pr.probability - Probability vertices are adjacent in a polygon

With regard to my original question:

A subset of k vertices is chosen from the vertices of a regular N-gon. What is the probability that two vertices are adjacent?

I suppose that the responses that were elicited to my question were to be expected. You see, I have been in your position, and thought what you did, on many occasions. But mostly not in the area of mathematics.

By way of providing background, I am not a student at all. In fact, I am a biochemist and part-time university instructor. I have often provided answers to students who post chemistry/biochemistry questions (samples will be provided upon request). And, like you, I hope that I have not (or had not!) become a vehicle for students to avoid thinking through THEIR chemistry homework assignments.

The above question, believe it or not, comes from the local, Boston-based television program "extrahelp", hosted by a somewhat sarcastic character who went by the name "Mr. Math". It was intended for the K-12 demographic, but, according to the story behind the video, an M.I.T. student was listening, and asked the question, ostensibly to give this man his comeuppance. The video is no longer available online, but I can send you a copy, if you don't mind that it is 25.1 MB, and that I don't have access to any FTP server at the university where I teach. It is hilarious.

When I saw this video (and after I stopped laughing), I was interested by the question itself. And since at least one of you asked for any work that I have done on my own, here is as far as I got before I made my original post:

1) N must, of course, be a positive integer >= 3. k must be a positive integer <= N.
2) When k = 1, the solution is trivial (p = 0). In fact, the non-trivial values of k are: 2 <= k <= (N 2), where "" is integer division. For other values of k, p = 1.
3) For non-trivial values of k, the denominator is C(N,k).
4) For k = 2, the numerator is N.
5) For k = 3, the numerator is N(N-k).
6) For N even, and k = N / 2, the numerator is N – 2. For N odd, and k = N 2, the numerator is C(N,k) – N.
7) For k = (N 2) -1, the numerator may be C(N,k) – N(N-k).

Where I am having trouble is, obviously, getting from here to a general solution. It has been suggested to me that I take the approach of finding the general expression for the probability of NOT selecting adjacent vertices, but the general answer of p = 1 - [C(N-k,k) / C(N,k)] is not confirmed by the examples that I am sure about (that is, where N is so small that the answers can be confirmed by enumeration)

I am sure that you would agree that, by now, enough time has passed that, if this really were a homework question, that the due date for such homework would have passed. Further, I hope that I have convinced you beyond a reasonable suspicion that a) this is definitely not a homework question, and b) that I have worked on the problem myself to such a degree that I haven't just passed this off on the respondents without making some effort.

real analysis - Decomposition of Hölder continuous functions

I have carried out the suggestion in the last paragraph of Yemon Choi's answer. Choose $phiin C^infty(mathbb{R})$, $phige0$ and $int_{mathbb{R}}phi(x)dx=1$, and let $phi_R(x)=Rphi(Rx)$. Define

$$ f=phi_Rstareta,quad g=eta-f.$$

Then it is easy to see that

$$ |f|_{C^n}=O(R^{n-alpha}),quad |g|_infty=O(R^{-alpha}),$$

but this is not what you are asking for.

My feeling is that the constant $C$ must show some dependence on $n$.

In response to your last comment, let me prove the estimate on $|f|_{C^n}$. We have

$$f^{(n)}=(phi_R)^{(n)}stareta=R^n(phi^{(n)})_Rstareta.$$

Since $(phi^{(n)})_R$ has mean zero, for any $xinmathbb{R}$:

$$ |f^{(n)}(x)|le R^nint_{mathbb{R}}|phi^{(n)}(y)||eta(x-frac{y}{R})-eta(x)|dyle HR^{n-alpha}int_{mathbb{R}}|phi^{(n)}(y)||y|^alpha dy,$$

where $H$ is $eta$'s Hölder constant.

big list - Examples of common false beliefs in mathematics

The first thing to say is that this is not the same as the question about interesting mathematical mistakes. I am interested about the type of false beliefs that many intelligent people have while they are learning mathematics, but quickly abandon when their mistake is pointed out -- and also in why they have these beliefs. So in a sense I am interested in commonplace mathematical mistakes.

Let me give a couple of examples to show the kind of thing I mean. When teaching complex analysis, I often come across people who do not realize that they have four incompatible beliefs in their heads simultaneously. These are

(i) a bounded entire function is constant;
(ii) $sin z$ is a bounded function;
(iii) $sin z$ is defined and analytic everywhere on $mathbb{C}$;
(iv) $sin z$ is not a constant function.

Obviously, it is (ii) that is false. I think probably many people visualize the extension of $sin z$ to the complex plane as a doubly periodic function, until someone points out that that is complete nonsense.

A second example is the statement that an open dense subset $U$ of $mathbb{R}$ must be the whole of $mathbb{R}$. The "proof" of this statement is that every point $x$ is arbitrarily close to a point $u$ in $U$, so when you put a small neighbourhood about $u$ it must contain $x$.

Since I'm asking for a good list of examples, and since it's more like a psychological question than a mathematical one, I think I'd better make it community wiki. The properties I'd most like from examples are that they are from reasonably advanced mathematics (so I'm less interested in very elementary false statements like $(x+y)^2=x^2+y^2$, even if they are widely believed) and that the reasons they are found plausible are quite varied.

ag.algebraic geometry - How to characterize Abelian sheaves that are quasi-coherent?

1) There is a very simple example that shows that it is impossible to answer the question of whether $mathcal{A}$ comes from a quasi-coherent sheaf $mathcal{F}$ on $X$ if all one is given is the underlying topological space $|X|$ and $mathcal{A}$ as a sheaf on $|X|$. Namely, if $|X|$ is a point and $mathcal{A}$ is such that $mathcal{A}(|X|)=mathbf{Q}$, then either outcome is possible: the answer is YES if $X=operatorname{Spec} mathbf{Q}$, but NO if $X=operatorname{Spec} mathbf{F}_p$.

2) There are some nontrivial necessary conditions that one can state in terms of the topological space and the sheaf of abelian groups alone. For example, in order for $mathcal{A}$ to come from a quasi-coherent sheaf, there must exist an open covering $(U_i)$ of $|X|$ such that the sheaf $mathcal{A}|_{U_i}$ on $U_i$ is acyclic for every $i$.

3) The condition in 2) is definitely not sufficient, even if the scheme structure on $|X|$ is not specified in advance. For instance the constant sheaf $mathbf{Z}/6mathbf{Z}$ on a point is acyclic, but it cannot be a quasi-coherent sheaf for any scheme structure on the point.

pr.probability - Constraints for different probability measures to have the same expectation.

Take different $D_i in mathbb{R} rightarrow mathbb{R}$ functions $f_1, f_2$ (i.e. $exists x : f_1(x) neq f_2(x)$). We have

$E[f_1(x)] = E[f_2(x)]$

Are there conditions that $f_1, f_2$ must satisfy for this to happen?

I translated this problem to integral form as
$ int_{E_1} x dg_1 = int_{E_2} x dg_2$

$g_1, g_2$ being the probability measures of $f_1(x)$ and $f_2(x)$, which can be easily calculated and $E_i$ the corresponding domains. Now, while the domains may be different, they are "similar", so we don't want to just fix domains conveniently -- instead, we want to study $f_1$ and $f_2$. Maybe there's a measure-theoretical backdoor into this, because every lead takes me to functional equations territory, which I can't handle at all.

Disclaimer: Not a homework problem. So yes, I'll be profiting indirectly from the solution, even though it's a tiny piece to a large, mostly non-mathematical puzzle. Also, I hope I'm making myself clear and following the local etiquette. I'm not a native english person, and this is my first post on MO.

ac.commutative algebra - a question about flatness

In the book "étale cohomology" by Milne, proposition 2.5 at p.9, it said :

Let $B$ be a flat $A-$algebra where $A$ and $B$ are noetherian rings, and consider $b in B$. If the image of $b$ in $B/mB$ is not a zero-divisor for any maximal ideal $m$ of $A$, then $B/(b)$ is a flat $A-$ algebra.

At the beginning of the proof, he said we can reduce to the case where $phi : A rightarrow B$ is a local homomorphism of local noetherian rings. The proof in this case uses the fact that it's a local homomorphsim.

But I think that in order to reduce the general case to the local case, we need the following condition, which I can't get from the original condition.

For any maximal ideal $n$ of $B$, the image of $b$ in $B/pB$ is not a zero-divisor, where $p = phi^{-1} (n)$.

How do you think?

rt.representation theory - Why are the characters of the symmetric group integer-valued?

I just want to emphasize that this question points at the rationality theory of representations and characters that is exposed so beautifully in Chapters 12 and 13 of Serre's book Linear Representations of Finite Groups.

In particular, one has the following facts.

[Section 13.1, Corollary 1]: The following are equivalent:
(i) Every character of $G$ is $mathbb{Q}$-valued.
(ii) Every character of $G$ is $mathbb{Z}$-valued.
(iii) Every conjugacy class of $G$ is rational: for every $g in G$ and positive integer
$k$ prime to the order of $g$, $g^k$ is conjugate to $g$.

As noted above, since raising an element of the symmetric group $S_n$ to a power prime to its order does not change the cycle decomposition, condition (iii) holds and the implication (iii) $implies$ (ii) answers the question. [The proof is the basic Galois-theoretic argument given in some other answers. The implication (ii) $implies$ (iii) is deeper in that it uses the irreduciblity of the cyclotomic polynomials.]

Some others have said that the shortest or simplest proof arises from knowing that all of the irreducible representations of $S_n$ can be explicitly constructed and therefore seen to be realizable over $mathbb{Q}$. I respectfully disagree. This is a nontrivial theorem of Young which Serre refers to but does not prove in his book (Example 1, p. 103).

Moreover, Serre explains that the condition of rationality of characters is in general weaker than rationality of representations: there are obstructions here in the Brauer group of $mathbb{Q}$! Namely, by Maschke's Theorem the group ring $mathbb{Q}[G]$ is semisimple, say a product of simple $mathbb{Q}$-algebras $A_i$ which are in bijective
correspondence with the irreducible $mathbb{Q}$-representations $V_i$. By Schur's Lemma,
$D_i = End_G(V_i)$ is a division algebra, and one has $A_i cong M_{n_i}(D_i)$. Then:

[Section 12.2, Corollary]: The following are equivalent:
(i) Each $D_i$ is commutative.
(ii) Every $mathbb{C}$-representation of $G$ is rational over the abelian number field generated by its character values.

Thus just knowing that the character table is $mathbb{Z}$-valued is not enough. The standard example [Exercise 12.3] is the quaternion group $G = $ {$pm 1, pm i, pm j, pm k$} for which

$mathbb{Q}[G] cong mathbb{Q}^4 oplus mathbb{H}$,

where $mathbb{H}$ is a division quaternion algebra over $mathbb{Q}$, ramified at $2$ and $infty$. It corresponds to an irreducible $2$-dimensional $mathbb{C}$-representation with rational character but which cannot be realized over $mathbb{Q}$.

career - What to look for in applicants to graduate programs (in mathematics)?

Let me take a crack at the question, since I am currently on the graduate committee at UGA. [The University of Georgia is about the 50th best department in the country, so just a little shy of being a research 1 university. We are strongest in algebra, number theory, and algebraic geometry and are able to attract some excellent students in these areas.]

The two most important things for us are:

1) Very good to excellent grades, in courses which go beyond the minimum necessary for a math major and include, if possible, at least one graduate course. We are looking for all grades B or higher and at least as many A's as B's (which implies a GPA of at least 3.5). One or two poor grades will not concern us too much if they are in lower level courses, are followed by several years of better grades, or some explanation is given in the personal statement and/or the letters of recommmendation. A successful applicant will probably have taken real analysis, abstract algebra and topology. We certainly do take into consideration the student's school: e.g. a student from a liberal arts college may not have any graduate courses available.

2) High GRE scores. We like to see at least 700 on the GRE quantitative. It is not as criticial, but I would like to (and often do not!) see at least 600 on the GRE verbal; both scores are kept in mind by the university when it makes decisions about who will get prestigious fellowships. As for the GRE math subject exam: I am sorry to say that as of this year we do not require it. After looking at other universities of equal and greater status, we have decided to start requiring this exam next year (i.e., for students who are applying to start in Fall 2011), although we recognize that this may shrink our applicant pool. A score in the top 50% on the math subject exam looks good to us.

Next come the recommendation letters, which we use to gauge the student's enthusiasm, ability and preparedness for graduate school as compared to other aspiring graduate students. It is much better for us if the letters come from someone that we have heard of, or whom we can verify is a successful research mathematician (I have looked some recommenders up on MathSciNet). Good things to see in such letters are comparisons to other students who have gone on to be successful at research 1 graduate programs.

REU experience looks good, especially if accompanied by a recommendation letter from the REU supervisor who can be specific about what is accomplished. Sometimes students do enclose papers or preprints that are the result of REU work. Again we like this in general (more if the paper looks interesting, less if it looks rather trivial) and may forward this along to other faculty members to see if they are especially interested in the student.

I would say that the personal statement is in fact not very important, except perhaps to address/explain weaknesses in other parts of the application. [I accept that it might be more useful at a different institution. I have also advised graduate students and postdocs applying for academic jobs that their cover letter is not very important, and I know that some people -- especially at liberal arts colleges -- have said exactly the opposite.] It is useful as a writing sample, and a lack of spelling, grammatical and punctutation errors is evidence that the student is serious about their application.

In fact it is probably more likely that you will lose points in your personal statement than gain them. When I was a college senior, we had a Q&A session about applying to grad school. The head of the computer science department (Lance Fortnow, I think) told us the following story: he once had an application from a candidate who had very strong grades, GRE scores and recommendation letters. But in his personal statement he was asked "Why do you want to go to graduate school in computer science?" The candidate's response "Because I am trying to avoid working very hard" was exactly the opposite of what the department head wanted to hear. The rest of the application was so strong that, albeit with some misgivings, this candidate was admitted. The result was disastrous: the candidate really didn't want to do any work so was (of course!) a most unsuccessful graduate student, eventually getting kicked out of the program. The CS department head concluded that after this experience, he would never admit a candidate who said something like that on their personal statement. (And neither would I.)

gn.general topology - Pair of curves joining opposite corners of a square must intersect---proof?

Reposting something I posted a while back to Google Groups.

In his 'Ordinary Differential Equations' (sec. 1.2) V.I. Arnold says
"... every pair of curves in the square joining different pairs of
opposite corners must intersect".

This is obvious geometrically but I was wondering how one could go
about proving this rigorously. I have thought of a proof using
Brouwer's Fixed Point Theorem which I describe below. I would greatly
appreciate the group's
comments on whether this proof is right and if a simpler proof is
possible.

We take a square with side of length 1. Let the two curves be
$(x_1(t),y_1(t))$ and $(x_2(t),y_2(t))$ where the $x_i$ and $y_i$ are
continuous functions from $[0,1]$ to $[0,1]$. The condition that the
curves join different pairs of opposite corners implies,
$$(x_1(0),y_1(0)) = (0,0)$$
$$(x_2(0),y_2(0)) = (1,0)$$
$$(x_1(1),y_1(1)) = (1,1)$$
$$(x_2(1),y_2(1)) = (0,1)$$

The two curves will intersect if there are numbers $a$ and $b$ in $[0,1]$
such that

$$p(a,b) = x_2(b) - x_1(a) = 0$$
$$q(a,b) = y_1(a) - y_2(b) = 0$$

We define the two functions

$$f(a,b) = a + p(a,b)/2 + |p(a,b)| (1/2 - a)$$
$$g(a,b) = b + q(a,b)/2 + |q(a,b)| (1/2 - b)$$

Then $(f,g)$ is a continuous function from $[0,1]times [0,1]$ into itself and
hence must have a fixed point by Brouwer's Fixed Point Theorem. But at
a fixed point of $(f,g)$ it must be the case that $p(a,b)=0$ and $q(a,b)=0$
so the two curves intersect.

Figuring out what $f$ and $g$ to use and checking the conditions in the
last para is a tedious. Can there be a simpler proof?

fa.functional analysis - Boundedness of nonlinear continuous functionals

Ady, I don't have an answer to the new version of your question but let me make some remarks which might be useful.

The new version is about non-linear real-valued continuous functions on
$ell_infty(Gamma)$ where $Gamma$ has the cardinality of the continuum.
This can be slightly generalized as follows:

Let $kappa$ be an infinite cardinal and set $K$ to be
the closed unit ball of $ell_infty(kappa)$. Let
$f:Ktomathbb{R}$ be a continuous map. Does there exist
an infinite-dimensional subspace $E$ of $ell_infty(kappa)$
such that $f(Kcap E)$ is bounded?

If $kappa=aleph_0$, then a counterexample can be constructed.

On the other hand, if $kappa$ is a measurable cardinal, then
there exists a subspace $E$ of $ell_infty(kappa)$ which is isomorphic
to $c_0(kappa)$ and such that $f(Kcap E)$ is bounded. The argument
goes back to Ketonen. Let $FIN(kappa)$ be the set of all non-empty finite
subsets of $kappa$ and define a coloring $c:FIN(kappa)tomathbb{N}$
as follows. Let $c(F)$ be $n$ if $n$ is the least integer $m$ such that

$ max{ |f(x)|: xin span{e_t: tin F} and xin K } leq m $

where $e_t$ is the dirac function at $t$. Notice that $c$ is well-defined.
There exist $n_0inmathbb{N}$ and a subset $A$ of $kappa$ with $|A|=kappa$
and such that $c$ is constant on $FIN(A)$ and equal to $n_0$. If we set $E$ to be
the closed linear span of ${e_t: tin A}$, then $E$ is isomorphic to
$c_0(kappa)$ and $F(Kcap E)$ is in the interval $[-n_0, n_0]$.

Concerning the continuum: it might be that there are set-theoretic issues.
Firstly, let me recall that it is consistent that the the continuum is real-valued
measurable (R. M. Solovay). On the other hand, if CH holds, then there is heavy
(and quite advanced) machinery for ``killing" various Ramsey properties on
$omega_1$ (largely due to S. Todorcevic).

---

A quick remark: there exists a non-linear continuous map $f:Ktomathbb{R}$,
where $K$ is the closed unit ball of $c_0(kappa)$ and $kappa$ is the continuum,
such that for every infinite-dimensional subspace $E$ of $c_0(kappa)$
the set $f(Kcap E)$ is unbounded.

big list - Which popular games are the most mathematical?

I consider a game to be mathematical if there is interesting mathematics (to a mathematician) involved in

• the game's structure,


• optimal strategies,


• practical strategies,


• analysis of the game results/performance.

Which popular games are particularly mathematical by this definition?

---

Motivation: I got into backgammon a bit over 10 years ago after overhearing Rob Kirby say to another mathematician at MSRI that he thought backgammon was a game worth studying. Since then, I have written over 100 articles on the mathematics of backgammon as a columnist for a backgammon magazine. My target audience is backgammon players, not mathematicians, so much of the material I cover is not mathematically interesting to a mathematician. However, I have been able to include topics such as martingale decomposition, deconvolution, divergent series, first passage times, stable distributions, stochastic differential equations, the reflection principle in combinatorics, asymptotic behavior of recurrences, $chi^2$ statistical analysis, variance reduction in Monte Carlo simulations, etc. I have also made a few videos for a poker instruction site, and I am collaborating on a book on practical applications of mathematics to poker aimed at poker players. I would like to know which other games can be used similarly as a way to popularize mathematics, and which games I am likely to appreciate more as a mathematician than the general population will.

Other examples:

• go


• bridge


• Set.

Non-example: I do not believe chess is mathematical, despite the popular conception that chess and mathematics are related. Game theory says almost nothing about chess. The rules seem mathematically arbitrary. Most of the analysis in chess is mathematically meaningless, since positions are won, drawn, or tied (some minor complications can occur with the 50 move rule), and yet chess players distinguish strong moves from even stronger moves, and usually can't determine the true value of a position.

To me, the most mathematical aspect of chess is that the linear evaluation of piece strength is highly correlated which side can win in the end game. Second, there is a logarithmic rating system in which all chess players say they are underrated by 150 points. (Not all games have good rating systems.) However, these are not enough for me to consider chess to be mathematical. I can't imagine writing many columns on the mathematics of chess aimed at chess players.

Non-example: I would exclude Nim. Nim has a highly mathematical structure and optimal strategy, but I do not consider it a popular game since I don't know people who actually play Nim for fun.

---

To clarify, I want the games as played to be mathematical. It does not count if there are mathematical puzzles you can describe on the same board. Does being a mathematician help you to learn the game faster, to play the game better, or to analyze the game more accurately? (As opposed to a smart philosopher or engineer...) If mathematics helps significantly in a game people actually play, particularly if interesting mathematics is involved in a surprising way, then it qualifies to be in this collection.

If my criteria seem horribly arbitrary as some have commented, so be it, but this seems in line with questions like Real world applications of math, by arxive subject area? or Cocktail party math. I'm asking for applications of real mathematics to games people actually play. If someone is unconvinced that mathematics says anything they care about, and you find out he plays go, then you can describe something he might appreciate if you understand combinatorial game theory and how this arises naturally in go endgames.

dg.differential geometry - Hodge Index theorem for Complex Manifolds

Complementing Andrea's posting: the answer to the question as it is stated is no. Indeed, the proof of the Hodge index formula for Kaehler manifolds uses the strong Lefschetz decomposition, which does not exist for arbitrary complex manifolds.

A counter-example in the analytic case is given by the Hopf surface $H$, which is the quotient of $mathbf{C}^2$ minus the origin by the group generated by $(x,y)mapsto (2x,2y)$. Indeed, it is not too difficult to show that $H$ does not admit a holomorphic 1-form, i.e. $h^{1,0}(H)=0$. It is a non-trivial theorem (Barth, Peters, van de Ven, Compact complex surfaces, p. 117) that the Hodge to de Rham spectral sequence of any complex surface degenerates at $E_1$. Using this and the Serre duality we can compute all other Hodge numbers $h^{p,q}(H)$, which turn out to be 1 for $(p,q)=(0,0),(0,1),(2,1),(2,2)$ and zero otherwise. Plugging this into the right hand side of the index formula, we get 4. On the other hand, $H$ is diffeomorphic to $S^1times S^3$ and so $H^2(H,mathbf{R})=0$.

rt.representation theory - Induction of tensor product vs. tensor product of inductions

Try using Frobenius reciprocity. Let $V$ and $W$ be two representations of $H$, and let $U$ be a representation of $G$. Consider first the space:
$$Hom_G left(U, Ind_H^G (V otimes W) right) cong Hom_H left( U, V otimes W right),$$
by Frobenius reciprocity.

On the other hand, one can consider the space:
$$Hom_G(U, Ind_H^G V otimes Ind_H^G W).$$
This is canonically isomorphic to
$$Hom_G(U otimes Ind_H^G V', Ind_H^G W),$$
where $V'$ denotes the dual representation of $V$. By Frobenius reciprocity again, this is isomorphic to:
$$Hom_H(U otimes Ind_H^G V', W).$$
This is canonically isomorphic to
$$Hom_H(U, (Ind_H^G V) otimes W).$$
Now, we are led to compare the two spaces:
$$Hom_H(U, V otimes W), quad Hom_H left( U, (Ind_H^G V) otimes W right).$$

There is a natural embedding of $V$ into $Res_H^G Ind_H^G V$. This gives a natural map:
$$iota: Hom_H(U, V otimes W) rightarrow Hom_H left( U, (Ind_H^G V) otimes W right).$$

Using complete reducibility, let us (noncanonically) decompose $H$-representations:
$$Res_H^G Ind_H^G V cong V oplus V^perp.$$
It follows that
$$Hom_H left( U, (Ind_H^G V) otimes W right) cong Hom_H left(U, V otimes W right) oplus Hom_H left( U, V^perp otimes W right).$$

It follows that $iota$ is injective. This explains (via Yoneda, if you like) why $Ind_H^G(V otimes W)$ is canonically a subrepresentation of $Ind_H^G V otimes Ind_H^G W$. It also explains that computation of "the rest" of $Ind_H^G V otimes Ind_H^G W$ -- the full decomposition into irreducibles -- requires Mackey theory: the decomposition of $Res_H^G Ind_H^G V$. There can be no neat answer, without performing this kind of Mackey theory.

ac.commutative algebra - How to design or create or generate a bijective ring map?

In generality (this is tagged "commutative algebra", so let's talk commutative rings) I wonder if there is more than taking generators of each side and writing the images as polynomials in the generators of the other side. This is a candidate for a bijection of rings, but so far isn't a homomorphism (you need to check the relations hold). In other words express each ring as a quotient of a polynomial ring, set up homomorphisms from the polynomial rings, and then show the kernels are what they should be.

nt.number theory - Factorization of elements vs. of ideals, and is being a UFD equivalent to any property which can be stated entirely without reference to ring elements?

Why exactly is the unique factorization of elements into irreducibles a natural thing to look for? Of course, it's true in $mathbb{Z}$ and we'd like to see where else it is true; also, regardless of whether something is natural or not, studying it extends our knowledge of mathematics, which is always good. But the unique factorization of elements - being specifically a question of elements - seems completely counter to the category theory philosophy of characterizing structure via the maps between objects rather than their elements. Indeed, I feel like unique factorization of ideals into prime ideals is less a generalization of unique factorization of elements into irreducibles than the latter is a messier, unnatural special case of the former, a "purer" question (ideals, being the kernels of maps between rings, I feel meet my criteria for being a category-theoretically acceptable thing to look at). Certainly, the common theme in algebra (and most of mathematics) is to look at the decomposition of structures into simpler structures - but quite rarely at actual elements.

Now, for nice cases like rings of integers in number fields, we can characterize being a UFD in terms of the class group and other nice structures and not have to mess around with ring elements, but looking at the Wikipedia page on UFDs and the alternative characterizations they list for general rings, they all appear to depend on ring elements in some way (the link to "divisor theory" is broken, and I don't know what that is, so if someone could explain it and/or point me to some resources for it, it'd be much appreciated).

Sorry about the rambling question, but I was wondering if anyone had any thoughts or comments? Is "being a UFD" equivalent to any property which can be stated entirely without reference to ring elements? Should we care whether it is or not?

---

EDIT: Here's a more straightforward way of saying what I was trying to get at: The structure theorem for f.g. modules over a PID, the Artin-Wedderburn theorem, the Jordan-Holder theorem - these are structural decompositions. Unique factorization of elements is not, because elements are not a structure. My feeling is that this makes it a fundamentally less natural question, and I ask whether being a UFD can be characterized in purely structural terms, which would redeem the concept somewhat, I think.

rt.representation theory - Is there a "categorical" description of Grothendieck's algebra of differential operators?

First, pick a commutative ring $k$ as the "ground field". Everything I say will be $k$-linear, e.g. "algebra" means "unital associative algebra over $k$". Then recall the following construction due to Grothendieck:

Let $A$ be a commutative algebra, and $X$ an $A$-module. Then the differential operators on $X$ is the filtered algebra $D = D(A,X)$ given inductively by:
$$ D_{leq 0} = text{image of $A$ in }hom_k(X,X) $$
$$ D_{leq n} = { phi in hom_k(X,X) text{ s.t. } [phi,a] in D_{leq n-1}\, forall ain A} $$
$$ D = bigcup_{n=0}^infty D_{leq n} $$
(Edit: In the comments, Michael suggests that $D_{leq 0} = hom_A(X,X)$ is the more standard definition, and the rest is the same.)

Then the following facts are more or less standard:

• If $A$ acts freely on $X$, then $D_{leq 1}$ acts on $A$ by derivations. (It's always true that $D_{leq 1}$ acts on $D_{leq 0}$ by derivations; the question is whether $D_{leq 0} = A$ or a quotient.) If $X = A$ by multiplication, then $D_{leq 1}$ splits as a direct sum $D_{leq 1} = text{Der}(A) oplus A$.


• If $k=mathbb R$, $M$ is a finite-dimensional smooth manifold, and $A = C^infty(M) = X$, then $D$ is the usual algebra of differential operators generated by $A$ and $text{Vect}(M) = Gamma(TM to M)$.


• If $A,X$ are actually sheaves, so is $D$.

Thus, at least in the situation where $A = X = C^{infty}(M)$, the algebra $D$ is acting very much like the universal enveloping algebra of $U (text{Vect}(M))$; in particular, the map $U(text{Vect}(M)) to D$ is filtered and is (almost) a surjection: it misses only the non-constant elements of $A$. So when $A = X = C^infty(-)$ are sheaves on $M$, it's very tempting to think of $D$ as a sheafy version of $U(text{Vect}(-))$. Note that $U(text{Vect}(-))$ is not a sheaf: its degree $leq 0$ part consists of constant functions, not locally constant, for example, and there are non-zero elements in $U_{leq 2}$ that restrict to $0$ on an open cover. I think that it cannot be true that the sheafification of $U(text{Vect}(-))$ is $D$, as the sheafification of $U_{leq 0}$ is the locally-constant sheaf, not $C^infty$.

So: is there a description of $D$ that makes it more obviously like a universal enveloping algebra? E.g. is there some adjunction or other categorical description? Is it really true that $D$ is a "sheafy" version of $U$ in a precise sense, or is this just a chimera?

ct.category theory - Using ends to construct categorical fixed points

Under advice from Toby Bartels, I am posting this question here; it falls under the general heading of constructing data types categorically as fixed points of functors.

The first question I have is a warm-up. There's a way to interpret a natural number n in any cartesian closed category C, as a dinatural transformation of the form

c^c --> c^c

which intuitively takes an element f: c --> c to its n-th iterate f^(n): c --> c. One may hope that if C is "nice", then every such dinatural transformation will be of this form, or better still, that the end

int_{c: C} (c^c)^(c^c)

(assuming it exists) behaves as a natural numbers object in C. So first, I am interested in what "nice" might mean here: what are some general conditions on C that ensure we can construct a natural numbers object as an end in this way?

Second, this end can be rewritten as

int_{c: C} c^(c^(1 + c))

provided that C has coproducts, and the assertion that this behaves as a natural numbers object is equivalent to saying it is an initial algebra for the endofunctor F(c) = 1 + c (that's algebra for an endofunctor, not for a monad), making it a "fixed point" of F by a famous old result of Lambek.

This suggests a second more general question: given an endofunctor F: C --> C on cartesian closed C with a strength (essentially, a structure of C-enrichment), I want to know what "nice" conditions on C and/or F guarantee that the end

e = int_{c: C} c^(c^F(c)),

if it exists, is an initial F-algebra. It's not hard to write down an an F-algebra structure for this end e, and show that it is weakly initial, i.e., show that if x is any F-algebra, then there at least exists an F-algebra map e --> x. The issue then is over the uniqueness of this map, or rather what nice conditions would guarantee that.

Discussion of specific cases like PER models would be alright, but I'd probably be a lot more excited if it led to consideration of more general abstract conditions on C or F.

gr.group theory - Maximum entry magnitudes for representations of symmetric groups

I've no idea on your general question, but as far as eliminating the easy answer that Specht matrices always have entries -1,0,1, I decided to check the computer. According to sage, the maximum entry's absolute value can be larger than 1. The first example appears to be on 7 points:

sage: chi=SymmetricGroupRepresentations(7);chi([2,2,2,1])([2,3,1,5,4,6,7])
[-1  0  0  0  0  1  0  0  0  0  0  0  0  0]
[-1  1  0  0  0  1 -1  0  0  0  0  0  0  0]
[ 0  0  0  0  0  1  0  0  0  0 -1  0  0  0]
[ 0  1  0  0  0  1  0  0  0  0  0  0 -1  0]
[-1  1  0  0  0  1  0  0  0  0  0 -1  0  0]
[-1  0  0  0  0  0  0  0  0  0  0  0  0  0]
[-1  1  0  0  0  0  0  0  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0  0  0  0  0  0]
[ 0  1  0  0 -1  0  0  0  0  0  0  0  0  0]
[-1  1  0 -1  0  0  0  0  0  0  0  0  0  0]
[-1  0  0  0  0  1  0 -1  0  0  0  0  0  0]
[-1  1  0  0  0  1  0  0  0 -1  0  0  0  0]
[-2  1  0  0  0  1  0  0 -1  0  0  0  0  0]
[-1  1  0  0  0  1  0  0  0  0  0  0  0 -1]

The first command gives you Irr(Sym(7)) in the variable chi, then chi(partition) gives you the specific irreducible Specht representation, and chi(partition)(permutation) gives you the matrix for that permutation. Permutations are specified in the combinatorics way by listing the images of [1,2,3,4,5,6,7] in order, also called one-line notation. The specific permutation is (1,2,3)(4,5)(6)(7).

According to magma one has:

> SymmetricRepresentation([2,2,2,1],Sym(7)!(1,2,7)(3,4));
[ 0 -1  0  1 -1 -1  0 -2 -1  0  0 -1  0 -1]
[ 0  1  0 -1  2  1 -1  1  1  0  0  0  0  1]
[ 0 -1  0  1 -1  0  1 -1  0  0  0  0  0  0]
[ 0  1  0  0  1  1  0  1  0  0  0  0  0  1]
[ 0  0  0  0 -1 -1  0 -1 -1  0  0  0  0  0]
[ 0  0  0  0  0  0  0  1  0  0  0  0  0  0]
[ 0  0  0  0  0 -1  0 -1  0  0  0  0  0  0]
[ 0  0  0  0 -1  0  1  0  0  0  0  0  0  0]
[ 0  0  0  0  1  1  0  0  0  0  0  0  0  0]
[ 0  1  0 -1  1  1  0  1  1  0  1  1  0  1]
[ 0 -1  0  1  0  0  0 -1  0  1  0 -1  0  0]
[ 1  1  0  0  0  1  0  1  0  0  0  1  0  1]
[ 0  1  0 -1  1  0 -1  0  0  0  0  0 -1  0]
[ 0 -1 -1  0 -1 -1  0  0  0  0  0  0  0 -1]

At any rate, both violate the bound, though requiring different conjugates.

At least when I was looking at the modular representations induced by Specht modules I noticed there is no convention in the published articles or software on which of the two modules is a Specht module and which is its dual. Over a field of characteristic 0 they are isomorphic, but over finite fields they are just dual. I suspect the entries of the matrices in the representation of the Specht module are not terribly well defined; you'll need to have some combinatorial definition to work from to even decide isomorphism over Z/pZ, much less exact entries.

nt.number theory - Possible values for differences of primes

OK, I just saw this...a conjecture I made a few years ago covers this area

It relates d to OEIS A129912 entries, which are derived from the primorials.

Note that the d spoken to would only be a subset of those possible but these are guaranteed. A quick example is the prime 189239, which is offset from A129912(24) by 9059. So, the primes 189239 and 9059 have d=180180.

Note the "adjacency" part of the conjecture. The conjecture reads as follows:

"Every prime number >2 must have an absolute distance to a sequence entry (primorials, 

primorial products) that is itself prime, aside from the special cases prime=2 and those primes immediately adjacent to a sequence entry (primorials, primorial products).
The property is required but not sufficient ...it considers distances no larger than the candidate"

Rephrasing, this merely means every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product. (the distance will be a prime smaller than the candidate)

Now obviously there are many other sets as you say ie 23-19,29-19,etc which lie outside the above.

soft question - intuition about the "section after base-change" for flat descent and exactness of the Amitsur complex

I think it feels like magic because there's something tautological going on. The story should really culminate with the definition of "faithfully flat" rather than begin with it.

As you suggested, let's consider the case where you cover an affine scheme Spec(A) by finitely many basic open affines Spec(A_f). The cover Spec(B)=Spec(∏A_f) has the following special property:

(∗) If you want to check exactness of any sequence of A-modules, it's enough to check exactness after restricting to Spec(B) (i.e. tensoring with B over A).

Now if you'd like to show that the Amitsur complex is exact, you know it's enough to find a section B→A. By (∗), we know what it's actually enough to find a section locally, and in the case Spec(B)=∐Spec(A_f)→Spec(A), it's obvious that there's a section locally. This is all very tautological, but it allows you to prove that schemes are sheaves in the Zariski topology, which maybe doesn't impress you so much.

But we have extra assumptions in this argument. We didn't really need B to be of the form ∏A_f, we just needed it to satisfy (∗). So let's make a new definition, saying that B is "faithfully flat" over A if (∗) holds. Now we get the result: "If B is faithfully flat over A, then the Amitsur complex is exact." As a consequence, we can show that schemes are sheaves in the faithfully flat topology on affine schemes, but all we've really done is shown that "schemes are sheaves in the strongest topology for which this proof works," and just defined "faithfully flat" to be that topology. Delightfully, we can prove (with commutative algebra) that lots of different properties of affine schemes and morphisms of affine schemes are local in this topology.

Okay, but we'd really like to prove things about schemes and morphisms of schemes, not about affine schemes and morphisms of affine schemes. I think part of your confusion arises from the desire to understand the etale topology (a good desire), rather than continuing the strategy of making definitions which make results trivial (or at least straight-forward). If you indulge your generality tooth, you'll ask the question,

What is the strongest topology on the category of schemes so that every cover can be understood as a combination of (a) Zariski covers and (b) faithfully flat covers of affine schemes by affine schemes?

The answer is the fpqc topology. Basically by construction, if a property of schemes (resp. morphisms of schemes) is local in the Zariski topology and is local in the faithfully flat topology when you restrict to the category of affine schemes, then it is local in the fpqc topology. Similarly, if a functor is a sheaf in the Zariski topology, and its restriction to the category of affine schemes is a sheaf in the faithfully flat topology, then it's a sheaf in the fpqc topology. In particular, we get that

This sounds very impressive given that nobody understands what general fpqc morphisms look like (I think), but that's just because we defined the fpqc topology to be whatever it has to be to make the straight-forward proofs work. One thing we do know is that fppf morphisms are fpqc (by EGA IV, Corollary 1.10.4). In particular, etale morphisms are fpqc.

gn.general topology - Can I detect the point of impact without looking at it?

Andrew's comments showed me that in my first answer I was misunderstanding several aspects of his question. Since I am still not entirely sure that I am capturing the spirit of the problem, let be begin this answer by stating in my own words (in very dry mathematical terms) what I interpret the question(s) to be, so that he or someone else can correct me if necessary.

---

Let $mathcal{F}$ be the set of smooth functions $f colon mathbf{R}^2 to mathbf{R}$ whose restriction to the $x$-axis is constant. Let $mathcal{C}$ be the set of smooth functions $c colon (0,1) to mathbf{R}$ such that for every $f in mathcal{F}$,

• the function $f circ c colon (0,1) to mathbf{R}$ extends to a smooth function $[0,1) to mathbf{R}$ (in the sense that all one-sided derivatives exist at the origin and equal the one-sided limits of the corresponding derivatives), and




• $lim_{t to 0^+} f(c(t)) = f(0,0)$.

Question 1: If $c in mathcal{C}$, must $lim_{t to 0^+} c(t)$ exist? (Actually, Andrew is also asking more generally what one can say about $c$ if the limit does not exist.)

Question 2: Is there a rule (function) $mathcal{C} to mathcal{F}^r$ for some positive integer $r$, say taking $c$ to $(f_{1,c},ldots,f_{r,c})$, such that $f_{1,c}$ is independent of $c$, and $f_{2,c}$ depends only on $f_{1,c} circ c$, and $f_{3,c}$ depends only on $f_{1,c} circ c$ and $f_{2,c} circ c$, and so on, together with a rule (function) $R$ that takes as input a sequence of smooth functions $(g_1,ldots,g_r)$ from $(0,1)$ to $mathbf{R}$ and outputs a point in $mathbf{R}^2$ such that for every $c in mathcal{C}$, we have $R(f_{1,c}circ c,ldots,f_{r,c} circ c) = lim_{t to 0^+} c(t)$ whenever the latter exists?

Question 3: Same as Question 2, but with $lim_{t to 0^+} c'(t)$ in place of $lim_{t to 0^+} c(t)$.

---

Answer to Question 1: No. A negative answer was essentially given already by Andrew himself. Namely, let $c(t) := (sin 1/t,e^{-1/t})$. This $c(t)$ has the following properties: the $x$-coordinate is bounded, the derivatives of the $x$-coordinate grow at most polynomially in $1/t$ as $t to 0^+$, and the $y$-coordinate and derivatives of the $y$-coordinate decay to $0$ exponentially as $t to 0^+$. As explained by Andrew, for any $f in mathcal{F}$, the chain rule shows that $f(c(t))$ extends to a smooth function $[0,1) to mathbf{R}$ whose value at $0$ is $f(0,0)$ and whose higher derivatives at $0$ are all $0$. $square$

Answer to Questions 2 and 3: Yes. In fact, we can do it with $r=2$, and with both $f_{1,c}$ and $f_{2,c}$ independent of $c$. Namely, use $f_1(x,y)=y$ and $f_2(x,y)=e^x y$. From $f_1 circ c$ and $f_2 circ c$, we may recover not only the $y$-coordinate of $c$ as $f_1 circ c$, but also the $x$-coordinate of $c$ as $log (f_2(c(t))/f_1(c(t)))$. So the whole function $c(t)$, and hence any property of $c(t)$, can be detected. $square$

gt.geometric topology - Knots that unknot in a manifold

Ryan gives a very nice answer in dimension 3, leaving the higher-dimensional case of the question open. I can't discuss the question with as much authority as I would like, but I can start to piece together an answer. My impression is that in any dimension $n$, an $(n-2)$-knot $K$ that is unknotted in $M$ is already unknotted in $B$ in the continuous category. In the smooth and PL categories, it should be the same except in $n=4$ dimensions, where the question runs into a lack of understanding of smooth (or equivalently PL) structures on 4-manifolds.

First, my impression from skimming some geometric group theory is that a finitely presented group cannot be an infinite free product. Moreover, that it if it is a finite free product, the factors have unique isomorphism types. If this impression is correct, then you can identify the knot group $pi_1(S^n setminus K)$ from the embedded fundamental group $pi_1(M setminus K)$. In order to be trivial in $M$, the knot group of $K$ would have to be $mathbb{Z}$.

Then, Aspherical manifolds and higher-dimensional knots, by Bruno Eckmann, begins:

E. Dyer and R. Vasquez proved that the complement of a higher-dimensional
knot $S^{n-2} subset S^n$, $n ge 4$, is never aspherical unless the knot group is infinite cyclic (and hence, for $n ge 5$, the imbedding is unknotted). [Footnote referring to papers of Levine, Stallings, Wall, Shaneson.] In the present note we give a simple proof of this fact based on some remarks concerning compact $partial$-manifolds.

My impression is that the surgery methods in dimension $n ge 5$ used to prove the unknottedness assertion are valid in the smooth, PL, and continuous categories. Dimension $n=3$ is of course a special case where you do not need to consider algebraic topology, but instead prove things with direct cut-and-paste arguments as Ryan describes. In dimension $n=4$, the surgery theory is (a) much more difficult in the continuous category, and (b) non-existent in the smooth/PL category.

The paper The algebraic characterization of the exteriors of certain 2-knots, by Jonathan Hillman, credits Freedman with the result that a 2-knot with knot group $mathbb{Z}$ is unknotted. If this requires Freedman's work, then I would think that it is open in the smooth/PL case. What would be open is whether the 4-manifold with boundary $B^3 times S^1$ has more than one smooth structure. If it does, I have seen a principle that smooth structures on a 4-manifold can merge together when you take a connected sum with manifolds such as $pm mathbb{C}P^2$ an $S^2 times S^2$. I do not know if this principle is established for manifolds with a fundamental group. But at the very least, I have heard that the semigroup of smooth structures on $S^4$ is completely unknown, and its action on the smooth structures on another 4-manifold is completely unknown. So if the smooth Poincare conjecture is sufficiently false, presumably you could have a non-trivial 2-knot in $B$ that becomes trivial in $M$.

computational complexity - Where does the game-theoretic characterization of PH come from?

I'll take a shot at explaining this. The canonical problem in PH is a problem of this sort: $exists x_1 forall x_2 ldots exists x_k f(x_1,x_2,ldots,x_k)$. (The last quantifier is exists or forall depending on whether k is odd or even. Let's take it to be exists for this example.)

The idea is to imagine two players, lets call them Eve (Player 1) and Adam (Player 2). (The names Eve and Adam were chosen so that Eve corresponds to the exists operator, and Adam corresponds to the forAll operator.)

Imagine that $x_1,x_3,ldots$ describe Eve's moves in this two player game, and $x_2,x_4,ldots$ describe Adam's moves. The function $f(x_1,x_2,ldots,x_k)$ evaluates if these moves lead to a win for Eve ($f=1$) or a loss ($f=0$). There is no possibility of a draw.

Now the idea is simple. The value of the boolean expression $exists x_1 forall x_2 ldots exists x_k f(x_1,x_2,ldots,x_k)$ exactly tells us if Eve has a winning strategy or not. In words, the boolean expression says this: "Is there a first move ($x_1$) that Eve can play so that no matter what Adam plays in his second move ($x_2$), there exists a move for Eve ($x_3$), so that no matter what Adam plays ........ there exists a $k^{th}$ move for Eve ($x_k$) so that she wins (i.e., $f(x_1,x_2,ldots,x_k) = 1$)."

So the Boolean expression exactly expresses whether Eve has a winning strategy in this two player game.

Ideal of "Compact Operators" in a W*-algebra which gives the sigma-strong-* topology.

I don't believe that (3) characterises the $sigma$-strong* topology. Here's what I think is a counter-example.

Let $H=ell^2$. Consider $ell^infty$ acting on $H$ in the obvious way, so $c_0$ also acts on $H$ as compact operators. Let $K$ be the compact operator induced by $(1/sqrt k)_{kgeq 1}in c_0$.
I'm going to build a net $(x_beta)$ in $ell^inftysubseteqmathcal B(H)$ such that $x_betarightarrow 0$ $sigma$-strong* but with $|x_beta K|geq 1$ for all $beta$, so that $(x_beta)$ does not tend strictly to 0.

To do this, observe that the $sigma$-strong* topology, restricted to $ell^infty$, is given by seminorms of the form $x=(x_k) mapsto sum_k |x_k|^2 |b_k|$ where $(b_k)inell^1$. So, given $b^{(1)},cdots,b^{(n)}inell^1$, let $c_k = sum_{j=1}^n b^{(j)}_k$. Then $(c_k)inell^1$, and if $sum_k |x_k|^2 |c_k|$ is small, then certainly $sum_k |x_k|^2 |b^{(j)}_k|$ is small for each $j$.

So it suffices to show that for each $(c_k)inell^1$ and $epsilon>0$, we can find $xinell^infty$ with $sum_k |x_k|^2|c_k|<epsilon$, but with $|x_k|/sqrt kgeq 1$ for some $k$. We can do this by setting $x_k=0$ unless $k=N$ in which case $x_N = sqrt N$. This follows, as if we cannot do this, then $N|c_N|geqepsilon$ for all $N$, and so $sum_k |c_k| geq epsilonsum_k 1/k = infty$, a contradiction.

Did you mean to restrict to bounded sets?? Anyway, I wonder what happens for, say, $L^infty([0,1])$; I'd be surprised if this contained non-trivial ideals which were its own multiplier algebra.

What is the standard reference on "infinitesimal space" in algebraic geometry??

infinitesimal 'spaces' is a serious issue in noncommutative (and commutative) geometry: they serve as a base of a Grothendieck-Berthelot crystalline theory and are of big importance
for the D-module theory.

Can anybody point out the standard reference for this topic? I tried to look for it at nLab, but it seems it did not tell the reference in the language of algebraic geometry.

I am not very familiar with French,so the English manuscript is better, however, French one is fine.

pr.probability - Markov chain on Groups

Let $G$ be a permutation group on the finite set $Omega$. Consider the Markov chain where you start with an element $alpha in Omega$ chosen from some arbitrary starting probability distribution. One step in the Markov chain involves the following:

• Move from $alpha$ to a random element $gin G$ that fixes $alpha$.
  
• Move from $g$ to a random element $betain Omega$ that is fixed by $g$.
  

Since it is possible to move from any $alpha$ to any $beta$ in $Omega$ in a single step (through the identity of $G$) the Markov chain is irreducible and aperiodic. This implies that there is a unique distribution that is approached by iterating the procedure above from any starting distribution. It's not hard to show that the limiting distribution is the one where all orbits are equally likely (i.e. the probability of reaching $alpha$ is inversely proportional to the size of the orbit containing it).

---

I read about this nice construction in P.J. Cameron's "Permutation Groups", where he brings up what he calls a slogan of modern enumeration theory: "...the ability to count a set is closely related to the ability to pick a random element from that set (with all elements equally likely)."

One special case of this Markov chain is when we let $Omega=G$ and the action be conjugation, then we get a limiting distribution where all conjugacy classes of $G$ are equally likely. Now, except for this nice result, it would also be interesting to know something about the rate of convergence of this chain. Cameron mentions that it is rapidly mixing (converges exponentially fast to the limiting distribution) in some important cases, but examples where it's not rapidly mixing can also be constructed. My question is:

Question: Can we describe the rate of convergence of the Markov chain described above in terms of group-theoretic concepts (properties of $G$)?

While giving the rate of convergence in terms of the properties of $G$ might be a hard question, answers with sufficient conditions for the chain to be rapidly mixing are also welcome.

ag.algebraic geometry - Why is the decomposition theorem awesome?

My answer is maybe more to praise the glory of perverse sheaves than the decomposition theorem exactly, but bear with me. To appreciate the theorem, I'd say first get a sense of what Hodge theory for smooth projective algebraic varieties says: hard Lefschetz etc. (already the fact that the proofs you're likely to see involve harmonic forms and analysis should convince you this is serious stuff). Then try to get a sense of what it means to understand this theorem in families, where things like Hodge filtrations start to appear.

Finally despair of what it might mean to even consider this picture if the "family" you were looking at was just a projective morphism $fcolon Y to X$, where $Y$ is smooth: the local systems you need for the families version of Hodge theory break down. However, enter perverse sheaves, as sort of singular local systems, and the decomposition theorem says the whole picture is miraculously saved. Viewed this way I think you get a proper sense of how amazing the theorem (and the discovery of perverse sheaves) really is.

P.S. This answer is a poor attempt to convey what others have told me: a better attempt is made in de Cataldo and Migliorini's article

pr.probability - Local view of setting p*n out of n bits to 1

You want $frac15 = sum_t |P_1(count=t) - P_2(count=t)|$.

where $P_1$ has a binomial distribution and $P_2$ is hypergeometric.

The difference between these distributions is shown in this Mathematica demonstration.

I believe both are reasonably well approximated by normal distributions. Both have mean $pk$. The variance for the binomial distribution is $kp(1-p)$, while it is $frac{n-k}{n-1}*k(p)(1-p)$ for the hypergeometric distribution.

So, the value of k should be so that the normal distributions $N(0,1)$ and $N(0,sqrt{frac{n-k}{n-1}})$ have total variation distance $frac1{10}$. That should be at about $k=(1-c)n$ where $N(0,1)$ and $N(0,sqrt{c})$ are $frac1{10}$ apart. Numerically, it seems that $c$ should be about 0.6605 so $sqrt{c}$ should be about 0.8127. $k = 0.3395n$.

It appears this is not sensitive to the value of $p$.

soft question - Favorite popular math book

Title: What is mathematics?

Author: Herbert Robbins and Richard Courant

This book is a very nice introduction to mathematics, it covers basic number theory, analysis, algebra, geometry and topology.

I'am very surprised that i couldn't find it on this list already.

(from a duplicate answer - feel free to edit) This would be for someone who has some mathematical ability, and really wants to understand what math is. Courant goes through essentially all of mathematics, starting at a very elementary level, but getting to some very deep and important stuff. He often does real proofs, and doesn't dumb it down, but does explain things conceptually very well, including sometimes giving just ideas or justifications for really difficult things, like the prime number theorem. I use this when I teach our senior proof seminar, just to force the math majors to own a copy.

pr.probability - Two-dimensional random walk

Let $a$ and $b$ be fixed points in the integer lattice, and let $f(p)$ be the probability that a random walk starting at the point $p$ will arrive at $a$ before $b$. Then for every point in the plane other than $a$ and $b$, we have,
$$
f(p) = frac{f(p+i)+f(p-i)+f(p+j)+f(p-j)}{4}
$$
where $i$ and $j$ are the basis unit vectors. That is, the value of $f$ at a point is equal to the average of the values of $f$ at the neighboring points.

A function on the square lattice with this property is called harmonic, and satisfies a discrete version of Laplace's equation:
$$
Delta f = 0
$$
where $Delta$ is the discrete Laplace operator. Unfortunately, the function $f$ is not quite harmonic, since the equation above need not hold when $p=a$ or $p=b$.

In particular, the function $f$ actually satisfies the Poisson equation
$$
Delta f(p) = C_1 delta_a(p) + C_2 delta_b(p),
$$
where $delta_a$ is the function which is $1$ at $a$ and zero elsewhere, $delta_b$ is the same for $b$, and $C_1$ and $C_2$ are unknown constants.

Since Poisonn's equation is linear, it suffices to solve the equations
$$
Delta f(p) = delta_a(p)qquadtext{and}qquadDelta f(p) = delta_b(p)
$$
independently, and then take an appropriate linear combination of the solutions. Solutions to equations such as these are called lattice Green's functions. For the integer lattice, the lattice Green's functions cannot be written in a closed form, but there are definite integral formulas that can be used to compute the function to arbitrary precision (see here).

Once you know the values of the lattice Green's functions, you ought to be able to solve for the constants $C_1$ and $C_2$ by using the boundary conditions $f(a) = 1$ and $f(b) = 0$.

ct.category theory - How to characterize good "models" of a category

Let ${bf Cat}$ denote the category of small categories. Recall that for a category $mathcal{C}$ and a functor $Fcolonmathcal{C}to{bf Cat}$, the Grothendieck construction of $F$, which I'll denote $int F$, is a category and it comes equipped with a natural fibration $int Ftomathcal{C}$.

[For reference: The objects of $int F$ are pairs $(c,x)$ where $cin{bf Ob}(mathcal{C})$ and $xin{bf Ob}(F(mathcal{C}))$, and a morphism $(c,x)to(c',x')$ is a pair $(f,g)$ where $fcolon cto c'$ in $mathcal{C}$ and $gcolon F(f)(x)to x'$ in the category $F(c')$.]

Now, given a category ${mathcal D}$, I'll define a model of ${mathcal D}$ to be a pair $({mathcal C},F,e)$ where $mathcal{C}$ is a category, $Fcolon mathcal{C}to{bf Cat}$ is a functor, and $ecolonint Ftomathcal{D}$ is a natural isomorphism. I will sometimes leave out $e$ if it is obvious. Allow me to leave morphisms of models undefined, as I'm not sure what I want here. [Supplying an appropriate definition of morphisms between models of $mathcal{D}$ should be part of giving a good answer to this overflow question.]

Every category $mathcal{D}$ has two canonical models which I'll denote by $(mathcal{D},{ast})$ and $({ast},mathcal{D})$. The first is the functor $mathcal{D}to{bf Cat}$ that sends every object of $mathcal{C}$ to the terminal set, and the second is the functor $*to{bf Cat}$ that sends the terminal category to $mathcal{D}$.

But there may be many models $Fcolonmathcal{C}to{bf Cat}$ of $mathcal{D}$ that lie "in between" these two extreme cases, and some are "better than others" in the sense that the fibers of $mathcal{D}=int Ftomathcal{C}$ are non-trivial yet "comprehensible" in some human sense.

Question: What can you say about the (yet-undefined) category of models of $mathcal{D}$ that will clarify the above ideas?

real analysis - a unique solution ? iteration involving conditional distributions

The transformation $L=TG$ is defined on vectors $x$ with positive coordinates by
$$
Lx(s)=sum_uq(u|s)mathrm{e}^{-r(u)}Mx(u),quadmbox{where}
Mx(s)=prod_ux(u)^{p(u|s)}.
$$
Thus $M$ and $L$ are homogenous and nondecreasing on the positive orthant. This means that one considers vectors $x$ such that $x(s)>0$ for every $s$, that $M(lambda x)=lambda Mx$ and $L(lambda x)=lambda L(x)$ for every positive scalar $lambda$, and that $Mxle Mtilde x$ and $Lxle Ltilde x$ if $xletilde x$ in the sense that $x(s)letilde x(s)$ for every $s$.

For every vector $x$ with positive coordinates, let $u(x)$ and $ell(x)$ denote the supremum and the infimum of its coordinates $x(s)$, hence $ell(x)le x(s)le u(x)$ for every $s$.

Since $p$ is a transition kernel, $displaystylesum_up(u|s)=1$ for every $s$ hence $ell(x)le Mx(s)le u(x)$ for every $s$ and $ell(x)a(s)le Lx(s)le u(x)a(s)$ with
$$
a(s)=sum_uq(u|s)mathrm{e}^{-r(u)}.
$$
More generally, for every positive $t$,
$$
ell(x)ell(a)^{t-1}a(s)le L^tx(s)le u(x)u(a)^{t-1}a(s),
$$
hence
$$
ell(x)ell(a)^{t}le ell(L^tx)le u(L^tx)le u(x)u(a)^{t}.
$$
Furthermore, $u(a)le u(mathrm{e}^{-r})$ and $ell(a)geell(mathrm{e}^{-r})$. Now everything depends on the hypothesis made on $r$.

If $r(s)>0$ for every $s$ (and I believe this is what the OP wanted to write), then $u(a)<1$ hence $L^tx$ converges geometrically to $0$. If $r(s)<0$ for every $s$ (and this is what the OP actually wrote), then $ell(a)>1$ hence $L^tx$ diverges geometrically to $+infty$.

For $(x_t)$ to converge to a nondegenerate limit, one should assume that $r$ has positive and negative coordinates.

set theory - "$kappa$ strongly inaccessible" = "every function $f:V_kappato V_kappa$ can be self-applied"?

Unfortunately, your characterizations of the strongly inaccessible cardinals are not quite correct. The correct definition is that κ is strongly inaccessible (also known as just plain inaccessible), if κ is an uncountable regular strong limit cardinal. The cardinal κ is regular if it is not the union of fewer than κ many sets of size less than κ. And κ is a strong limit cardinal if whenver β < κ, then the power set of β also has size less than κ.

This is not equivalent to the assertion that V_κ is a model of ZFC. (Although, to be sure, this false assertion has appeared surprisingly often in print and I have even heard a famous proof theorist make this assertion to a very large audience of hundreds of logicians.) The reason is that if κ is strongly inaccessible, then a Lowenheim-Skolem argument shows that there will be many γ < κ for which V_γ is elementary in V_κ, and so these also will be models of ZFC. It is an exercise to show that the least γ for which V_γ is a model of ZFC has cofinality ω, and so is definitely not inaccessible.

Also, since ZFC is a first order theory in a countable language, if it has any models at all, then it has models in every infinite cardinality. So it is not correct to characterize inaccessible cardinals as the sizes of models of ZFC in that way either.

It is also not equivalent to asserting that κ is regular and not the size of a power set of a smaller set. The reason is that if, say, CH failed, then ω₁ would be regular and also not be the size of the power set of any smaller set (since 2^ω would be already too large). But ω₁ is not an inaccessible cardinal.

Your remark that V_κ is closed under pairs when κ is inaccessible actually doesn't need any amount of inaccessibility. If x and y are sets in any V_α, then the pair (x,y) appears just a few steps later (and actually, one can use flat pairing function that do not increase rank at all, for infinite rank sets), and so every V_λ is closed under pairing for any limit ordinal λ. If one uses a flat pairing function (instead of the common Kuratowski pairing function), then every V_α for every infinite ordinal α will be closed under pairing.

Finally, yes, if V_λ is closed under pairing, then you can apply such functions to themselves, and this idea is used quite often when we have elementary embeddings defined on models of ZFC. For example, if j:V to M, then j(j) is a function defined on M, into some structure j(M), which will be the union of j(V_α^M). This operation is called application.

There is a famous result of Laver concerning the left distributive algebra of nontrivial elementary embeddings j:V_λ to V_λ. The first results characterizing normal forms in the free algebra with one generator used such embeddings, with the accompanying very large large cardinal hypothesis. For example, Laver produced a decision procedure, which was only known to work under these enormous large cardinal assumptions. Later, the large cardinal hypotheses were removed and the algebra became studied apart from the large cardinals, but the basic properties were definitely inspired and discovered by knowledge of what the large caridnals were like. The basic operation in this algebra is known as application, and is exactly the operation that you mention.

nt.number theory - Motivation for uniform surjectivity of mod l representations associated to elliptic curves

Background

Let $E$ be an elliptic curve over $mathbb{Q}$ and let $G_{mathbb{Q}}$ be the absolute Galois group $Aut(overline{mathbb{Q}})$. For any positive integer $n$ the $n$-torsion subgroup $E[n](overline{mathbb{Q}})$ is stable under the $G_{mathbb{Q}}$-action. Since $E[n](overline{mathbb{Q}})$ is isomorphic to $(mathbb{Z}/n\mathbb{Z})^2$ one gets a continuous (with respect to the profinite topology on the left and the discrete on the right) homomorphism
$$
overline{rho_{E,, n}}colon G_{mathbb{Q}} to GL_2(mathbb{Z}/nmathbb{Z})
$$
which one calls the mod $n$ representation associated to $E$. As $n$ varies these are compatible and taking limits gives representations $rho_{E,ell^{infty}}$ and $rho_E$ with values in $GL_2(mathbb{Z}_l)$ and $GL_2(widehat{mathbb{Z}})$ which one calls respectively the $ell$-adic and adelic representations associated to $E$.

Alternatively, $rho_{E,n }$ is isomorphic to the representation induced by the action of $G_{mathbb{Q}}$ on the etale cohomology $H^1_{text{et}}(E_{overline{mathbb{Q}}}; mathbb{Z}/nmathbb{Z})$; the description of $rho_{E,n}$ via torsion generalizes to give representations $rho_{A,n}$ for higher dimensional abelian varieties, but for a general variety one must instead use cohomlogy.

Serre famously proved that for $E$ an elliptic curve with $End(E) = mathbb{Z}$, $rho_E(G_{mathbb{Q}})$ has finite index in $GL_2(widehat{mathbb{Z}})$. In particular for $ell$ large $rho_{l^{infty}}$ is surjective; how large one must take $ell$ depends on $E$.

Conjecture

This last fact does not depend on $E$. I.e. there exists a constant $N$ such that for every $E/mathbb{Q}$ with $End(E) = mathbb{Z}$ and $ell geq N$, the mod $ell$ representation $overline{rho_{E,ell}}$ is surjective; equivalenty there is an upper bound (independent of $E$) on the index of $rho_E(G_{mathbb{Q}})$.

By a recent paper of Bilu and Parent, one knows that the image is either surjective or contained in a non-split cartan subgroup.

---

My Question

Why do people expect this to be true? Does it follow from other believable conjectures? Is there some heuristic that predicts this?

One fact is that one can phrase this question as the failure of various modular curves (e.g. $X_{ns}(l)$, which have increasingly large genus) to have non-trivial rational points. Is there a geometric reason that one expects these modular curves to have no non-trivial rational points?

topos theory - Universal property for collection of epimorphisms

Question Is there a nice universal property which captures the notion of "collection of all epimorphisms out of a given object". Of course I will have to consider two epimorphisms $X rightarrow Y$ the same if they are isomorphic over $X$. The answer to the dual question is yes, at least in a topos: The power object $P(X)=Omega^X$ , where $Omega$ is the subobject classifier can be thought of as "the collection of all subobjects of X". The universal property is just the property for exponentials.

Background (Not strictly necessary for the question): I have been reading Sheaves in Geometry and Logic by Mac Lane and Moerdijk. Their definition of an elementary topos is this: A category with pullbacks, a terminal object (i.e. all finite limits), a subobject classifier, and a power object for every object. They construct all other exponential objects from these axioms. The construction they use is to basically consider the "collection" of all graphs of morphisms. This is just the standard construction in set theory suped up to toposes.

This construction agrees with the set theoretic convention that a function should be regarded as a set of ordered pairs, i.e. if $f:A rightarrow B$, then the set theorist will define $f$ as the image of the map $A rightarrowtail A times B$ induced by the $1_A$ and $f$ (this may be the most convoluted sentence I have ever written). Why not define functions dually? There is also a map $A+B twoheadrightarrow B$ induced by $1_B$ and $f$. Then we could define $f$ as the partition of $A$ induced by this epimorphism, which seems like a perfectly nice way to define functions.

I was wondering if this construction could be used to construct exponential objects if I was given finite colimits and some kind of epimorphism classifier, or collection of epimorphisms out of a given object.

Comment if it turns out that there is no really nice answer to this question, do you think that has bearing on the fact that the formula for the number of subsets of a set is easy ($2^{|X|}$) but the formula for the number of partitions of a set is relatively hard (http://en.wikipedia.org/wiki/Partition_of_a_set)?

gromov witten theory - Are Fukaya categories Calabi-Yau categories?

At first sight, the Fukaya category has obvious cyclic symmetry, because the $A_infty$ structure maps count points in spaces of rigid pseudo-holomorphic polygons subject to Lagrangian boundary conditions, and these spaces depend only on the cyclic order of the Lagrangians. This indeed proves that the cohomological Fukaya category, in which the hom-spaces are Floer cohomology spaces, is cyclically symmetric.

The trouble comes when these Lagrangians don't intersect one another transversely - for instance, the same Lagrangian occurs more than once - because then the morphism spaces and structure maps invoke Hamiltonian perturbations which need not be cyclically symmetric. The problem which Fukaya has solved over the reals (see Matthew Ballard's answer) is, I presume, to find a way to make these perturbations cyclically symmetric whilst also achieving the necessary coherence between them, as well as transversality for compactified moduli spaces of inhomogeneous pseudo-holomorphic polygons (or worse, their abstract perturbations). These are the things which actually define the $A_infty$-structure.

FOOO worked extremely hard to get geometrically-meaningful units in their Fukaya endomorphism algebras, where other authors are content to define units by tweaking the $A_infty$-structure algebraically. My hope would be that algebra will also give a cheaper approach to cyclic symmetry, particularly since I'm told that for Costello's theorem to hold, one only needs "derived" cyclic symmetry.

By the way, let's be clear that Costello's theorem, suggestive as it may be, is not about GW invariants! It's about theories over $M_{g,n}$, not over $overline{M}_{g,n}$.

Dimension of module

Does dimension of a module (say, dimension of its support) have anything to do with the supremum length of chains of prime submodules like rings?
Let's restrict to finitely generated modules over Noetherian ring.
Prime submodules are defined analogously to primary submodules: a submodule P in M is prime if P$neq$M and $M/P$ has no zero divisors, i.e. $amin P$ implies $min P$ or $a in mbox{Ann}(M/P)$.

rt.representation theory - Generalized Haar Measures and Semiring-Valued Integrals on Lie Groups

In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete groups, this is pretty easy to set up. Given a semiring, $(S, oplus, odot)$, and a discrete group $G$, define the semialgebra $S[G]$ as the collection of maps $S^G$ equipped with the bilinear operator $star : S[G] times S[G] to S[G]$ such that for all $f, h in S[G]$:

$(f star h)(x) = bigoplus limits_{y in G} f(y) odot h(y^{-1}x)$

This works well, and is consistent with the usual definition of a group algebra when $S$ is a ring. However, in almost all of the interesting situations I can think of, I would like to take $G$ to be a Lie group. To make this happen, it seems that the right thing to do should be to define some kind of 'semiring valued Haar measure', $mu : 2^G to S$ which would allow one to compute $S$-valued integrals over $G$. Assuming this works just like it ought to in fields of characteristic 0, then convolution becomes some $S$-valued integral over $G$ like the following thing:

$(f star h)(x) = bigoplus limits_{y in G} f(y) odot g(y^{-1}x) odot d mu(y)$

Where the symbol $bigoplus$ means something like a 'semiring' Lebesgue integral over $G$. Unfortunately this definition is not very robust. One can easily pick plausible values of $S, G, mu$ which catastrophically fail. For instance, if $S = mathbb{Z} / n mathbb{Z}$, then it seems to me that the convolution integral is divergent for any function with measurable support!

However, all is not lost. I can think of a few easy cases where one can construct a measure which is convergent and gives 'reasonable' results (and by reasonable, I mean that they are intuitively similar to the results one would get in the case of a discrete group). Consider, for example, the case where $S$ is the idempotent boolean semifield, $B = ( mathbf{2}, OR, AND )$. Then take:

$mu(emptyset) = 0$

$mu(S neq emptyset) = 1$

Now the elements $f, g in B^G$ can be identified with subsets $X_f, X_g subseteq G$, and moreover the convolution on $G$ over $B$ gives rise to a so-called generalized Minkowski product:

$f star g cong X_f oplus X_g = bigcup limits_{x in X_f} x X_g$

Which is indeed a useful calculation! Similarly, one can define a measure like this over the $(max, +)$ semiring via another ad-hoc construction. This begs the question: when does this convolution actually work? More specifically, what are the conditions on $S, G, mu$ such that we can guarantee that convolution is convergent?

(Note: I am probably not being as careful as I should be with definitions. It is probably a good idea to restrict the elements of $S[G]$ to 'Haar-like' measurable functions, but I really don't have a good grasp of what this should mean until I know what the proper conditions on $mu$ should be...)

lo.logic - Extensional theorems mostly used intensionally

Georges Gonthier and François Garillot are doing interesting things with phantom types and unification in Coq to allow one to write, for example, directv (V + W) to mean the proposition that $V oplus W$ is a direct sum.

I haven't fully grasped how it works yet, but let me give you a simplified explanation of what I think is going on. What is happening is that directv X is really notation for directv_def _ (Phantom _ X).

Phantom is a constructor of a very trivial inductive type

Inductive phantom (A:Type) (a:A) : Type := Phantom : phantom A a

The function Phantom is a polymorphic constructor of type forall (A:Type)(a:A), phantom A a. The purpose of Phantom is to lift values to the type level so that type inference can operate on these values.

directv_def doesn't even use the (Phantom _ X) argument (because it contains no data). The only purpose of this argument is to drive the type inference engine to fill in the first argument. directv_def has type forall (VW : addv_expr) (_ : phantom _ (Vadd VW)), Prop. addv_expr is a record type.

Record addv_expr := build_addv_expr {
 V1 : VectorSpace; 
 V2 : VectorSpace;
 Vadd : VectorSpace }

The definition of directv_def is

directv_def (VW : addv_expr) _ := dim (V1 VW) + dim (V2 VW) = dim (Vadd VW)

The final ingredient is that fun V1 V2 => (build_addv_expr V1 V2 (V1 + V2)) is declared as a Canoncial Structure.

So what does Coq read when you write directv (V + W)? Well it parses this as notation for

directv_def _ (Phantom _ (V + W))

The first parameter to Phantom is the type of (V + W) so we can quickly fill that in to get

directv_def _ (Phantom VectorSpace (V + W))

Phantom VectorSpace (V + W) has type phantom VectorSpace (V + W), but directv_def is expecting something of type phantom _ (Vadd _) so it tries to unify (V + W) with (Vadd _). Because Vadd is a record projection, Coq tries to look up in its list of canonical structures to see if there are any declared whose Vadd field is of the form (V + W). It says, "ahha! there is! I can use build_addv_expr V W (V + W)" (notice the intensional behaviour of canonical inference here). So Coq successfully unifies (V + W) with (Vadd (build_addv_expr V W (V + W)), and this forces the first parameter of directv_def:

directv_def (build_addv_expr V W (V + W)) (Phantom VectorSpace (V + W))

And that is it for type inference. Later on this expression might be used, so it will start normalizing:

dim (V1 (build_addv_expr V W (V + W))) + dim (V2 (build_addv_expr V W (V + W))) = dim (Vadd (build_addv_expr V W (V + W))) 

and then to

dim V + dim W = dim (V + W)

If you try to write something else like directv 0 then the canonical structure inference will fail and you will get a (probably obtuse) type error.

---

This has been as simplified example. In reality, directv is much more complicated and allows one to write directv (sum_(0 <= i < n) V i) to mean $bigoplus_{i=0}^n V_i$ is a direct sum and accepts things like directv 0 to mean a trivial direct sum.

Matita allows you to write unification hints directly without the necessarily building canonical structures. I suspect doing this sort of intentional inference would be easier in such a system.

differential topology - Smooth homotopy theory

Let us define the nth smooth homotopy group of a smooth manifold $M$ to be the group $pi_n^infty(S^k)$ of smooth maps $S^n to S^k$ modulo smooth homotopy. Of course, some care must be taken to define the product, but I don't think this is a serious issue. The key is to construct a smooth map $S^n to S^n lor S^n$ (regarded as subspaces of $mathbb{R}^{n+1}$) which collapses the equator to a point; we then define the product of two (pointed) maps $f, g: S^n to S^k$ to be the map $S^n lor S^n to S^k$ which restricts to $f$ on the left half and $g$ on the right half. To accomplish this, use bump functions to bend $S^n$ into a smooth "dumbell" shape consisting of a cylinder $S^{n-1} times [0,1]$ with two large orbs attached to the ends, and retract $S^{n-1}$ to a point while preserving smoothness at the ends. Then retract $[0,1]$ to a point, and we're done.

Question: is the natural "forget smoothness" homomorphism $phi: pi_n^infty(S^k) to pi_n(S^k)$ an isomorphism? If not, what is known about $pi_n^infty(S^k)$ and what tools are used?

In chapter 6 of "From Calculus to Cohomology", Madsen and Tornehave prove that every continuous map between open subsets of Euclidean spaces is homotopic to a smooth map. Thus every continuous map $f$ between smooth manifolds is "locally smooth up to homotopy", meaning that every point in the source has a neighborhood $U$ such that $f|_U$ is homotopic to a smooth map. However it is not clear to me that the local homotopies can be chosen in such a way that they glue together to form a global homotopy between $f$ and a smooth map. This suggests that $phi$ need not be surjective.

In the same reference as above, it is shown that given any two smooth maps between open subsets of Euclidean spaces which are continuously homotopic, there is a smooth homotopy between them. As above this says that two smooth, continuously homotopic maps between smooth manifolds are locally smoothly homotopic, but I again see no reason why the local smooth homotopies should necessarily glue to form a global smooth homotopy. This suggests that $phi$ need not be injective.

I am certainly no expert on homotopy theory, but I have read enough to be surprised that this sort of question doesn't seem to be commonly addressed in the basic literature. This leads me to worry that my question is either fatally flawed, trivial, useless, or hopeless. Still, I'm retaining some hope that something interesting can be said.

gt.geometric topology - Topological results from geometry

A nice topic to read about is Chern-Weil theory. This is the generalisation of Gauss-Bonnet to higher dimensions and to vector bundles other than the tangent bundle. Put very briefly, topological invariants of a vector bundle over a manifold (its characteristic classes - certain classes in the cohomology of the base) can be computed using the curvature tensor of any choice of connection in the bundle.

The prototype is Gauss-Bonnet in which, as you know, the Euler characteristic of a (compact orientable) surface is equal to a fixed constant times the integral of the scalar curvature of any Riemannian metric on the surface.

ag.algebraic geometry - Positivity properties of virtual Hodge numbers of Calabi-Yaus

Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ $D$ enjoy some sort of positivity property?

Being virtual, defined by inclusion-exclusion from complete varieties, they're not individually positive. What I'd most like is to hear "The Euler characteristic is nonnegative". But that's not true for $X$ a quintic 3-fold, $D$ empty.

How do we study the theory of reductive groups?

Sit at a table with the books of Borel, Humphreys, and Springer. Bounce around between them: if a proof in one makes no sense, it may be clearer in the other. For example, Springer's book develops everything needed about root systems from scratch, and has lots of nice exercises relate to that stuff. On the other hand, Borel is better about systematically allowing general ground fields from early on (so one doesn't have to redo the proofs all over again upon discovering that it is a good idea to allow ground fields like $mathbf{R}$, $mathbf{Q}$, $mathbf{F}_ p$, and $mathbf{F} _p(t)$). Pay attention to the power of inductive arguments with centralizers and normalizers (especially of tori).

Unfortunately, none makes good use of schemes, which clarifies and simplifies many things related to tangent space calculations, quotients, and positive characteristic. (For example, the definition of central isogeny in Borel's book looks a bit funny, and if done via schemes becomes more natural, though ultimately equivalent to what Borel does.) So if some proofs feel unnecessarily complicated, it may be due to lack of adequate technique in algebraic geometry. (Everyone has to choose their own poison!) Waterhouse's book has nothing serious to say about reductive groups, but the theory of finite group schemes that he discusses (including Cartier duality and structure in the infinitesimal case) is very helpful for a deeper understanding isogenies between reductive groups in positive characteristic. The exposes in SGA3 on quotients and Grothendieck topologies (etale, fppf, etc.) are helpful a lot too (some of which is also developed in the book "Neron Models"). Galois cohomology is also useful when working with rational points of quotients.

gt.geometric topology - Ramified cover of 4-sphere

The answer is yes, at least if we interpret your phrase "ramification of order 2" to mean "simple branched covering". See Piergallini, R., Four-manifolds as $4$-fold branched covers of $S^4$. Topology 34 (1995), no. 3, 497--508. Any closed, orientable PL 4-manifold can be expressed as a 4-fold simple branched covering of S⁴ branched along an immersed surface with only transverse double points. It is apparently still an open question whether the branch set can be chosen to be nonsingular. A simple branched covering of degree d is a branched covering in which each branch point is covered by d-1 points, only one of which is singular, of local degree 2.

Is there an analogue of curvature in algebraic geometry?

An algebraic analog of Chern-Weil theory (explicitly taking symmetric polynomials of curvature) is given by the Atiyah class.
Given a vector bundle $E$ on a smooth variety we can consider the short exact sequence
$$ 0to End(E) to A(E) to T_Xto 0$$
where $T_X$ is the tangent sheaf and $A(E)$ is the "Atiyah algebroid" --- differential operators of order at most one acting on sections of $E$, whose symbol is a scalar first order diffop (hence the map to the tangent sheaf). A (holomorphic or algebraic) connection is precisely a splitting of this sequence, and a flat connection is a Lie algebra splitting. Now algebraically such splittings will often not exist (having a holomorphic connection forces your characteristic classes to have type $(p,0)$ rather than the $(p,p)$ you want..) but nonetheless we can define the extension class, which is the Atiyah class
$$a_Ein H^1(X, End(E)otimes Omega^1_X).$$
This is the analog of the curvature form in the Riemannian world -- we now can take symmetric polynomials in the $End(E)$ factor to get the characteristic classes of $E$ in $H^p(X,Omega_X^p)$ as desired.

This answer and Mariano's agree of course in the sense that Atiyah classes can be interpreted via Hochschild and cyclic (co)homology and generalized to arbitrary coherent sheaves (or complexes) on varieties (or stacks) (let me stick to characteristic zero to be safe). Namely the Atiyah class of the tangent sheaf can be used to define a Lie algebra structure (or more precisely $L_infty$) on the shifted tangent sheaf $T_X[-1]$, and Hochschild cohomology is its enveloping algebra. This Lie algebra acts as endomorphisms of any coherent sheaf (which is another way to say Hochschild cohomology is endomorphisms of the identity functor on the derived category), and one can take characters for these modules, recovering the characteristic classes defined concretely above.

(In fact the notion of characters is insanely general... for example an object of any category - with reasonable finiteness - defines a class (or "Chern character") in the Hochschild homology of that category, which is cyclic and so descends to cyclic homology. An example of this is the category of representations of a finite group, whose HH is class functions, recovering usual characters, or coherent sheaves on a variety, recovering usual Chern character. or one can go more general.)

rt.representation theory - How many ways are there to globalize Harish Chandra modules?

Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie group. Denote by $mathfrak{g}$ the complexified Lie algebra, and denote by $K$ a maximal compact in the complexification of $G$.

Denote by $mathbf{HC}(mathfrak{g},K)$ the category of admissible $(mathfrak{g},K)$-modules or (Harish Chandra modules), and $(mathfrak{g},K)$-module homomorphisms. Denote by $mathbf{Rep}(G)$ the category of admissible representations of finite length (on complete locally convex Hausdorff topological vector spaces), with continuous linear $G$-maps.

The Harish Chandra functor $mathcal{M}colonmathbf{Rep}(G)tomathbf{HC}(mathfrak{g},K)$ assigns to any admissible representation $V$ the Harish Chandra module of $K$-finite vectors of $V$. This is a faithful, exact functor. Let us call an exact functor $mathcal{G}colonmathbf{HC}(mathfrak{g},K)tomathbf{Rep}(G)$ along with a comparison isomorphism $eta_{mathcal{G}}colonmathcal{M}circmathcal{G}simeqmathrm{id}$ a globalization functor.

Our first observation is that globalization functors exist.

Theorem. [Casselman-Wallach] The restriction of $mathcal{M}$ to the full subcategory of smooth admissible Fréchet spaces is an equivalence. Moreover, for any Harish Chandra module $M$, the essentially unique smooth admissible representation $(pi,V)$ such that $Mcongmathcal{M}(pi,V)$ has the property $pi(mathcal{S}(G))M=V$, where $mathcal{S}(G)$ is the Schwartz algebra of $G$.

If we do not restrict $mathcal{M}$ to smooth admissible Fréchet spaces, then we have a minimal globalization and a maximal one.

Theorem. [Kashiwara-Schmid] $mathcal{M}$ admits both a left adjoint $mathcal{G}_0$ and right adjoint $mathcal{G}_{infty}$, and the counit and unit give these functors the structure of globalization functors.

Construction. Here, briefly, are descriptions of the minimal and maximal globalizations. The minimal globalization is

$$mathcal{G}_0=textit{Dist}_c(G)otimes_{U(mathfrak{g})}-$$

where $textit{Dist}_c(G)$ denotes the space of compactly supported distributions on $G$, and the maximal one is

$$mathcal{G}_{infty}=mathrm{Hom}_{U(mathfrak{g})}((-)^{vee},C^{infty}(G))$$

where $M^{vee}$ is the dual Harish Chandra module of $M$ (i.e., the $K$-finite vectors of the algebraic dual of $M$).

For any Harish Chandra module $M$, the minimal globalization $mathcal{G}_0(M)$ is a dual Fréchet nuclear space, and the maximal globalization $mathcal{G}_{infty}(M)$ is a Fréchet nuclear space.

Example. If $Psubset G$ is a parabolic subgroup, then the space $L^2(G/P)$ of $L^2$-functions on the homogeneous space $G/P$ is an admissible representation, and $M=mathcal{M}(L^2(G/P))$ is a particularly interesting Harish Chandra module. In this case, one may identify $mathcal{G}_0(M)$ with the real analytic functions on $G/P$, and one may identify $mathcal{G}_{infty}(M)$ with the hyperfunctions on $G/P$.

[I think other globalizations with different properties are known or expected; I don't yet know much about these, however.]

Consider the category $mathbf{Glob}(G)$ of globalization functors for $G$; morphisms $mathcal{G}'tomathcal{G}$ are natural transformations that are required to be compatible with the comparison isomorphisms $eta_{mathcal{G}'}$ and $eta_{mathcal{G}}$. Since $mathcal{M}$ is faithful, this category is actually a poset, and it has both an inf and a sup, namely $mathcal{G}_0$ and $mathcal{G}_{infty}$. This is the poset of globalizations for $G$.

I'd like to know more about the structure of the poset $mathbf{Glob}(G)$ — really, anything at all, but let me ask the following concrete question.

Question. Does every finite collection of elements of $mathbf{Glob}(G)$ admit both an inf and a sup?

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[Added later]

Emerton (below) mentions a geometric picture that appears to be very well adapted to the study of our poset $mathbf{Glob}(G)$. Let me at introduce the main ideas of the objects of interest, and what I learned about our poset. [What I'm going to say was essentially outlined by Kashiwara in 1987.] For this, we probably need to assume that $G$ is connected.

Notation. Let $X$ be the flag manifold of the complexification of $G$. Let $lambdainmathfrak{h}^{vee}$ be a dominant element of the dual space of the universal Cartan; for simplicity, let's assume that it is regular. Now one can define the twisted equivariant bounded derived categories $D^b_G(X)_{-lambda}$ and $D^b_K(X)_{-lambda}$ of constructible sheaves on $X$. Now let $mathbf{Glob}(G,lambda)$ denote the poset of globalizations for admissible representations with infinitesimal character $chi_{lambda}$, so the objects are exact functors $mathcal{G}colonmathbf{HC}(mathfrak{g},K)_{chi_{lambda}}tomathbf{Rep}(G)_{chi_{lambda}}$ equipped with natural isomorphisms $eta_{mathcal{G}}:mathcal{M}circmathcal{G}simeqmathrm{id}$.

Matsuki correspondence. [Mirkovic-Uzawa-Vilonen] There is a canonical equivalence $Phicolon D^b_G(X)_{-lambda}simeq D^b_K(X)_{-lambda}$. The perverse t-structure on the latter can be lifted along this correspondence to obtain a t-structure on $D^b_G(X)_{-lambda}$ as well. The Matsuki correspondence then restricts to an equivalence $Phicolon P_G(X)_{-lambda}simeq P_K(X)_{-lambda}$ between the corresponding hearts.

Beilinson-Bernstein construction. There is a canonical equivalence $alphacolon P_K(X)_{-lambda}simeqmathbf{HC}(mathfrak{g},K)_{chi_{lambda}}$, given by Riemann-Hilbert, followed by taking cohomology. [If $lambda$ is not regular, then this isn't quite an equivalence.]

Now we deduce a geometric description of an object of $mathbf{Glob}(G,lambda)$ as an exact functor $mathcal{H}colon P_G(X)_{-lambda}tomathbf{Rep}(G)_{chi_{lambda}}$ equipped with a natural isomorphism $mathcal{M}circmathcal{H}simeqalphacircPhi$, or equivalently, as a suitably t-exact functor $mathcal{H}colon D^b_G(X)_{-lambda}to D^bmathbf{Rep}(G)_{chi_{lambda}}$ equipped with a functorial identification between the (complex of) Harish Chandra module(s) of $K$-finite vectors of $mathcal{H}(F)$ and $mathrm{RHom}(mathbf{D}Phi F,mathcal{O}_X(lambda))$ for any $Fin D^b_G(X)_{-lambda}$. In particular, as Emerton observes, the maximal and minimal globalizations can be expressed as

$$mathcal{H}_{infty}(F)=mathrm{RHom}(mathbf{D}F,mathcal{O}_X(lambda))$$

and

$$mathcal{H}_0(F)=Fotimes^Lmathcal{O}_X(lambda)$$

Note that Verdier duality gives rise to an anti-involution $mathcal{H}mapsto(mathcal{H}circmathbf{D})^{vee}$ of the poset $mathbf{Glob}(G,lambda)$; in particular, it exchanges $mathcal{H}_{infty}$ and $mathcal{H}_0$.

I now expect that one can show the following (though I don't claim to have thought about this point carefully enough to call it a proposition).

Conjecture. All globalization functors are representable. That is, every element of $mathbf{Glob}(G,lambda)$ is of the form $mathrm{RHom}(mathbf{D}(-),E)$ for some object $Ein D^b_G(X)_{-lambda}$.

Question. Can one characterize those objects $Ein D^b_G(X)_{-lambda}$ such that the functor $mathrm{RHom}(mathbf{D}(-),E)$ is a globalization functor? Given a map between any two of these, under what circumstances do they induce a morphism of globalization functors (as defined above)?

In particular, note that if my expectation holds, then one should be able to find a copy of the poset $mathbf{Glob}(G,lambda)$ embedded in $D^b_G(X)_{-lambda}$.

rt.representation theory - Classifying strata for the adjoint representation of GL from first principles

I'm going to give a partial answer here for two reasons: (1) I am lazy and (2) this is starting to feel a little homeworky to me. Obviously, no one would assign this material as homework, but part of reading a math paper is taking the time to work out lots of simple examples and see how the definitions work. I feel like you are pushing the boundaries of how much of this work it is reasonable to ask other people to do. Not a major criticism, certainly not a vote to close the question, but my input.

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On to the math. I've scanned the first 3 pages of Hesselink's paper. He make the following definitions. G acts on V, v is a point of V and $star$ a chosen base point of V fixed by G. In your setting, G is $GL_n$, V is the $n times n$ matrices where G acts by conjugation, and $star$ is zero. Hesselink writes Y(G) for what is essentially $mathrm{Hom}(mathbb{C}^*, G)$. More precisely, Hesselink tensors with $mathbb{Q}$, so that he can talk about maps like $t mapsto left( begin{smallmatrix} t^{1/3} & 0 \\ 0 & t^{-2/7} end{smallmatrix} right)$. I'll ignore this detail.

For $lambda in Y(G)$, Hesselink defines a rational number $m(lambda)$. We talked about this in your previous question. In this setting, where V is an $N$-dimensional vector space, Hesselink gives an explicit formula for m on the bottom of page 142/top of page 143: Diagonalize the action of $lambda$ as $t mapsto mathrm{diag}(t^{m_1}, cdots, t^{m_N})$ and write $v = sum v_i e_i$.. Then $m(lambda) = min(m_i : v_i neq 0)$ if this number is nonnegative, and is $- infty$ if this minimum is negative.

Let's see what this definition means in your setting. We can conjugate any $lambda$ into diagonal form as $t mapsto mathrm{diag}(t^{c_1}, cdots, t^{c_n})$. I've replaced $m_i$ by $c_i$ to point out that these $c$'s are not the $m$'s of the previous paragraph. In our notation, the $N$ of the previous paragraph is $n^2$. The vector space $V$ has dimension $n^2$ with basis $e_{ij}$. The action of $lambda(t)$ on $e_{ij}$ is by $t^{c_i - c_j}$. (Exercise!).

So $m(lambda) > 0$ if and only if $c_i leq c_j$ implies $v_{ij} =0$.

We may as well order our basis such that $c_1 geq c_2 geq cdots geq c_n$.
If $c_1 > c_2 > cdots >c_n$ then we see that $m(lambda) > 0$ if and only if $v$ is a strictly upper triangular matrix. When there are some equalities among the $c$'s, you want $v$ to be strictly block upper triangular. For such a $v$, $m(lambda) = min(c_i - c_j : v_{ij} neq 0)$. In particular, notice that there exists a $lambda$ such that $m(lambda) > 0$ if and only if $v$ is nilpotent.

Hesselink defines $Lambda(v)$ to be the locus in ${ lambda : m(lambda) = 1 }$ where $q(lambda)$ is minimized, where $q$ is the inner product from your previous question. What you want to show is that $Lambda(v)$ determines the Jordan normal form of $v$.

I must admit that I haven't thought out how to prove this. But I hope this makes things explicit enough that you can attack it.