universal algebra - Do non-associative objects have a natural notion of representation?

A magma is a set $M$ equipped with a binary operation $* : M times M to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional axioms that "symmetries of things" should satisfy. This is made precise in the sense that for any object $A$ in a category $C$, the invertible morphisms $A to A$ have a group structure again given by composition. An alternate definition of "group," then, is "one-object category with invertible morphisms," and then the additional axioms satisfied by groups follow from the axioms of a category (which, for now, we will trust as meaningful). Groups therefore come equipped with a natural notion of representation: a representation of a group $G$ (in the loose sense) is just a functor out of $G$. Typical choices of target category include $text{Set}$ and $text{Hilb}$.

It seems to me, however, that magmas (and their cousins, such as non-associative algebras) don't naturally admit the same interpretation; when you throw away associativity, you lose the connection to composition of functions. One can think about the above examples as follows: there is a category of groups, and to study the group $G$ we like to study the functor $text{Hom}(G, -)$, and to study this functor we like to plug in either the groups $S_n$ or the groups $GL_n(mathbb{C})$, etc. on the right, as these are "natural" to look at. But in the category of magmas I don't have a clue what the "natural" examples are.

Question 1: Do magmas and related objects like non-associative algebras have a "natural" notion of "representation"?

It's not entirely clear to me what "natural" should mean. One property I might like such a notion to have is an analogue of Cayley's theorem.

For certain special classes of non-associative object there is sometimes a notion of "natural": for example, among not-necessarily-associative algebras we may single out Lie algebras, and those have a "natural" notion of representation because we want the map from Lie groups to Lie algebras to be functorial. But this is a very special consideration; I don't know what it is possible to say in general.

(If you can think of better tags, feel free to retag.)

Edit: Here is maybe a more focused version of the question.

Question 2: Does there exist a "nice" sequence $M_n$ of finite magmas such that any finite magma $M$ is determined by the sequence $text{Hom}(M, M_n)$? (In particular, $M_n$ shouldn't be an enumeration of all finite magmas!) One definition of "nice" might be that there exist compatible morphisms $M_n times M_m to M_{n+m}$, but it's not clear to me that this is necessarily desirable.

Edit: Here is maybe another more focused version of the question.

Question 3: Can the category of magmas be realized as a category of small categories in a way which generalizes the usual realization of the category of groups as a category of small categories?

Edit: Tom Church brings up a good point in the comments that I didn't address directly. The motivations I gave above for the "natural" notion of representation of a group or a Lie algebra are in some sense external to their equational description and really come from what we would like groups and Lie algebras to do for us. So I guess part of what I'm asking for is whether there is a sensible external motivation for studying arbitrary magmas, and whether that motivation leads us to a good definition of representation.

Edit: I guess I should also make this explicit. There are two completely opposite types of answers that I'd accept as a good answer to this question:

• One that gives an "external" motivation to the study of arbitrary magmas (similar to how dynamical systems motivate the study of arbitrary unary operations $M to M$) which suggests a natural notion of representation, as above. This notion might not look anything like the usual notion of either a group action or a linear representation, and it might not answer Question 3.




• One that is "self-contained" in some sense. Ideally this would consist of an answer to Question 3. I am imagining some variant of the following construction: to each magma $M$ we associate a category whose objects are the non-negative integers where $text{Hom}(m, n)$ consists of binary trees with $n$ roots (distinguished left-right order) and $m$ "empty" leaves (same), with the remaining leaves of the tree labeled by elements of $M$. Composition is given by sticking roots into empty leaves. I think this is actually a 2-category with 2-morphisms given by collapsing pairs of elements of $M$ with the same parent into their product. An ideal answer would explain why this construction, or some variant of it, or some other construction entirely, is natural from some higher-categorical perspective and then someone would write about it on the nLab!

ag.algebraic geometry - Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed.

Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian group on the orbits of X. When X is a nonsingular algebraic curve over the algebraic closure of a finite field k and f is the Frobenius map, Div(X) is naturally isomorphic to the group of fractional ideals of k(X) (at least, I think; correct me if I'm wrong). There is a distinguished subgroup Prin(X) consisting of the preimage of the principal ideals, and Div(X)/Prin(X) is the divisor class group.

Is there a canonical definition of Prin(X) for general dynamical systems? If not, how much extra structure does X need to have for a construction like this to make sense and give some kind of useful information about X?

The case I'm interested in is that X is the set of aperiodic closed walks on a finite graph with a distinguished point and f moves the distinguished point.

rt.representation theory - Classification of adjoint orbits for orthogonal and symplectic Lie algebras?

"Classification" can mean more than one thing, but it's useful to be aware of the extensive development of adjoint quotients by Kostant, Steinberg, Springer, Slodowy, and others. This makes sense for all semisimple (or reductive) groups and their Lie algebras over any algebraically closed field, perhaps avoiding a few small prime characteristics. Older sources include Steinberg's 1965 IHES paper on regular elements and the Springer-Steinberg portion of the
1970 Lecture Notes in Math. vol. 131. (For an overview with references, based partly on Steinberg's Tata lectures, see Chapter 3 of my 1995 AMS book Conjugacy Classes in Semisimple Algebraic Groups.) While the general focus has been on developing a picture of the collection of all classes or orbits as some kind of "quotient", quite a few special features of the classical types are also brought out in the Springer-Steinberg notes. As suggested by Victor, Roger Carter's book Finite Groups of Lie Type also has a lot of related material but with special emphasis on nilpotent orbits. The Jordan decomposition does reduce many classification questions to the nilpotent case, at least in principle, if you are willing to deal with various centralizers along the way.

[ADDED] The paper in J. Math. Physics gives a nice concrete answer to the original question, building on some of the older theory but using mainly tools from linear algebra and basic group theory. This is the traditional approach of most physicists, though papers in this mixed journal are sometimes unreliable and contain mathematics of the sort probably not usable in physics but also not publishable in math journals. Djokovic and his collaborators are more reliable than most, fortunately, and he has written many papers using parts of Lie theory as well. One downside is the narrower perspective than found in the notes of Springer-Steinberg, for instance. But it all depends on whether you want to work over other fields or want to organize the classes/orbits more conceptually.

Here is a MathSciNet reference: MR708648 (85g:15018) 15A21 (17B99 17C99)
Djokovic´,D. Zˇ . [¯Dokovic´, Dragomir Zˇ .] (3-WTRL); Patera, J. [Patera, Jirˇ´ı] (3-MTRL-R);
Winternitz, P. (3-MTRL-R); Zassenhaus, H. (1-OHS)
Normal forms of elements of classical real and complex Lie and Jordan algebras.
J. Math. Phys. 24 (1983), no. 6, 1363–1374 (review by R.C. King). Their references include the work by Burgoyne-Cushman, Milnor, Springer-Steinberg mentioned by Bruce and me.

reference request - Polynomial invariants of the exceptional Weyl groups

Keeping in mind that a generating set of invariant polynomials having the required degrees is not unique, various computations have been recorded in the literature. Those I was aware of before 1990 are listed in the references to section 3.12 of my book, but there may have been others I overlooked. A later paper of interest discusses "canonical" choices of generators, with details for the classical cases as well as dihedral groups (including $G_2$):

MR1469638 (98j:13007) 13A50 (20F55),
Iwasaki, Katsunori (J-KYUS),
Basic invariants of finite reflection groups.
J. Algebra 195 (1997), no. 2, 538–547.

Physicists usually look for very explicit expressions, though their notation and approach may be hard for mathematicians to decipher. Their interest comes from the direction of Casimir operators as Jose points out in his literature citations. But those operators live in the center of the universal enveloping algebra, which by Harish-Chandra is isomorphic to the Weyl group invariants asked about here. The complication is that expressions for Casimir operators get much more elaborate-looking in terms of the Lie algebra notation. (Also, the reflection group theory shows that polynomial invariants and degrees play a uniform role even in
non-crystallographic cases like $H_3$ and $H_4$ as well as dihedral groups which are not Weyl groups.)

ADDED: I'd emphasize that writing suitable generators (Casimir operators) for the center of $U(mathfrak{g})$ should involve a choice of PBW or other basis, though the initial approach might not start with such a basis but rather with the Killing form. However this is done (non-uniquely), it takes some care to realize from these operators a set of basic polynomial invariants for the Weyl group. The latter calculation by itself can be done much more straightforwardly, though hardly anyone has taken the trouble to write down (for example) a basic invariant polynomial in 8 variables of degree 30 for $E_8$. The 1988 paper by M.L. Mehta in Communications in Algebra seems to be a good attempt at giving a comprehensive treatment. Unfortunately, the journal itself is not so easy to access, and my own copy of the paper photographed from typescript by the journal is barely readable.

I have had less success in deciphering the physics literature, which may or may not all be mathematically reliable. In particular, I haven't yet reached any conclusions about what is in the JMP paper cited by Jose. (That journal is sometimes quite useful but can also be quite frustrating to extract information from for mathematical purposes.) My only experience has been with the literature on finite (mostly real) reflection groups and their invariants, where the degrees themselves are most important for most applications. One concrete source I should mention is the added Chapter 7 in the second edition of Grove-Benson Finite Reflection Groups (GTM 99, Springer, 1985). Their book was first developed as an advanced undergraduate text, then expanded somewhat, and gives more details than my book --- where for instance I left the computation of basic invariants for dihedral groups as an exercise.

sequences and series - Uniquely generate all permutations of three digits that sum to a particular value?

Visualizing this problem, as unique ways to hand out ninja stars to ninjas. This also shows how each larger solution is made up of its neighboring, more simple solutions.

Here is how to implement it in php: (might help you understand it too)

function multichoose($k,$n)
{
 if ($k < 0 || $n < 0) return false;
 if ($k==0) return array(array_fill(0,$n,0));
 if ($n==0) return array();
     if ($n==1) return array(array($k));
     foreach(multichoose($k,$n-1) as $in){ //Gets from a smaller solution -above as (blue)
  array_unshift($in,0);  //This prepends the array with a 0 -above as (grey)
      $out[]=$in;
     }
     foreach(multichoose($k-1,$n) as $in){ //Gets the next part from a smaller solution too. -above as (red and orange)
  $in[0]++; //Increments the first row by one -above as (orange)
      $out[]=$in;
     }
     return $out;
}

print_r(multichoose(3,4)); //How many ways to give three ninja stars to four ninjas?

Not optimal code: Its more understandable that way.

Our output:

(0,0,0,3)
(0,0,1,2)
(0,0,2,1)
(0,0,3,0)
(0,1,0,2)
(0,1,1,1)
(0,1,2,0)
(0,2,0,1)
(0,2,1,0)
(0,3,0,0)
(1,0,0,2)
(1,0,1,1)
(1,0,2,0)
(1,1,0,1)
(1,1,1,0)
(1,2,0,0)
(2,0,0,1)
(2,0,1,0)
(2,1,0,0)
(3,0,0,0)

Fun use to note: Upc relies upon this exact problem in barcodes. The sum of the whitespace and blackspace for each number is always 7, but is distributed in different ways.

//Digit   L Pattern  R Pattern  LR Pattern (Number of times a bit is repeated)
    0   0001101    1110010    2100
    1   0011001    1100110    1110
    2   0010011    1101100    1011
    3   0111101    1000010    0300
    4   0100011    1011100    0021
    5   0110001    1001110    0120
    6   0101111    1010000    0003
    7   0111011    1000100    0201
    8   0110111    1001000    0102
    9   0001011    1110100    2001

Note only 10 of the 20 combinations are used, which means the code can be read upside-down just fine. All 20 can be used however, and are in EAN13, with a bit more complexity.

http://en.wikipedia.org/wiki/EAN-13

http://en.wikipedia.org/wiki/Universal_Product_Code

http://www.freeimagehosting.net/uploads/58531735d3.png

lo.logic - Are there natural examples of mathematical statements which follow from consistency statements?

Vitali famously constructed a set of reals that is not Lebesgue measurable by using the Axiom of Choice. Most people expect that it is not possible to carry out such a construction without the Axiom of Choice.

Solovay and Shelah, however, proved that this expectation is exactly equiconsistent with the existence of an inaccessible cardinal over ZFC. Thus, the consistency statement Con(ZFC + inaccessible) is exactly equivalent to our inability to carry out a Vitali construction without appealing to AC (beyond Dependent Choice).

Thus, if $T$ is the theory $ZFC+$inaccessible, then T+Con(T) can prove "You will not be able to perform a Vitali construction without AC", but $T$, if consistent, does not prove this.

I find both this theory and the statement to be natural (even though the statement can also be expressed itself as a consistency statement). Most mathematicians simply believe the statement to be true, and are often surprised to learn that it has large cardinal strength.

---

There is another general observation to be made. For any consistent theory $T$ whose axioms can be computably enumerated, and this likely includes most or all of the natural theories you might have in mind, there is a polynomial $p(vec x)$ over the integers such that $T$ does not prove that $p(vec x)=0$ has no solutions in the integers, but $T+Con(T)$ does prove this. So if you regard the question of whether these diophantine equations have solutions as natural, then they would be examples of the kind you seek. And the argument shows that every computable theory has such examples.

The proof of this fact is to use the MRDP solution of Hilbert's 10th problem. Namely, Con(T) is the assertion that there is no proof of a contradiction from $T$, and the MRDP methods show that such computable properties can be coded into diophantine equations. Basically, the polynomial $p(vec x)$ has a solution exactly at a Goedel code of a proof of a contradiction from $T$, so the existence of a solution to $p(vec x)=0$ is equivalent to $Con(T)$. If $T$ is consistent, then it will not prove $Con(T)$, and so will not prove there are no integer solutions, but $T+Con(T)$ does prove that there are no integer solutions.

By the way, it is not true in general that if $T$ is consistent, then so is $T+Con(T)$. Although it might be surprising, some consistent theories actually prove their own inconsistency! For example, if PA is consistent, then so is the theory $T=PA+neg Con(PA)$, but this theory $T$ proves $neg Con(T)$. Thus, there are interesting consistent theories $T$, such as the one I just gave, such that $T+Con(T)$ proves any statement at all!

lo.logic - Using consistency to create new axioms in set theory

What you propose is very reasonable, of course, since when we believe in a theory T, then it is natural for us also to believe that T is consistent. And the axioms that you propose to add to ZFC formalize this process. The (philosophical) question here is, does this process somehow find a completion?

(Let me quibble with your remark that we can formalize ZFC_α for any ordinal α. We need that the ordinal α is somehow representable in the theory in order for the assertion Con(ZFC_α) to be expressible. Of course, in a countable language, we have only countably many statements, and so we must eventually run out of representable ordinals.)

The answer is that your axioms are the pre-beginnings of the large cardinal hierarchy, as hinted at by Kristal Cantwell and Dorais. If there is a (strongly) inaccessible cardinal κ, then V_κ is a model of ZFC, and so your theory ZFC₁ holds.

But I claim much more, and from a weaker hypothesis. One doesn't need an inaccessible cardinal even to know that all the expressible ZFC_α are consistent.

I claim that if there is an ω model of ZFC, then all the expressible ZFC_α are true and consistent.

To see this, suppose that M is an ω model of ZFC. This means that M has the standard natural numbers. From this, it follows that the ordinals of M are well-founded for some distance above ω, but may become ill-founded much higher up. Since M has the same natural numbers as we do in the meta-theory, it follows that M has exactly the same formulas in the language of set theory and, more importantly, exactly the same proofs. Thus, for any theory T that exists in M, it will be consistent in M if and only if it is consistent.

This is enough to perform an interesting ramping-up argument. Namely, since M is a model of ZFC, it follows that ZFC is consistent for us, and so M agrees, and so M is a model of ZFC+Con(ZFC), which is to say, of ZFC₁. Thus, ZFC₁ is consistent, and so M agrees that ZFC₁ is consistent, and so M is a model of ZFC₂. Thus, ZFC₂ is consistent, and so M agrees, and so ZFC₃ is consistent, and so on. Do you see how it works? If ZFC_α is consistent, then M will agree (if α is in M), and so ZFC_α+1 is also consistent. (And limit stages are basically free, since proofs are finite.)

So the scheme of theories ZFC_α forms a hierarchy of consistency strength that sits very low below the beginning of the large cardinal hierarchy. I think much of the sense of your question is this:

• We know by the Incompleteness theorem that no theory can prove its own consistency, and so we want to consider theories that transcend this consistency in the way you describe.

And this is exactly what the large cardinal hierarchy provides. Each level of the large cardinal hierarchy implies the consistency of the lower levels, and the consistency of the consistency and so on, iterating in the style of your questions. But the large cardinals are able to jump higher than these small steps of consistency, by finding natural axioms that imply the consistency of all iterations of the consistency process that you describe for the lower levels.

---

I just noticed the bit at the end of your question, about whether independence results also hold for ZFC_α. This is a very interesting question, and the answer is Yes, they all work just the same. The reason is that all the independence results, proved either by forcing or by the method of inner models, have the property that the resulting models have the same arithmetic truths as the original model. Since the consistency statements you are considering are arithemtic statements, they are not affected by forcing or inner models. In particular, Cohen's proof that Con(ZFC) implies Con(ZFC+¬CH) turns directly into a proof that Con(ZFC_α) implies Con(ZFC_α+¬CH). If one formalizes a version of (ZFC+¬CH)_α, it follows that it will be equivalent to ZFC_α+¬CH. And the same holds for all the other indpendence results of which I am aware.

Is there a non-trivial knot with trivial Homfly polynomial?

All I know is that in their 2003 paper Eliahou, Kauffman and Thistlethwaite
write that they did not find any links with trivial HOMFLY-PT. Although they do find links
with both trivial Jones and Alexander.

http://www.math.uic.edu/~kauffman/ekt.pdf

My guess would be there exist links with trivial HOMFLY-PT but no such knots.

Although as Qiaochu mentions it is still open whether there are knots with trivial Jones, Joergen Andersen claims on his website that there are no knots with trivial
Colored Jones. (a.k.a Jones of all the cables of the knot)

http://home.imf.au.dk/andersen/

enumerative geometry - Interaction of topology and the Picard group of Algebraic surfaces

This answer addresses the cubic surface case. It may not tell you anything you don't already know, but just in case: one way to obtain $X$, which is useful for analyzing the lines,
and also the topology, is as the blow-up of ${mathbb P}^2$ at six (generically positioned)
points.

Each point blows up to a ${mathbb P}^1$, or topologically, an $S^2$, giving the six extra
spheres in $X$ that make the $H^2$ have dimension 7 (1 from the class of a line in
${mathbb P}^2$, and the rest from the six blow-ups).

Lable the points $P_i$, and the $S^2$s on $X$ obtained by blowing up $E_i$ (for ``exceptional divisor'').

Now the other 21 of the 27 lines are constructed as follows:

(a) draw lines between distinct pairs of six points (giving 15 lines), and take the proper
transform of each of these. Here "proper transform'' means the following: if $H_{ij}$ is the line joining $P_i$ and $P_j$ (here $H$ is for "hyperplane'', which in this context
is just a line), we pull-back $H_{ij}$ to $X$, which gives the union of
a line $l_{ij}$ intersecting each of $E_i$ and $E_j$ together with $E_i$ and $E_j$. Now
subtract $E_i$ and $E_j$, to obtain just the line $l_{ij}$.

Now in the cohomology ring of $X$, we see that the pull-back of $H_{ij}$ lies in
the ``first'' copy of ${mathbb Z}$ in ${mathbb Z}^7$, namely the one coming from
the class of a line in ${mathbb P}^2$, and so $l_{ij}$ maps to the class which
is $1$ in the first copy of ${mathbb Z}$, $-1$ in the copies spanned by $E_i$ and
$E_j$, and $0$ in the other copies.

More succinctly, the cohomology class of $l_{ij}$ is a linear combination of some of
the seven classes we already know.

(b) draw a conic through five of the six points, say all but $P_i$; denote
this $C_i$. Now take the proper transform of $C_i$; i.e. pull-back $C_i$
to obtain a line $l_i$ together with the five $E_j$ ($jneq i$), and then subtract
off these five $E_j$.

I have now accounted for 6 + 15 + 6 = 27 lines (the $E_i$, the $l_{ij}$, and the $l_i$)
using just the 7 independent cohomology classes.

Since in the cohomology of ${mathbb P}^2$ the class of a conic is just twice that
of a line, we see that again the class of $l_i$ is in the span of the known classes,
and we don't get anything beyond the ${mathbb Z}^7$ we already had.

In the case of a quartic surface, the situation will be similar; if one has a line
$l$ on the surface, the class of this line in cohomology will coincide with some
(linear combinination of) class(es) that we already know.

algorithms - Checking whether a set family forms a matroid.

Here's a probabilistic approach.

First check if your set family $mathcal{I}$ is closed under taking subsets. If not, then it is not a matroid. Next assign a 'random' weight function $w: S to mathbb{R}_{+}$, to the ground set $S$. Now run the greedy algorithm. If $mathcal{I}$ is indeed a matroid, then the greedy algorithm will output a member $I$ of $mathcal{I}$ of maximum weight. However, we can directly compute the weight of each maximal (under inclusion) member of $mathcal{I}$. So, if $I$ does not have maximum weight, then $mathcal{I}$ is not a matroid. If $I$ does have maximum weight, then this does not mean that $mathcal{I}$ is a matroid, we may have just gotten lucky. So, we choose another random weight function and repeat. I haven't analyzed how many times we need to do this to be reasonably 'sure' that $mathcal{I}$ is a matroid.

ag.algebraic geometry - Reference Request for Drinfeld and Laumon Compactifications

Background

Let $X$ denote a smooth projective curve over $mathbb{C}$ and let $G$ denote a semi-simple simply connected algebraic group over $mathbb{C},$ which has associated flag variety $G/B.$

Then we can consider the variety $Maps^d(X, G/B)$ of maps from $X$ to $G/B$ of fixed degree $d$ where $d$ is an $mathbb{N}$-linear combination of coroots of $G.$ See the top of page 2 of this paper by Alexander Kuznetsov Kuznetsov for the definition of degree. The Plucker embedding of the flag variety into projective space gives an alternative formulation of $Maps^d(X, G/B)$ which can be found in section 1.2 of Kuznetsov or in this survey article of Alexander Braverman Braverman.

In general, $Maps^d(X, G/B)$ is not compact, but there is a compactification due to Drinfeld, which is referred to as the variety of quasi-maps and denoted $QMaps^d(X, G/B).$ See Kuznetsov or Braverman.

On the other hand, when $G = SL_n,$ there is a second compactfication due to Laumon. This is because when $G = SL_n,$ we have both the Plucker embedding description of the flag variety, but also the description of the flag variety as flags of vector spaces. This latter description gives another formulation of $Maps^d(X, G/B)$ but leads to a compactification known as quasi-flags. Once again, see Kuznetsov. When $n>2,$ varieties of quasi-maps and of quasi-flags are different. It turns out that quasi-flags are always smooth, while quasi-maps have singularities.

Broadening our focus somewhat, we could instead consider the representable map of stacks $Bun_B(X) to Bun_G(X),$ and note that the fiber over the trivial $G$-bundle is the union of all the $Maps^d(X, G/B)$ for all possible degrees (note that the degree just tells us which connected component of $Bun_B$ we live in).

Just as the variety of maps above was not compact, the map $Bun_B to Bun_G$ is not proper. But there exists a relative compactification of $Bun_B,$ also referred to as the Drinfeld compactification, which I will denote $Bun_B^D.$ This compactification still maps to $Bun_G,$ but the map is now proper. The fiber over the trivial bundle of this map coincides with the union of all $QMaps^d(X, G/B).$

As before, when $G = SL_n,$ there is a second compactification of $Bun_B$ which I will denote $Bun_B^L$ whose fiber over the trivial bundle coincides with the union of all the quasi-flags varieties. See this paper by Braverman and Gaitsgory BG or this follow-up paper by Braverman, Gaitsgorgy, Finkelberg, and Mirkovic BGFK for more details.

Question

In Kuznetsov, Kuznetsov proves that when $X = mathbb{P}^1$ and $G = SL_n,$ there is a map from the space of quasi-flags of degree $d$ to the space of quasi-maps of degree $d$ which is a small resolution of singularities.

Later, in BG, it is asserted that Kuznetsov proved that $Bun_B^L(X)$ is a small resolution of singularities of $Bun_B^D(X)$ for any smooth projective curve $X.$

It seems to me that there are two discrepancies here. One has to do with an arbitrary smooth projective curve versus $mathbb{P}^1.$ The second has to do with moving from the varieties of quasi-maps and quasi-flags to the stacks $Bun_B^D$ and $Bun_B^L.$

Does anyone know a reference which explains the bridge between Kuznetsov and the assertions of BG? Or perhaps this was just something clear to the experts which never warranted an explanation?

References for logarithmic geometry

I believe that Arthur Ogus has been working on a book on this topic for many years. I don't know if it (or at least some version of it) has appeared. (I looked on his web-page and found what looks like a nice set of slides from a talk, 62 pages of them, but no actual book.)

In any event, Ogus certainly has many papers on the topic. My recommendation, if you have gotten through Kato's article, would be to start reading some of Ogus's and others' articles.
A lot of them are reasonably foundational, and should be accessible if you have Kato's article under your belt. In addition to the names already mentioned in the various comments and answers, Kisin has a couple of nice papers using log-schemes on his web-page.

One nice application, arithmetic in nature (and the first place that I saw log schemes), is the paper of Coleman--Voloch on companion forms. Kisin's paper on the Galois action on the
prime-to-p-etale fundamental group is another nice application to arithmetic geometry that I know of.

algebraic groups - What is the difference between PSL_2 and PGL_2?

As Kevin says, the "right" definition of ${rm{PSL}}_n$ is as representing the quotient sheaf ${rm{SL}}_n/mu_n$, just as one defines ${rm{PSO}}(q) = {rm{SO}}(q)/Z_{{rm{SO}}(q)}$ (with $Z_G$ denoting the scheme-theoretic center of a smooth group $G$). So really, there is no difference between ${rm{PSL}}_n$ and ${rm{PGL}}_n$ when defined correctly, and likewise ${rm{PSO}}(q) = {rm{PGO}}(q)$. Personally, I avoid the notation ${rm{PSL}}_n$ like the plague, since it creates too much confusion.

Lest this seem like a flippant answer, let me point out that for a general ring $R$ with nontrivial Picard group, it is likewise not true that ${rm{PGL}}_n(R)$ is the "naive" thing either! For example, if $R$ is a Dedekind domain whose Picard group has nontrivial 2-torsion then ${rm{PGL}} _2(R)$ is generally bigger than
${rm{GL}} _2(R)/R^{times}$. And this is not a quirk with algebraic geometry. The same thing happens with Lie groups: if a smooth manifold $M$ has nontrivial 2-torsion line bundles it can and does happen that there are $C^{infty}$ maps $f:M rightarrow {rm{PGL}} _2(mathbf{R})$ which do not arise from a map to ${rm{GL}} _2(mathbf{R})$ (concretely, pulling back the quotient map ${rm{GL}} _2(mathbf{R}) rightarrow {rm{PGL}} _2(
mathbf{R})$ along $f$ yields a line bundle on $M$ that may be non-trivial).

And the "weirdness" of it all (based on experience over an algebraically closed field) is also seen by the fact that the concrete definition of the group scheme ${rm{PGL}}_n$ is as a basic affine open in the projective space of $n times n$ matrices, and we know that "points" of projective spaces are a subtle thing (compared with the case of geometric points) when the source has nontrivial line bundles (whether a scheme or manifold).

Since one cannot get by with field-valued points when doing representability arguments, the same "problem" which one sees for the naive viewpoint on ${rm{PSL}} _n$ is also relevant when doing proofs for ${rm{PGL}} _n$. The difference is that for the latter one has to work more "globally" to see the surprise because the degree-1 Zariski and fppf cohomologies for $mathbf{G} _m$ coincide (so for local rings nothing funny happens, as they have vanishing higher Zariski sheaf cohomology) whereas for the former there is already a funny thing happening for local rings and even fields (which can have nontrivial degree-1 fppf cohomology for $mu_n$). In more concrete terms, it is equivalent to define the functor ${rm{PGL}} _n$ as a quotient sheaf for either the Zariski or fppf topologies, whereas for ${rm{PSL}} _n$ one has to sheafify for the fppf topology (etale ok when $n$ is a unit on the base), and in either case the naive functor on rings (inspired by the case of algebraically closed fields) is not even a Zariski sheaf and hence beyond local rings something has to be done to get the right functor (e.g., one that is representable).

Many books on linear algebraic groups use a version of algebraic geometry that is not well-suited to the subtleties of quotient considerations (e.g., Borel uses Serre's clever method "quotient by $p$-Lie algebra" to handle quotients by infinitesimal groups without saying "infinitesimal group", and some of his quotient arguments would be much shorter if he could have used flatness systematically). In particular, Springer's book has some serious errors when the ground field is not algebraically closed (in the later parts, where he discusses $F$-reductive groups and related things). For example, he uses the incorrect argument that surjectivity of an $F$-map between smooth $F$-varieties can be checked on $F_s$-points, which is not true when $F$ is not perfect (remove an inseparable point from the affine line and consider the inclusion into the line). So be careful in that part of his book. (Some statements are false, not just proofs.)

st.statistics - statistical analysis between 3 groups of data - what is the approptiate post hoc test for them?

I measured one dependent variable between three groups of data (control, group1, group2).The result of t-test indicates significant difference between control and group 1 (p vale =.04), control and group 2(p value .01), but not significant between group 2 and group 3.Do I still need to use the post hoc test? If so, which test is the most appropriate?

gn.general topology - In a locally contractible space can we find local bases of contractible sets whose closures are locally contractible?

In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_ini x$ such that the closure of each set $overline{U_i} subset X$ is also locally contractible?

More precisely, for this question we may assume $X$ is compact and is an ANR (for the class of separable metric spaces), we can even assume that $X$ is embedded as a subspace of $mathbb{R}^n$ if that makes the question easier.

ag.algebraic geometry - Algebraic varieties which are topological manifolds

The answer from Dmitri motivates this partial answer from the topological side of the question.

It is a theorem of Mark Goresky and others that every stratified space, and in particular every complex variety $V$, has a smooth triangulation. Moreover, I would bet (although I don't know that Goresky's paper has it) that the associated piecewise linear structure is unique. This means that the PL homeomorphism type of the link of a singular point $p$ of $V$ is a local invariant. I don't know how to compute this local invariant in general, but there must be some way to do it from the local ring at $p$. There can't be a simple calculation of this invariant that is fully general. As a special case, $V$ can be the cone of a projective variety $X$. If so, then the link at the cone point $p$ is the total space of the tautological bundle on $X$. $X$ and therefore the link can be all sorts of things. If $p$ is an isolated singularity, then the type of this link is obtained by "intersecting with a small sphere", as Dmitri says.

The variety $V$ is a PL manifold if and only if the link of every vertex is a PL sphere. This is the case for the Brieskorn examples.

On the other hand, a theorem of Edwards (or maybe Cannon and Edwards) says that a polyhedron is a topological $n$-manifold (for $n ge 3$) if and only if the link of every vertex is simply connected and the link of every point is a homology $(n-1)$-sphere. In particular, the link of a simplex which is not a point does not have to be simply connected! For example, if $Gamma subseteq text{SU}(2)$ is the binary icosahedral group, then $mathbb{C}^2/Gamma$ is not a manifold, because the link of the singular point is the Poincaré homology sphere. But $(mathbb{C}^2 / Gamma) times mathbb{C}$ is a topological manifold, even though it is not a PL manifold.

So for the question as stated, you would want to combine Goresky's theorem with Edwards' theorem, and with a method to compute the topology of the link of a singular point. On the other hand, whether a variety $V$ is a PL manifold could be a more natural question than whether it is a topological manifold.

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At least in the case of isolated singularities, the possible topology of the link of a singular point has been studied in the language of complex analytic geometry rather than complex algebraic geometry. I found this paper by Xiaojun Huang on this topic. The link of the singular point is in general a strictly pseudoconvex CR manifold. This is a certain kind of odd-dimensional analogue of a complex manifold and you could study it with algebraic geometry tools. (I think that strict pseudoconvexity also makes it a contact manifold?) But the analytic style seems to be more popular, maybe because a CR manifold is not a scheme.

Sometimes, for instance in the case of a Brieskorn-Pham variety, such a CR manifold has a circle action whose quotient is a complex algebraic variety. At a smooth point, this quotient is just the usual Hopf fibration from $S^{2n-1}$ to $mathbb{C}P^{n-1}$. In the famous Brieskorn examples, the link is a topological sphere with a circle action, but the circle action yields a non-trivial Seifert fibration over an orbifold-type complex variety. On the other hand, I don't think that this circle action always exists.

mg.metric geometry - Algorithm for finding the volume of a convex polytope

Here's a fairly straightforward solution for polyhedra (3 dimensions), with running time O(v+ve), where v is the number of vertices and e is the number of edges. I suppose it could be extended to higher dimensions, but it would probably have much worse running time (I fear roughly exponential as in O(vⁿ), where n is the number of dimensions).

Let our polyhedron have n vertices, defined by their x,y,z coordinates: v₁, v₂, ..., v_n and let the lowest point be v₁ and the origin (modify the values for the others accordingly), and let it have e edges, defined by the vertices which they connect. Then, since we have coordinates for the vertices (as that is how we defined them), there must be a "ground level" plane p₀ running through the x and z axes (the y-axis being height, and the ground never having an elevation). Then, let v₂ be the point closest to the ground plane (shortest line perpendicular to the plane), and let v₃ be the next closest, etc, through v_n.

Through each of the points v₂ through v_n, draw a plane perpendicular to the ground, and let them be numbered p_m, where m is the subscript of the vertex through which it was drawn. Then, the volume of our polyhedron is equal to the sum of the volumes of the figures between the planes. We should have something resembling this:

Let the heights between the segments be h₁ through h_n-1, where height h_j is the height between planes p_j and p_j+1.

Now, through each plane, we have a polygon (or more, if the figure is concave), whose vertices' coordinates can be calculated easily as follows:
Let the edge that runs through the plane p_j have endpoints v_a and v_b. Then, the displacement vector is v_b - v_a (assuming the coordinates of v are in vector-form), and the percentage travelled up is $frac{h_j-h_a}{h_{b-1}-h_a}$. Multiply this by v_b - v_a and add to v_a to calculate the new point of intersection for that edge:
Intersection point = $(v_b-v_a)frac{h_j-h_a}{h_{b-1}-h_a}+v_a$
The area of these polygons can be determined using triangles, or a simplification of this very process in just 2 dimensions.

PlanetMath says that the volume of a prismatoid (which is the type of figure contained between sequential planes) is $hfrac{B_1 + B_2 + 4M}{6}$, where the Bs are the areas of the parallel polygons and M is the area of the midway polygon, which is exactly halfway between them (and parallel to them). Since we already know the area of each of the end polygons, and we can easily calculate the vertices of the midway polygon (using the previous paragraph's method), we can calculate the volume of the resulting prismatoids. Adding them up yields the total volume of the polyhedron.

I suppose that the only real issue in this case, then, is, via code, determining which edges run through any particular plane, but if we were to actually look at it, we could tell very easily.

A simpler version of this can be used to figure out the area of any polygon; simply draw lines through the vertices parallel to the x-axis and calculate the area of the resulting trapezoids as (b₁+b₂)/2

ac.commutative algebra - Can an infinite commutative ring have a finite (but nonzero) number of non-nilpotent zero-divisors?

No, there is no such example.

Recall that the nilradical $N$ of $R$ is the ideal of nilpotent elements. It equals the intersection of all prime ideals of $R$.

On the other hand, the set $D$ of zero-divisors of $R$ can be expressed as the union of the radicals of the annihilators of individual nonzero elements of $R$ (Atiyah-MacDonald Prop. 1.15):

$$D = bigcup_{xneq 0} sqrt{(0:x)}$$

Here $(0:x)$ is an ideal, and its radical is the intersection of all the primes containing it. Thus $D$ is a union of ideals, each of which contains the nilradical $N$. If any of these ideals $I$ properly contains $N$, then if $N$ is infinite we conclude $Isetminus N$ is also infinite (since it contains a whole coset of $N$), and hence $Dsetminus N$ is infinite.

EDIT: Here's an easier proof in a different spirit, motivated by the preceding argument.

Suppose $x,yin R$, such that $x$ is nilpotent and $y$ is a zerodivisor. I claim $x+y$ is a zerodivisor. Let $zneq 0$ be such that $yz=0$. If $xz=0$, we are done. Otherwise, let $n$ be the smallest number such that $x^nz=0$ (which happens for some $n$ since $x$ is nilpotent). Then $x^{n-1}zneq 0$ but $x(x^{n-1}z)=0$, so $(x+y)x^{n-1}z=0$. Thus $x+y$ is a zerodivisor.

Now if $y$ is not nilpotent, $x+y$ is not nilpotent since the nilradical $N$ is an ideal. It follows that the coset $N+y$ consists entirely of nonnilpotent zerodivisors, so if $N$ is infinite then there are infinitely many nonnilpotent zerodivisors.

mg.metric geometry - Unit triangles with vertices on circles

There seems to be an obvious geometric approach. Let T be the 3-torus, and take the smooth function F on it to $mathbb{R}^3$ like this: use three angles on three given circles as the parameters on T, and from the points P, Q, R on the respective circles construct F as the squares of the Euclidean distances from P to Q, Q to R, R to P. So we are interested in the cases where F maps a point of T to the lattice point (1, 1, 1). The inverse image of the lattice point is a closed subset of T. To make it finite, we need by compactness of T only to understand the derivative of F: where it is invertible the inverse function theorem will work for us. So it seems to come down to computing the derivative of F, in explicit terms of the centres and radii of the circles. (The margin here is too small for so much notation.)

Edit: I now understand the problem a bit better, having manipulated the Jacobian of F. It appears to vanish under the following (sufficient) condition. Write x(1), x(2) and x(3) for the centres of the circles, and v(1) etc. for the corresponding "velocity vectors", i.e. the tangent vectors at a given point of a circle of length given by the radius, which are what one finds as the derivative of the position of a rotating point. Key quantities are the scalar products (x(1) - x(2)).(v(1) - v(2)), and so on. Where all three of these scalar products vanish, the derivative of F is not invertible. This can be seen to happen in particular configurations where the centres are at the vertices of an equilateral triangle, and the circles have equal radius. This condition is not clearly necessary, however.

arithmetic topology - Zeta function for curves in a manifold

Motivation

In the analogy between prime numbers and knots, the prime number is thought sometimes as the circle of length $l([p]) = text{log},p$. This is so you can express the zeta function as
$$ zeta(s) = sum_{Dge0} e^{-l(D)s}$$

where the sum goes over effective divisors on $text{Spec},mathbb Z$ and length is extended there by additivity. Similarly, you can do it to rewrite Dedekind zeta function for other number fields.

Question

I wonder, what is the right analogue of above formula for a manifold with metric? Perhaps:

1. integration over all closed curves of the expression $e^{-l(D)s}$


2. summation over positive sums of classes of closed geodesics.

I think I've heard something about definition 2, but I suspect if the two above are defined correctly they will be the same. Is it possible to formalize this definition? Do different formalizations lead to the same zeta-function?

Updates

Yes, I think this should be related to Laplacians, Selberg trace formula and dynamical system zetas. What I said I've heard about definition 2 was probably the Selberg zeta, but I can't say it clearly, hence questions.

st.statistics - Question about orthogonal matching pursuit

Let y be a n-vector, X a n-by-p matrix of full rank (p < n) and b a p-vector, so that y = Xb + e, for some noise vector e. I am not sure how to show reduction of error in orthogonal matching pursuit method at some step k.

More precisely, let H_C be a projection matrix defined by the columns of X indexed by the set $C subset {1, ..., p}$ of cardinality k, i.e. $H&#95;C = X&#95;C (X&#95;C' X&#95;C)^{-1} X&#95;C'$. Squared error using columns indexed by C can be computed as $RSS(C) := y'(I&#95;n - X&#95;C)y$, where I_n is the identity matrix. Next, the procedure selects the column of X (say column j) that is not in C, so that $RSS(C cup {j})$ is minimized over j. Let $D := C cup {j}$.

My question is how to rigorously show that
$$ RSS(C) - RSS(D) = Vert((X&#95;j'X&#95;j)^{-1} (I&#95;n - H&#95;C) X&#95;j X&#95;j' (I&#95;n -H&#95;C)')(I&#95;n - H&#95;C)yVert^2 ?$$

Intuitively, y is projected into the space orthogonal to the space spanned by columns indexed by C, which gives residuals after the step k. Next, a new column is chosen so that the decrease in RSS is maximal. This decrease in RSS is computed by projecting residuals onto the space spanned by $X_j(I&#95;n - H&#95;C)$.

ag.algebraic geometry - Possible singularities of the base of a Mori fiber space

Dear Eugene,

I think I can answer one of your questions, namely about $Z$ having rational singularities. It does and here is why:

$Z$ is indeed normal by construction.

Since $f$ is a Mori-contraction, $-K_X$ is $f$-nef and $f$-big. (In fact, since the relative Picard number is $1$ it is $f$-ample). I claim that $R^if_*mathcal O_X=0$ for all $i>0$. If $f$ were birational, this would be a simple consequence of Kawamata-Viehweg vanishing. In this case we need to work a little more.

First, by Kollár's torsion-freeness theorem (see [Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki. Introduction to the minimal model problem. In Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., pages 283-360. North-Holland, Amsterdam, 1987.] for the format needed here and see the references there for proofs) these sheaves are torsion-free. (Not in general, just in this case!).

Second, let $Usubseteq Z$ be a non-empty subset over which $f$ is flat. The fibers over $U$ are Fano varieties and hence $H^i(F,mathcal O_F)=0$ for $i>0$ for a fiber $F$. I probably should have taken a resolution to start with and then one would not have to worry about singularities, but at least for a general $F$ one can say that $(F,Delta_{|F})$ is klt and hence Kawamata-Viehweg vanishing applies, so we get the above vanishing. Anyway, then by Grauert's theorem [Hartshorne, III.12] it follows that $(R^if_*mathcal O_X)_{|U}=0$ for all $i>0$. However, we have already established that these sheaves are torsion-free, so if they are zero on a non-empty open set then they are zero everywhere.

OK, so we get that $R^if_*mathcal O_X=0$ for all $i>0$. Now we are only almost there because the original definition of rational singularities would require this for a birational morphism and $f$ is decidedly not birational. But this still implies that $Z$ has rational singularities by SJK: A characterization of rational singularities.

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Now, regarding whether $Z$ can be klt, canonical, etc. One simplification I can suggest is that you do not need the $Delta'$ there. If you find a $Delta'$ such that $(Z,Delta')$ is klt, dlt, lc, etc, then so is $Z$ since by the $mathbb Q$-factoriality $Delta'$ will be $mathbb Q$-Cartier. See [Kollár-Mori, 2.35].

soft question - Yet another graph invariant: the similarity matrix

Preliminaries

Let $n in mathbb{N}$ and $v$ be a vertex of a graph $G$. Let the $n$-neighbourhood of $v$, $N_n(v)$, be the induced subgraph of $G$ containing $v$ and all vertices at most $n$ edges away from $v$.

With $epsilon(v)$ the eccentricity of $v$, $N_{epsilon(v)}(v)$ is obviously nothing but the connected component of $G$ containing $v$, so it is natural to restrict $n$ for a given $v$ to the values $0,1, ..., epsilon(v)$.

Consider for any two vertices $v$, $w$ the greatest $s = s(v,w)$ such that $N_s(v)$ and $N_s(w)$ are isomorphic [added:] with the isomorphism sending $v$ to $w$ (for short: $N_s(v) cong_{vw} N_s(w)$). If $s(v,w) = epsilon(v) = epsilon(w)$, $v$ and $w$ are conjugate. For non-conjugate vertices $v$, $w$ the number $s= s(v,w)$ reflects the size of the smallest neighbourhood that is needed to distinguish $v$ and $w$, since $N_{s+1}(v) ncong_{vw} N_{s+1}(w)$ by definition.

Let's call the positive number

$$sigma(v,w) = frac{2 cdot s(v,w)}{epsilon(v) + epsilon(w)}$$

the similarity index of $v$ and $w$.

$sigma(v,w) = 0$ indicates that $v$, $w$ have different $1$-neighbourhoods (and are maximally dissimilar), $sigma(v,w) = 1$ indicates that $v$, $w$ are conjugate (i.e. maximally similar = indistinguishable by their neighbourhoods).

The matrix $Sigma(G) = lbrace sigma(v,w) rbrace_{v,w in V(G)}$ reflects the symmetry of the graph $G$:

• If $sigma(v,w) = 1$ only if $v = w$, the graph is asymmetric.


• If the $1$'s of the matrix come in square blocks along the diagonal, these blocks indicate the orbits of the graph.

$Sigma(G)$ is a graph invariant up to matrix equivalence. (Is this the right wording?)

Definition: An $n times n$-matrix $S$
is a similarity matrix iff there is
a graph $G$ such that $S = Sigma(G)$.

Questions

Is the notion of $n$-neighbourhood treated in other contexts, maybe under another name?

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Is there already research on this concept of similarity (or a related one)?

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How might similarity matrices be
characterized (sufficient/necessary conditions)? ("A matrix is a similarity matrix if ...") Any idea?

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How will graphs with the same similarity matrix (plus same eccentricity vector) be related? (They will probably not be don't have to be isomorphic, but maybe something weaker?)

order theory - When does a Galois connection induce a topology?

Some comments, unrelated to the other answer, on what happens in the case of the Zariski topology. Here the extra axiom is

$$V(I(S_1 cup S_2)) = V(I(S_1)) cup V(I(S_2))$$

(where, to fix notation, the $S_i$ are subsets of $k^n$, $I$ sends a subset of $k^n$ to the ideal of functions vanishing on it in $k[x_1, ..., x_n]$, and $V$ sends a subset of $k[x_1, ..., x_n]$ to its zero locus in $k^n$.) Now, we have $I(S_1 cup S_2) = I(S_1) cap I(S_2)$ for abstract reasons (the adjointness implies that $V$ and $I$ both send colimits to limits, or equivalently suprema to infima), but we don't have $V(T_1 cap T_2) = V(T_1) cup V(T_2)$ for arbitrary subsets $T_1, T_2$ of $k[x_1, ..., x_n]$ for rather silly reasons, e.g. $T_1$ and $T_2$ could be disjoint but generate the same ideal. Hence the condition mentioned in Nacho Lopez's answer doesn't apply here.

At best we can hope to have $V(T_1 cap T_2) = V(T_1) cup V(T_2)$ for $T_1, T_2$ ideals, which would still be enough to get the extra axiom, but note that this fails when $k$ has zero divisors, e.g. if $k = mathbb{Q}[a, b]/(ab)$ then we could take $T_1 = (ax_1), T_2 = (bx_2)$, and then $T_1 cap T_2 = (0)$ has zero locus $k^n$ but $V(T_1) cup V(T_2)$ is much smaller, e.g. it doesn't contain $(1, 1)$. That strongly suggests it can't be true for abstract reasons: at the very least we need to use somewhere the fact that $k$ is an integral domain.

What is true for abstract reasons is that $T_1 cap T_2 subseteq T_1, T_2$, hence $V(T_1), V(T_2) subseteq V(T_1 cap T_2)$, hence $V(T_1) cup V(T_2) subseteq V(T_1 cap T_2)$. In particular, we get the inclusion

$$V(I(S_1)) cup V(I(S_2)) subseteq V(I(S_1) cap I(S_2)) = V(I(S_1 cup S_2))$$

for abstract reasons. But to get the other inclusion we need to actually do something. For example:

Suppose $v in V(I(S_1 cup S_2))$, so $f(v) = 0$ for all $f in I(S_1 cup S_2)$. In particular $g(v) h(v) = 0$ for all $g in I(S_1), h in I(S_2)$ (since $IJ subseteq I cap J$ for ideals). If $g(v) = 0$ for all $g in I(S_1)$ then $v in V(I(S_1))$; otherwise, there is some $g$ such that $g(v) neq 0$, hence from $g(v) h(v) = 0$ for all $h$ we deduce that $h(v) = 0$ for all $h$, and then $v in V(I(S_2))$.

So, abstractly, here is what we did:

• In the domain of $V$ and the range of $I$ we restricted our attention to ideals.


• On ideals we introduced a new operation, namely ideal product, satisfying $I(S_1) I(S_2) subseteq I(S_1 cup S_2)$.


• We showed that $V(IJ) subseteq V(I) cup V(J)$. (For this step we needed to use the fact that $k$ is an integral domain.)

This is enough: once these conditions are met we then have

$$V(I(S_1 cup S_2)) subseteq V(I(S_1) I(S_2)) subseteq V(I(S_1)) cup V(I(S_2)).$$

To formalize this argument, we can think of ideal product as a monoidal operation on the poset of ideals making it a monoidal category. The poset of subsets of $k^n$ also naturally has a monoidal operation given by union. The condition on $I$ above is that $I$ is a lax monoidal functor with respect to these monoidal operations, and the condition on $V$ above is that $V$ is an oplax monoidal functor with respect to these monoidal operations. All told, we get the following:

Let $F : P to Q$ and $G : Q to P$ be an antitone Galois connection between posets $P, Q$. Suppose furthermore that $P$ has finite coproducts and that $Q$ is equipped with a monoidal operation $otimes$ with respect to which $F$ is lax monoidal (where $P$ is equipped with the coproduct) and $G$ is oplax monoidal. Then the closure operator $GF$ on $P$ preserves binary coproducts.

But this is somehow unsatisfying.

differential equations - Criterion for finite time blowup of an ODE

(This used to be a comment, but I think it deserves to be an answer, after mulling over it a bit.)

I don't think your criteria are quite correct. Some counterexamples:

Let $f(x) = - x^2$, and $x(0) = -1$. This ODE blows up in finite time toward $-infty$. But $int_{-1}^infty dx / f(x) $ diverges due to the singularity at $x = 0$.

Similarly, for any $f(x)geq 0$ such that $f(0) = 0$, for any initial value $x(0) < 0$ we must have $x(0) leq x(t) leq 0$ for any $t > 0$. Hence we have global existence (no blow up). But if we take $f(x) = sqrt{|x|}$ if $|x| leq 1$ and $f(x) = x^2$ if $|x| > 1$, then $1/f(x)$ is integrable, and in particular $int_{x_0}^infty dx/f(x) < infty$ for any $x_0$.

One key point used in your examples $f(x) = x^2$ and $f(x) = sqrt{x}$ with initial data positive, is that there are no stationary points. And so for any positive initial datum the evolution eventually goes toward $xto infty$, and so the question of blowup reduces to a question of how fast that happens. And for that the integral test is a good one.

Another way of saying this is that you wanted to use the equality

$$ int_0^s dt = int_{x(0)}^{x(s)} frac{dx}{f(x)} $$

but you falsely presupposed that the end state is necessarily $x(s) = infty$.

rt.representation theory - When is a generalized Cartan matrix invertible?

Let $A=(a_{ij})$ be a generalized Cartan matrix, i.e. $a_{ij} in Z, a_{ii}=2$, $a_{ij}leq 0$ for $i neq j$ and $a_{ij}=0$ iff $a_{ji}=0.$
If $A$ is classical Cartan matrix or hyperbolic, it is known that $A$ is invertible, while if $A$ is affine it has a 1-dimensional kernel.

What is known about general (indecomposable) $A$ of indefinite type?

EDIT: If $A in Z^n$ is invertible, then one clearly can find $v,w in Z^n$
such that $$A':=
begin{pmatrix} A & v \
w^t & 2 end{pmatrix}$$

is again an indecomposable gCM. So the question should be: Given $A$ invertible, how do you produce $A'$ such that $A$ is the upper-left corner of $A'$ and $rank(A)=rank(A')$?

na.numerical analysis - Finding all roots of a polynomial

This argument is problematic; see Andrej Bauer's comment below.

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Sure. I have no idea what an efficient algorithm looks like, but since you only asked whether it's possible I'll offer a terrible one.

Lemma: Let $f(z) = z^n + a_{n-1} z^{n-1} + ... + a_0$ be a complex polynomial and let $R = text{max}(1, |a_{n-1}| + ... + |a_0|)$. Then all the roots of $f$ lie in the circle of radius $R$ centered at the origin.

Proof. If $|z| > R$, then $|z|^n > R |z|^{n-1} ge |a_{n-1} z^{n-1}| + ... + |a_0|$, so by the triangle inequality no such $z$ is a root.

Now subdivide the disk of radius $R$ into, say, a mesh of squares of side length $epsilon > 0$ and evaluate the polynomial at all the lattice points of the mesh. As the mesh size tends to zero you'll find points that approximate the zeroes to arbitrary accuracy.

There are also lots of specialized algorithms for finding roots of polynomials at the Wikipedia article.

plane geometry - Why does the area function of a parallelogram have a nonintuitive geometric solution?

I was reading a blog post on a simple derivation of the cross product. I learned how to determine the area of a parallelogram enclosed by two vectors $A$ and $B$.

First, here is the proof of the solution ($area = A_x B_y - B_x A_y$)

And here's the geometric implication of the solution.

It bewilders me that the geometric solution is simple but nonintuitive.

One commenter (Tim Poston) on the blog offered an explanation by using the following lemma, which he states can be easily shown using elementary plane geometry, without the use of right angles or base*height to find areas:

$C(u+w,v) = C(u,v) + C(w,v)$

where $u, v, w$ are arbitrary vectors, and $C$ is the area function of the parallelogram between 2 vectors.

Here's my illustration of an example implication of this lemma (visually simplified to triangles without loss of generality)

I was unable to prove this lemma, so I couldn't follow the argument. Any thoughts or direction with this lemma or the original problem would be appreciated.

ag.algebraic geometry - When does direct image with proper support have a right adjoint?

I assume the question holds in contexts where we can glue open immersions and proper morphisms to produce $f_!$ for $f$ separated of finite type. In particular, we shall have $f_!=f_ast$ for $f$ proper.

Non derived setting ---
If $f:Xrightarrow Y$ is proper, one can ask if $f_ast$ has a right adjoint.
Note that, if $Y=mathrm{Spec}(k)$ for an algebraically closed field, then $f_ast$ is essentially the global section functor on $X$ (if we work with reasonnable topologies like Zariski, Nisnevich, or étale). This suggests that $f_ast$ does not have a right adjoint in general (otherwise, it would be exact, and this would say that proper schemes have not any interesting cohomology).
If $f$ is quasi-compact and quasi-separed, then $f_ast$ preserves filtered colimits (this is the case if $f$ is proper). Hence the obstruction for $f_ast$ to have a right adjoint is only its left exactness. There are still cases where $f_ast$ has a right adjoint at the level of sheaves: when f is a closed immersion (for the Zariski topology), and when f is finite (for the Nisnevich topology and for the étale topology): the right exactness of $f_ast$ is proved by contemplating the behaviour of $f_ast$ on stalks of $Y$: for the Zariski topology, one uses that the points are the local rings, and that any quotient of a local ring is local; similarly, for the Nisnevich (resp. étale) topology, one uses the fact that the points are the henselian (resp. strictly henselian) rings, and that any finite extension of an henselian ring is still henselian. So in conclusion, it seems that we might expect $f^!$ to exists when $f$ is an immersion (for the Zariski topology), or when $f$ is quasi-finite (for the Nisnevich or étale topology).

Derived setting --- As, for an open immersion $j, j_!$ is still the extension by zero, it always has a right adjoint $j^ast$. The problem is then again to get an adjoint of $f_!=f_ast$ for $f$ proper. The problem is similar to the one in the non derived case, except that the obstruction is smaller: as $f_ast$ is an exact functor between well generated triangulated categories (in the sense of Neeman; for this we have to work with unbouded complexes), $f_ast$ has a right adjoint if and only if it preserves small direct sums (this is an instance of the Brown representability theorem). There is a nice sufficient condition for this, which I will recall. Remember that an object $X$ in a triangulated category is compact if, for any integer $n$, the functor $mathsf{Hom}(X[n],-)$ preserves small direct sums. A triangulated category $T$ is compactly generated if it admits small sums, and if there exists a small generating family $G$ in $T$ which consists of compact objects (i.e. all the element of $G$ are compact in $T$, and for an object $M$ in $T$, one has $M=0$ iff $mathsf{Hom}(X[n],M)=0$ for any integer $n$ and any $X$ in $G$). A compactly generated triangulated category is a basic example of a well generated triangulated category. In particular, if $F:Trightarrow T'$ is functor between compactly generated triangulated categories, then it has a right adjoint iff it preserves small direct sums. The good news are that, for such an $F$, a sufficient condition for this is the following: assume that $F$ has a left adjoint $L:T'rightarrow T$ and that there exists a small family $G'$ of compact generators of $T'$ such that $L$ sends the elements of $G'$ to compact objects of $T$. Then $F$ preserves small direct sums (hence, has a right adjoint).

If we come back to our functor $f_ast$ (with $f$ proper), a sufficient condition for $f^!$ to exist is then that our derived categories of sheaves are compactly generated and that the representable sheaves form a family of compact generators: as $f^ast$ obviously preserves representables, these assumptions give the existence of $f^!$. As for the conditions under which representables are compact or not (which is a finiteness assumption on $X$ and $Y$ themselves), some sufficient conditions have already been suggested here (the fact that representables form a generating family is obviously always true).

lo.logic - Abstract Thought vs Calculation

A beautiful classical example from Functional Analysis is the Hausdorff moment problem: characterize the sequences
$m:=(m_0,m_1,dots)$ of real numbers that are moments of some positive, finite Borel measure on the unit interval $I:=[0,1]$:
$$m_k=int_I x^kdmu(x).$$
A necessary condition immediately comes from $int_I x^{ j}(1-x)^{ k} dmu(x)geq0$, and is expressed saying that $m$ has to be a "completely monotone" sequence, that is
$$(I-S)^k mge0,$$
where $S$ is the shift operator acting on sequences (in other words, the $k$-th discrete difference of $m$ has the sign of $(-1)^k$: $m$ is positive, decreasing, convex,...). The nontrivial fact is that this is also a sufficient condition, thus caracterizing the sequences of moments. Moreover, the measure is then unique.

I'll quote two proofs, both very nice. The first is close to the original one by Hausdorff; the second is a consequence of the Choquet's theorem.

Proof I, with computation (skipped). Bernstein polynomials
$$B_nf(x):=sum_{k=0}^n fbig(frac{k}{n}big)Big( {n atop k}Big)x^k(1-x)^{n-k}, qquad ninmathbb{N},; fin C^0(I), ; xin I ,$$
define a sequence of linear positive operators strongly convergent to the identity
$$B_n:C^0(I)to C^0(I) .$$

Therefore the transpose operators $$B_n ^*:C^0(I)^ *to C^0(I)^ *$$ give a sequence of operators weakly convergent to the identity. If you write down what is $B_n^ *(mu)$ for a Radon measure $muin C^0(I)^ *$ you'll observe that it is a linear combinations of Dirac measures located at the points ${k/n}_{0leq kleq n}$, and with coefficients only depending on the moments of $mu$. This gives a uniqueness result and a heuristic argument: if $m$ is a sequence of moments for some measure $mu$, then $mu$ can be reconstructed by its moments as a weak* limit of discrete measures $mu_n:=B_n^*(mu)$. This observation leads to a constructive solution of the problem. Indeed, given a completely monotone sequence $m$, consider the corresponding sequence of measures $mu_n$ suggested by the experssion of $B_n^*(mu)$ in terms of the $(m_k)$. Due to the assumption of complete monotoniticy they turns out to be positive measures, and with some more computations one shows that they converges weakly* to a measure $mu$ with moment's sequence $m$.

Proof II, no or little computation. Completely monotone sequences with $m_0=1$ are a closed convex, thus weakly* compact and metrizable subset $M$ of $l^infty$. A one-line, smart computation shows that the extremal points of $M$ are exactly the exponential sequences, $m^{ (t)}:=(1,t,t^2,dots)$, for $0leq t leq1$ (these turn out to be the moments of Dirac measures in points of $I$, of course). By the Choquet's theorem, for any given $min M$ there exists a probability measure on $mathrm{ex}(M),$ that we identify with $I,$ such that $m=int_I m^{ (t) } dmu(t).$ But this exactly means $m_k=int_I t^{ k} d mu(t)$ for all $kinmathbb{N}.$

set theory - Are there as many real-closed fields of a given cardinality as I think there are?

Hi Pete!

There's been a lot of study of this and similar problems. I believe that Shelah's theorem, from his 1971 paper "The number of non-isomorphic models of an unstable first-order theory" (Israel J. of Math) answers your question about real closed fields in the positive.

The best big result on such questions that I know of is in the 2000 Annals paper "The uncountable spectra of countable theories." by Hart, Hrushovski, Laskowski.

To answer the question on real closed fields specifically (and somewhat cautiously since I'm not a model-theorist):

The theory of real closed fields is a complete first order theory, with countable language. It is an unstable (an easy fact, I think, and explained better on wikipedia than I could explain) theory as well. Hence Shelah's result applies, and the bound $2^kappa$ is realized as you surmised.

Bonus points should go to Shelah (and perhaps also to Hart, Hrushovski, Laskowski, whose paper mentions the result of Shelah and proves other things) for proving that this bound is realized (for uncountable cardinals), except for theories $T$ which have all of the following properties:

1. $T$ has infinite models.


2. $T$ is superstable.


3. $T$ has prime models over pairs.


4. $T$ does not have the dimensional order property.

I have no clue what the fourth property means. But there are plenty of non-superstable theories to which Shelah's theorem applies, and hence which realize your bound (for uncountable cardinals).

For countable cardinality, I think there are still some open problems about how many non-isomorphic models there can be of a given theory, with cardinality $aleph_0$.

set theory - Is the collection of isomorphism classes of groups a proper class?

All the other answers are more than satisfactory. I have some lecture notes on basic set theory which also answer these questions, so I might as well post links to them:

To see that the cardinals do not form a set, see Fact 20 on page 10 of

http://math.uga.edu/~pete/settheorypart1.pdf

This first handout is super-naive set theory (countable choice is assumed without comment) which is meant to be accessible to an undergraduate math major. In particular I don't say "cardinal" there but rather spell things out in a more explicit way.

This seems a little simpler than the Burali-Forti paradox (which on the other hand is manifestly choice-free), for which see Section 1.4 of

http://math.uga.edu/~pete/settheorypart3.pdf

To see that there is an X of any given infinite cardinality (where X is: a field, a noncommutative group, etc.) see

http://math.uga.edu/~pete/settheorypart4.pdf

My argument for the case of fields -- which implies that of groups by taking the additive group of the field -- uses (only) that for any infinite cardinal $kappa$,
$kappa times aleph_0 = kappa$. I'm pretty sure that this is a lot weaker than AC.

If you know the Skolem-Lowenheim theorem in model theory, then it is silly to do all of these cases individually: a consistent theory in a language of cardinality $kappa$ admits models of any infinite cardinality which is greater than or equal to $kappa$. This is equivalent to AC (see Bell and Slomson), but the special case of countable languages is presumably not.

graph theory - Why do we use priority queues when implementing Dijkstra's Algorithm?

Consider the running time for adding a new element to a sorted list, keeping the list sorted. If the list is an array, you can find the insertion point in $O(log n)$ steps, where $n$ is the current size of the list. But then you have to make room for the new element by shifting all the elements behind it one step back, and that takes $n/2$ steps on the average. Or you could use a linked list, but then binary search is not available, and it takes $n/2$ steps (on the average) to find the insertion point (and $O(1)$ to do the actual insertion). For a properly implemented priority queue, insertion is $O(log n)$, and so is fixing up the queue after removing the smallest member.

ag.algebraic geometry - Why "smooth Gelfand duality" does not involve a topology on the algebras?

The following question naturally originates from this question
and this one.

While the usual $C^{0}$ Gelfand duality involves a topology on the function algebras considered (it relates compact Hausdorff topological spaces to unital $C^{*}$-algebras, which in particular are Banach algebras), why the "smooth Gelfand duality" seems, according to what I understood from the above questions, to see only the "pure" algebraic structure of certain algebras over $mathbb{R}$ ?

Edit: I've just read the introduction of this. The topology actually enters the picture, but not in the form of a structure of topological algebra on the function spaces that locally model those $C^{infty}$-differentiable spaces; it enters the picture when defining a "differentiable algebra" as the quotient of the algebra of smooth functions on $mathbb{R}^n$ by a Fréchet-closed ideal.

But a question still stands: would it be possible to define compact Hausdorff topological spaces in the analogous way?
Perhaps the answer is "no because of a lack of a universal local model $C^{0}(...)$", but "yes in the case of topological manifolds".
Does it make sense?

soft question - Most helpful math resources on the web

Sloane's OEIS has already been mentioned.

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

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

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

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

na.numerical analysis - Numerical integration over 2D disk

You might like this one, I do not know if the entire link will fit.

Wait, it is available for download from her website!

http://www.math.tamu.edu/~gpetrova/

Journal of Approximation Theory
Volume 104, Issue 1, (May 2000)

Uniqueness of the Gaussian Quadrature for a Ball
Pages 21-44

Borislav Bojanov and Guergana Petrova

Department of Mathematics, University of Sofia, Boulevard James Boucher 5, 1164, Sofia, Bulgariaf1

Department of Mathematics, University of South Carolina, Columbia, South Carolina, 29208, U.S.A., f2
Received 8 June 1999;
accepted 22 October 1999. ;
Available online 26 March 2002.

Abstract

We construct a formula for numerical integration of functions over the unit ball in Image d that uses n Radon projections of these functions and is exact for all algebraic polynomials in Image d of degree 2n−1. This is the highest algebraic degree of precision that could be achieved by an n term integration rule of this kind. We prove the uniqueness of this quadrature. In particular, we present a quadrature formula for a disk that is based on line integrals over n chords and integrates exactly all bivariate polynomials of degree 2n−1.

Author Keywords: Gauss quadrature formula; orthogonal polynomials; highest degree of precision

Different article by same people:

http://www.math.tamu.edu/~gpetrova/CAM7238.pdf

Best Algebraic Geometry text book? (other than Hartshorne)

I've found something extraordinary and of equally extrordinary pedigree online recently. I mentioned it briefly in response to R. Vakil's question about the best way to introduce schemes to students. But this question is really where it belongs and I hope word of it spreads far and wide from here.

Last fall at MIT, Micheal Artin taught an introductory course in algebraic geometry that required only a year of basic algebra at the level of his textbook. The official text was William Fulton's Algebraic Curves, but Artin also wrote an extensive set of lecture notes and exercise sets. I found them quite wonderful and very much in the spirit of his classic textbook(By the way, simply can't wait for the second edition.).

Not only has he posted these notes for download, he's asked anyone working through them to email him any errors found and suggestions for improvements.All the course materials can be found at the MIT webpage. I've also posted the link at MathOnline, of course.

I don't know if most of the hardcore algebraic geometers here would recommend these materials for a beginning course. But for any student not looking to specialize in AG, I can't think of a better source to begin with. That's just my opinion. But it certainly belongs as a possible response to this question. Then again, it may be too softball for the experts,particularly those of the Grothendieck school.

Here's keeping our fingers crossed that this is the beginning of the gestation of a full blown text on the subject by Artin.

nt.number theory - Is there an elementary proof of a result about the parity of the period of the repeating block in the continued fraction expansion of square roots

This doesn't answer your question, but it might gives some intuition as to why this is true. Given the fact that there exists a solution to the equation $x^2-Py^2=-1$ when $Pequiv 1(mod 4)$, one sees that the matrix
$$
begin{pmatrix}x & Py \ y & xend{pmatrix}
$$
fixes $pmsqrt{P}$ (under the action of $PGL_2(mathbb{Z})$ on $mathbb{RP}^1=partial_{infty} mathbb{H}^2$). The conjugacy class of a primitive matrix in $GL_2(mathbb{Z})$ (which is not reducible or finite order) is determined by the closed geodesic on the modular orbifold that it represents. This in turn is determined by a sequence of triangles which the geodesic crosses in the Farey graph:

These triangles come in bunches sharing a common vertex, where the number in each bunch corresponds to coefficients of the continued fraction expansion.
The matrix is conjugate to $$pm left[begin{array}{cc}1 & a_1 \ 0 & 1end{array}right] left[begin{array}{cc}1 & 0 \ a_2 & 1end{array}right] cdots left[begin{array}{cc}1 & a_{2n} \ 0 & 1end{array}right]$$ if the determinant is 1, and to
$$pm left[begin{array}{cc}1 & a_1 \ 0 & 1end{array}right] left[begin{array}{cc}1 & 0 \ a_2 & 1end{array}right] cdots left[begin{array}{cc}1 & 0 \ a_{2n-1} & 1end{array}right] left[begin{array}{cc}0 & 1 \ 1 & 0end{array}right] $$ if the determinant is $-1$.

[Remark: the labels in this figure don't quite correspond to the matrices - it should be $a_i$'s instead of $alpha_i$'s, and $alpha_{pm}$ should be $pmsqrt{P}$]

The number of such factors corresponds precisely to the period of the continued fraction expansion of fixed points of the matrix, since the closed geodesic is asymptotic in $mathbb{H}^2$ to the geodesic connecting $infty$ to $sqrt{P}$, whose Farey sequence gives rise to the continued fraction expansion of $sqrt{P}$.
This number is even if and only if the matrix is orientation preserving, which is if and only if the determinant is 1. So the determinant is $1$ if and only if the continued fraction has even period, and the determinant is $-1$ if and only if the continued fraction has odd period, corresponding to $Pequiv 1(mod 4)$.

set theory - When $2^alpha = 2^beta$ implies $alpha=beta$ ($alpha,beta$ cardinals)

François gives the correct affirmative answer. For the negative side, the usual method of proving that the negation of the Continuum Hypothesis is consistent with ZFC is to use the method of forcing to add Aleph₂ many Cohen reals, so that 2^ω = ω₂ in the forcing extension V[G]. In this model V[G], it is also true that 2^ω₁ = ω₂. Thus, this model shows that it is not necessarily true that different-sized sets have different sized power sets. The case of symmetric groups is likely more interesting than free groups, because in this model, the symmetric groups S_ω and S_ω1 have the same cardinality ω₂. Nevertheless, these two groups are not isomorphic, as explained in this MO question.

The general answer about what can be true for the continuum function κ --> 2^κ is exactly provided by Easton's Theorem. This remarkable theorem states that if you have any class function E, defined on the regular cardinals κ, with the properties that

• κ ≤ λ implies E(κ) ≤ E(λ)


• κ < E(κ)


• κ < Cof(E(κ))

then there is a forcing extension V[G] in which 2^κ = E(κ) for all regular cardinal κ. In particular, this shows that the sizes of the power sets (on regular cardinals) are restricted to obey only and exactly the hypotheses listed explicitly above. Each of these properties corresponds to a well-known fact about cardinal exponententiation.

Using Easton's theorem, we can build models of set theory where 2^κ = κ⁺⁺ for all regular κ. The added power of the Woodin/Foreman result mentioned by François is that they also get this for singular cardinals.

The point now is that there are innumerable examples provided by Easton's theorem that satisfy your hypothesis that the continuum function is one-to-one. If one begins with a model of GCH and selects any injective Easton function E, then the resulting model of set theory V[G] will have E as it's continuum function κ --> 2^κ = E(κ) for regular κ, and the GCH will continue to hold at singular κ, preserving injectivity. So one is quite free to satisfy your hypothesis while having any kind of crazy failures of GCH.

pr.probability - Is there a theory on two sequences of measures weakly asymptotic to each other?

Suppose $(P_n)_{nge 1}$ is a sequence of probability measures on a metric space $E$. Everybody knows what weak convergence of $P_n$ to a measure $P$ is.

Instead, let $(Q_n)_{nge 1}$ be another sequence of probability measures, and I want to ask the following question: when are $Q_n$ and $P_n$ asymptotically close in the sense of weak convergence? Of course, I can say that it means that $d(P_n,Q_n)to 0$ where $d$ is one of the metrics compatible with weak convergence of measures. However, is there a useful theory helping to prove $d(P_n,Q_n)to 0$? For the usual case i.e. weak convergence $P_nto P$, there is a number of technologies based on tightness, and here we may have a situation where none of the two sequences is tight, but they still "converge to each other".

This could be useful if one of the two sequences is much simpler or better understood than the other one, and I am mostly interested in situations where the metric space $E$ is a functional space like $C$ or $D$.

UPD. It turns out that "merging" is the keyword and there is some literature on the issue
(thanks to Mark Meckes for pointing to one of these papers), but it looks like there is still no tool to prove merging in functional spaces.

It seems that the most recent paper related to this topic is

Davydov & Rotar: On asymptotic proximity of distributions. J. Theoret. Probab. 22 (2009), no. 1, 82--98.

gt.geometric topology - Failure of smoothing theory for topological 4-manifolds

This is in response to John's addendum. As I understand it, one has the following hierarchy:

• Any Poincare complex $X$ has a Spivak normal spherical fibration $S$.


• If $X$ carries a topological manifold structure then $S$ has a microbundle reduction.


• If $X$ carries a smooth manifold structure then $S$ has a vector bundle reduction refining the microbundle reduction.

I'm going to concentrate on simply connected Poincare 4-complexes $X$ with even intersection form. These have Kirby-Siebenmann smoothing obstruction $ksin H^4(X;mathbb{Z}/2)=mathbb{Z}/2$ equal to $sigma(X)/8$ mod 2, where $sigma$ is the signature. This is just the obstruction coming from Rochlin's theorem: $sigma$ is divisible by $16$ if $X$ is smoothable.

Freedman tells us that $X$ has a unique topological manifold structure, and hence $S$ has a canonical microbundle structure. So, to ask whether there is a vector bundle reduction of the microbundle is the same as asking whether $S$ has a vector bundle reduction.

Let $BG$ be the classifying space for stable spherical fibrations. To solve the obstruction-theory problem of lifting $Xto BG$ to a map $Xto BO$, we need to know the low-dimensional homotopy groups of $BO$ and $BG$ - specifically, whether $pi_i(BO)to pi_i(BG)$ is surjective. I read off from a table in Ranicki's book "Algebraic and geometric surgery" that this is so for $i=1$ and $2$, but that $pi_3(BG)=mathbb{Z}/2$ whereas $pi_3(BO)=0$. So there is an obstruction $oin H^4(X;mathbb{Z}/2)$ to finding a vector bundle reduction.

I'm a bit nervous of $ks$ due to my ignorance of topological manifold theory, but I think it should then be the case that $o=ks$ (they seem to be similar beasts; I'm thinking of $ks$ as coming from $pi_3 (BTOP)$, where $o$ comes from $pi_3(BG)$). What I actually want to use is the corollary, which if true should have a direct proof - that $o=sigma/8$ mod 2. Anyone?

Given any unimodular matrix $Q$, I can build a Poincare 4-complex with $Q$ as its intersection matrix (plumb together disc-bundles over $S^2$ according to $Q$, cone off the homology 3-sphere boundary). If it's correct that $o=sigma/8$, then when $Q=E_8$, I get a complex with no tangent bundle, whereas when $Q=E_8oplus E_8$ I get a complex which has a tangent bundle but which is not smoothable by Donaldson's diagonalizability theorem.

noncommutative geometry - Gluing perverse sheaves?

Ryan Reich has a paper called Notes on Beilinson's "How to glue perverse sheaves" -- it's also available off of Dennis Gaitsgory's Geometric Representation Theory page which is an amazing resource in the area. The paper by MacPherson and Vilonen, Elementary construction of perverse sheaves. Invent. Math. 84 (1986), no. 2, 403--435
provides another perspective on the same construction of perverse sheaves by gluing.

I don't know of a way to construct perverse sheaves in any noncommutative generality, and it seems like an unlikely proposition to me. You can certainly define modules over the de Rham complex and the like in great generality, but the condition of constructibility (not to mention the perverse t-structure) seems very commutative, involving restricting supports and dimensions etc. You can define holonomicity homologically, so once you have a notion of D-modules you can almost reverse engineer perverse sheaves (not sure how you'd describe regularity truly noncommutatively), but I don't see how you'd find a non-definitional version of the Riemann-Hilbert correspondence unless you're secretly in an almost commutative setting. (For that matter D-modules themselves are an almost-commutative notion..)

ag.algebraic geometry - Number of sheaves in a full exceptional collection

As a far as I know, there is no obvious relation between the number of objects in a full exceptional collection and the dimension of $X$, unless you impose extra conditions on the full exceptional collection, as in Bondal and Polishchuk's 'Homological properties of associative algebras', where they prove that when the collection is 'geometric', then ${rm rank}(K_{0})(X)={rm dim} X + 1$. Their hypotheses apply for example to $mathbb{P}^{n}$, where the standard full exceptional collection is $mathcal{O},mathcal{O}(1), ..., mathcal{O}(n)$.

But for examples like $mathbb{P}^{1} times mathbb{P}^{1}$ these hypotheses do not apply, since the standard full exceptional collection is $mathcal{O}, mathcal{O}(1,0), mathcal{O}(0,1), mathcal{O}(1,1)$, so ${rm rank}(K_{0}(X))={rm dim} X + 2$.

In general, a variety $X$ will not admit a full exceptional collection of objects in its derived category of coherent sheaves, for instance because $K_{0}(X)$ is usually not free of finite rank. And some varieties cannot carry any exceptional objects at all. For instance, if $X$ is smooth projective with trivial canonical bundle, then if you checked that $Hom(E,E)$ were one dimensional as needed for $E$ to be exceptional, Serre duality would imply that $Ext^{n}(E,E)$ is also one dimensional.

To me it seems more natural to ask for a nice 'compact generator' of the derived category of a variety. This is a perfect complex $E$ on $X$ (one locally quasi-isomorphic to a finite length complex of vector bundles) that generates the category $Perf(X)$ of perfect complexes in finitely many steps for each object or $D_{QCoh}(X)$, the unbounded derived category of quasi-coherent sheaves, in infinitely many steps. This is a generalization
of the sum $E$ of the objects in a full exceptional collection, with no extra conditions imposed on the $Ext$ algebra of $E$.

Such compact generators exist in great generality, due to work of Bondal and van den Bergh. Given such a compact generator and letting $A=RHom(E,E)$, the derived endomorphism dg algebra of $E$, we get equivalences $RHom(E,?): D_{QCoh}(X) rightarrow D(A)$ and $RHom(E,?): Perf(X) rightarrow Perf(A)$, just as we would if $E$ were the sum of objects in a full exceptional collection.

One could ask if you can find concrete examples of such compact generators, and indeed you can. If $X$ is projective of dimension $d$, let $L$ be an ample, globally generated line bundle. Then it is easy to show using a standard criterion for compact generation that
$E=mathcal{O}oplus L oplus cdots oplus L^{d}$ is a compact generator of the derived category of $X$.

So one can always say that if $X$ is projective, then the minimal number of line bundles (not necessarily exceptional) needed to generate the derived category of $X$ is bounded by ${rm dim} X + 1$.

I would also guess that fewer line bundles will not suffice, but I'm not sure.

A learning roadmap for algebraic geometry

FGA Explained. Articles by a bunch of people, most of them free online. You have Vistoli explaining what a Stack is, with Descent Theory, Nitsure constructing the Hilbert and Quot schemes, with interesting special cases examined by Fantechi and Goettsche, Illusie doing formal geometry and Kleiman talking about the Picard scheme.

For intersection theory, I second Fulton's book.

And for more on the Hilbert scheme (and Chow varieties, for that matter) I rather like the first chapter of Kollar's "Rational Curves on Algebraic Varieties", though he references a couple of theorems in Mumfords "Curves on Surfaces" to do the construction.

And on the "algebraic geometry sucks" part, I never hit it, but then I've been just grabbing things piecemeal for awhile and not worrying too much about getting a proper, thorough grounding in any bit of technical stuff until I really need it, and when I do anything, I always just fall back to focus on varieties over C to make sure I know what's going on.

EDIT: Forgot to mention, Gelfand, Kapranov, Zelevinsky "Discriminants, resultants and multidimensional determinants" covers a lot of ground, fairly concretely, including Chow varieties and some toric stuff, if I recall right (don't have it in front of me)

gn.general topology - What does the property that path-connectedness implies arc-connectedness imply?

A space X is path-connected if any two points are the endpoints of a path, that is, the image of a map [0,1] to X. A space is arc-connected if any two points are the endpoints of a path, that, the image of a map [0,1] to X which is a homeomorphism on its image. If X is Hausdorff, then path-connected implies arc-connected.

I was wondering about the converse: What properties must X have if path-connected implies arc-connected? In particular, what are equivalent properties?

ag.algebraic geometry - Frobenius splitting of tangent bundles of flag varieties

BACKGROUND

Let $X$ be a variety over an algebraically closed field $k$ of positive characteristic $p$. Let $F : X to X$ denote the absolute Frobenius morphism, i.e. the morphism that is the identity on points and whose comorphism is the $p^{th}$ power map. We say that $X$ is Frobenius split if the $p^{th}$ power morphism $ mathcal O_X to F_* mathcal O_X $ splits (in the category of sheaves of $mathcal O_X$-modules).

Let $G$ be a semisimple algebraic group over $k$ and let $B subseteq G$ be a Borel subgroup. A number of varieties associated to $G$ are known to be Frobenius split: the flag variety $G/B$, the cotangent bundle $T^*(G/B)$, etc.

QUESTION

Is it known in general whether or not the tangent bundle $T(G/B)$ to $G/B$ is Frobenius split? I think that the splitting of $T(G/B)$ would be implied by a splitting of $X times X$ that maximally compatibly splits the diagonal, so by recent work of Lauritzen and Thomsen (in fact, the proof in type $C$ was just posted on the arXiv), I believe one does know that $T(G/B)$ is Frobenius split in types $A$ and $C$. Does anyone know of any other work exploring this topic?

soft question - Describe a topic in one sentence.

Representation theory of Lie groups: there is a whole world between $mathrm{Sym}^n V$ and $wedge^n V$. (Okay, this is an oversimplication - I am talking about the representations of $mathrm{GL}left(Vright)$ here, but this is the fundament of all other classical groups.)

Constructive logic: if you can't compute it, shut up about it. (At least some forms of constructive logic. Brouwer seemed to have a different opinion iirc.)

Homological algebra: How badly do modules fail to behave like vector spaces?

Gröbner basis theory: polynomials in $n$ variables can be divided with rest (at least if you have some $Oleft(N^{N^{N^{N}}}right)$ of time)

Finite group classification: what works for Lie groups will surely be even simpler for finite groups, right? ;)

Algebraic group theory: In order to differentiate a function on a Lie group, we just have to consider the group over $mathbb Rleft[varepsilonright]$ for an infinitesimal $varepsilon$ ($varepsilon^2=0$).

Semisimple algebras: The representations of a sufficiently nice algebra mirror a structure of the algebra itself, namely how it breaks into smaller algebras.

$n$-category theory: all the obvious isomorphisms, homotopies, congruences you have always been silently sweeping under the rug are coming back to have their revenge.

Modern algebraic geometry (schemes instead of varieties): let's have the beauty of geometry without its perversions.

How many of these did I get totally wrong?