sg.symplectic geometry - Hamiltonian circle actions and Lefschetz pencils

I think, that in order to answer this question it is worth to conisder the complex algebraic analog of this question. Namely, suppose we have a $mathbb C^*$ action on a projective manifold $V^n$. Can we find an invariant Lefshetz pencil?

It is sufficient to conisder the case of (complex) surfaces to spot some problems. Namely, the action of $mathbb C^*$ in a neighborhood of an isolated fixed point should be of one of the following $3$ types:
$$(z,w)to (tz,tw), ;; (z,w)to (tz,t^{-1}w),;; (z,w)to (tz,w)$$

where $tin mathbb C^*$ and $z,w$ are local coordinates at the fixed point.

There are not so many examples of $mathbb C^*$ actions on surfaces that have only these three types of fixed points. But here are two examples: first is $mathbb CP^2$ with the action that fixes one point and a separate line. Second is the action on $mathbb CP^1times mathbb CP^1$ that fixes $4$ points. In both cases there is an (obvious) invariant Lefshetz pencil. You can also take the first example and blow up several distinct points on the fixed line in $mathbb CP^2$. Maybe this is the complete list...
Surelly, from all these examples one gets also the symplectic Lefshetz pencil of the kind that you wish to have.

I don't see why there will be much more examples if you will go into symplectic cathegory.

soft question - Most helpful heuristic?

A very interesting heuristic is a principle in complex variables called "Bloch's heuristic principle".

Bloch's principle is about families of analytic functions called "normal". A family F of analytic functions on a domain D is called normal on D if every sequence of functions of F has a subsequence that converges uniformly on compact subsets (either to an analytic function or to infinity). Normal families are very studied for their applications in complex dynamics.

Bloch's principle goes as follows :

A family of analytic functions on a domain D having a property in common is most likely to be normal if there is no non-constant entire function having this property on the whole complex plane.

There are many examples of Bloch's principle. For example, take the property of being bounded : a well known theorem of Montel says that a family of analytic functions on a domain D which is uniformly bounded is necessarily normal on D, and Liouville's theorem says that there is no non-constant entire bounded function.

Or, take the property of omitting two distinct complex values. Again, a theorem of Montel says that a family of analytic functions on a domain D such that each function omits a,b in C, a different than b, is normal on D. The version for the whole complex plane is a well known theorem of Picard, that says that there is no non-constant entire function that omits two distinct complex values.

However, there are many counter-examples to Bloch's principle as it is stated, but it can be transformed into a rigourous theorem that goes like "If a property satisfies these conditions, then bloch's principle is respected".

I wouldn't qualify Bloch's principle as "most helpful", but it is certainly interesting.

Malik

gr.group theory - Orbit structures of conjugacy class set and irreducible representation set under automorphism group

By Brauer's permutation lemma, the permutation characters are always equal, but the representations need not be isomorphic. For instance, the non-abelian group of order 27 and exponent 9 provides an example. One condition for an equivalence for subgroups of the automorphism group is given in Isaacs's Character Theory textbook as theorem 13.24 on page 230–231:

If S is a solvable subgroup of Aut(G), and gcd(|S|,|G|)=1, then the permutation representations of S on Irr(G) and Cl(G) are isomorphic.

This will rarely directly answer your question as Aut(G) and G usually have common prime divisors, but perhaps the ideas will be useful to you. In particular, it describes a strengthening of your #2 which implies #1.

Let me know if you would like GAP code to verify the order 27 example. The action on classes has orbits of sizes 1, 1, 1, 2, 6 and the action on the irreducibles has orbits of sizes 1, 2, 2, 3, 3.

GAP code to check permutation isomorphism:

OnCharactersByGroupAutomorphism := function( pnt, act )
  return Character( UnderlyingCharacterTable( pnt ),
  pnt{FusionConjugacyClasses(act^-1)} );
end;;
OnCBGA := OnCharactersByGroupAutomorphism;;

g := ExtraspecialGroup(27,9);;
a := AutomorphismGroup(g);;
gensIrr := List( GeneratorsOfGroup(a), f ->
  PermListList( Irr(g), List( Irr(g), chi -> OnCBGA( chi, f ) ) ) );
gensCcl := List( GeneratorsOfGroup(a), f ->
  PermList( FusionConjugacyClasses(f) ) );
# perm iso?
fail <> RepresentativeAction( SymmetricGroup( NrConjugacyClasses( g ) ),
  gensCcl, gensIrr, OnTuples );

Some of what you asked for might be more along the lines of asking if the permutation groups generated by gensIrr and gensCcl are conjugate, so I chose an example where even the images are not conjugate. The example given below of G=2×2×2 is the smallest if you only want strict permutation (non-)isomorphism.

Hopf algebra structure on $prod_n A^{otimes n}$ for an algebra $A$

For a finite dimensional $k$-algebra $A$, each $A^{otimes n}, n geq 0$ is a $k$-algebra ($A^0 = k $). Let $T= prod_{n geq 0} A^{otimes n}$. This is a $k$-alegbra with unit $(1,1,dots)$ and multiplication is component-wise. Let $Delta^{(n)} : A^{otimes n} to T otimes T$ be the deconcatenation map
$$ Delta^{(n)}(a_1 otimes dots otimes a_n) = sum_{i=0}^n (a_1 otimes dots otimes a_i) otimes (a_{i+1} otimes dots otimes a_n ). $$

I want to extend these $Delta^{(n)}$ to a comultiplication $Delta : T to T otimes T $. This does not seem to work in a straightforward way because if $t = ( t_0, t_1, dots ) in T, $ then $sum_n Delta^{(n)}(t_n)$ may not be a finite sum of pure tensors in $T otimes T$ (I have not shown this sum can be infinite, but suspect it can be).

Is there a way to make $T$ into a Hopf algebra so that $Delta(t) = Delta^{(n)}(t)$ when $t_i=0$ for $i ne n$? If not, is there an algebra similar to $T$ where this does work?

Is there a standard way to complete the tensor product and instead get a map $Delta$ from $T$ to the completion? Does this give rise to a genuine Hopf algebra or some generalization of Hopf alegbras?

ag.algebraic geometry - What are the Benefits of Using Algebraic Spaces over Schemes?

Here is one intuitive way to think about it:

A scheme is something which is Zariski-locally affine, whereas an algebraic space is something which is etale-locally affine.

One way to make this precise: a scheme is the coequalizer of a Zariski-open equivalence relation, whereas an algebraic space is the coequalizer of an etale equivalence relation.

Another way is to say that, as a functor on rings, a scheme has a Zariski-open covering by affine functors, whereas an algebraic space has an etale covering by affine functors (thus bypassing reference to locally ringed spaces, regarding your second question).

Why do we care? A priori, if you want to work in the etale topology anyway, why not fix the definition of scheme to say "etale-locally affine" instead of "Zariski-locally affine". This is just one motivation for studying algebraic spaces, which you can read more about in Champs algébriques.

(Edit: For "a fortiori" reasons to study algebraic spaces, I'll just say read the other answers :)

lo.logic - Solutions to the Continuum Hypothesis

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain there.

You can find the slides here, under "Recent results about the Continuum Hypothesis, after Woodin". (In true Bourbaki fashion, I heard that the talk was not well received.)

Roughly, Woodin's approach shows that in a sense, the theory of $H(omega_2)$ decided by the usual set of axioms, ZFC and large cardinals, can be "finitely completed" in a way that would make it reasonable to expect to settle all its properties. However, any such completion implies the negation of CH.

It is a conditional result, depending on a highly non-trivial problem, the $Omega$-conjecture. If true, this conjecture gives us that Cohen's technique of forcing is in a sense the only method (in the presence of large cardinals) that is required to established consistency. (The precise statement is more technical.)

$H(omega_2)$, that Dehornoy calls $H_2$, is the structure obtained by considering only those sets $X$ such that $Xcupbigcup Xcupbigcupbigcup Xcupdots$ has size strictly less than $aleph_2$, the second uncountable cardinal.

Replacing $aleph_2$ with $aleph_1$, we have $H(omega_1)$, whose theory is completely settled in a sense, in the presence of large cardinals. If nothing else, one can think of Woodin's approach as trying to build an analogy with this situation, but "one level up."

Whether or not one considers that settling the $Omega$-conjecture in a positive fashion actually refutes CH in some sense, is a delicate matter. In any case (and I was happy to see that Dehornoy emphasizes this), Woodin's approach gives strength to the position that the question of CH is meaningful (as opposed to simply saying that, since it is independent, there is nothing to decide).

(2) There is another approach to the problem, also pioneered by Hugh Woodin. It is the matter of "conditional absoluteness." CH is a $Sigma^2_1$ statement. Roughly, this means that it has the form: "There is a set of reals such that $phi$", where $phi$ can be described by quantifying only over reals and natural numbers. In the presence of large cardinals, Woodin proved the following remarkable property: If $A$ is a $Sigma^2_1$ statement, and we can force $A$, then $A$ holds in any model of CH obtained by forcing.

Recall that forcing is essentially the only tool we have to establish consistency of statements. Also, there is a "trivial" forcing that does not do anything, so the result is essentially saying that any statement of the same complexity as CH, if it is consistent (with large cardinals), then it is actually a consequence of CH.

This would seem a highly desirable ''maximality'' property that would make CH a good candidate to be adopted.

However, recent results (by Aspero, Larson, and Moore) suggest that $Sigma^2_1$ is close to being the highest complexity for which a result of this kind holds, which perhaps weakens the argument for CH that one could do based on Hugh's result.

A good presentation of this theorem is available in Larson's book "The stationary tower. Notes on a Course by W. Hugh Woodin." Unfortunately, the book is technical.

(3) Foreman's approach is perhaps the strongest opponent to the approach suggested by Woodin in (1). Again, it is based in the technique of forcing, now looking at small cardinal analogues of large cardinal properties.

Many large cardinal properties are expressed in terms of the existence of elementary embeddings of the universe of sets. This embeddings tend to be "based" at cardinals much much larger than the size of the reals. With forcing, one can produce such embeddings "based" at the size of the reals, or nearby. Analyzing a large class of such forcing notions, Foreman shows that they must imply CH. If one where to adopt the consequences of performing these forcing constructions as additional axioms one would then be required to also adopt CH.

---

I had to cut my answer short last time. I would like now to say a few details about a particular approach.

(4) Forcing axioms imply that $2^{aleph_0}=aleph_2$, and (it may be argued) strongly suggest that this should be the right answer.

Now, before I add anything, note that Woodin's approach (1) uses forcing axioms to prove that there are "finite completions" of the theory of $H(omega_2)$ (and the reals have $aleph_2$). However, this does not mean that all such completions would be compatible in any particular sense, or that all would decide the size of the reals. What Woodin proves is that all completions negate CH, and forcing axioms show that there is at least one such completion.

I believe there has been some explanation of forcing axioms in the answer to the related question on GCH. Briefly, the intuition is this: ZFC seems to capture the basic properties of the universe of sets, but fails to account for its width and its height. (What one means by this is: how big should power sets be, and how many ordinals there are.)

Our current understanding suggests that the universe should indeed be very tall, meaning there should be many many large cardinals. As Joel indicated, there was originally some hope that large cardinals would determine the size of the reals, but just about immediately after forcing was introduced, it was proved that this was not the case. (Technically, small forcing preserves large cardinals.)

However, large cardinals settle many questions about the structure of the reals (all first order, or projective statements, in fact). CH, however, is "just" beyond what large cardinals can settle. One could say that, as far as large cardinals are concerned, CH is true. What I mean is that, in the presence of large cardinals, any definable set of reals (for any reasonable notion of definability) is either countable or contains a perfect subset. However, this may simply mean that there is certain intrinsic non-canonicity in the sets of reals that would disprove CH, if this is the case.

(A word of caution is in order here, and there are candidates for large cardinal axioms [presented by Hugh Woodin in his work on suitable extender sequences] for which preservation under small forcing is not clear. Perhaps the solution to CH will actually come, unexpectedly, from studying these cardinals. But this is too speculative at the moment.)

I have avoided above saying much about forcing. It is a massive machinery, and any short description is bound to be very inaccurate, so I'll be more than brief.

An ordinary algebraic structure (a group, he universe of sets) can be seen as a bi-valued model. Just as well, one can define, for any complete Boolean algebra ${mathbb B}$, the notion of a structure being ${mathbb B}$-vaued. If you wish, "fuzzy set theory" is an approximation to this, as are many of the ways we model the world by using a probability density to decide the likelihood of events. For any complete Boolean algebra ${mathbb B}$, we can define a ${mathbb B}$-valued model $V^{mathbb B}$ of set theory. In it, rather than having for sets $x$ and $y$ that either $xin y$ or it doesn't, we assign to the statement $xin y$ a value $[xin y]in{mathbb B}$. The way the construction is performed, $[phi]=1$ for each axiom $phi$ of ZFC. Also, for each element $x$ of the actual universe of sets, there is a copy $check x$ in the ${mathbb B}$-valued model, so that the universe $V$ is a submodel of $V^{mathbb B}$. If it happens that for some statement $psi$ we have $[psi]>0$, we have established that $psi$ is consistent with ZFC. By carefully choosing ${mathbb B}$, we can do this for many $psi$. This is the technique of forcing, and one can add many wrinkles to the approach just outlined. One refers to ${mathbb B}$ as a forcing notion.

Now, the intuition that the universe should be very fat is harder to capture than the idea of largeness of the ordinals. One way of expressing it is that the universe is somehow "saturated": If the existence of some object is consistent in some sense, then in fact such object should exist. Formalizing this, one is led to forcing axioms. A typical forcing axiom says that relatively simple (under some measure of complexity) statements that can be shown consistent using the technique of forcing via a Boolean algebra ${mathbb B}$ that is not too pathological, should in fact hold.

The seminal Martin's Maximum paper of Foreman-Magidor-Shelah identified the most generous notion of "not too pathological", it corresponds to the class of "stationary set preserving" forcing notions. The corresponding forcing axiom is Martin's Maximum, MM. In that paper, it was shown that MM implies that the size of the reals is $aleph_2$.

The hypothesis of MM has been significantly weakened, through a series of results by different people, culminating in the Mapping Reflection Principle paper of Justin Moore. Besides this line of work, many natural consequences of forcing axioms (commonly termed reflection principles) have been identified, and shown to be independent of one another. Remarkably, just about all these principles either imply that the size of the reals is $aleph_2$, or give $aleph_2$ as an upper bound.

Even if one finds that forcing axioms are too blunt a way of capturing the intuition of "the universe is wide", many of its consequences are considered very natural. (For example, the singular cardinal hypothesis, but this is another story.) Just as most of the set theoretic community now understands that large cardinals are part of what we accept about the universe of sets (and therefore, so is determinacy of reasonably definable sets of reals, and its consequences such us the perfect set property), it is perhaps not completely off the mark to expect that as our understanding of reflection principles grow, we will adopt them (or a reasonable variant) as the right way of formulating "wideness". Once/if that happens, the size of the reals will be taken as $aleph_2$ and therefore CH will be settled as false.

The point here is that this would be a solution to the problem of CH that does not attack CH directly. Rather, it turns out that the negation of CH is a common consequence of many principles that it may be reasonable to adapt in light of the naturalness of some of their best known consequences, and of their intrinsic motivation coming from the "wide universe" picture.

(Apologies for the long post.)

---

Edit, Nov. 22/10: I have recently learned about Woodin's "Ultimate L" which, essentially, advances a view that theories are "equally good" if they are bi-interpretable, and identifies a theory ("ultimate L") that, modulo large cardinals, would work as a universal theory from which to interpret all extensions. This theory postulates an $L$-like structure for the universe and in particular implies CH, see this answer. But, again, the theory is not advocated on grounds that it ought to be true, whatever this means, but rather, that it is "richest" possible in that it allows us to interpret all possible "natural" extensions of ZFC. In particular, under this approach, only large cardinals are relevant if we want to strengthen the theory, while "width" considerations, such as those supporting forcing axioms, are no longer relevant.

Since the approach I numbered (1) above implies the negation of CH, I feel I should add that one of the main reasons for it being advanced originally depended on the fact that the set of $Omega$-validities can be defined "locally", at the level of $H({mathfrak c}^+)$, at least if the $Omega$-conjecture holds.

However, recent results of Sargsyan uncovered a mistake in the argument giving this local definability. From what I understand, Woodin feels that this weakens the case he was making for not-CH significantly.

Added link: slides of a 2010 lecture by Woodin on "Ultimate L."

ag.algebraic geometry - Ample line bundles, sections, morphisms to projective space

1. Are there simple examples (say on a curve or surface) of line bundles that are globally generated but not ample, of ample line bundles with no sections, of ample line bundles that are globally generated but not very ample, and of very ample line bundles with higher cohomology?

On a curve of genus $g$, a general divisor of degree $d le g-1$ has no sections. Of course, if $d>0$ then it is ample.

$K_X$ on a hyperelliptic curve is globally generated but not very ample.

Look at $L=mathcal O(1)$ on a plane curve of genus $d$. Then from
$$ 0to mathcal O_{mathbb P^2}(1-d) to mathcal O_{mathbb P^2}(1) to mathcal O_C(1)to 0$$

you see that $H^1(mathcal O_C(1))=H^2(mathcal O_{mathbb P^2}(1-d))$ which is dual to $H^0(mathcal O_{mathbb P^2}(d-4))$. So that's nonzero for $ge4$.

2. Given an ample line bundle $L$, what is the minimal number $k$ so that I can be sure $L^k$ has sections, is globally generated, is very ample? Is $k$ related to the dimension of $X$?

Again, just look at the divisor of a degree 1 on a curve of genus $g$. You need $kge g$, so you see that there is no bound in terms of the dimension.

It turns out that a better right question to ask is about the adjoint line bundles $omega_Xotimes L^k$ ($K_X+kL$ written additively). Then the basic guiding conjecture is by Fujita, and which says that for $kge dim X+1$ the sheaf is globally generated, and for $kge dim X+2$ it is very ample. This is proved for $dim X=2$, proved with slightly worse bounds for $dim X=3$. For higher dimensions the best result is due to Angehrn-Siu who gave a quadratic bound on $k$ instead of linear. There are some small improvements for some special cases.

3. If $L$ is very ample, I can use it to embed $X$ into some projective space. Then by projecting from points off of $Xsubset mathbb P^N$, I can eventually get a finite morphism $Xto mathbb{P}^d$, where $d$ is the dimension of $X$. But what if I just know that $L$ is ample and globally generated? Can I also use it to get such a finite morphism to $mathbb P^d$?

But of course $L$ gives a morphism $f$, and it follows that $f$ is finite: $f$ contacts no curve so $f$ is quasifinite, and $f$ is projective (since $X$ was assumed to be projective). And quasifinite + proper = finite.

pr.probability - Estimating the mean of a truncated gaussian curve

OK, let me fully address the question since there is no easy way out. The
normal approach is to maximize the "likelihood" of the data under the
parameter. The key question here is how to define likelihood for a
mixed distribution. Let's use the standard approach as our guide.

Parameter estimation is usually based on the idea that we want to
choose parameters that make our data "the most likely." For a discrete
probability distribution, we interpret this to mean that our data is
the most probable. But this breaks down in the case of continuous
probability distributions, where, no matter our choice of parameters,
our data has probability zero.

Statisticians thus replace the probability with the probability
density for continuous distributions. Here is the justification for
this. Instead of actually having a set of numbers drawn from the
probability distribution, you have a highly accurate
measurement---say, your sequence ${x_i}$ for $i = 1,dots,n$ tells
you that the true value of the (still unknown) sequence ${g_i}$ satisfies $|x_i -
g_i| < varepsilon$ for all $i$. When $varepsilon$ is sufficiently
small, the replacement
$$
mathbb{P}(|x_i - g_i|) < varepsilon )approx varepsilon p_{g}(x_i)
$$
is very accurate, where $p_g$ is the pdf of $g_i$. Assuming that
your sequence is iid, we are led to the approximation
$$
mathbb{P}(|x_i - g_i| < varepsilon text{ for all } i)
approx varepsilon^n prod_{i=1}^n p_g(x_i).
$$
We thus choose the pdf from our family which maximizes the right
hand side of the above equation, reproducing the standard maximum
likelihood method.

Now the question is, what do we do with mixed distributions? When
there is a mass at a point $x_i$, that is $mathbb{P}(x_i=g_i) > 0$,
our first approximation is incorrect; for very small $varepsilon$, we
have the approximation
$$ mathbb{P}(|x_i - g_i| < varepsilon) approx mathbb{P}(x_i = g_i)
$$
If we let $mathcal{N}$ be the index set of the "massless" samples, we can approximate
the probability of our data as
$$ mathbb{P}(|x_i - g_i| < varepsilon) approx varepsilon^n
prod_{i in mathcal{N}} p_g(x_i) prod_{i notin mathcal{N}} mathbb{P}(x_i = g_i).
$$
where $n$ is the number of elements in $mathcal{N}$. That is, we can reasonably define our maximum likelihood estimate
for a parameter $m$ as the value of the parameter that maximizes
$$
prod_{i in mathcal{N}} p_g(x_i) prod_{i notin mathcal{N}} mathbb{P}(x_i = g_i).
$$

In your case, it is fairly simple to write down the value of the likelihood function above. First, note that
$$mathbb{P}(x=0) = frac{1}{sqrt{2pi}} int_{-infty}^{-m}
e^{-x^2/2}dx.$$
For $x>0$, you have the standard Gaussian pdf
$p_g(x) = tfrac{1}{sqrt{2pi}} e^{-(x-m)^2/2}$.

I won't do any more here; suffice it to say that the standard approach
to maximizing the likelihood involves taking the logarithm of the likelihood function
and setting its derivative to zero. You will probably get a
transcendental equation that you will need to solve numerically.

lo.logic - Solutions to the Continuum Hypothesis

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain there.

You can find the slides here, under "Recent results about the Continuum Hypothesis, after Woodin". (In true Bourbaki fashion, I heard that the talk was not well received.)

Roughly, Woodin's approach shows that in a sense, the theory of $H(omega_2)$ decided by the usual set of axioms, ZFC and large cardinals, can be "finitely completed" in a way that would make it reasonable to expect to settle all its properties. However, any such completion implies the negation of CH.

It is a conditional result, depending on a highly non-trivial problem, the $Omega$-conjecture. If true, this conjecture gives us that Cohen's technique of forcing is in a sense the only method (in the presence of large cardinals) that is required to established consistency. (The precise statement is more technical.)

$H(omega_2)$, that Dehornoy calls $H_2$, is the structure obtained by considering only those sets $X$ such that $Xcupbigcup Xcupbigcupbigcup Xcupdots$ has size strictly less than $aleph_2$, the second uncountable cardinal.

Replacing $aleph_2$ with $aleph_1$, we have $H(omega_1)$, whose theory is completely settled in a sense, in the presence of large cardinals. If nothing else, one can think of Woodin's approach as trying to build an analogy with this situation, but "one level up."

Whether or not one considers that settling the $Omega$-conjecture in a positive fashion actually refutes CH in some sense, is a delicate matter. In any case (and I was happy to see that Dehornoy emphasizes this), Woodin's approach gives strength to the position that the question of CH is meaningful (as opposed to simply saying that, since it is independent, there is nothing to decide).

(2) There is another approach to the problem, also pioneered by Hugh Woodin. It is the matter of "conditional absoluteness." CH is a $Sigma^2_1$ statement. Roughly, this means that it has the form: "There is a set of reals such that $phi$", where $phi$ can be described by quantifying only over reals and natural numbers. In the presence of large cardinals, Woodin proved the following remarkable property: If $A$ is a $Sigma^2_1$ statement, and we can force $A$, then $A$ holds in any model of CH obtained by forcing.

Recall that forcing is essentially the only tool we have to establish consistency of statements. Also, there is a "trivial" forcing that does not do anything, so the result is essentially saying that any statement of the same complexity as CH, if it is consistent (with large cardinals), then it is actually a consequence of CH.

This would seem a highly desirable ''maximality'' property that would make CH a good candidate to be adopted.

However, recent results (by Aspero, Larson, and Moore) suggest that $Sigma^2_1$ is close to being the highest complexity for which a result of this kind holds, which perhaps weakens the argument for CH that one could do based on Hugh's result.

A good presentation of this theorem is available in Larson's book "The stationary tower. Notes on a Course by W. Hugh Woodin." Unfortunately, the book is technical.

(3) Foreman's approach is perhaps the strongest opponent to the approach suggested by Woodin in (1). Again, it is based in the technique of forcing, now looking at small cardinal analogues of large cardinal properties.

Many large cardinal properties are expressed in terms of the existence of elementary embeddings of the universe of sets. This embeddings tend to be "based" at cardinals much much larger than the size of the reals. With forcing, one can produce such embeddings "based" at the size of the reals, or nearby. Analyzing a large class of such forcing notions, Foreman shows that they must imply CH. If one where to adopt the consequences of performing these forcing constructions as additional axioms one would then be required to also adopt CH.

---

I had to cut my answer short last time. I would like now to say a few details about a particular approach.

(4) Forcing axioms imply that $2^{aleph_0}=aleph_2$, and (it may be argued) strongly suggest that this should be the right answer.

Now, before I add anything, note that Woodin's approach (1) uses forcing axioms to prove that there are "finite completions" of the theory of $H(omega_2)$ (and the reals have $aleph_2$). However, this does not mean that all such completions would be compatible in any particular sense, or that all would decide the size of the reals. What Woodin proves is that all completions negate CH, and forcing axioms show that there is at least one such completion.

I believe there has been some explanation of forcing axioms in the answer to the related question on GCH. Briefly, the intuition is this: ZFC seems to capture the basic properties of the universe of sets, but fails to account for its width and its height. (What one means by this is: how big should power sets be, and how many ordinals there are.)

Our current understanding suggests that the universe should indeed be very tall, meaning there should be many many large cardinals. As Joel indicated, there was originally some hope that large cardinals would determine the size of the reals, but just about immediately after forcing was introduced, it was proved that this was not the case. (Technically, small forcing preserves large cardinals.)

However, large cardinals settle many questions about the structure of the reals (all first order, or projective statements, in fact). CH, however, is "just" beyond what large cardinals can settle. One could say that, as far as large cardinals are concerned, CH is true. What I mean is that, in the presence of large cardinals, any definable set of reals (for any reasonable notion of definability) is either countable or contains a perfect subset. However, this may simply mean that there is certain intrinsic non-canonicity in the sets of reals that would disprove CH, if this is the case.

(A word of caution is in order here, and there are candidates for large cardinal axioms [presented by Hugh Woodin in his work on suitable extender sequences] for which preservation under small forcing is not clear. Perhaps the solution to CH will actually come, unexpectedly, from studying these cardinals. But this is too speculative at the moment.)

I have avoided above saying much about forcing. It is a massive machinery, and any short description is bound to be very inaccurate, so I'll be more than brief.

An ordinary algebraic structure (a group, he universe of sets) can be seen as a bi-valued model. Just as well, one can define, for any complete Boolean algebra ${mathbb B}$, the notion of a structure being ${mathbb B}$-vaued. If you wish, "fuzzy set theory" is an approximation to this, as are many of the ways we model the world by using a probability density to decide the likelihood of events. For any complete Boolean algebra ${mathbb B}$, we can define a ${mathbb B}$-valued model $V^{mathbb B}$ of set theory. In it, rather than having for sets $x$ and $y$ that either $xin y$ or it doesn't, we assign to the statement $xin y$ a value $[xin y]in{mathbb B}$. The way the construction is performed, $[phi]=1$ for each axiom $phi$ of ZFC. Also, for each element $x$ of the actual universe of sets, there is a copy $check x$ in the ${mathbb B}$-valued model, so that the universe $V$ is a submodel of $V^{mathbb B}$. If it happens that for some statement $psi$ we have $[psi]>0$, we have established that $psi$ is consistent with ZFC. By carefully choosing ${mathbb B}$, we can do this for many $psi$. This is the technique of forcing, and one can add many wrinkles to the approach just outlined. One refers to ${mathbb B}$ as a forcing notion.

Now, the intuition that the universe should be very fat is harder to capture than the idea of largeness of the ordinals. One way of expressing it is that the universe is somehow "saturated": If the existence of some object is consistent in some sense, then in fact such object should exist. Formalizing this, one is led to forcing axioms. A typical forcing axiom says that relatively simple (under some measure of complexity) statements that can be shown consistent using the technique of forcing via a Boolean algebra ${mathbb B}$ that is not too pathological, should in fact hold.

The seminal Martin's Maximum paper of Foreman-Magidor-Shelah identified the most generous notion of "not too pathological", it corresponds to the class of "stationary set preserving" forcing notions. The corresponding forcing axiom is Martin's Maximum, MM. In that paper, it was shown that MM implies that the size of the reals is $aleph_2$.

The hypothesis of MM has been significantly weakened, through a series of results by different people, culminating in the Mapping Reflection Principle paper of Justin Moore. Besides this line of work, many natural consequences of forcing axioms (commonly termed reflection principles) have been identified, and shown to be independent of one another. Remarkably, just about all these principles either imply that the size of the reals is $aleph_2$, or give $aleph_2$ as an upper bound.

Even if one finds that forcing axioms are too blunt a way of capturing the intuition of "the universe is wide", many of its consequences are considered very natural. (For example, the singular cardinal hypothesis, but this is another story.) Just as most of the set theoretic community now understands that large cardinals are part of what we accept about the universe of sets (and therefore, so is determinacy of reasonably definable sets of reals, and its consequences such us the perfect set property), it is perhaps not completely off the mark to expect that as our understanding of reflection principles grow, we will adopt them (or a reasonable variant) as the right way of formulating "wideness". Once/if that happens, the size of the reals will be taken as $aleph_2$ and therefore CH will be settled as false.

The point here is that this would be a solution to the problem of CH that does not attack CH directly. Rather, it turns out that the negation of CH is a common consequence of many principles that it may be reasonable to adapt in light of the naturalness of some of their best known consequences, and of their intrinsic motivation coming from the "wide universe" picture.

(Apologies for the long post.)

---

Edit, Nov. 22/10: I have recently learned about Woodin's "Ultimate L" which, essentially, advances a view that theories are "equally good" if they are bi-interpretable, and identifies a theory ("ultimate L") that, modulo large cardinals, would work as a universal theory from which to interpret all extensions. This theory postulates an $L$-like structure for the universe and in particular implies CH, see this answer. But, again, the theory is not advocated on grounds that it ought to be true, whatever this means, but rather, that it is "richest" possible in that it allows us to interpret all possible "natural" extensions of ZFC. In particular, under this approach, only large cardinals are relevant if we want to strengthen the theory, while "width" considerations, such as those supporting forcing axioms, are no longer relevant.

Since the approach I numbered (1) above implies the negation of CH, I feel I should add that one of the main reasons for it being advanced originally depended on the fact that the set of $Omega$-validities can be defined "locally", at the level of $H({mathfrak c}^+)$, at least if the $Omega$-conjecture holds.

However, recent results of Sargsyan uncovered a mistake in the argument giving this local definability. From what I understand, Woodin feels that this weakens the case he was making for not-CH significantly.

Added link: slides of a 2010 lecture by Woodin on "Ultimate L."

von neumann algebras - Is there a finite-index finite-depth II$_1$ subfactor which is more than $7$-super-transitive?

Background: See Noah and Emily's posts on subfactors and planar algebras on the Secret Blogging Seminar.

There are plenty of examples of $3$-super-transitive (3-ST) subfactors; Haagerup, $S_4 < S_5$, and others. There's exactly one known example of a $5$-ST subfactor, the Haagerup-Asaeda subfactor, and one $7$-ST subfactor, the extended Haagerup subfactor.

Below index $4$ there are the $A_n$ and $D_n$ families, which are arbitrarily super-transitive. Ignore those; I'm just interested above index $4$.

Is there anything that's even more super-transitive?

What is the universal enveloping algebra?

As you say, given a symmetric monoidal category $mathcal C$ enriched in abelian groups, the words "Lie algebra object in $mathcal C$" and "associative algebra object in $mathcal C$" make sense. (Actually, the latter does not depend on the symmetric structure nor the ab-gp enrichment.) In particular, there are natural categories $text{LieAlg}_{mathcal C}$ and $text{AssocAlg}_{mathcal C}$ — it makes sense to say whether an arrow in $mathcal C$ between Lie/associative algebra objects is a homomorphism — and there is a natural "forgetful" functor from associative algebra objects to Lie algebra objects. If this functor has a left adjoint, said adjoint deserves to be called "free" or "universal enveloping" (but see below).

Of course, you are not guaranteed such an adjoint. For example, in the category of finite-dimensional vector spaces you cannot build (most) UEAs. You can see this very explicitly: working over characteristic $0$, the Lie algebra $mathfrak{sl}(2)$ acts faithfully and transitively on representations of arbitrary dimension, and so $U(mathfrak{sl}(2))$ cannot be finite-dimensional.

The minimum extra structure that I know of to guarantee the existence of a left-adjoint to $text{Forget}: text{AssocAlg}_{mathcal C} to text{LieAlg}_{mathcal C}$ is:

1. Existence of arbitrary countable direct sums in $mathcal C$.


2. Existence of cokernels in $mathcal C$.

If you have these, then you can do the usual construction to define $Umathfrak g$.

---

If you are working in a category in which all hom sets are vector spaces over $mathbb Q$, then you can also define $Umathfrak g$ as a deformation of the symmetric algebra $Smathfrak g$, provided this symmetric algebra exists. Namely, pretend for a moment that our category is just the usual category of $mathbb K$-vector spaces for $mathbb K$ a field of characteristic $0$. Then there is a "symmetrization" map $Smathfrak g to Umathfrak g$ given on monomials by $x_1cdots x_n mapsto frac1{n!} sum_{sigma in S_n} x_{sigma(1)}cdots x_{sigma(n)}$, where $S_n$ is the symmetric group in $n$ letters. This is a (filtered) vector space isomorphism (and also a coalgebra isomorphism, and also a $mathfrak g$-module isomorphism), and so you can use it to pull back the algebra structure on $Umathfrak g$ to one on $Smathfrak g$, which you should think of as some sort of "star product".

So do this in $mathbb K$-vector spaces, and then interpret the formulas on $mathcal C$.
For details, and in particular for an explicit formula for the star product in terms of the usual monomial basis on $Smathfrak g$, see:

• Deligne, Pierre; Morgan, John W.
  Notes on supersymmetry (following Joseph Bernstein). Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 41--97, Amer. Math. Soc., Providence, RI, 1999. MR1701597

But I see no conditions weaker than 1–2 above to guarantee the existence of the symmetric algebra.

---

Finally, I should mention that in general, even if $text{Forget}$ has a left adjoint $U$, it does not necessarily deserve to be called the "universal enveloping algebra". Namely, simply by being an adjoint, there is a canonical Lie algebra map $mathfrak g to Umathfrak g$. For $Umathfrak g$ to "envelop" $mathfrak g$, this map should be a monomorphism in $mathcal C$.

The following example is due to:

• Cohn, P. M.
  A remark on the Birkhoff-Witt theorem.
  J. London Math. Soc. 38 1963 197--203. MR0148717

Let $mathbb K$ be a field of characteristic $p neq 0$, and consider the free associative (noncommutative) algebra $mathbb K langle x,yrangle$. Then $Lambda_p(x,y) overset{rm def}= (x+y)^p - x^p - y^p$ is a non-zero Lie polynomial — it is a sum of compositions of brackets. For example, $Lambda_2(x,y) = [x,y]$ and $Lambda_3(x,y) = [x,[x,y]] + [y,[y,x]]$.

Let $R = mathbb K[alpha,beta,gamma]/(0 = alpha^p = beta^p = gamma^p)$; it is a commutative ring. Let $mathcal C = Rtext{-mod}$ be the category of $R$-modules, with the usual symmetric tensor structure $otimes_R$. Let $mathfrak f_3$ be the free Lie algebra in $mathcal C$, with the generators $x,y,z$, and let $mathfrak g = mathfrak f_3 / (alpha x = beta y + gamma z)$.

Then $Lambda_p(beta y,gamma z)$ is non-zero in $mathfrak g$, but is $0$ in $Umathfrak g$. Hence, internal to $mathcal C$, $mathfrak g$ does not embed into its universal enveloping algebra. (Of course, it does if we were just working over $mathbb K$, as then the original PBW proof applies. And we always have an embedding in characteristic $0$, as there we can define $Umathfrak g$ as a deformation of $Smathfrak g$.)

simplicial stuff - What are the endofunctors on the simplex category?

To carry Charles' train of thought further:

By an 'interval' let us mean a finite ordered set with at least two elements; let $Int$ be the category of intervals, where a morphism is a monotone map preserving both endpoints.

The simplicial set $Delta^1$ can be viewed as a simplicial interval. That is, this functor $Delta^{op}rightarrow Set$ factors through the forgetful functor from $Int$ to $Set$. In fact, the resulting functor $Delta^{op}rightarrow Int$ is an equivalence of categories.

This extra structure (ordering and endpoints) on $Delta^1$ is inherited by Charles' $K_1=F^*Delta^1$; it, too, is a simplicial interval.

There aren't that many things that a simplicial interval can be. Its realization must be a compact polytope with a linear order relation that is closed. That makes it at most one-dimensional, and makes each component of it either a point or a closed interval. Simplicially each of these components can be either a $0$-simplex, or a $1$-simplex with its vertices ordered one way, or a $1$-simplex with its vertices ordered the other way, or two or more $1$-simplices each ordered one way or the other and stuck together end to end.

The three simplest things that a simplicial interval can be are: two points, a forward $Delta^1$, and a backward $Delta^1$. These arise as $F^*Delta^1$ for three examples of functors $F:Deltarightarrow Delta$, the only examples that satisfy $F([0])=[0]$, namely the constant functor $[0]$, the identity, and "op".

It's clear that any functor with $F([0])=[n]$ has the form $F_0coproddotscoprod F_n$ where $F_i[0]=[0]$ for each $i$. This means that the corresponding simplicial interval can be made by sticking together those which correspond to the $F_i$. For example, the 'shift' functor mentioned in the question is $idcoprod [0]$; Reid mentioned $idcoprod id$ and $opcoprod id$. These correspond respectively to: a $1$-simplex with an extra point on the right, two $1$-simplices end to end, and two $1$-simplices end to end one of which is backward. As another example, the constant functor $[n]$ corresponds to $n+1$ copies of (two points) stuck together end to end, or $n+2$ points.

In short, every functor $F:Deltarightarrow Delta$ is a concatenation of one or more copies of $[0]$, id, and op. I can more or less see how to prove this directly (without toposes or ordered compact polyhedra).

ag.algebraic geometry - Find Vandermonde data to satisfy V*1=p

I would like to state something about the existence of solutions $x_1,x_2,dots,x_n in mathbb{R}$ to the set of equations

$sum_{j=1}^n x_j^k = np_k$, $k=1,2,dots,m$

for suitable constants $p_k$. By "suitable", I mean that there are some basic requirements that the $p_k$ clearly need to satisfy for there to be any solutions at all ($p_{2k} ge p_k^2$, e.g.).

There are many ways to view this question: find the coordinates $(x_1,dots,x_n)$ in $n$-space where all these geometric structures (hyperplane, hypersphere, etc.) intersect. Or, one can see this as determining the $x_j$ necessary to generate the truncated Vandermonde matrix $V$ (without the row of 1's) such that $V{bf 1} = np$ where ${bf 1} = (1,1,dots,1)^T$ and $p = (p_1,dots,p_m)^T$.

I'm not convinced one way or the other that there has to be a solution when one has $m$ degrees of freedom $x_1,dots,x_m$ (same as number of equations). In fact, it would be interesting to even be able to prove that for finite number $m$ equations $k=1,2,dots,m$ that one could find $x_1,dots,x_n$ for bounded $n$ (that is, the number of data points required does not blow up).

A follow on question would be to ask if requiring ordered solutions, i.e. $x_1 le x_2 le dots le x_n$, makes the solution unique for the cases when there is a solution.

Note: $m=2$ is easy. There is at least one solution = the point(s) where a line intersects a circle given that $p_2 ge p_1^2$.

Any pointers on this topic would be helpful -- especially names of problems resembling it.

mirror symmetry - Do you understand SYZ conjecture

Hi-

Just saw this thread. Maybe I should comment. The conjecture
can be viewed from the perspective of various categories:
geometric, symplectic, topological. Since the argument is
physical, it was written in the most structured (geometric)
context -- but it has realizations in the other categories
too.

Geometric: this is the most difficult and vague, mathematically,
since the geometric counterpart of even a conformal field theory
is approximate in nature. For example, a SUSY sigma model with
target a compact complex manifold X is believed to lie in the
universality class of a conformal field theory when X is CY,
but the CY metric does not give a conformal field theory on
the nose -- only to one loop. Likewise, the arguments about
creating a boundary conformal field theory using minimal (CFT) +
Lagrangian (SUSY) are only valid to one loop, as well.
To understand how the corrections are organized, we should
compare to (closed) GW theory, where "corrections" to the classical
cohomology ring come from worldsheet instantons -- holomorphic
maps contributing to the computation by a weighting equal
to the exponentiated action (symplectic area). The "count"
of such maps is equivalent by supersymmetry to an algebraic
problem. No known quantity (either spacetime metric or
Kahler potential or aspect of the complex structure) is
so protected in the open case, with boundary. That's why
the precise form of the instanton corrections is unknown,
and why traction in the geometric lines has been made
in cases "without corrections" (see the work of Leung, e.g.).
Nevertheless, the corrections should take the form of
some instanton sum, with known weights. The sums seem
to correspond to flow trees of Kontsevich-Soibelman/
Moore-Nietzke-Gaiotto/Gross-Siebert, but I'm already running
out of time.

Topological: Mark Gross has proven that the dual torus
fibration compactifies to produce the mirror manifold.

Symplectic: Wei Dong Ruan has several preprints which
address dual Lagrangian torus fibrations, which come
to the same conclusion as Gross (above). I don't know
much more than that.

Also-

Auroux's treatment discusses the dual Lagrangian
torus fibration (even dual slag, properly understood)
for toric Fano manifolds, and produces the mirror
Landau-Ginzburg theory (with superpotential) from this.

With Fang-Liu-Treumann, we have used T-dual fibrations
for the same fibration to map holomorphic sheaves
to Lagrangian submanifolds, proving an equivariant version of
homological mirror symmetry for toric varieties.
(There are many other papers with similar results
by Seidel, Abouzaid, Ueda, Yamazaki, Bondal, Auroux,
Katzarkov, Orlov -- sorry for the biased view!)

Reversing the roles of A- and B-models, Chan-Leung
relate quantum cohomology of a toric Fano to the
Jacobian ring of the mirror superpotential via T-duality.

Help or hindrance?

dg.differential geometry - What is the easiest way to classify all possible smooth orientable closed 2-manifolds?

Consider harmonic functions $f$ with exactly 2 log-singularities of weight $pm 1.$ (locally $f(z)=a logparallel zparallel+g,$ $g$ being smooth at $z=0$, $a$ being the weight) on your compact surface equipped with a Riemann metric. They exists by standard elliptic theory ( the two weights $a_1$ and $a_2$ have two add to zero). Consider $partial f,$ the complex linear part of the differential. This is a meromorphic section of your canonical bundle. Then $degpartial f=-frac{1}{2pi i}int KdA,$ as the Levi-Civita connection defines a complex linear connection on the canonical bundle.

This shows that if the total curvature is large enough $geq 4pi,$ $f$ will not have critical points (only two singularities). Moreover $e^f$ is the real part of a holomorphic bijection onto $CP^1.$

If $f$ has a critical point, then you can easily construct a non-sepreating loop, as in Morse theoretic proofs. You cut your surface, and add two disc (with the right orientation). One, can easily see, that this must increase the total curvature by $4pi,$ and you end up with the two-sphere after a finite number of steps.

combinatorial game theory - Do there exist chess positions that require exponentially many moves to reach?

With n/100 black kings, one white king, its possible.

The idea is to make a "switch", a room which is in one of two states, has 1 entrance, e, and 2 exits; x and y, the white king enters the room through a tunnel of pawns, if its in state 1, it must exit through x, while switching the state of the room to 0, in state 0 it must exit through y leaving the room in state 1.

With k switches labeled 1 to k, connect all the x_i with e_1, and y_j to e_(j+1) for all j. Start it with king in e_1=x_i, and all switches at 0, to turn the last switch on youll see that you need to transition thorugh all possible states, giving a minimum 2^k moves.

The switch really needs a diagram, but the idea is to have a black king for each switch, the black king will block a black rook preventing check, so the white king can get to a next room, where it will block a white rook, so that the black king can get to a third room, blocking a black rook, having the white king exit, the black king cant go back without the white king, but it can continue to the black entrance of a second such sequence of rooms, the black exit of which is connected to the black entrance of this sequence. The two white entrances are connected, and the two white exits are x and y. So the black king is in one of two tunnels, corresponding to the states 0/1, which determines which exit is available for the white king.

There might be some way to construct the switches without kings aswell.

There should be a row of pawns below and above the diagram of the switch component.
Couple 4 of these of each color to make 1 switch.

ag.algebraic geometry - Quotients of Abelian Varieties by Finite Groups

The quotients of abelian surfaces (over $mathbb C$) by finite groups are classified by Yoshihara. In particular he determines the possible Kodaira dimensions.
For instance, if the holonomy part of the group ( quotient by its maximal translation subgroup ) has cardinality greater than $24$ then he shows the quotient is rational.

The precise reference is

• Yoshihara, Hisao. Quotients of abelian surfaces. Publ. Res. Inst. Math. Sci. 31 (1995), no. 1, 135--143.

Unfortunately, I am not aware of any electronic version of this paper.

For higher dimensional abelian varieties I am not aware of any work studying the
finite quotients. But for 3-dimensional complex tori there is for instance this paper by Birkenhake, González-Aguilera, and Lange which classifies the possible finite subgroups. The same
authors also have a paper dealing with finite subgroups of the $3$-dimensional abelian varieties (over $mathbb C$ if I remember correctly).

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EDIT (May 18)

You may want to take a look at Complex crystallographic groups I and II. The authors study compact quotients $X$ of $mathbb C^n$ by discrete subgroups $Gamma subset rm{Aff}(mathbb C^n)$.

In dimension two they obtain classification results for the pairs $(X, Gamma)$ assuming
$Gamma$ (more precisely its holonomy part) is generated by reflections (paper I); or $X$ is rational (paper II).

ac.commutative algebra - Gaps in Dimension Polynomials

There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a lower bound for the Lascar Rank, an invariant coming from model theory. It turned out that the question of finding a lower bound is related to the following question.

Let $Ksubseteq Klangle etarangle$ be a partial differential field extension of characteristic zero. Suppose that the Kolchin polynomial $omega_{eta/K}(t)$ is of degree $n>0$. Is it true that for any $k < n$ there is a $nu$ in $Klangle etarangle$ such that the degree of $omega_{nu/K}(t)$ is $k$? (Here $eta$ and $nu$ are finite tuples).

Now, as differential algebra is not a very popular subject and most people do not know what a Kolchin polynomial is, I will also ask a very similar question in commutative algebra.

Let $S$ be a graded commutative algebra over $K[X_1,ldots,X_d]$ where $K$ is a field of characteristic zero and let $H_S(t)$ be its Hilbert polynomial. Suppose that $deg (H_S(t))=n>0$. Is it true that for any $k< n$ there is a graded subalgebra $T$ of $S$ such that $deg(H_T(t))=k$?

gn.general topology - Is the long line paracompact?

For connected locally Euclidean spaces, paracompactness is equivalent to second countable. Therefore if we truly insisted that our manifolds be second countable then the only thing that we would lose would be arbitrary coproducts. However, we don't insist that. Although it's often stated in the definition early on in a differential topology course or book, in my experience that's just because it's easier to explain than paracompactness (I tend to pick metrisable, myself). Of course, that early in a course we're probably not too concerned about arbitrary coproducts either.

Moreover, because manifolds split into a coproduct of their connected components, we tend to deal with connected manifolds unless we really can't avoid it. And even when they aren't connected, they most often have countably many components. So in practise the distinction doesn't arise.

I don't have it in front of me here, but I believe that an appendix to the first volume of Spivak's "Introduction to differential geometry" contains a proof of four equivalent conditions for locally Euclidean spaces (perhaps requiring Hausdorff). If I remember aright, the conditions are: paracompact, second countable, metrisable, and σ-compact.

There was a paper on the arxiv on Monday, 0910.0885, which lists 107 conditions for a connected locally Euclidean Hausdorff space equivalent to that it be metrisable. Amongst them are paracompactness and second countable.

ca.analysis and odes - Zeta-function regularization of determinants and traces

I will answer some of my questions in the negative.

3.
First consider the case of rescaling an operator A by some (positive) number λ. Then ζ_λA(s) = λ^-sζ_A(s), and so TR λA = λ TR A. This is all well and good. How does the determinant behave? Define the "perceived dimension" DIM A to be log_λ[ (DET λA)/(DET A) ]. Then it's easy to see that DIM A = ζ_A(0). What this means is that DET λA = λ^ζ_A(0) DET A.

This is all well and good if the perceived dimension of a vector space does not depend on A. Unfortunately, it does. For example, the Hurwitz zeta functions ζ(s,μ) = Σ₀^∞(n+μ)^-s (-μ not in N) naturally arise as the zeta functions of differential operators — e.g. as the operator x(d/dx) + μ on the space of (nice) functions on R. One can look up the values of this function, e.g. in Elizalde, et al. In particular, ζ(0,μ) = 1/2 - μ. Thus, let A and B be two such operators, with ζ_A = ζ(s,α) and ζ_B = ζ(s,β). For generic α and β, and provided A and B commute (e.g. for the suggested differential operators), then DET AB exists. But if DET were multiplicative, then:

DET(λAB) = DET(λA) DET(B) = λ^{1/2 - α} DET A DET B

but a similar calculation would yield λ^{1/2 - β} DET A DET B.

This proves that DET is not multiplicative.

1.
My negative answer to 1. is not quite as satisfying, but it's OK. Consider an operator A (e.g. x(d/dx)+1) with eigenvalues 1,2,..., and so zeta function the Reimann function ζ(s). Then TR A = ζ(-1) = -1/12. On the other hand, exp A has eigenvalues e, e², etc., and so zeta function ζ_{exp A}(s) = Σ e^-ns = e^-s/(1 - e^-s) = 1/(e^s-1). This has a pole at s=0, and so DET exp A = lim_s→0 e^s/(e^s-1)² = ∞. So question 1. is hopeless in the sense that A might be zeta-function regularizable but exp A not. I don't have a counterexample when all the zeta functions give finite values.

5.
As in my answer to 3. above, I will continue to consider the Hurwitz function ζ(s,a) = Σ_n=0^∞ (n+_a_)^-s, which is the zeta function corresponding, for example, to the operator x(d/dx)+a, and we consider the case when a is not a nonpositive integer. One can look up various special values of (the analytic continuation) of the Hurwitz function, e.g. ζ(-m,a) = -B_m+1(a)/(m+1), where B_r is the _r_th Bernoulli polynomial.

In particular,

TR(x(d/dx)+a) = -ζ(-1,a)/2 = -a²/2 + a/2 - 1/12

since, for example (from Wikipedia):

B₂(a) = Σ_n=0² 1/(n+1) Σ_k=0ⁿ (-1)^k {n choose k} (a+_k_)² = a² - a + 1/6

Thus, consider the operator 2_x_(d/dx)+a+_b_. On the one hand:

TR(x(d/dx)+a) + TR(x(d/dx)+b) = -(a²+b²)/2 + (a+_b_)/2 - 1/6

On the other hand, TR is "linear" when it comes to multiplication by positive reals, and so:

TR(2_x_(d/dx)+a+_b_) = 2 TR(x(d/dx) + (a+_b_)/2) = -(a²+2_ab_+b²)/4 + (a+_b_)/2 - 1/6

In particular, we have TR(x(d/dx)+a) + TR(x(d/dx)+b) = TR( x(d/dx)+a + x(d/dx)+b ) if and only if a=_b_; otherwise 2_ab_ < a²+b² is a strict inequality.

So the zeta-function regularized trace TR is not linear.

0./2.
My last comment is not so much to break number 2. above, but to suggest that it is limited in scope. In particular, for an operator A on an infinite-dimensional vector space, it is impossible for A^-s to be trace-class for s in an open neighborhood of 0, and so if the zeta-function regularized DET makes sense, then det doesn't. (I.e. it's hopeless to say that det A = DET A.) Indeed, if the series converges for s=0, then it must be a finite sum.

Similarly, it is impossible for A to be trace class and also for A^-s to be trace class for large s. If A is trace class, then its eigenvalues have finite sum, and in particular cluster near 0 (by the "divergence test" from freshman calculus). But then the eigenvalues of A^-s tend to ∞ for positive s. I.e. it's hopeless to say that tr A = TR A.

My proof for 2. says the following. Suppose that dA/dt A^-1 is trace class, and suppose that DET A makes sense as above. Then

d/dt [ DET A ] = (DET A)(tr dA/dt A^-1)

I have no idea what happens, or even how to attack the problem, when dA/dt A^-1 has a zeta-function-regularized trace.

co.combinatorics - Is there any meaning to a "nice bijective proof?"

From Zeilberger's PCM article on enumerative combinatorics:

The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal appeal of a general principle... On the other hand, its universality is also a major drawback, since involution-principle proofs usually do not give any insight into the specific structures involved, and one feels a bit cheated. [... O]ne still hopes for a really natural, "involution-principle-free proof."

The quest for bijective proofs really is, at heart, the quest to categorify an equation proved via "decategorified" arithmetic, to provide an explicit family of isomorphisms in the category of finite sets (or the category of finite sets and bijections, I don't think it matters). So categorifying multiplication and addition is easy -- they just correspond to product and coproduct of sets, respectively. Categorifying subtraction is trickier, but that's what the involution principle does for us. (On a tangential note, I know that categorifying division is [in]famously hard in the category of sets -- is it anything like as hard when we talk about finite sets? ETA: Looking at Conway's paper, the answer seems to be "yes and no." No, because with finite sets it's kosher to use a bijection between a set of size n and {0, 1, ..., n-1}; yes, because this is sort of like a "finitary Axiom of Choice," and in particular it's not canonical.)

So is there any real meaning, in a categorical sense, to the enumerative-combinatorial dream of "really nice proofs?" Or will there necessarily be identities that can only be proved bijectively by categorifying, in a general and universal way, their "manipulatorics" proofs?

Edit: So philosophically this is a category-theory question, and it'd be nice to have it as a "real" category-theory question. Here's my (very rough) attempt at phrasing it as such.

Let T be a topos where we can categorify addition, multiplication, and subtraction of natural numbers. Then, are there functors between T and FinSet that preserve these decategorifications? If so, then I think maybe we can ask the question in terms of topos theory, although maybe not -- I don't really know much topos theory, so I'm certainly having trouble.

soft question - How to write math well?

One trick that my advisor, Ronnie Lee, advocated was to use a descriptive term before using the symbolic name for the object. Thus write, "the function $f$, the element $x$, the group $G$, or the subgroup $H$. Most importantly, don't expect that your reader has internalized the notation that you are using. If you introduced a symbol $Theta_{i,j,k}(x,y,z)$ on page 2 and you don't use it again until page 5, then remind them that the subscripts of the cocycle $Theta$ indicate one thing while the arguments $x,y,z$ indicate another.

Another trick that is suggested by literature --- and can be deadly in technical writing --- is to try and find synonyms for the objects in question. A group might be a group for a while, or later it may be giving an action. In the latter case, the set of symmetries $G$ that act on the space $X$ is given by $ldots$. Context is important.

Vary cadence. Long sentences that contain many ideas should have shorter declarative sentences interspersed. Read your papers out loud. Do they sound repetitive?

My last piece of advice is one I have been wanting to say for a long time. Don't write your results up. Write your results down. You figure out what I mean by that.

Example of an algebra finite over a commutative subalgebra with infinite dimensional simple modules

Let $A$ be an algebra over an algebraically closed field $k.$ Recall that if $A$ is
a finitely generated module over its center, and if its center is a finitely generated
algebra over $k,$ then by the Schur's lemma all simple $A$-modules are finite dimensional
over $k.$

Motivated by the above, I would like an example of a $k$-algebra $A,$ such that:

1) $A$
has a simple module of infinitie dimension over $k,$

2) $A$ contains
a commutative finitely generated subalgebra over which $A$ is a finitely generated
left and right module.

Thanks in advance.

gr.group theory - Can we bound degrees of complex irreps in terms of the average conjugacy class size?

Let b(G) be the maximum degree of an irreducible character of the finite group G.

This answer is mostly to address the solvable, but not nilpotent case. I already up-voted the other two excellent answers: Jim Humphreys mentioned how the groups of Lie type behave (showing that K_G≈q^2N can be enormous compared to b(G)≈q^N), and Bugs Bunny gave the very clear example of the family of extraspecial p-groups showing the opposite extreme (with K_G = p⁽²ⁿ⁺¹⁾/(p⁽²ⁿ⁾+2) < p, and b(G) = pⁿ).

---

One idea is to bound b(G) using important characteristic subgroups:

For any finite group G, b(G) ≤ √[G:Z(G)] and b(G) divides [G:Z(G)].

If G is solvable, then Gluck's conjecture is that √[G:Fit(G)] ≤ b(G), and this has been verified for solvable G such that G/Φ(G) has an Abelian Sylow 2-subgroup or G such that G″ = 1. (If G is non-abelian simple, then Fit(G)=1, and so the bound cannot hold).

---

Another idea is to bound b(G) using abelian subgroups:

If A is an abelian subgroup of G, then b(G) ≤ [G:A]. If A is abelian and subnormal, then in fact b(G) divides [G:A].

If G is solvable, then min(√lg([G:A])) ≤ lg(b(G)). If G is solvable of odd order, then min([G:A]^1/6) ≤ b(G). If γ_∞(G) is abelian, then min([G:A]^1/4) ≤ b(G).

---

Versions of most of these results are in Isaacs's book (page 28, 30, 84, 190, 212, 216), but I tried to include the 21st century versions where I found them.

I would also caution against thinking "having an abelian subgroup of small index" as "close to abelian". Perhaps there is some truth to it, but the p-groups that are furthest from being abelian (maximal class, that is, having the largest nilpotency class possible amongst groups of the same order) also tend to have the largest abelian subgroups, often index p. Poor extra-special groups are nilpotency class 2 and nearly abelian (they are understood as symplectic vector spaces), yet their abelian subgroups are tiny, |A| ≤ p√[G:Z]. In other words extremely not-abelian groups may have large abelian subgroups, and extremely abelian groups may not.

linear algebra - Quantum observables

If I understood you right:

If we have two-commuting bounded linear operators (if they Hermitian), in quantum physics it's mean, if I'm right, that we can't make such observation and determine the state of both parameters of the system. But the suggestion, that we have such operator that commute to both can possibly make both states — A and B determined simultaneously.

So, for my small opinion, Com(A, B) didn't exist or equal to zero.

computer algebra - Can Gröbner bases be used to compute solutions to large, real-world problems?

First, the Gröbner basis is not sparse. I am speaking a little off-the-cuff, but empirically when I ask SAGE for the Gröbner basis of $(y^n-1,xy+x+1)$ in the ring $mathbb{Q}[x,y]$, it gets worse and worse as $n$ increases. Any bound would have to be in terms of the degrees of the original generators as well as their sparseness, and I suspect that the overall picture is bad.

Overall your questions play to the weaknesses of Gröbner bases. You would need new ideas to make not just the computations of the bases, but also the actual answer numerically stable. You would also need new ideas to make Gröbner bases sparse.

You are probably better off with three standard ideas from numerical analysis: Divide and conquer, chasing zeroes with an ODE, and Newton's method. If you have the generators for the variety in an explicit polynomial form, then you are actually much better off than many uses for these methods that involve messy transcendental functions. Because you can use standard analysis bounds, specifically bounding the norms of derivatives, to rigorously establish a scale to switch between divide-and-conquer and Newton's method, for instance. Moreover you can subdivide space adaptively; the derivative norms might let you stop much faster when you are far away from the variety.

---

To explain what I mean by derivative bounds, imagine for simplicity finding a zero of one polynomial in the unit interval $[0,1]$. If the polynomial is $100-x-x^5$, then a simple derivative bound shows that it has no zeroes. If the polynomial is $40-100x+x^5$, then a simple derivative bound shows that it has a unique zero and Newton's method must converge everywhere within the interval. If the polynomial is something much more complicated like $1-x+x^5$, then you can subdivide the interval and eventually the derivative bounds become true. Also, with polynomials, you can make bounds to know that there are no zeroes in an infinite interval under suitable conditions.

You can do something similar in higher dimensions. You can divide space into rectangular boxes, and just median subdivisions. It's not very elegant, but it works well enough in low dimensions. In high dimensions, the whole problem can be intractable; you need to say something about why you think that the solution locus is well-behaved to know what algorithm is suitable.

big picture - Are there any "homotopical spaces"?

This is a somewhat vague question; I don't know how "soft" it is, and even if it makes sense.

[Edit: in the light of the comments, we can state my question in a formally precise way, that is: "Is the homotopy category of topological spaces a concrete category (in the sense, say, of Kurosh and Freyd)?". You may still want to read what follows, for a bit of motivation]

Historically, I'd bet people started to look at concrete metric spaces $(X,d)$ before exploring the utility of the abstraction given by general topological spaces $(X,tau)$ ($tau$ is a topology). The heuristic idea captured by the concept of a topological space is endowing a set $X$ with a "geometry" that forgets the rigidity of a hypothetical metric structure $(X,d)$, though retaining the "qualitative" aspects of the geometry given by the metric.

Of course there are non-metrizable topological spaces, but let stick to metrizable ones for the moment. I think it should be possible to see a topological space $(X,tau)$ as an equivalence class of metric spaces: $(X,[d])$, where $[d]$ is the class of all metrics on the set $X$ that give rise to the same topology. So, an $(X,tau)$ just has several "rigid" models $(X,d)$, and a morphism of topological spaces $f:(X,tau) rightarrow (Y,tau')$ is given by taking any map $(X,d) rightarrow (Y,d^{'})$ of "representatives" which is continuous according to the "metric ball" definition.
[Please correct the above picture if it is imprecise or even just wrong!!]

The (perhaps naive) way I have always thought about homotopy is that it is an even "further step" in making the geometry more "qualitative" and less rigid: you can "collapse positive dimensional appendices" of a space as far as they are contractible, and so forth. When trying to make this formal, you consider "homotopy types", which are equivalence classes $[(X,tau)]$ of topological spaces, where $(X,tau) sim (X',tau')$ if there is a homotopy equivalence $varphi:(X,tau) rightarrow (X',tau')$. What are morphisms in the homotopy category? Just morphisms $f$ between "representatives", but now you have also to consider them up to homotopy, i.e. you take $[f]$ where $f sim f'$ if there's a homotopy $alpha: f rightarrow f'$.

It's ugly to think of topological spaces as $[(X,d)]$ (or, rather, $(X,[d])$): it's better to use the simpler and more expressive abstraction $(X,tau)$.

So, the question is:

• Is there some kind of "homotopology" $h$ (whatever it is) that one can put on sets $S$ so that each homotopy type $[(X,tau)]$ is fully described by a "homotopical space" (whatever it is) $(S,h)$ and homotopy classes of morphisms $[f]$ correspond to "$h$-compatible" (whatever it means) set-theoretic maps $F:S rightarrow S'$?

(I don't even dare asking about the existence of "non-topologizable homotopical spaces" because the above question is already by far too vague!)

set theory - Does Cantor-Bernstein hold for classes?

This is an interesting question. I think there are some issues which the others did not mention yet. But I'm not an expert at all, I might be wrong. Please leave me a comment!

In the following, I work in $ZF$. Thus, a class is just a formula. There are some constructions and relations of sets which directly carry over to classes. For example $A=B$ means that the formulas of $A,B$ are equivalent.

Let's sketch the proof for classes. Let $A,B$ classes, $f : A to B$, $g : B to A$ injective maps. Define the classes $A_n subseteq A, B_n subseteq B$ recursively by $A_0=A, B_0 = B, A_{n+1} = g[B_n], B_{n+1}=f[A_n]$. Then $h : A to B$, defined by $f$ on $cap_n A_n cup (cup_n A_{2n} setminus A_{2n+1})$ and by $g^{-1}$ in the rest, is well-defined and a bijection.

Unions, cuts, images of functions etc. are not the problem. But what about the recursion? What we really need here is a recursion scheme for classes. Actually there is a theorem which might be called the transfinite recursion scheme for classes:

Let $R$ be a well-founded and set-like relation of the class $A$ and $F : A times V to V$ a function. Then there is a function $G : A to V$, such that for all $x in A$

$G(x)=F(x,G|_{{y in A : y R x}})$.

However, note that the images of $G$ are sets, not proper classes. We can't use that theorem here.

I think we need a meta-theorem stating that the above also holds when $V$ is replaced by the set of formulas and $R=mathbb{N}$. Also, the meta-world should be able to talk about functions. But this is not plausible at all, since the resulting formula won't have to be finite, right?

For example, try to define a formula $G(n)$ recursively by $G(0)=phi_0$ (doesn't matter what $phi_n$ is) and $G(n)=G(n-1) wedge phi_n$. Why should there be a formula $psi(n,x)$ such that $psi(n,x) Leftrightarrow wedge_{i=0}^{n} phi_i$? I think we need a rather mighty logical calculus for that.

Also note that Francois' great answer here (proving Schröder-Bernstein without the existence of the set $omega$) also causes problems when you want to write down the formula for the part "...$exists s : {0,...,n} to A$ ...". Perhaps there is really no bijection between the two classes mentioned above (class of all singletons, and class of all 2-element sets)?

geometric invariant theor - Are irregular points of an action necessarily in the closure of a larger orbit?

Suppose G is an affine algebraic group acting linearly on a vector space V. A point v∈V is stable if the orbit Gv is closed and v is regular (the dimension of the stabilizer of v is locally constant, or equivalently, locally minimum). I would really like to say this is equivalent to the orbit Gv being closed and not being in the closure of another orbit.

Since the orbit of a regular point has locally maximum dimension, it can't be in the closure of another orbit. But is the converse true? If a point is not in the closure of an orbit larger than its own, is it regular?

The answer is no ... we have to throw in some hypotheses. If you consider the action of G_a on A² given by t(x,y)=(x,tx+y), then all the orbits are closed, but points of the form (0,y) are irregular. So let's throw in the hypothesis that G is linearly reductive. I feel like we might also want to insist that v≠0, but I'm not sure about that.

Linearly reductive seems like a strange hypothesis, so feel free to modify it. I was thinking that you could somehow show that if v is not in the closure of a larger-dimensional orbit, then span(Gv) would be an invariant subspace with no complement, but I haven't been able to get this argument to work.

ag.algebraic geometry - Tannaka formalism and the étale fundamental group

For quite a while, I have been wondering if there is a general principle/theory that has
both Tannaka fundamental groups and étale fundamental groups as a special case.

To elaborate: The theory of the étale fundamental group (more generally of
Grothendieck's Galois categories from SGA1, or similarly of the fundamental group of a
topos) works like this: Take a set valued functor from the category of finite étale
coverings of a scheme satisfying certain axioms, let $pi_1$ be its automorphism group and
you will get an equivalence of categories ( (pro-)finite étale coverings) <-> ( (pro-)finite cont.
$pi_1$-sets).

The Tannaka formalism goes like this: Take a $k$-linear abelian tensor category $mathbb{T}$ satisfying certain axioms (e.g. the category of finite dim. $k$-representations of an abstract group), and a $k$-Vector space
valued tensor functor $F$ (the category with this functor is then called neutral Tannakian), and let $Aut$ be its tensor-automorphism functor, that is the functor that assigns to a $k$-algebra $R$ the set of $R$-linear tensor-automorphisms $F(-)otimes R$. This will of
course be a group-valued functor, and the theory says it's representable by a group scheme $Pi_1$,
such that there is a tensor equivalence of categories $Rep_k(Pi_1)cong mathbb{T}$.

Both theories "describe" under which conditions a given category is the (tensor) category of
representations of a group scheme (considering $pi_1$-sets as "representations on sets" and $pi_1$ as constant group scheme). Hence
the question:

Are both theories special cases of some general concept? (Maybe, inspired by
recent questions, the first theory can be thought of as "Tannaka formalism for
$k=mathbb{F}_1$"? :-))

Maximal localizations of von Neumann algebras

Suppose M is a von Neumann algebra.
Denote by L its maximal noncommutative localization,
i.e., the Ore localization with respect to the set of all left and right regular elements,
i.e., elements whose left and right support equals 1.

Denote by A the set of all closed unbounded operators with dense domain
affiliated with the standard representation of M on a Hilbert space, i.e., L^2(M),
also known as the standard form of M.

Von Neumann proved that if M is finite, then L and A are canonically isomorphic.

What can we say about the relationship of L and A when M has type III?

I am also interested in the properly infinite semifinite case.

etale cohomology - Cohomological dimension-doubling

I'm not sure I can give you a morally satisfying answer. To my mind, this sort of theorem should be true because it works over $mathbb{C}$, for topological reasons. Of course, that isn't a proof over other fields. But, to my limited understanding, the intuition comes from $mathbb{C}$ and the proofs are motivated by taking proofs which work over $mathbb{C}$ and seeing whether we can generalize them to an arbitrary field.

That said, I can sketch two proofs of this theorem. The first proof is by induction on dimension. Write $X$ as a family over $Y$, with fibers of dimension $dim X - dim Y$. A spectral sequence shows that, if the cohomology of $Y$ vanishes is in degree $>i$ and the cohomology of the fibers vanishes in dimension $>j$, then the cohomology of $X$ vanishes in degree $>i+j$. This reduces us to showing that curves have no cohomology above $H^2$, which can be done by hand; I think this is in Chapter 14 of Milne's lectures.

The other method is for deRham-like cohomology theories (rigid, crystalline, etc.). Roughly speaking, those methods compute cohomology as the hypercohomology of a complex $Omega^0 to Omega^1 to Omega^2 to cdots$. If your ground field is characteristic zero, and your variety is smooth, these are actually the familiar sheaves of differentials. If one of these two conditions fails, you have to adjust in some manner; the details of this adjustment describe which of the theories you are working in. In any case (very roughly speaking) the $2n$ appears here as $n+n$: the complex has length $n$ and each of the terms in the complex has no cohomology above degree $n$. Again, a spectral sequence finishes the proof from here.

pr.probability - analog of principle of inclusion-exclusion

I'll take a stab at answering my own question.

The missing “something” in the edited version of my question appears to be the mutual
information of one or more events, denoted I(A,B,C,...). More precisely,
it appears to be the quantity e^{−I(A,B,C,...)}. It would be good if
someone knowledgeable about information theory could weigh in here—I haven't
found a definition in the literature of the mutual information of more than
two events, but the one I give below seems to be the “right” one.

In the information theory textbooks I looked at, the information of a single
event E is defined as I(E) = −log P(E). The mutual information of two
events, E and F, is defined as
I(E,F) = −log [P(E) P(F) / P(E ∩ F)],
and is a measure of the degree to which E and F fail to be independent. That is, I(E,F)
is zero if E and F are independent, positive if they are positively correlated, and negative if
they are negatively correlated.

The appropriate generalization to three events seems (following the
suggestion of Kenny Easwaran) to be
I(E,F,G) = −log [P(E) P(F) P(G) P(E ∩ F ∩ G)
/ P(E ∩ F) P(E ∩ G) P(F ∩ G)]
which makes some sense as a measure of the failure of independence since,
in order for E, F, and G to be independent, it is required, not only that
P(A ∩ B) = Pr(A) P(B) for all pairs of events, but also that
P(E ∩ F ∩ G) = P(E) P(F) P(G).
(Does anyone know if the definition of I(E,F,G) given above is standard?)

Now define C({A,B,C,...}) = P(A ∩ B ∩ C ∩ ...). The appropriate
generalization of the mutual information to an arbitrary number of events
seems to be
I(E,F,G,...) = −log [Π_S C(S) / Π_T C(T)]
where the product over S runs over all subsets of {E,F,G,...} of odd
cardinality and the product over T runs over all subsets of
{E,F,G,...} of even cardinality. (Again, does anyone know if this definition
is standard?)

With these definitions, we get the inclusion-exclusion-like rule,

−log P(E ∩ F ∩ G ∩ ...) = I(E) + I(F) + I(G) + ...
− I(E,F) − I(E,G) − ... + I(E,F,G) + ... − ...

This can be proved by a counting argument identical to the one used to
prove the principle of inclusion-exclusion. Negating and exponentiating
both sides produces an identity of the desired form. It would be nice
to also find a Möbius-inversion style proof.

As to whether this is “useful” in the same sense that the principle
of inclusion-exclusion is useful, I can't say. The definition of I(E,F,G,...)
is itself an inclusion-exclusion-like rule, so it's tautological that when you
invert it to find −log P(E ∩ F ∩ G ∩ ...) you will get an
inclusion-exclusion-like rule. I suppose the significance of all this depends
on how fundamental the mutual information is.

Addendum:
The book Elements of Information Theory by Cover and Thomas mentions the problem of defining the mutual information
of three random variables in Problem 2.25. The mutual information of random variables is related to but somewhat different from the mutual information of events since, for example, I(X,Y,Z) involves taking the expectation over all events (X=x, Y=y, Z=z). The problem notes that, in contrast with the two random variable case, the mutual information of three random variables is not a non-negative quantity in general. This perhaps explains why not much theory has been developed around it. Interestingly, however, the mutual information of three random variables can be expressed in terms of the entropy of one, two, or three random variables via the inclusion-exclusion principle.

at.algebraic topology - singular cohomology of SO(4)

I'm trying to compute the singular cohomology of SO(4), just as practice for using spectral sequences. I got H⁰=Z, H¹=0, H²=Z/2Z, H³=Z⊕Z, H⁴=0, H⁵=Z/2Z, and H⁶=Z. Are these correct? I'm not sure if I'm reading it right, but these calculations seem to disagree with this pretty cool little note on the cohomology ring of SO(n) (check out the crazy pictures at the bottom!).

Also, in the spirit of "teaching a man to fish", does anyone know of some place where people have collected all these sorts of calculations (and possibly also homology and homotopy calculations)?

Lastly, how can I determine the ring structure on H*(SO(4)) from these calculations? Supposedly the isomorphism of whatever your usual cohomology is (de Rham, singular, whatever) with the cohomology of the double complex respects the ring structure, but is it really just as easy as saying that the product of a (p,q)-element with an (r,s)-element lives in (p+r,q+s) and is the thing you'd expect it to be?

ADDENDUM:

Since it's likely that they're incorrect, I'll lay out my process here and hopefully someone with some spare time on their hands can tell me where I went wrong. (I apologize in advance for trying to describe the spectral sequence of a double complex without any diagrams! I'm working out of Bott & Tu. I'm using the standard fibration of SO(4) over S³ with fiber SO(3). They have Leray's theorem (15.11, at least in my very old edition) giving that, since the base is simply-connected, E₂^p,q=H^p(S³,H^q(SO(3);Z)). We know that H⁰(SO(3))=H³(SO(3))=Z, H²(SO(3))=Z/2Z, and Hⁿ(SO(3))=0 otherwise. By the universal coefficient theorem, the singular cohomology of S³ with coefficients in an abelian group G is just G in dimensions 0 and 3, and 0 elsewhere. So I've got E₂ only nonzero in columns 0 and 3, where it's Z, 0, Z/2Z, Z, 0, 0, 0... This is in fact E_∞, since the only potentially nonzero map from here on out is from the (0,2) entry to the (3,0) entry, but this has to be a homomorphism from Z/2Z to Z, which is necessarily zero. Summing along the diagonals yields the results I gave above.

fa.functional analysis - Factoring operators $L_infty longrightarrow L_2$ as the composition of $n$ strictly singular operators, $nin mathbb{N}$

Motivation and background This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $mathcal{B}(L_infty)$ of all (bounded, linear) operators on $L_infty$ $(=L_infty[0,1])$. As far as I am aware, the only known nontrivial ideals in $mathcal{B}(L_infty)$ are the ideals $mathcal{K}(L_infty)$ of compact operators and $mathcal{W}(L_infty)$ of weakly compact operators. Most of the other well-known closed operator ideals not containing the identity operator of $L_infty$ seem to coincide with one of these two operator ideals on $L_infty$. Let me mention explicitly the following further relevant pieces of background information:

• Any nontrivial closed ideal $mathcal{J}$ of $mathcal{B}(L_infty)$ must satisfy $mathcal{K}(L_infty)subseteq mathcal{J} subseteq mathcal{W}(L_infty)$.




• We have $mathcal{S}(L_infty) = mathcal{W}(L_infty)$, where $mathcal{S}$ denotes the (closed) operator ideal of strictly singular operators. Moreover, the ideal of operators $L_infty longrightarrow L_infty$ that factor through $L_2$ $(=L_2[0,1])$ is a norm dense subset of $mathcal{W}(L_infty)$. With regards to this latter fact, we have that if $mathcal{J}$ is any closed ideal of $mathcal{B}(L_infty)$ satisfying $mathcal{K}(L_infty)subsetneq mathcal{J} subsetneq mathcal{W}(L_infty)$, then there exists an element of $mathcal{W}(L_infty)setminus mathcal{J}$ that factors through $L_2$.

One possible avenue towards discovering more closed ideals in $mathcal{B}(L_infty)$ is to consider products of closed operator ideals, and my question below concerns but one approach along these lines. We recall now that for operator ideals $mathcal{A}$ and $mathcal{B}$, their product $mathcal{B}circmathcal{A}$ is the class of all operators of the form $BA$, where $Ainmathcal{A}$, $Binmathcal{B}$ and the codomain of $A$ coincides with the domain of $B$ (so that the composition is defined). It is well-known that $mathcal{B}circmathcal{A}$ is an operator ideal and that $mathcal{B}circmathcal{A}$ is a closed operator ideal whenever $mathcal{A}$ and $mathcal{B}$ are, the latter fact being due to Stefan Heinrich. For $n$ a natural number, one may define the power of an operator ideal $mathcal{A}^n$ as the product $mathcal{A}circ ldots circ mathcal{A}$ with $n$ factors, and $mathcal{A}^n$ is closed for all $n$ whenever $mathcal{A}$ is closed. Moreover, we may define $mathcal{A}^infty:= bigcap_{ninmathbb{N}}mathcal{A}^n$, with $mathcal{A}^infty$ being a closed operator ideal whenever $mathcal{A}$ is.

It is well-known that every operator from $L_infty$ to $L_2$ $(=L_2[0,1])$ is strictly singular. However, the answer to the following question is not so clear to me:

Question: Is $mathcal{B} (L_infty, L_2) = mathcal{S}^infty (L_infty, L_2)$? If no, what is the least $n$ for which $mathcal{B} (L_infty, L_2) neq mathcal{S}^n (L_infty, L_2)$?

Further comments: A negative answer to my question above will yield through easy arguments that $mathcal{K}(L_infty) subsetneq mathcal{S}^infty (L_infty) subsetneq mathcal{W}(L_infty)$ (note that $mathcal{K}(L_infty) subsetneq mathcal{S}^infty (L_infty)$ in any case since there are noncompact operators $L_infty longrightarrow L_infty$ that have the formal inclusion operator $ell_2 hookrightarrow ell_infty$ as a factor, and this inclusion operator belongs to $mathcal{S}^infty$).

It seems to me that the best chance of obtaining an element of $mathcal{B} (L_infty, L_2) setminus mathcal{S}^infty (L_infty, L_2)$ would be to consider a surjective element of $mathcal{B} (L_infty, L_2)$, for instance the adjoint of an isomorphic embedding of $L_2$ into $L_1$. A further possibility along these lines would be to factor such an embedding as the product of $n$ operators whose adjoints are strictly singular, for each $nin mathbb{N}$. However, it seems to me that such an approach will require a far deeper knowledge of the subspace structure of $L_1$ than I have at the present time. Interpolation methods may work too, but I am also too ignorant of that theory to know whether it's a genuinely feasible approach.

linear algebra - Geometric interpretation of characteristic polynomial

A rather simple response is to differentiate the characteristic polynomial and use your interpretation of the determinant.

$$det(I-tf) = {t^n}det(frac{1}{t}I-f) = (-t)^ndet(f-frac{1}{t}I)= {(-t)^n}chi(f)(1/t)$$

So if we let $chi(f)(t) = Sigma_{i=0}^n a_it^i$, then ${(-t)^n}chi(f)(1/t) = (-1)^nSigma_{i=0}^n a_it^{n-i}$

But $I-tf$ is the path through the identity matrix, and $Det(A)$ measures volume distortion of the linear transformation $A$.

$$det(I-tf)^{(k)}(t=0) = (-1)^nk!a_{n-k}$$

and a change of variables ($tlongmapsto -t$) gives (and superscript $(k)$ indicates $k$-th derivative)

$$det(I+tf)^{(k)}(t=0) = (-1)^{n+k}k!a_{n-k}$$

So the coefficients of the characteristic polynomial are measuring the various derivatives of the volume distortion, as you perturb the identity transformation in the direction of $f$.

$$a_k = frac{det(I+tf)^{(n-k)}(t=0)}{(n-k)!}$$

nt.number theory - Upper bound on Brun's constant

This seems to me like a simple matter of enumerating all the small twin primes, and then estimating the resulting error using a sieve bound. In particular, if $pi_2(x)$ is the number of twin primes $leq x$ then we know $pi_2(x) ll x (log x)^{-2}$. By a simple summation-by-parts exercise, this gives

$B=sum_{p,twin,p < X}frac{1}{p}+frac{1}{p+2}+O(log{X}^{-1})$.

I'm not sure what the numerical constant in the O-term is, but presumably it can be computed.

rt.representation theory - Representations of products of groups (and monoids)

I have very little knowledge of representation theory, but the following has come up in my summer undergrad research project (relates to conformal field theory and geometric function theory).

Suppose we have a group $G$ and subgroups $A$ and $B$ such that $A cap B = {1}$ and for every $g in G$, there exists $a in A, b in B$ such that $g = ab$. Then we can express $G = A bowtie B$ as a Zappa–Szép product. This of course reduces to the semidirect or direct product in the nice cases.

Then suppose, we have sufficiently nice representations of $A$ on an $F$-vector space V, and $B$ on an $F$-vector space W, then can we find a representation of $G$ which in some sense preserves the representations of $A$ and $B$?

I've been told that the solution for semidirect products uses something called Clifford Theory, but we don't have a semidirect product here.

Our problem involves a monoid, not a group, but the Zappa-Szep product is constructed the same way there.