.number theory - On periods of algebraic integers modulo rational primes

I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.

Let $K$ be a number field, which we may assume Galois if it helps, $cal O$ its ring of integers and for each prime number $p$ let $R_p={cal O}/p{cal O}$ (a finite product of finite fields of characteristic $p$ for almost all $p$). Fix $lambdain{cal O}$, $lambdaneq0$ or a root of $1$. Then $barlambdain R_p^times$ for almost all $p$ and the period $pi_p=pi_p(lambda)$ of $lambda$ is defined as the smallest positive $d$ such that $barlambda^d=1$.

It is obvious that if we fix an integer $n$ the number of $p$'s such that $pi_pleq n$ is finite, since $pi_p=d$ implies that $p|(lambda^d-1)$ and there are only finitely many of those.

On the other hand, if $ngeq2$ the number of $p$'s such that
$max{text{Supp}(pi_p)}leq n$ (the support $text{Supp}(N)$ of an integer $N$ is the set of prime divisors of $N$) is infinite. For instance, the set of elements $lambda^{2^k}-1$ has an infinite set of rational prime divisors because $lambda^{2^{k+1}}-1=(lambda^{2^k}-1)(lambda^{2^k}+1)$ and the only common prime divisors in $cal O$ to the 2 factors are primes of residual characteristic 2. Thus, as k grows, a new set of primes adds up at each step, so to speak.

Now the question is: fix an arithmetic progression ${cal P}:a,a+d,a+2d,dots$ with $(a,d)=1$, is it true that there are infinitely many primes in $cal P$ such that $max{text{Supp}(pi_p)}leq n$? Conditionally on $n$?

In particular: suppose $K$ quadratic, and let $ell>2$ a prime. Are there infinitely many primes $pequiv 1bmodell$ such that $max{text{Supp}(pi_p)}leqell-1$?

ag.algebraic geometry - Extending vector bundles on a given open subscheme, reprise

The simplest example is the following. Take $X = A^3$ with coordinates $(x,y,z)$, and let $E = Ker(O_X oplus O_X oplus O_X stackrel{(x,y,z)}to O_X)$. Let $U$ be the complement of the point $(0,0,0) in X$. Then $E_{|U}$ is a vector bundle. On the other hand, $E$ is not a vector bundle, but $E^{**} cong E$, hence $E$ is the reflexive envelope of $i_*i^*E$, and thus there is no vector bundle on $X$ extending $E_{|U}$.

---

[Edit by Anton: I just spent some time digesting some pieces of the above answer, so figured I'd include the results for future readers similar to me.]

("$E$ is not a vector bundle") The sequence $O_Xxrightarrow{pmatrix{z\ y \ x}}O_X^3xrightarrow{pmatrix{y & -z & 0\ -x & 0 & z\ 0 &x&-y}}O_X^3xrightarrow{pmatrix{x& y& z}}O_X$ is exact, so $E$ is the cokernel of the first map. Since taking fibers commutes with taking cokernels, we compute that $E$ has 2-dimensional fibers away from the origin, and 3-dimensional fiber at the origin.

("$E^{**}cong E$") Note that $E$ is $S_2$ (i.e. sections defined away from codimension 2 extend uniquely) since it is the kernel of a map from an $S_2$ sheaf to a torsion-free sheaf (the section of $O_X^3$ extends uniquely, and its image is zero away from codimension 2, so must be zero, so the extended section is in $E$). Note also that the dual of any sheaf is $S_2$ (if $phicolon Fto O_X$ is defined on an open set $V$ with codimension 2 complement and $s$ is a section, $phi(s)$ must be the unique extension of $phi(s|_V)$ as a section of $O_X$), so $E^{**}$ is $S_2$. The canonical map $Eto E^{**}$ is then a map of $S_2$ sheaves which is an isomorphism away from codimension 2, so it must be an isomorphism.

("and thus there is no vector bundle on $X$ extending $E|_U$") If $F$ is an $S_2$ extension of $E|_U$ (i.e. $i^*F=i^*E$), then there is a map $Fto i_*i^*Eto (i_*i^*E)^{**}=E$ which is an isomorphism over $U$, so is an isomorphism by the argument in the previous paragraph. A vector bundle extension would be a different $S_2$ extension.

co.combinatorics - What is the relationship between the Bell numbers, the Bell polynomials, and the partition numbers?

As you already remarked, integer partitions can be regarded as the
isomorphismtypes of set partitions. The relationship between their generating
functions is in my opinion best understood in the language of species. Let $E$
be the species of sets (ensembles in french :-), $E_{>0}$ the species of
non-empty sets. Then the species of set partitions $P$ is

$P=Ecirc E_{>0}$

(read: sets of non-empty sets). To understand the relationship between there
generating functions, we need the cycle index series $Z_F(p_1, p_2,dots)$ of a
species F. The main points are:

1) The exponential generating function for the structures of $F$ (in our case:
set partitions) $F(x)$ is

$$Z_F(x,0,0,dots).$$

2) The ordinary generating function for the isomorphism types of $F$ (in our case: integer partitions) $tilde F(x)$ is
$$Z_F(x,x^2,x^3,dots).$$

3) The cycle index series of $Fcirc G$ is

$$Z_{Fcirc G}(p_1, p_2,dots) = Z_F(Z_G(p_1, p_2,dots),
Z_G(p_2, p_4,dots),
Z_G(p_3, p_6,dots)dots).$$

4) The cycle index series of sets is

$$Z_E(p_1,p_2,dots)=exp(p_1+frac{p_2}{2}+frac{p_3}{3}+dots).$$

5) The cycle index series of the empty set 1, therefore the cycle index series of nonempty sets is

$$Z_{E_{>0}}(p_1,p_2,dots)=exp(p_1+frac{p_2}{2}+frac{p_3}{3}+dots)-1.$$

6) Combining 3), 4) and 5) we obtain

$$Z_P(p_1,p_2,dots)=expsum_{kgeq 1}frac{1}{k}(exp(p_k+frac{p_{2k}}{2}+frac{p_{3k}}{3}+dots)-1).$$

7) According to 1), we obtain

$$P(x)=exp(exp(x)-1).$$

8) According to 2), we obtain

$$tilde P(x)=prod_{kgeq 1}frac{1}{1-x^k}.$$

See Bergeron, Labelle, Leroux, "Combinatorial Species and Tree-like Structures", Section 1.4, page 45.

But I suppose the real question is: what is the cycle index series? I'm afraid I can only point you to the wikipedia article, if you don't have access to the book by Bergeron, Labelle and Leroux...

Maybe some intuition helps: a species is a functor from the category of finite sets and bijections into the same category, that is -- roughly -- a machine that produces a finite set of objects (called "combinatorial structures") given a finite set (of "labels"), together with another machine that, given a bijection on the labels (a "relabelling") produces a bijection on the set of objects (that relabels all the objects).

The cycle index series captures the information which structures can be obtained from a given structure by permuting some of the labels, such that the permutation has given cycle type. But really, I think you need to look into the book.

nt.number theory - Lowest Unique Bid

Each of n players simultaneously choose a positive integer, and one of the players who chose [the least number of [the numbers chosen the fewest times of [the numbers chosen at least once]]] is selected at random and that player wins.

For n=3, the symmetric Nash equilibrium is the player chooses m with probability 1/(2^m).

What is the symmetric Nash equilibrium for n=4? Is it known for general n?

co.combinatorics - Remove unnecessary dependencies in a task graph?

For each vertex x, make a set that contain each vertex y that can reach x. This sets also includes x.

If you have two edges b -> a and c -> a, then if the set associated with b is a subset of the set associated with c, then the edge b -> a can be removed.

Example:

a -> b
b -> c
a -> c

The set are:
a: { a }
b: { a, b } // Can be reached from a and b
c: { a, b, c}

If you look at the edges:
b -> c
a -> c

Then you see that the set of a is a subset of b. So, the edge a -> c can be removed.

Lucas

supersymmetry - Does the super Temperley-Lieb algebra have a Z-form?

Background Let V denote the standard (2-dimensional) module for the Lie algebra sl₂(C), or equivalently for the universal envelope U = U(sl₂(C)). The Temperley-Lieb algebra TL_d is the algebra of intertwiners of the d-fold tensor power of V.

TL_d = End_U(V⊗…⊗V)

Now, let the symmetric group, and hence its group algebra CS_d, act on the right of V⊗…⊗V by permuting tensor factors. According to Schur-Weyl duality, V⊗…⊗V is a (U,CS_d)-bimodule, with the image of each algebra inside End_C(V⊗…⊗V) being the centralizer of the other.

In other words, TL_d is a quotient of CS_d. The kernel is easy to describe. First decompose the group algebra into its Wedderburn components, one matrix algebra for each irrep of S_d. These are in bijection with partitions of d, which we should picture as Young diagrams. The representation is faithful on any component indexed by a diagram with at most 2 rows and it annihilates all other components.

So far, I have deliberately avoided the description of the Temperley-Lieb algebra as a diagram algebra in the sense that Kauffman describes it. Here's the rub: by changing variables in S_d to u_i = s_i + 1, where s_i = (i i+1), the structure coefficients in TL_d are all integers so that one can define a ℤ-form TL_d(ℤ) by these formulas.

TL_d = C ⊗ TL_d(ℤ)

As product of matrix algebras (as in the Wedderburn decomposition), TL_d has a ℤ-form, as well: namely, matrices of the same dimensions over ℤ. These two rings are very different, the latter being rather trivial from the point of view of knot theory. They only become isomorphic after a base change to C.

---

There is a super-analog of this whole story. Let U = U(gl_1|1(C)), let V be the standard (1|1)-dimensional module, and let the symmetric group act by signed permutations (when two odd vectors cross, a sign pops up). An analogous Schur-Weyl duality statement holds, and so, by analogy, I call the algebra of intertwiners the super-Temperley-Lieb algebra, or STL_d.

Over the complex numbers, STL_d is a product of matrix algebras corresponding to the irreps of S_d indexed by hook partitions. Young diagrams are confined to one row and one column (super-row!). In that sense, STL_d is understood. However, idempotents involved in projecting onto these Wedderburn components are nasty things that cannot be defined over ℤ

---

Question 1: Does STL_d have a ℤ-form that is compatible with the standard basis for CS_d?

Question 2: I am pessimistic about Q1; hence, the follow up: why not? I suspect that this has something to do with cellularity.

Question 3: I care about q-deformations of everything mentioned: U_q and the Hecke algebra, respectively. What about here? I am looking for a presentation of STL_d,q defined over ℤ[q,q^-1].

at.algebraic topology - Understand Cech Cohomology

For Question #2: Look at chapter 2 of Bott, R. & Lu, T. "Differential forms in Algebraic Topology", Springer GTM vol 82. There you will find a nice introduction to Cech cohomology through a Mayer-Vietoris general point of view.

For Question #4: Yes. Look at Bredon, G. "Topology and Geometry", Springer GTM vol 139, p. 289. There is a Mayer-Vietoris set-construction which is a nice way to avoid presheaf calculations.

ct.category theory - Determinant of a pullback diagram

Suppose that X and Y are finite sets and that f : X → Y is an arbitrary map. Let PB denote the pullback of f with itself (in the category of sets) as displayed by the commutative diagram

PB → X
↓      ↓
X   → Y

Terence Tao observes in one the comments on his weblog that the product of |PB| and |Y| is always greater than or equal to |X|². (This is an application of the Cauchy-Schwarz inequality.) This fact may be rephrased as follows: If we ignore in the above diagram all arrows and replace the sets by their cardinalities we obtain a 2x2 matrix with a non-negative determinant.

The question is whether this is a general phenomenon. Suppose that n is a positive integer and that X₁, X₂, ... ,X_n are finite sets; furthermore we are given maps f₁ : X₁ → X₂, f₂ : X₂ → X₃, ... , f_n-1 : X_n-1 → X_n. We construct a pullback diagram of size nxn. The diagram for n=4 is shown below.

PB → PB → PB → X₁
↓       ↓       ↓     ↓
PB → PB → PB → X₂
↓       ↓       ↓     ↓
PB → PB → PB → X₃
↓       ↓       ↓     ↓
X₁ →  X₂ →  X₃ → X₄

Here, the maps between the X_i in the last row and column are the corresponding f_i and the PBs denote the induced pullbacks. (Of course, although they are denoted by the same symbol, different PBs are different objects.) The PBs can be constructed recursively. First, take the pullback of X₃ → X₄ ← X₃; it comes with maps X₃ ← PB → X₃. Having constructed this, take the pullback of X₂ → X₃ ← PB and so forth.

Ignore all arrows and replace sets by their cardinalities. Is the determinant of the resulting nxn matrix always non-negative?

gt.geometric topology - Constructing 4-manifolds with fundamental group with a given presentation.

Related to question (1), suppose you wanted to get a 4-manifold with boundary as a submanifold of $mathbb{R}^4$ with fundamental group a finitely presented group, with presentation complex $K$. It is a result of Curtis that any 2-complex $K$ is
homotopy equivalent to a 2-complex which embeds in $mathbb{R}^4$ (see also the Stallings
reference in the comment on this MO question and Dranishnikov-Repovs - in fact this works if the 2-complex $K$ has only one vertex). However, to thicken up $Ksubset mathbb{R}^4$ to get a tubular neighborhood which is a 4-manifold (with boundary) in $mathbb{R}^4$ which retracts to $K$, $K$ must be nicely embedded. For example, the 2-cells should be locally flat. I'm not sure if this holds true for the embeddings of Curtis or for the other constructions.

To answer (2), think about what happens when you thicken up the complex. Thickening some points in $mathbb{R}^5$ gives some 5-balls, whose boundary is a union of 4-spheres. Thickening an interval attached to some points, one gets a 1-handle $D^1times D^4$, whose boundary removes two 4-balls from the 4-spheres and attaches in $D^1 times S^3$, giving a 4-manifold with free fundamental group (a connect sum of $S^3times S^1$'s). The 2-cells thicken up to 2-handles $D^2times D^3$, which remove $S^1times D^3$ from the 4-manifold (which doesn't change the fundamental group, since $S^1$ is codimension 3), and replaces it with $D^2 times S^2$, which has the effect of killing the element in the free group corresponding to the circle by Van Kampen. So you see that this gives the same thing as the construction in Henry Wilton's answer to the other question.

ct.category theory - Motivation for equivariant sheaves?

If you know that the sections of a vector bundle form a standard example of a sheaf, then the corresponding example of a $G$-equivariant sheaf on a space $X$ with $G$-action is a vector bundle $V$ over $X$ with a $G$-action compatible with the projection (i.e. making the projection $G$-equivariant, i.e. intertwining the actions). Such actions on vector bundles over homogeneous spaces were considered in representation theory by Borel, Weil, Bott and Kostant ("homogeneous vector bundles"; and later many generalizations to sheaves by Beilinson-Bernstein, Schmid, Miličić etc.). David Mumford introduced $G$-equivariant structures on sheaves under the name $G$-linearization for the purposes of geometric invariant theory.

While for a function on a $G$-space the appropriate notion is the $G$-invariance, for sheaves the invariance is useful only up to a coherent isomorphism, what spelled out yields the definition of the $G$-equivariant sheaf. This is an example of a categorification (recall that functions form a set and sheaves form a category). Using the Yoneda embedding one can indeed consider the $G$-equivariant objects as objects in some fibered category of objects on $X$ with an action on each hom-space (see the lectures by Vistoli).

While for a function to be invariant is a property, for a sheaf the $G$-equivariance entails the additional coherence data, so it is a structure.

Category of $G$-equivariant sheaves on $X$ is not a quotient of the category of usual sheaves on $X$, but rather equivalent to the category of sheaves on the geometric quotient $G/X$, in the case when the action of $G$ on $X$ is principal; or in general if we replace the geometric quotient by the appropriate stack $[G/X]$.

mp.mathematical physics - A Poisson Geometry Version of the Fukaya Category

The fundamental technique of symplectic topology is the theory of pseudo-holomorphic curves. One studies maps $u$ from a Riemann surface into a symplectic manifold, equipped with an almost complex structure tamed by the symplectic form, such that $Du$ is complex-linear. Numerous algebraic structures can be built from such maps: Gromov-Witten invariants, Hamiltonian Floer cohomology, Floer cohomology of pairs of Lagrangian submanifolds, and most elaborate of all, $A_infty$-structures on Lagrangian Floer cochains (Fukaya categories).

Though the basic theory of pseudo-holomorphic curves makes sense on more general almost-complex manifolds, the presence of the symplectic structure is vital for Gromov compactness to be applicable. Without this, your curves are liable to vanish into thin air. None of the algebraic structures I mentioned have been developed on almost complex manifolds, nor on Poisson manifolds. It's conceivable that leafwise constructions can be made to work in the Poisson context, but there are basic analytic and geometric questions to be addressed.

There are situations where one might reasonably hope to find relations between Poisson geometry and symplectic topology, but in those situations it may be wise to go via intermediate constructions. For instance, a version of the derived Fukaya category of $T^{ast} L$ was shown by Nadler to be equivalent to the derived category of constructible sheaves on $L$, and I'm told that that category is related to deformation quantization of $T^{ast} L$ - something which truly does belong to Poisson geometry.

soft question - How much reading do you do before you attack a problem?

When I attacked the problem of finding a quantum factoring algorithm, I had read four or five papers on the subject, which constituted nearly all the literature on quantum algorithms at the time. However, there were lots of other relevant papers that I didn't even know about in the field which would later be called quantum information theory, and I didn't feel compelled to do a literature search to find them.

This is one extreme. If you try to go this route, you may very likely miss some important techniques that are commonly used in the new field, so I would actually recommend substantial reading in the new field. I had to do that when I started working on quantum information theory.

It would make a lot of sense if you worked on your problem while you did this substantial reading (even though you're liable to go in the wrong direction), because that will be a good guide for choosing which papers you should read. It also helps if you have a colleague in the new area you can talk to or collaborate with. So, as Deane says in the comments, "It depends."

ag.algebraic geometry - parameterizing polynomial loops in $mathbb{C}^times$

This is mostly a series of comments, but guided by the questions you asked.

First of all, I will only talk about $X_n$, interpreting it as the space of non-zero complex polynomials $p$ of degree at most $n$ such that no root of $p$ lies on the unit circle, taken up to non-zero scaling. We may as well think of the polynomials as homogeneous of degree exactly $n$ in two variables, so that each point of $X_n$ defines a subset of $n$ points of the complex projective line $mathbb{CP}^1$ (the set of roots of the polynomial) that is disjoint from the unit circle. Therefore, $X_n$ is certainly a non-empty open subset in the standard topology of the projective space of homogeneous polynomials of degree $n$, and therefore $X_n$ has (real) dimension $2n$. Observe that the unit circle disconnects $mathbb{CP}^1$, and that the points of $X_n$ are likewise distributed into (at least) $n+1$ connected components, corresponding to how the $n$ points in $mathbb{CP}^1$ are distributed with respect to the two halves obtained by removing the unit circle (recall that $mathbb{CP}^1$ is topologically a two-dimensional sphere and that the unit circle can be identified with the equator of the sphere, so that, among the $n$ points we are talking about, there are some that are in one hemisphere and some that are in the other). On the other hand, a non-empty Zariski open subset of an irreducible algebraic variety is connected. Thus, as a subset of the space of homogenous polynomials of degree $n$, the space $X_n$ is certainly not a complex algebraic subvariety.

But, we may decide to analyze further the space $X_n$, zooming in on the locus $X_n^k$ where, for a fixed integer $k$, there are $k$ points on the half containing the origin and $n-k$ points on other half. Clearly, within each hemisphere, the points are free to roam around! Thus $X_n^k$ is homeomorphic (and in fact diffeomorphic with the natural choice of differentiable structure) to the space of ordered pairs $(p_1,p_2)$ where $p_1$ is a monic polynomial of degree $k$ and $p_2$ is a monic polynomial of degree $n-k$: expand each hemisphere to a whole complex plane and "code" the $k$ points on one half by the unique monic polynomial having them as a root (and do the same to the other half). Thus each space $X_n^k$ is connected and in fact diffeomorphic to $mathbb{C}^k times mathbb{C}^{n-k}$. As a complex manifold you can also say that $X_n^k$ is the product of the symmetric product of $k$ copies of the unit disk with the symmetric product of $n-k$ copies of the unit disk. Thus, again using a structure induced from the ambient space of homogeneous polynomials, any complex algebraic subvariety of $X_n$ would be a complex algebraic subvariety of a symmetric product of unit disks: I think that this means that the only complex algebraic subvarieties of $X_n$ are the points.

Finally, let me make a small stab at getting your hand on $X$, by mentioning one description of the "glueing" of $X_n$ inside $X_{n+1}$. From the point of view of non-homogeneous polynomials, this corresponds to simply realizing that a polynomial of degree at most $n$ is also a polynomial of degree at most $n+1$. From the point of view of their homogenizations, the inclusion corresponds to replacing $z^i$ by $x^iy^{n+1-i}$ instead of $x^iy^{n-i}$. Effectively, we are adding the point at infinity as one of our roots (namely the extra root $y=0$). Thus in terms of the description above, we observe that the two hemispheres are not "identical": one of them has a point that is special, namely the point at infinity. The points of $X_{n+1}$ that come from points of $X_n$ are the points that correspond to $(n+1)$-tuples one of whose elements is the point at infinity.

gr.group theory - Finitely presented sub-groups of GL(n,C)

I just wanted to make a comment on Mal'cev's theorem (if I could leave this as a comment, I would).

Mal'cev's paper is a great exposition of the theorem, as well as a lot of other related material, all written in a basic yet enlightening style.

Also, if you know a little commutative algebra (as in the Nullstellensatz, the one given in Eisenbud pg. 132), there is quick and easy proof of Mal'cev's theorem. I could sketch it if necessary, but I am right now in the process of LaTeX-ing it, so I'll probably just come back and post a link.

Steve

EDIT - a sketch of the argument:
Mal'cev's theorem says a finitely generated linear group is residually finite. So let $Xsubset GL(n,F)$ be a finite subset of the general linear group over some field $F$, and $G=langle X rangle$. First, make $X$ symmetric, so that if $xin X$ then also $x^{-1}in X$. Each $xin X$ is an $ntimes n$ matrix, so we can assemble all entries from all elements of $X$, getting a finite subset of $F$. Let $R$ denote the subring of $F$ generated by this subset (along with $1$). Then $R$ is a Jacobson ring, and since it's a subring of $F$, it's Jacobson radical is $0$. Now $G$ is a subgroup of $GL(n,R)$; let $gin G$ be a non-identity element, so that $g-I_nneq 0$, where $I_n$ is the identity matrix. Thus $g-I_n$ has a non-zero element, and thus there is some maximal ideal $msubset R$ not containing this non-zero element. The matrix ring homomorphism $M_n(R)rightarrow M_n(R/m)$ (reducing everything mod $m$) induces a group homomorphism $Grightarrow GL(n,R/m)$, where $g$ is not in the kernel. But $R/m$ is finite (by the Nullstellensatz), so $GL(n,R/m)$ is a finite group.

at.algebraic topology - Relation between $KO$ and $K$

Every real bundle can be complexified so there's a natural transformation $KO(X) to K(X)$. Going all the way around multiplies by $2$ each time so if you localise at the prime $2$, you get a nice splitting.

More generally, these fit into a long exact sequence. I don't remember if the third term has a special name or not. (As this is a community wiki question, someone should feel free to edit this answer to add it in.) I've seen its classifying space written as $B(U/O)$ which suggests not.

reference request - Largest pair of homometric Golomb rulers?

A Golomb ruler is a set of $n$ integers that determines $binom{n}{2}$ distinct differences.
Two sets are homometric if they determine the same (multiset) of differences.
For example,
$${0,1,4,10,12,17} ;,; {0,1,8,11,13,17}$$
are a homometric pair of Golomb rulers, determining 15 distinct differences (excluding only
${14,15}$).
Although there are arbitrarily large Golomb rulers, and arbitrarily large pairs of homometric
sets (allowing multipiclity), it is unclear (from my searching) if there are arbitrarily large pairs of Golomb rulers.

In the 1994 paper, "There Are No New Homometric Golomb Ruler Pairs with 12 Marks or Less,"
the authors say that they "are divided on whether any additional nontrivial homometric rulers are to be found." The nice paper "Reconstructing sets from interpoint distances" does not seem to
attend to the special case where all distances are distinct. Nor does the Rosenblatt-Seymour paper "The structure of homometric sets" (inferring from secondary sources—I don't have that paper yet).

My question is: What is the largest pair of homometric Golomb rulers known? Is it still open whether or not there are arbitrarily large pairs? Thanks for any pointers on this topic!

Addendum. Thanks to Yota Otachi for uncovering the 2007 paper by Bekir and Golomb he cites below.
As he says, it proves that there are no homometric Golomb rulers of more than six marks.
The proof uses Golumb's "polynomial model."

sheaf theory - Functoriality of base change

Let $a:Wrightarrow X$, $c:Xrightarrow Z$, $b:Wrightarrow Y$ and $d:Yrightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $kappa$ between two functors $b_! circ a^*$ and

$d^* circ c_!$. Usually this natural isomorphism is called base change.

Suppose we have another pull-back diagram, $d:Yrightarrow Z$, $f:Zrightarrow U$, $e:Yrightarrow V$ and $g:Vrightarrow U$. Then we have another natural isomorphism $kappa'$ between $e_! circ d^*$ and

$g^*circ f_!$.

By the universal property of pull-back, one can see that $a:Wrightarrow X$,$f circ c:Xrightarrow U$, $ecirc b:Wrightarrow V$ and $g:Vrightarrow U$ is also a pull-back diagram. Denote the corresponding natural isomorphism by $kappa''$.

Is it true that $kappa''=kappa'circ kappa$?

Probably the equality is a little confusing, but the formulation is clear if one thinks of it.

ag.algebraic geometry - Bad Categorical Quotients

Let G be an algebraic group acting on a scheme X. Then f: X --> Y is a categorical quotient if it is constant on G-orbits and any other G-invariant morphism factors through it in a unique fashion. We say f is a 'good' categorical quotient if:

1) f is a surjective open submersion (i.e. the topology on Y is induced from X).

2) for any open U ⊂ Y, the induced map from functions on U to G-invariant functions on f^-1(U) is an isomorphism.

Does anyone know an example of a 'bad' categorical quotient (by which I mean...well...a not good one).

co.combinatorics - Graphs and hypercubes

Thank you Theo for the corrections in the source text. It is my first time on MO and I am still trying to understand what it's possible to do. I do not yet understand why some latex instructions work well at a moment and not later.

About your answer : I don't think that it is exactly what I want, since I consider my hypercubes in an (additive) category C. So, if I understand well your suggestion, I should embed the diagram $X overset{f}rightarrow Y overset{g}rightarrow Z$ into the diagram
$begin{array}
X & overset{f}rightarrow & Y \
downarrow && downarrow \
Y & overset{g}rightarrow & Z
end{array}$
where the vertical arrows are 0. This square is commutative, but where is the composition $g circ f$? Even if you consider the identity from Y on the top to Y on the bottom, the diagram wouldn't be commutative.

I hope it is clear that I think to all the diagrams, both `I' and hypercubes, as living in an additive category C, and in the embedding I want to preserve compositions and commutativity (indeed, it gives a functor which is a presheaf from I to C).

oc.optimization control - Simplex method for SDP?

[the first part of this answer is similar to Dinakar Muthiah's]

When optimizing a linear function on a convex set, it can always be assumed that the optimal solution lies on an extreme point of the feasible region (if there are several optimal solutions, at least one is an extreme point).

In the case of linear programming, these extreme points are vertices of a polyhedron, with the nice property that

• there are a finite number of vertices


• every vertex admits a simple algebraic description (this is the notion of basis, which is essentially a list of active inequalities)

However, for semidefinite programming, the feasible region, altough convex, typically admits an infinite number of extreme points, for which there is no clear equivalent to the concept of basis.

Note that simplex-type methods (also called active-set methods) can be generalized to quadratic programming (minimization of a convex quadratic function over a polyhedron). On the other hand, I am not aware of any such generalization for quadratically constrained quadratic programming (QCQP, i.e. quadratic programming with convex quadratic constraints) or second-order cone programming (a slight extension of QCQP), two problem classes whose instances can be posed as semidefinite programming problems.

co.combinatorics - Counting graphs whose vertex induced subgraphs are members of a fixed set

Let $C$ be the class of graphs of girth $g$ or more. $C$ can alternatively be characterized as: $G in C$ iff each of $G$'s vertex induced subgraphs on less than $g$ nodes is a forest.

We can generalize this as follows. Let $A$ be a class of graphs. Define $C_A$ as: $G in C_A(k)$ iff all of $G$'s vertex induced subgraphs on less than $k$ nodes are members of $A$.

A natural question is then:

Given a class of graphs $A$ and integer $k$ how many graphs on $n$ vertices are in $C_A(k)$? Or more generally, can one bound the number of graphs on $n$ vertices in $C_A(k)$ given $k$ and $A$?

The particular situation I am interested in is where we allow $k$ to be a function of the number of vertices in the graph. That is let $C_A(f(n))$ (where $f(n) < n$) be defined as: $G in C_A(f(n))$ iff all of $G$'s vertex induced subgraphs on less than $f(n)$ vertices are members of $A$ (where $n$ is the number of vertices in $G$).

My actual question is: Is there a class of graphs $A$ such that the number of graphs in $A$ on $n$ vertices is $2^{O(g(n))}$ but the number of graphs on $n$ vertices in $C_A(c cdot g^{-1}(n))$ is $2^{omega(g(n))}$, for some constant $c$? Stated another way: all vertex induced subgraphs on $n$ vertices of a graph on $g(n)$ vertices in $C_A(c cdot g^{-1}(n))$ are in $A$.

In particular I am interested in the situation where $g(n) = nlog{n}$ and $A$ is hereditary (every vertex induced subgraph of a graph in $A$ is also in $A$). This implies that all vertex induced subgraphs of size $n$ or less of a graph on $nlog{n}$ vertices in $C_A(g^{-1}(n))$ are in $A$.

EDIT: My motivation for this question is that it is related to an open problem on implicit graph representation in: S. Kannan, M. Naor, S. Rudich, Implicit Representation of Graphs, SIAM Journal on Discrete Mathematics 5, 596-603, 1992

gr.group theory - Cohomology analogue for central series of length more than two

This looks like a (slightly) non-additive version of Grothendieck's theory of
"extensions panachées" (SGA 7/I, IX.9.3). There he considers objects (in some
abelian category) $X$ together with a filtation $0subseteq X_1subseteq
X_2subseteq X_3=X$. In the first version he also fixes (just as one does for
extensions) isomorphisms $Prightarrow X_1$, $Qrightarrow X_2/X_1$ and
$Rrightarrow X_3/X_2$. However, in the next version he fixes the isomorphism
class of the two extensions $0rightarrow Prightarrow X_2rightarrow
Qrightarrow0$ and $0rightarrow Qrightarrow X_3/X_1rightarrow Rrightarrow0$
so that if $E$ is an extension of $P$ by $Q$ and $F$ is an extension of $Q$ by
$R$, then the category $mathrm{EXTP}(F,E)$ has as objects filtered objects $X$
as above together with fixed isomorphisms of extensions $Erightarrow X_2$ and
$Frightarrow X_3/X_1$ and whose morphisms are are morphisms of $X$'s preserving
the given structures. The morphisms of $mathrm{EXTP}(F,E)$ are necessarily
isomorphisms so we are dealing with a groupoid. Similarly for objects $A$ and
$B$ $mathrm{EXT}(B,A)$ is the groupoid of extensions of $B$ by $A$.
Grothendieck then shows that $mathrm{EXTP}(F,E)$ is a torsor over
$mathrm{EXT}(R,P)$ (in the category of torsors, Grothendieck had previously
defined this notion). The action on objects of an extension $0rightarrow
Prightarrow Grightarrow Rrightarrow0$ is given by first taking the pullback
of it under the map $X/X_1rightarrow R$ and then using the obtained action by
addition on extensions of $P$ by $F$. To more or less complete the picture,
there is an obstruction to the existence of an object of $mathrm{EXTP}(F,E)$:
We have that $E$ gives an element of $mathrm{Ext}^1(Q,P)$ and $F$ one of
$mathrm{Ext}^1(R,Q)$ and their Yoneda product gives an obstruction in
$mathrm{Ext}^2(P,Q)$.

The case at hand is similar (staying at the case of $n=3$ and with the caveat
that I haven't properly checked everything): We choose fixed isomorphisms with
$K_2$ and a given central extension and with $K_3/K_1$ and another given central
extension (assuming that we have three groups $P$, $Q$ and $R$ as before)
getting a category $mathrm{CEXTP}(F,E)$ of central extensions. We shall shortly
modify it but to motivate that modification it seems a good idea to start with
this. We get as before an action of $mathrm{CEXT}(R,P)$ on
$mathrm{CEXTP}(F,E)$ as we can pull back central extensions just as before. It
turns however that the action is not transitive. In fact we can analyse both the
difference between two elements of $mathrm{CEXTP}(F,E)$ and the obstructions
for the non-emptyness of it by using the Hochschild-Serre spectral sequence. To
make it easier to understand I use a more generic notation. Hence we have a
central extension $1rightarrow Krightarrow Grightarrow G/Krightarrow1$ and
an abelian group $M$ with trivial $G$-action. There is then a succession of two
obstructions for the condition that a given central extension of $M$ by $G/K$
extend to a central extension of $M$ by $G$. The first is $d_2colon
H^2(G/K,M)rightarrow H^2(G/K,H^1(K,M))$, the $d_2$-differential of the H-S
s.s. Now, we always have a map $H^2(G/K,M)rightarrow H^2(G/K,H^1(K,M))$ given
by pushout of $1rightarrow Grightarrow G/Krightarrow1$ along the map
$Krightarrow mathrm{Hom}(K,M)=H^1(K,M)$ given by the action by conjugation of
$K$ on the given central extension of $M$ by $K$ (equivalently this map is given
by the commutator map in that extension). It is easy to compute and identify
$d_2$ but I just claim that it is equal to that map by an appeal to the What Else
Can It Be-principle (which works quite well for the beginnings of spectral
sequences with the usual provisio that the WECIB-principle only works up to a
sign).

This means that we can cut down on the number of obstructions by redefining
$mathrm{CEXTP}(F,E)$. We add as data a group homomorphism $varphicolon
K_3/K_1rightarrowmathrm{Hom}(Q,P)$ that extends $Qrightarrow
mathrm{Hom}(Q,P)$ which describes the conjugation action on $K_2$ and only look
the elements of $mathrm{CEXTP}(F,E)$ for which the action is the given
$varphi$ to form $mathrm{CEXTP}(F,E;varphi)$. Now the action of
$mathrm{CEXT}(R,P)$ on $mathrm{CEXTP}(F,E;varphi)$ should make
$mathrm{CEXTP}(F,E;varphi)$ a
$mathrm{CEXT}(R,P)$-(pseudo)torsor. Furthermore, there is now only a single
obstruction for non-emptyness which is given by $d_3colon H^2(R,M)rightarrow
H^3(P,M)$.

Going to higher lengths there are two ways of proceeding in the original
Grothendieck situation: Either one can look at the the two extensions of one
length lower, one ending with the next to last layer (i.e., $X_{n-1}$) and the
other being $X/X_1$. This reduces the problem directly to the original case
(i.e., we look at filtrations of length $n-2$ on $Q$). One could instead look at
the successive two-step extensions and then look at how adjacent ones build up
three-step extensions and so on. This is essentially an obstruction theory point
of view and quickly becomes quite messy. An interesting thing is however the
following: We saw that in the original situation the obstruction for getting a
three step extension was that $ab=0$ for the Yoneda product of the two twostep
filtrations. If we have a sequence of three twostep extensions whose three step
extensions exist then we have $ab=bc=0$. The obstruction for the existence of
the full fourstep extension is then essentially a Massey product $langle
a,b,crangle$ (defined up to the usual ambiguity). The messiness of such an
iterated approach is well-known, it becomes more and more difficult to keep
track of the ambiguities of higher Massey products. The modern way of handling
that problem is to use an $A_infty$-structure and it is quite possible (maybe
even likely) that such a structure is involved.

If we turn to the current situation and arbitrary $n$ then the first approach
has problems in that the midlayer won't be abelian anymore and I haven't looked
into what one could do. As for the second approach I haven't even looked into
what the higher obstructions would look like (the definition of the first
obstruction on terms of $d_3$ is very asymmetric).

ac.commutative algebra - Neusis constructions

There are various flavors of neusis construction. In the weakest flavor, in addition to having the marked straightedge pass through the pole point, the two marks on it are required to lie, one each, on two specified lines; we might call that tool a line-line neusis. For a line-circle neusis, one mark must lie on a specified line while the other must lie on a specified circle. For a circle-circle neusis, the two marks must lie, one each, on two specified circles. (We view a line as a special case of a circle; so each of these tools is at least as powerful as its predecessors.)

If we allow ourselves a straightedge, a compass, and a line-line neusis, then, as Stillwell tells us, we get precisely the closure of the rationals under complex square roots and cube roots. The Alperin paper that Stillwell mentions is a high-level reference. Here are some more details.

In one direction, consider the line through the pole point that has slope $s$. We can intersect that line with the two specified lines. The distance between the two resulting intersections equals the fixed distance between the two marks on the straightedge just when a certain quartic equation in $s$ holds. And, of course, any quartic can be solved using complex square roots and cube roots.

In the other direction, the compass allows us to bisect any angle and to extract any real square root; so we can take complex square roots. To show that the line-line neusis can take complex cube roots, we need to show two things: that it can trisect any angle and that it can extract any real cube root.

Trisecting first: There is a well-known neusis angle-trisection credited to Archimedes; but that construction uses a line-circle neusis, and hence doesn't help us here. But the Greeks also knew of a trisection using a line-line neusis. Alperin credits that construction to Apollonius, but gives no details. For the details, see either A History of Greek Mathematics, Volume 1: From Thales to Euclid, by Sir Thomas Heath, reprinted by Dover in 1981, pages 236-238. Or see Exercise 10 on page 245 of Michael O'Leary's Revolutions in Geometry, published by Wiley in 2010. (Note that, in this construction, of the four slopes for the neusis straightedge that satisfy the distance requirement, all four are real; one is trivial and should be ignored, while the other three are the three trisectors.)

Now for real cube roots: The Greeks also knew a line-line neusis construction, credited to Nicomedes, for extracting real cube roots. One source for that construction, pointed out by Gerry Myerson, is the article "Constructions using a compass and twice-notched sraightedge", by Arthur Baragar, pages 151-164 in volume 109, number 2 of the American Mathematical Monthly. (In this construction, of the four slopes mentioned above, one is trivial and should be ignored, a second gives the required real cube root, and the remaining two are complex.)

One side remark: The Nicomedes cube-root construction is a bit subtle. But Conway and Guy give a dead-simple line-line neusis construction for the special case of the cube root of 2, on page 195 of The Book of Numbers, published by Springer in 1996.

So the line-line neusis gives us precisely the power to solve quadratic and cubic (and hence also quartic) equations, which makes it equivalent to "conic constructability" or, as the Greeks called it, "solid constructability".

What about the other flavors of neusis construction? Baragar shows that, for either a line-circle neusis or a circle-circle neusis, there are in general six slopes for the line through the pole that have the proper distance relationship -- six, rather than the four of the line-line case. He then gives an explicit example of a line-circle neusis construction in which one of these slopes is real and trivial, three others are real, and the final two are complex. Furthermore, the five nontrivial slopes are the roots of a irreducible quintic equation whose Galois group is all of $S_5$, and which hence cannot be solved with radicals. Thus, the line-circle neusis is a strictly more powerful tool than the line-line neusis.

As an upper bound on the power of these more general neusis constructions, Baragar shows that any point generated by either the line-circle neusis or the circle-circle neusis lies in an extension field of the rationals that can be reached by a tower of fields in which each adjacent pair has index either 2, 3, 5, or 6. So the jump in power over the line-line neusis, where the adjacent-pair indices are either 2 or 3, is not too great.

Baragar's paper closes with some interesting open problems. One problem that he doesn't mention is this: Is the circle-circle neusis strictly more powerful than the line-circle neusis?

correspondence between invariant forms and Lie groups

I don't have a full answer yet. Some notation: let's write $mathrm{Inv}(G)$ for the collection (algebra) of all invariant tensors for $G subseteq GL_n$, and $mathrm{Grp}(I)$ for the group of matrices that leave invariant some collection $I$ of tensors.

Then a necessary condition for $G = mathrm{Grp}(mathrm{Inv}(G))$ is for $G$ to be Zariski-closed in $GL_n$. (Recall that $GL_n$ is a codimension-one Zariski-closed subset of affine $(n^2 +1)$-space, where the first $n^2$ coordinates are the matrix coefficients, and the last one is the inverse determinant; $GL$ is cut out by the condition that the actual determinant times the last coefficient is unity.) Indeed, $mathrm{Grp}(I)$ is Zariski-closed for any $I$, because it is the intersection of $mathrm{Grp}(i)$ over all $iin I$, and fixing a tensor is a Zariski-closed condition, because $GL_n to mathrm{End}(V)$ is polynomial for $V$ and tensor product of the $n$-dimensional representation and its dual.

So this rules out things like the irrational line in the torus (diagonal two-by-two matrices with eigenvalues $exp(x)$ and $exp(pi x)$ as $x$ ranges over the field, and $pi$ is your favorite irrational number).

I think that a sufficient condition is for $G$ to be compact. This is because if you know all the tensor invariants, then you know the full subcategory of representations that are tensor-generated by the defining representations, and in fact all the subrepresentations of these, and if $G$ is compact then this category is equivalent to the full category of representations and knows the group by Tannakian arguments. But this is much too strong --- $SL_n(mathbb{R})$ is not compact, for example.

qa.quantum algebra - What are the correct axioms for a "weakly associative monoidal functor"?

Definitions and the main question

Recall that a category $mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties):

• a functor $otimes: mathcal C times mathcal C to mathcal C$,


• a functor $1: {text{pt}} to mathcal C$,


• a natural transformation $alpha: (Xotimes Y)otimes Z oversetsimto Xotimes(Yotimes Z)$ between functors $mathcal C^{times 3} to mathcal C$ (natural in $X,Y,Z$),


• natural transformations $lambda: 1 otimes X oversetsimto X$, $rho: Xotimes 1 oversetsimto X$ between functors $mathcal C to mathcal C$ (natural in $X$; this uses the canonical isomorphisms of categories ${text{pt}} times mathcal C cong mathcal C cong mathcal C times {text{pt}}$),


• such that $alpha$ satisfies a pentagon,


• and $alpha,lambda,rho$ satisfy some other equations.

I tend to be less interested in the unit laws $lambda,rho$, which is my excuse for knowing less about their technicalities. In my experience, it's the associativity law $alpha$ that can have interesting behavior.

Let $mathcal C,mathcal D$ be monoidal categories. Recall that a functor $F: mathcal C to mathcal D$ is (strong) monoidal if it comes with the following data (two natural transformations, and three properties):

• a natural isomorphism $phi: otimes_{mathcal D} circ (Ftimes F) oversetsimto Fcirc otimes_{mathcal C}$ of functors $mathcal C times mathcal C to mathcal D$,


• a natural isomorphism $varphi: 1_{mathcal D} oversetsimto Fcirc 1_{mathcal C}$ of functors ${text{pt}} to mathcal D$,


• satisfying some properties, the main one being that the two natural transformations $(FX otimes_{mathcal D} FY) otimes_{mathcal D} FZ oversetsimto F(Xotimes_{mathcal C} (Y otimes_{mathcal C}Z))$ that are built from $phi, alpha_{mathcal C}, alpha_{mathcal D}$ agree. This property expresses that the associators in $mathcal C$, $mathcal D$ are "the same" under the functor $F$.

My question is whether there is a (useful) weakening of the axioms for a monoidal functor that expresses the possibility that the associators might disagree.

An example: quasiHopf algebras

Here is my motivating example. Let $A$ be a (unital, associative) algebra (over a field $mathbb K$), and let $Atext{-rep}$ be its category of representations. I.e. objects are pairs $V in text{Vect}_{mathbb K}$ and an algebra homomorphism $pi_V: A to text{End}_{mathbb K}(V)$, and morphisms are $A$-linear maps. Then $Atext{-rep}$ has a faithful functor $Atext{-rep} to text{Vect}_{mathbb K}$ that "forgets" the map $pi$.

Suppose now that $A$ comes equipped with an algebra homomorphism $Delta: A to A otimes_{mathbb K} A$. Then $Atext{-rep}$ has a functor $otimes: Atext{-rep} times Atext{-rep} to Atext{-rep}$, given by $pi_{(Votimes W)} = (pi_V otimes pi_W) circ Delta: A to text{End}(Votimes_{mathbb K}W)$. Just this much data is not enough for $Atext{-rep}$ to be monoidal. (Well, we also need a map $epsilon: A to mathbb K$, but I'm going to drop mention of the unit laws.) Indeed: there might not be an associator.

A situation in which there is an associator on $(Atext{-rep},otimes)$ is as follows. Suppose that there is an invertible element $p in A^{otimes 3}$, such that for each $ain A$, we have
$$ pcdot (Delta otimes text{id})(Delta(a)) = (text{id} otimes Delta)(Delta(a))cdot p $$
and $cdot$ is the multiplication in $A^{otimes 3}$. Then for objects $(X,pi_X), (Y,pi_Y), (Z,pi_Z) in Atext{-rep}$, define:
$$ alpha_{X,Y,Z} = (pi_X otimes pi_Y otimes pi_Z)(p) :
((Xotimes_{Atext{-}{rm rep}} Y) otimes_{Atext{-}{rm rep}} Z) to (X otimes_{Atext{-}{rm rep}} (Y otimes_{Atext{-}{rm rep}} Z)) $$
You can check that it is in fact a isomorphism in $Atext{-rep}$. Moreover, supposing that $p$ satisfies:
$$ (text{id} otimes text{id} otimes Delta)(p) cdot (Delta otimes text{id} otimes text{id})(p) = (1 otimes p) cdot (text{id} otimes Delta otimes text{id})(p) cdot (p otimes 1) $$
where now $cdot$ is the multiplication in $A^{otimes 4}$, then $alpha$ is an honest associator on $Atext{-rep}$.

Then (provided also that $A$ have some sort of "antipode"), the data $(A,Delta,p)$ is a quasiHopf algebra.

Anyway, it's clear from the construction that the forgetful map $text{Forget}: Atext{-rep} to text{Vect}_{mathbb K}$ is a faithful exact functor which is weakly monoidal in the sense that $text{Forget}(X otimes_{Atext{-}{rm rep}} Y) = text{Forget}(X) otimes_{mathbb K} text{Forget}(Y)$ — indeed, this is equality of objects, so perhaps it is "strictly" monoidal — but it is not "monoidal" since it messes with the associators.

Actual motivation

My actual motivation for asking the question above is the understand the Tannaka duality for quasiHopf algebras. In general, we have the following theorem:

Theorem: Let $mathcal C$ be an abelian category and $F: mathcal C to text{FinVect}_{mathbb K}$ a faithful exact functor, where $text{FinVect}_{mathbb K}$ is the category of finite-dimensional vector spaces of $mathbb K$. Then there is a canonical coalgebra $text{End}^{vee}(F)$, and $mathcal C$ is equivalent as an abelian category to the category of finite-dimensional corepresentations of $text{End}^{vee}(F)$.

For details, see A Joyal, R Street, An introduction to Tannaka duality and quantum groups, Category Theory, Lecture Notes in Math, 1991 vol. 1488 pp. 412–492.

The Tannaka philosophy goes on to say that if in addition to the conditions in the theorem, $mathcal C$ is a monoidal category and $F$ is a monoidal functor, then $text{End}^{vee}(F)$ is a bialgebra, and $mathcal C$ is monoidally equivalent to $text{End}^{vee}(F)text{-corep}$. If $mathcal C$ has duals, $text{End}^{vee}(F)$ is a Hopf algebra. If $mathcal C$ has a braiding, then $text{End}^{vee}(F)$ is coquasitriangular. Etc.

My real question, then, is:

What is the statement for Tannaka duality for (co)quasiHopf algebras?

It seems that the standard paper to answer the real question is:
S. Majid, Tannaka-Krein theorems for quasi-Hopf algebras and other results. Contemp. Math. 134 (1992), pp. 219–232.
But I have not been able to find a copy of this paper yet.