ag.algebraic geometry - Generic fiber of morphism between non-singular curves

Here is a complete proof: as remarked in the answer by Norondion, we can reduce to
the case when $C_1 rightarrow C_2$ is generically separable, i.e. $k(C_1)$ is separable
over $k(C_2)$. Let $A subset k(C_1)$ be a finite type $k$-algebra consisting of the regular
functions on some non-empty affine open subset $U$ of $C_2$ (it doesn't matter which one
you choose), so that $k(C_2)$ is the fraction field of $A$.

By the primitive element theorem, we may write $k(C_1) = k(C_2)[alpha]$, where
$alpha$ satisfies some polynomial $f(alpha) = alpha^n + a_{n-1}alpha^{n-1} + cdots + a_1 alpha + a_0 = 0,$ for some $a_i$ in $K(C_2)$.

Now the $a_i$ can be written as fractions involving elements of $A$, i.e. each
$a_i = b_i/c_i$ for some $b_i,c_i in A$ (with $c_i$ non-zero). We may replace
$A$ by $A[c_0^{-1},ldots,c_{n-1}^{-1}]$ (this corresponds to puncturing $U$ at
the zeroes of the $c_i$), and thus assume that in fact the $a_i$ lie in $A$.

The ring $A[alpha]$ is now integral over $A$, and of course has fraction field equal
to $k(C_2)[alpha] = k(C_1)$. It need not be that $A[alpha]$ is integrally closed,
though. We are going to shrink $U$ further so we can be sure of this.

By separability of $k(C_1)$ over $k(C_2)$, we know that the discriminant $Delta$
of $f$ is non-zero, and so replacing $A$ by $A[Delta^{-1}]$ (i.e. shrinking $U$
some more) we may assume that $Delta$ is invertible in $A$ as well.

It's now not hard to prove that $A[alpha]$ is integrally closed over $A$. Thus
$text{Spec }A[alpha]$ is the preimage of $U$ in $C_1$ (in a map of smooth curves,
taking preimages of an affine open precisely corresponds to taking the integral closure
of the corresponding ring).

In other words, restricted to $U subset C_2$, the map has the form
$text{Spec }A[alpha] rightarrow text{Spec }A,$ or, what is the same,
$text{Spec }A[x]/(f(x)) rightarrow text{Spec A}$.

Now if you fix a closed point $mathfrak m in text{Spec }A,$ the fibre over this point
is equal to $text{Spec }(A/mathfrak m)[x]/(overline{f}(x)) = k[x]/(overline{f}(x)),$
where here $overline{f}$ denotes the reduction of $f$ mod $mathfrak m$.
(Here is where we use that $k$ is algebraically closed, to deduce that $A/mathfrak m = k,$
and not some finite extension of $k$.)

Now we arranged for $Delta$ to be in $A^{times}$, and so $bar{Delta}$ (the reduction
of $Delta$ mod $mathfrak m$, or equivalently, the discriminant of $bar{f}$)
is non-zero, and so $k[x]/(bar{f}(x))$ is just a product of copies of $k$,
as many as equal to the degree of $f$, which equals the degree of $k(C_1)$ over
$k(C_2)$. Thus $text{Spec }k[x]/(bar{f}(x))$ is a union of that many points,
which is what we wanted to show.

st.statistics - Correlation of Statistical Tests

Suppose I have a sequence ${x_i}_{i=1}^infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are conjectured to be generated by independent coin flips). I can use a statistical test --say to see if the sequence 11 occurs as much as it should up to some bound $x$, or I can use a different test --say to see if $1r1$, where the $r$ is anything--occurs as much as it should up to the same bound (in general I'll be interested as $x rightarrow infty$).

The question is, does knowing that one test suggests that this is a randomly generated set guarantee (or somehow influence) the other test's results? Are they strongly correlated? Are there tests which would be correlated? I believe this study falls under the subject of 0-1 laws, but do not know where to begin looking. Any references, books, papers, answers would be appreciated--I'm still trying to find my way around the field.

Also please retag if you know of better tags to use.

ag.algebraic geometry - Why is Riemann-Roch an Index Problem?

This is discussed in detail in chapter IV of the RRT notes on my web page (roy smith at math dept, university of georgia). Briefly, the index point of view is a valuable simplification, but the Riemann Roch theorem is more than an index statement in general. Moreover the content of the index statement depends on the definition of the "index". In the answer by Spinorbundle above, the analytic index is defined in such a way that stating it as he does gives a complete statement of the RRT theorem. I.e. his definition of the index includes the statement of Serre duality as well, which in the case of curves also implies Kodaira vanishing.

Usually an index of a linear operator is the difference between the kernel and cokernel of that operator. In Riemann's original formulation of his theorem, that operator is a matrix of periods of integrals, and the RR problem is that of computing just its kernel. In the sheaf theoretic version of his approach, the index of the relevant operator associated to a divisor D is the difference chi(D) = h^0(D) - h^1(D). An easy long exact sheaf cohomology sequence implies immediately that chi(D) - chi(O) = deg(D).

Then if one simply defines the genus to be 1-chi(O) as is sometimes done, the result is the formula chi(D) = 1-g + deg(D), a sort of "computation" of the index chi(D), sometimes called the Riemann Roch theorem. This however is not very useful unless one can also compute g, i.e. chi(O). The real RR theorem should thus relate chi(O) to some more illuminating definition of the genus, such as h^0(K) or the topological genus. I.e. the very weak formula in this paragraph does not reveal that the index chi(D) is a topological invariant. Finally conditions should be given when chi(D) = h^0(D), the actual Riemann Roch number.

In dimension one, defining the index as Spinorbundle does, and computing it, does solve all these problems at once. But that computation is correspondingly more difficult. In higher dimensions even that computation does not give a criterion for the index to equal the Riemann Roch number.

When the index chi(L) is defined as the alternating sum of the dimensions of sheaf cohomology groups of a line bundle L, as is more common in algebraic geometry, there are then several steps to the full RR theorem:

1) compute chi(L) - chi(O), the difference of the indices of L and of O, as a topological invariant. This is the relatively easy part, by sheaf theory.

2) compute chi(O), also a topological invariant. this is sometimes called the Noether formula (at least for surfaces). One then has a topological formula for the index chi(L).

3) relate chi(L) to h^0(L), and perhaps h^0(K-L). this involves the vanishing criteria of serre and kodaira and mumford, and duality. This is the hardest part.

Moral: computing the index chi(L) is topological, hence relatively easy. Then one tries to go from chi(L) to h^0(L), using the deep results of Serre duality and Kodaira vanishing. Saying the RRT is (just) an index problem is like saying you can compute the number of vertices of a polyhedron just from knowing its Euler characteristic. But I confess to pretending otherwise at times. Indeed one of my t - shirts reads "Will explain Riemann Roch for gianduia: chi(D)-chi(O) = deg(D), and chi(O) = 1-g", which is merely the index statement.

In the RRT notes on my webpage, pp.37-42 there is an easy proof of steps 1 and 2, for curves, inspired by the introduction to one of Fulton's papers. Basically, once you have identified a topological invariant, you can compute it by degeneration to a simpler case. These notes also discuss Riemann's original proof, as well as generalizations of the index point of view to the case of surfaces, and a little about the Hirzebruch RR theorem in higher dimensions. Serre's proof of the duality theorem is also sketched. Briefly Serre lumps all the relevant cohomology spaces for all divisors D together into two infinite dimensional complex vector spaces, which he then shows are both one dimensional over the larger field of rational functions. It is then easier to prove they are isomorphic over that field, by showing the natural map between them is non zero, hence all their individual components are isomorphic over the complex numbers.

Algebraic geometry examples

The gluing along closed subscheme examples are a nice exercise for playing with Spec. That is, computing the spec of a pushout of affine schemes like

$$(X leftarrow Z rightarrow Y)$$
where one of the arrows is a closed immersion (or both are, for simplicity). It's a decent exercise to show that the Spec of such a pushout is what you expect pointwise.

Some examples that are worth trying as exercises are:

[1.] take a copy of $mathbb{A}^1$ for $X$, a pair of points for $Z$ and a single point for $Y$. You get a nodal singularity. Alternately, glue two copies of $mathbb{A}^1$ together at the origin.

[2.] $X = Spec k[x]$ as above, $Z = Spec k[x]/x^2$, $Y = k$, producing a cusp.

[3.] If $X = mathbb{A}^2$, $Z$ is an axis and the map $Z to Y$ is the usual 2-to-1 cover, then you get a pinch point.

[4.] You can make the map $Z to Y$ non-finite and produce non-noetherian affine schemes too. For example, $X = mathbb{A}^2$, $Z$ one of the axes (or any curve), and $Y$ a point.

You can later point out why you can't always do this for general (non-affine) schemes. That is, $X = mathbb{P}^2$, $Z$ an elliptic curve, and look at doing that sort of pinch point construction for various maps $Z to mathbb{P}^1$. This can lead to things like algebraic spaces if you are so inclined.

Another direction you can go with this sort of stuff is normality (ie, what is the geometric meaning of normality, all the examples you just computed with gluing are non-normal).

at.algebraic topology - complex structure on S^n

It is known that $S^4$ doesn't even have an almost complex structure, and the case for $S^6$ is open. The lack of almost complex structure can be proved a number of ways, one way is by showing that an almost complex, compact, four manifold with $dim_{mathbb{Q}}H^2(X,mathbb{Q})=0$ has $chi(X)=0$, but the four sphere doesn't. (It follows from the index theorem, here's a quick reference, first result.)

ag.algebraic geometry - tamely branched cover over P^1

k is an algebraically closed field, X is a smooth, connected, projective curve over k. f: X-->P^1 is a finite morphism. Let t be a parameter of P^1, suppose f is etale outside t=0 and t=infty, and tamely ramified over these two points. Prove that f is a cyclic cover, i.e., K(X)=k(t)[h]/(h^n-ut), u is a unit in field k.

set theory - first-order definability transitive closure operator

As Mike Shulman and François G. Dorais correctly point out, the official language of set theory has only the binary relation ε, and so there are no terms to speak of in that language beyond the variable symbols.

But no set theorist remains inside that primitive language, and neither is it desirable or virtuous to do so. Rather, as in any mathematical discourse, we introduce new terminology, define notions and introduce terms. What gives? I think the substance of your question is really:

• How can a set theorist (or any mathematician) sensibly and legitimately use terms that are not expressible as terms in the official language of the subject?

The answer is quite general. In any first order theory T, if one can prove that there is a unique object with a certain property, then one may expand the language by adding a term for that object, plus the defining axiom that that term has the desired property. The resulting theory T⁺ will be a conservative extension of T, meaning that the consequences of T⁺ that are expressible in the old language are exactly the same as the consequences of T. The reason is that any model M of T can be (uniquely) expanded to a model of T⁺, simply by interpreting the new term in M by its definition. This is why we may freely introduce symbols for emptyset or ω (or Q and R) and so on to set theory. Similarly, if T proves that for every x, there is a unique object y such that φ(x,y), then we may introduce a corresponding symbol f_φ(x), with the defining axiom ∀x φ(x,f_φ(x)). This new theory, in the expanded language with f_φ, is again conservative over T.

This is what is going on with the term TC(x) for the transitive closure of x. Although there is no official term for the transitive closure of x in the basic language of set theory, we may introduce such a term, once we prove that every set x does indeed have a transitive clsoure. And once having done so, the term becomes officially part of the expanded language.

To see that every set x has a transitive closure, one needs very little of ZFC, and as Dorais mentions in the comments to your question, you don't need to build the V_α hierarchy. For example, every set has a transitive closure even in models of ZF- (and much less), where the power set axiom fails and so the V_α hierarchy does not exist. Simply define a sequence x₀ = x and x_n+1 = U x_n. By Replacement, the set { x_n | n ε ω } exists, and the union of this set is precisely TC(x).

In summary, we should feel free to introduce defined terms, and there is absolutely no reason not to write TC(x) on the chalkboard, as you mentioned. In particular, we should not feel compelled to express our beautiful mathematical ideas in a primitive language with only ε, like some kind of machine code, just because it is possible in principle to do so.

fa.functional analysis - Request for reference: Banach-type spaces as algebraic theories.

Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight at finding that one of my old favourites (functional analysis) and one of my new fads (category theory, and in particular algebraic theories) are actually very closely connected!

I was going to ask about the state of play of these things as it's a little unclear exactly what stage has been achieved. Reading the paper On the equational theory of $C^ast$-algebras and its review on MathSciNet then it appears that although it's known that $C^ast$-algebras do form an algebraic theory, an exact presentation in terms of operations and identities is still missing (at least at the time of that paper being written), though I may be misreading things there. It's possible to do a little reference chasing through the MathSciNet database, but the trail does seem to go a little cold and it's very hard to search for "$C^ast$ algebra"!

But now I've decided that I don't want to just know about the current state of play, I'd like to learn what's going on here in a lot more detail since, as I said, it brings together two seemingly disparate areas of mathematics both of which I quite like.

So my real question is

• Where should I start reading?

Obviously, the paper Yemon pointed me to is one place to start but there may be a good summary out there that I wouldn't reach (in finite) time by a reference chase starting with that paper. So, any other suggestions? I'm reasonably well acquainted with algebraic theories in general so I'm looking for specifics to this particular instance.

Also, I'll write up my findings as I find them on the n-lab so anyone who wants to join me is welcome to follow along there. I probably won't actually start until the new year though.

translation - Transcription of a panel discussion about G.Perelman

Surprisingly, this was the only instance I heard the voice of Grisha Perelman. I thought his lectures in Stony Brook in April 2003 were videotaped and asked Dr.Christina Sormani whether that was so---she attended the lectures and had also posted her transparencies here. She told me that she believed he refused his lectures being videotaped. I still do not know what happened to his lectures in MIT…

Have you come across any other?

So, coming to the transcription, here is the translation of Perelman’s replies copied verbatim from Science 2.0 There you can find the whole story.

Grisha has picked up the phone. Here is approximately the dialog which followed:

---

Teacher: "Hello, can you hear me, do you recognize me ?"

Grisha: "Sure, of course, nice to hear from you."

Teacher: "Grisha, I am sorry but I am really overwhelmed by all these journalists who would like to know why you have refused to accept your prizes. May be you can tell me ?"

Grisha: "But I have already told all of them that I would not like to give any kind of interview."

Teacher: "But you know, I am not journalist !"

Grisha: "I will not tell you as well."

Teacher: "It is OK, sure I will conform to the common rule. But, please, you'd definitely benefit a lot, if you'd try not to be so adversarial to the people surrounding you ... Well, OK, tell me please, what are you doing right now ?"

Grisha: "Couldn't you please call me any other time ? I have a feeling that our present conversation is being recorded somehow ...".

Teacher: "No problem, I will definitely call you again soon and possibly even come to see you. Is it OK ?"

Grisha: "That's fine, thank you. Bye."

Teacher: "Yes, never mind, Grisha ! Bye !"

---

I admit that the above answer remains the best.

gt.geometric topology - Topological scaling (?)

Let $x_i$ be the coordinate of point $i$ and imagine there is an edge $e_{ij}$ between any pair of neighbors. The optimization problem

$$begin{array}{rl} underset{x}{mathrm{argmin}} & displaystylesum_{e_{ij}} left( x_j - x_i right)^2 \ mathrm{s.t.} 0 leq x_i leq 1 end{array}$$

is a strongly convex optimization problem (namely, a linearly constrained quadratic program or LCQP), hence it has a unique global minimum. Hence, the solution can be computed in polynomial time using an interior point method such as the barrier method. In fact, box-constrained quadratic programs can be solved quite efficiently and implementations are readily available (in MATLAB, for instance). In order to avoid trivial solutions, imagine that some of the $x_i$ are constrained to lie on the boundary, e.g., you can either add constraints like $x_i^1=1$ (where the superscript denotes the component) or simply hold some of the $x_i$ fixed.

Note that this problem attempts to minimize the pairwise distance rather than maximize it. On a compact domain the two problems will yield similar results, but the latter problem is nonconvex and hence it is harder to make guarantees about global optimality.

lo.logic - Can you tell if you have escaped from a recursive definition?

Most people define a function, f(n) on N recursively. I think I can calculate f(n) without dealing with f(n-r) for any 0 < r < n. How do I know that my method isn't still going through the same calculations needed to find f(n-1) (or whatever previous terms are required to find f(n) recursively) -- ?

1. If my method takes many fewer calculations than the recursive way of calculating it does that show that I am not relying on f(n-r) for any 0 < r < n? What would "many fewer" have to mean for this to be significant?




2. The number of calculations my method takes still depends on n, just like the recursive way of calculating f(n), does that alone mean that the methods are pretty much the same?




3. If my method takes the same number (or more) calculations than recursive way of calculating f(n) is there any other way of telling if my method is not, in some way, duplicating the recursive way of calculating f(n)?

Examples:

f(n) is recursively defined to be f(n) = f(n-1) + 1 and f(1) = 1
Then f(n) = n. Clearly, f(n) = n is a much faster way to find f(2876) rather than counting up from 1.

f(n) is recursively defined to be f(n) = f(n-1) + f(n-2) This is a linear recurrence and has a closed-form solution. $Fleft(nright) = {{varphi^n-(1-varphi)^n} over {sqrt 5}}={{varphi^n-(-1/varphi)^{n}} over {sqrt 5}},$
(from wikipedia)

$S(n,k)=kS(n−1,k)+S(n−1,k−1)$ with $S(n,n)=S(n,1)=1$ (Stirling numbers of the second kind) Almost seems like it's a linear recurrence... but we need to know about k-1. These numbers are defined as "the number of ways to partition a set of n objects into k groups" so, if I have written a few programs to find S(n,k) from that definition I want to know if I "must" find the values in the linear recurrence along the way...

But, I was trying to keep it more general to make it interesting?

One more example:

$C(n,k)=C(n−1,k)+C(n−1,k−1)$ with $C(n,n)=C(n,1)=1$ but $C(n,k) = frac{n!}{k!(n-k!)}$, most people like the 2nd one better of you want to look at large values of n and k>1...

pr.probability - How to fill a simplex with almost disjoint cuboids?

OK, since we finally have figured out what Andres is asking and since 600 characters is a bit too restrictive, I'll post this as an answer.

The following Asymptote code will draw the filling except I used the size 4 simplex instead of size 1 one here:

size(400);
import three;
import graph3;


pen[] q={red,green,magenta,blue,black};
q.cyclic(true);

int N=4;
triple O=(0,0,0),A=(4,0,0),B=(0,4,0),C=(0,0,4);
draw(O--A--B--C--A--C--O--B);
for(int n=0;N>n;++n)

{
real s=1/2^n;
for(real x=4-3*s;x>=0;x-=2s)
for(real y=4-x-3*s;y>=0;y-=2s)
{
draw(shift((x,y,4-x-y-3*s))*scale3(s)*unitcube,q[n]);
draw(shift((x-s,y,4-x-y-3*s))*scale3(s)*unitcube,q[n]);
draw(shift((x-s,y+s,4-x-y-3*s))*scale3(s)*unitcube,q[n]);
draw(shift((x-s,y,4-x-y-2*s))*scale3(s)*unitcube,q[n]);
}
}

The unitcube is just $[0,1]^3$, the rest should be self-explanatory.

gn.general topology - Why should algebraic objects have naturally associated topological spaces? (Formerly: What is a topological space?)

I will take the question at face value, but not in the sense of justifying the definition.

A topological space is a convenient way of encoding, or perhaps better, organising, certain types of information. (Vague but true! I will give some instances. the data is sometimes `spatial' but more often than not, is not.)

Perhaps we should not think of spaces as 'god given' merely 'convenient', and there are variants that are more appropriate in various contexts.

A related question, coming from an old Shape Theorist (myself) is : when someone starts a theorem with 'Given a space $X$...', how is the space 'given'? As an algebraic topologist I sometimes need to use CW-complexes, but face the inconvenience that if I could give the CW structure precisely I could probably write down an algebraic model for its homotopy type precisely, and vice versa, so a good model is exactly the same as the one I started with. I hoped for more insight into what the space 'was' from my modelling. Giving the space is the end of the process, not the beginning. Strange. A space is a pseudo-visual way of thinking about 'data', which encodes important features, or at leastsome features that we can analyse, partially.

If someone gives me a compact subspace of $mathbb{R}^n$, perhaps using some equations and inequalities, can I work out algebraic invariants of its homotopy type, rather than just its weak homotopy type? The answer will usually be no. Yet important properties of $C^*$ algebras on such a space, can sometimes be related to algebraic topological invariants of the homotopy type.

Spaces can arise as ways of encoding actual data as in topological data analysis, where there is a 'cloud' of data points and the practitioner is supposed to say something about the underlying space from which the data comes. There are finitely many data points, but no open sets given, they are for the data analyst to 'divine'.

Not all spatial data is conveniently modelled by spaces as such and directed spaces of various types have been proposed as models for changing data. Models for space-time are like this, but also models for concurrent systems.

Looking at finite topological spaces is again useful for encoding finite data (and I have rarely seen infinite amounts of data). For instance, relations between finite sets of data can be and are modelled in this way. Finite spaces give all homotopy types realisable by finite simplicial complexes. Finite spaces can be given precisely (provided they are not too big!) How do invariants of finite spaces appear in their structure? (Note the problem of infinite intersections does not arise here!!!)

At the other extreme, do we need points? Are locales not cleaner beasties and they can arise in lots of algebraic situations, again encoding algebraic information. Is a locale a space?

I repeat topological spaces are convenient, and in the examples you cite from algebraic geometry they happen to fit for good algebraic reasons. In other contexts they don't. Any Grothendieck topos looks like sheaves on a space, but the space involved will not usually be at all `nice' in the algebraic topological sense, so we use the topos and pretend it is a space, more or less.

dg.differential geometry - What is a section?

To your first question, "function on a space" $X$ usually means a morphism from $X$ to one of several "ground spaces" of choice, for example the reals if you work with smooth manifolds, Spec(A) if you work with schemes over a ring, etc. (This is a fairly selective use of the word "function" which used to confuse me.) A section $gamma$ of a (some-kind-of) bundle $Eto X$ is thought of as a "generalized function" on $X$ by thinking of it as a funcion with "varying codomain", i.e. at each point $xin X$, it takes value in the fibre
$E_xto x$. If one is talking about locally free / locally trivial bundles, meaning $E$ is locally (over open sets
$Usubset X$) isomorphic to some product $Utimes T$, then we can locally identify the fibres with $T$. Thus locally a section just looks like a function with codomain $T$, which is often required to be nice.

To your second question, I generally take the "right-inverse" or "pre-inverse" definition from category theory, because it relates back to others in the following precise way:

Say $pi: Yto X$ is a space over $X$ (intentionaly vague). The word "over" is used to activate the tradition of suppressing reference to the map $pi$ and refering instead to the domain $Y$. For $Usubseteq X$ open, the notation $Gamma(U,Y)$ denotes sections of the map $pi$ over $U$, i.e. maps $Uto Y$ such that the composition $U to Yto X$ is the identity (thus necessarily landing back in $U$). It's not hard to see that
$Gamma(-,Y)$ actually forms a sheaf of sets on $X$.

Conversely, given any sheaf of sets $F$ on a space $X$, one can form its espace étalé, a topological space over $X$, say $pi: acute{E}t(F) to X$. Then for an open $Usubseteq X$, the elements of $F(U)$ correspond precisely to sections of the map $pi$, which by the above notation is written $Gamma(U,acute{E}t(F)$. That is to say,
$F(-)simeqGamma(-,acute{E}t(F))$ as sheaves on $X$. This explains why people often refer to sheaf elements as "sections" of the sheaf.

Moreover, what we now denote by $acute{E}t(F)$ actually used to be the definition of a sheaf, so people tend to identify the two and write $Gamma(-,F)$ a instead of $Gamma(-,acute{E}t(F))$. This explains the otherwise bizarre tradition of writing $Gamma(U,F)$ instead of the the more compact notation $F(U)$.

$Big($Unfortunate linguistic warning: Many people incorrectly use the term "étale space". However, the French word "étalé" means "spread out", whereas "étale" (without the second accent) means "calm", and they were not intended to be used interchangeably in mathematics. This is unfortunate, because the espace étalé has very little to with with étale cohomology. More unfortunate is the annoying coincidence that when dealing with schemes the projection map from the espace étalé happens to be an étale morphism, because it is locally on its domain an isomorphism of schemes, a much stronger condition.$Big)$

To your third question, I think the observation that $Gamma(-,Y)$ forms a sheaf on $X$ gives a nice context in which to think of sections $X$ to $Y$: they "live in" the sheaf $Gamma(-,Y)$ as its globally defined elements.

ca.analysis and odes - What is the Implicit Function Theorem good for?

One thing I've learned recently (moving into symplectic geometry from topology) is that people often underestimate the value of regarding manifolds locally as graphs of functions, or submanifolds of Euclidean space. The IFT helps with this, and provides actual applications, but here's possible motivation.

--My recent favorite example for intuition: How does a surface of negative curvature behave? Like the graph of a harmonic function on $mathbb R^2$ (where the Hessian is not 0). That is to say, if you have trouble imagining a surface in $mathbb R^3$ which has no hills and valleys but only saddles, try graphing a function like $xy$ or $e^xcos y$.

--Related to this: use IFT to study the curvature of a surface in $mathbb R^3$ (or $n$-manifold in $mathbb R^{n+1}$) nearby a strict maximum in any one coordinate. I'm not sure what the nicest precise statement should be here to cover all non-generic cases, but it's fun route to differential geometry if someone hasn't had it.

--Less directly related to the IFT: What does a Morse critical point "look like''? We like to say that the Morse function can be given in some local coordinates by $x_1^2+ldots +x_k^2-x_{k+1}^2-ldots x_n^2$, but when we visualize this, we usually see the manifold (OK, surface) itself as the graph of that function over a piece of $mathbb R^n$. Thus making the Morse function "height" as usual.

arithmetic geometry - L-functions and random matrices

I am curious about the connection between properties of L-functions and random matrices, and about (if existent) function field versions of that. Do you know a survey or an other article where one could get an idea of those themes and possibly related issues (e.g. which of the many sorts of L-functions are related to random matrices)?

A very nice survey
on the function field case by Douglas Ulmer:
"The goal of this survey is to give some insight into how well-distributed sets of matrices in classical groups arise from families of L-functions in the context of the middle column of Weil’s trilingual inscription, namely function fields of curves over finite fields. The exposition is informal and no proofs are given; rather, our aim is illustrate what is true by considering key examples." There are several other very interesting articles on his website, BTW.

Dirichlet Approximation over a Number Field

For any $alpha in mathbb{R}$ and a parameter $Q$, we can write $alpha = a/q + theta$, for integers $a, q$ with $q leq Q$, and real $theta$ with $|theta|leq (qQ)^{-1}$, a simple application of the Dirichlet approximation theorem. I'm looking for a similar statement for number fields.

Setup: $K$ is a fixed number field of degree $n$ over $mathbb{Q}$, with ring of integers $O_K$. $omega_1, dots , omega_n$ is a fixed $mathbb{Z}$-basis for $O_K$.
$sigma_i, dots sigma_{n}$ are the distinct embeddings of $K$.

$V$ is the $n$- dimensional commutative $mathbb{R}$-algebra $K otimes_mathbb{Q} mathbb{R}$.

We define a distance function $| cdot |$ on $V$ as follows:

$$|x| = |x_1 omega_1 + cdots + x_n omega_n| = maxlimits_{i} | x_i |.$$ I would just like to point out that I am not necessarily attached to this distance function, if you can say anything sensible using another distance function, then I am interested.

Precise Statement: Given $alpha in V$ and a parameter $Q$, is it always possible to find $lambda, mu in O_K$, such that $|mu| ll Q$ and

$$|alpha - dfrac{lambda}{mu}| ll dfrac{1}{Q |mu|}?$$

Equivalent Statement: can we find $gamma in K$ such that $mathcal{N}=textrm{N}(bf{a}_gamma) ll Q^n$, and

$$|alpha - gamma| ll dfrac{1}{Q mathcal{N}^{1/n}},$$ where $bf{a}_gamma$ is the denominator ideal of $gamma$?

Note that it is easy to find an analogous statement to Dirichlet's original theorem, ie $exists lambda, mu in O_K$, such that $|mu| ll Q$ and

$$|alpha mu - lambda| ll dfrac{1}{Q},$$ by an application of the pigeonhole principle, but unlike in $mathbb{R}$, we cannot just divide through by $mu$ at this point, as the only decent bound (that I know of) for $|mu^{-1}|$ is $$|mu^{-1}| ll dfrac{|mu|^{n-1}}{textrm{N}(mu)}$$ .

Does anyone have a reference for dealing with the fractional form like this? The closest I have managed to find was a generalisation to number fields of the Thue-Siegel-Roth theorem by LeVeque.

Constructing Twisted K-theory

There is a simple, intuitive "construction" of twisted K-theory if we are allowed to ignore that many things only hold up to homotopy. We know that maps to $K(Z,2)$ give line bundles on a space and that $K(Z,2)$ forms a group corresponding to the tensor product of line bundles. Line bundles also act as endomorphisms of K-theory given by the tensor product. Thus, there is an action of $K(Z,2)$ on $F$ (where $F$ is the classifying space for $K^0$). $K(Z,2)$ principal bundles are classified by maps to $BK(Z,2) cong K(Z,3)$, ie, elements of $H^3$. Choosing such a map, we get a principal $K(Z,2)$ bundle, $E$, and we can form the associated bundle $E times_{K(Z,2)} F$. Twisted K-theory is then the homotopy classes of sections of this bundle.

The usual constructions of twisted K-theory that I have seen make the above precise by choosing representatives of the relevant objects so that all the needed relations hold on the nose. My question is whether you can avoid doing that. In other words, can you define all the various notions up to homotopy and obtain a definition of twisted K-theory that way?

homotopy theory - Are non-empty finite sets a Grothendieck test category?

A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $widehat{A}$ of presheaves of sets on $A$ admits a model category structure, which is Quillen equivalent to the usual model category structure on spaces.

The notion of test category was proposed by Grothendieck, and the above result was proved by Cisinski (Les préfaisceaux comme modèles des types d'homotopie. Astérisque No. 308 (2006)).
Examples of test categories include the category $Delta$ of non-empty finite ordered sets (i.e., the indexing category for simplicial sets), and $square$, the indexing category for cubical sets.

It's hard for me to give the precise definition of test category here: it involves the counit of an adjunction $i_A: widehat{A} rightleftarrows mathrm{Cat} :i_A^*$, where the left adjoint $i_A$ sends a presheaf $X$ to the comma category $A/X$ (where we think of $Asubset widehat{A}$ by yoneda). An online introduction to test categories, which includes the full definition and an account of Cisinski's results, is given in Jardine, "Categorical homotopy theory".

I don't really understand how one should try to prove that a particular category is a test category. The example I have in mind is $G$, the (skeleton of) the category of non-empty finite sets, and all maps between them. I believe this should be a test category; is this true?

Note that there is a "forgetful functor" $Deltarightarrow G$, which induces some pairs of adjoint functors between $widehat{Delta}$ and $widehat{G}$. If $G$ is really a test category, I would expect one of these adjoint pairs to be a Quillen equivalence.

Another note: $G$ is equivalent to the category of finite, contractible groupoids, which is how I am thinking about it.

co.combinatorics - Tiling A Rectangle With A Hint of Magic

Here's a a famous problem:

If a rectangle $R$ is tiled by rectangles $T_i$ each of which has at least one integer sidelength, then the tiled rectangle $R$ has at least one integer side length.

$mbox{}$

There are a number of proofs of this result (14 proofs in this particular paper). One would think this problem is a tedious exercise in combinatorics, but the broad range of solutions which do not rely on combinatorial methods makes me wonder what deeper principles are at work here. In particular, my question is about the proof using double integrals which I sketch out below:

Suppose the given rectangle $R$ has dimensions $atimes b$ and without loss of generality suppose $R$ has a corner at coordinate $(0,0)$. Notice that $int_m^nsin(2pi x)dx=0$ iff $mpm n$ is an integer. Thus, for any tile rectangle $T_i$, we have that:

$intint_{T_i}sin(2pi x)sin(2pi y)dA=0$

If we sum over all tile rectangles $T_i$, we get that the area integral over $R$ is also zero:

$intint_{R}sin(2pi x)sin(2pi y)dA=sum_iintint_{T_i}sin(2pi x)sin(2pi y)dA=0$

Since the cornor of the rectangle is at $(0,0)$, it follows that either $a$ or $b$ must be an integer.

My question is as follows: where exactly does such a proof come from and how does it generalize to other questions concerning tiling? There is obviously a deeper principle at work here. What exactly is that principle?

One can pick other functions to integrate over such as $x-[x]-1/2$ and the result will follow. It just seems like black magic that this works. It's as if the functions you are integrating over tease out the geometric properties of your shape in an effortless way.

EDIT: It's likely that one doesn't necessary need integrals to think in the same flavor as this solution. You're essentially looking at both side lengths in parallel with linear test functions on individual tiles. However, this doesn't really explain the deeper principles here, in particular how one could generalize this method to more difficult questions by choosing appropriate "test functions."

ac.commutative algebra - Primary decomposition for modules

I am quite curious about the definition and applications of the primary decomposition for modules.

1. The definition of a primary submodule. (Let's assume we work over a commutative noetherian ring $R$ and an $R$-module $M$) When I first worked on Atiyah-Macdonald I used this definition:

A submodule $N$ of $M$ is primary if any
zero divisor on $M/N$ is nilpotent.

But recently I saw the definition in Matsumura's commutative algebra, which is slightly different:

A submodule $N$ of $M$ is primary if any
zero divisor on $M/N$ is locally nilpotent, i.e. if $a$ is a zero divisor, then for any $x in M/N$, there exists $n$ possibliy depending on $x$ such that $a^n x = 0$.

Of course, these two definitions agree when $M$ is a finite $R$-module. (which I guess is the most interesting case) But what should be the "right" definition in the general situation?

1. The application of this. Is this generality of any use? If M is finite, then I know it admits a filtration with quotients being $R/{mathfrak{p}_i}$ where $mathfrak{p}_i$ are associated primes. This seems to be quite useful in some proofs. But what about the case where M is infinite?

3 Geometric meaning. Primary decomposition of an ideal I in R is related to the irreducible components of $mathrm{Spec}(R/I)$. Is there something similar for the module case?

Thanks very much!

Edit: As there still does not seem to be a clear consensus of answers, it would be great if experts could weigh in.

quantum field theory - photon propagator

Just multiply it out in Fourier, where $partial = ik$:

$$bigl[ (-k^2 +m^2) g^{munu} + k^mu k^nubigr] frac{ -g_{nulambda} + m^{-2} k_nu k_lambda }{k^2 - m^2} = frac1{k^2 - m^2} bigl( - (-k^2 + m^2) delta^mu_lambda + (-k^2 + m^2) m^{-2} k^mu k_lambda - k^mu k_lambda + k^mu k^2 k_lambda m^{-2} bigr) = delta^mu_lambda$$

which is a function of $k$. But converting back to position space, $1(k) = delta(x)$.

This proves that $D$ is a solution. To be the solution, you usually have to impose boundary conditions, etc. In this case, there are no solutions to $bigl[ (-k^2 +m^2) g^{munu} + k^mu k^nubigr] f_nu = 0$: the corresponding equation in Fourier is $(-k^2 + m^2) g^{munu} + k^mu k^nu = 0$, and contracting with $g_{mu nu}$ gives $0 = d(-k^2 + m^2) + k^2 = dm^2 - (d-1)k^2$, where $d$ is the dimension of spacetime, so $k^2 = frac{d}{d-1}m^2$, but $-frac1{d-1}m^2 g^{munu} + k^mu k^nu$ cannot equal $0$, as $k^mu k^nu$ cannot be an invertible matrix. So $D$ is the only solution.

Graphs with fractal properties?

Okay, so Steven's right -- there is a (countably infinite) graph which is totally self-similar, called the Rado graph. It's self-similar in the following way: If you partition its vertex set into two (or in general finitely many) parts, and consider the induced subgraph on each of those parts, one of those subgraphs will be isomorphic to the whole graph.

There are two other graphs with this property: the complete graph on countably infinitely many vertices, and the empty graph on countably many vertices. Up to isomorphism, these are the only countable such graphs. (or so Diestel tells me, and the proof actually isn't that hard.)

If you're looking for weaker forms of self-similarity... I'm not sure exactly what it is you want, then. You say something about "transforming nodes into subgraphs that are the same as the larger graph," but there's no reason you can't do something like that with a generic graph. There's a notion of graph product along these lines, where you replace the vertices of a graph by copies of another graph and then connect the new graphs according to the old one. So starting with a graph G, one could certainly construct $G cdot G$, and $G cdot G cdot G$, and so on and so forth... but I don't know if this sequence has a limit (or even in what category that would make sense -- presumably the category of graphs and embeddings, but...)

If you want "looks the same to certain graph invariants," then expander graphs are well worth looking at.

co.combinatorics - Covering of a graph via independent sets

I suspect that a topic such as this may have been considered before: if so, I hope that someone can point me to a reference on the subject.

I have a graph G with an upper bound d on its degree. Define an independent-set cover for G to be a family of independent sets in G, whose union is V(G). Is there a non-trivial upper bound for the smallest independent-set cover of G? [That is: I would like the number of independent sets in the family to be as small as possible; the size of any particular independent set within that family is unimportant.]

I'm also interested in the case where the degrees of the vertices may differ substantially from the maximum degree.

Edited to add: thanks to those who pointed out that the size of the minimum-size cover is just the chromatic number (by definition). I'm not sure how I managed to avoid noticing that this is what I was asking about, except that I probably think of colourings too much in terms of vertex-labellings and not often enough in terms of vertex partitions.

stable homotopy - Are there universe-indexed spectra over simplicial sets?

Yes to both interpretations of your question. It is not clear to me where you want to put pointed simplicial sets.

One interpretation of your question is that you want to replace pointed topological spaces with pointed simplicial giving the notion of a spectrum as a functor from supspaces of U to pointed sSet. This is a very common thing to do, and in some circles in homotopy theory is the standard definition of spectrum. Often "space" is interpreted as meaning simplicial set. This is because of the standard Quillen equivalence between Top and sSet.

The other possible interpretation of your question is that you are trying to replace vector spaces with simplicial sets. This is also (essentially) something which has been done. It gives a model of spectra known as W-spaces. This is one of the standard diagram category models of spectra. See the following paper for a comparison:

Model categories of diagram spectra, by M. A. Mandell, J. P. May, S. Schwede, and B. Shipley

linear algebra - A question on star-congruence.

We consider $ntimes n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $*-$congruent, then

$$(i_+(A), i_-(A), i_0(A))=(i_+(B), i_-(B), i_0(B)).qquad{(1)}$$
If two general matrices $A$ and $B$ are $*-$congruent, (1) may not hold (can you provide an example?).
Moreover, whether a matrix and its transpose are always $*-$congruent?

ag.algebraic geometry - Is tensoring with a module representable iff it is locally free of finite rank?

Motivation:

It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme
$Yto Spec(A)$. That is, maps $Spec(A)to Y$ such that $Spec(A)to Y to Spec(A)$ is the identity.

What's wrong with the "espace étalé":

One way to do this is to consider the associated sheaf $tilde{M}$, and form its "espace étalé" $acute{E}t(tilde{M})$. Observe that this topological space is naturally an $X$-scheme (essentially by its construction, as for $acute{E}t$ of any sheaf of sets), and that $Gamma(U,acute{E}t(tilde{M})) = tilde{M}(U)$ for opens $Usubseteq Spec(A)$.

I'm not happy with this construction in that it has "the wrong fibres": for $Itriangleleft A$, the sections of $acute{E}t(tilde{M})$ over (base changed to) a closed subscheme $Z(I)$, e.g. a point, do not correspond to $widetilde{M/IM}$. This is just an instance of the fact that $acute{E}t$ doesn't respect base change: given $f:Spec(B)to Spec(A)$, in general $acute{E}t(f^* tilde{M})neq f^* acute{E}t(tilde{M})$.

Conclusion:

I want a construction that does respect base change. That is, for any module $M$ on $X$, I want an $X$-scheme $Y$ such that for any $X'to X$, $Gamma(X',Y_{X'}) = tilde{M}_{X'}(X')$. This amounts to finding a scheme which represents the functor $Bmapsto Botimes_A M$ from $A$-algebras to sets.

The question: (updated, thanks to some comments from a fortiori and buzzard)

EGA I (1971) 9.4.10 mentions in passing, without proof, that this functor is representable by a scheme if and only if $M$ is locally free of finite rank.

• If this is correct, does anyone know where to find the proof?




• If not, does anyone know a correct (and useful) equivalent condition on $M$?

So far, I gather that:

• It is not always representable if $M$ is not finitely generated; see this earlier question.




• If $M$ has a pre-dual, say $N^vee = M$, $mathbb{V}(N)=Spec(Sym(N))$ does not generally work (see a fortiori's comment below)

(This may not have a useful answer, or perhaps it has several...)

fa.functional analysis - Yet more on distortion

I would like to elaborate a little bit on my previous question which can be found
here.

Firstly, let me recall that a separable Banach space $(X, | cdot |)$ is said to be
arbitrarily distortable if for every $r > 1$ there exists an equivalent norm $| cdot |$
on $X$ such that for every infinite-dimensional subspace $Y$ of $X$ we can find a pair of
vectors $x, y$ in $Y$ such that $|x|=|y|=1$ and $|x| / |y| >r$. If $X$ has a Schauder
basis $(e_n)$, then this definition is equivalent to the following:

For every $r > 1$ there exists an equivalent norm $| cdot |$ on $X$ such that for every
normalized block sequence $(v_n)$ of $(e_n)$ there exist a non-empty finite subset $F$
of $mathbb{N}$ and a pair of vectors $x, y$ in $span{v_n: nin F}$ such that $|x|=|y|=1$
and $|x| / |y| >r$.

This equivalence gives us no hind of where the finite set $F$ is located. In other words:

if a Banach space $X$ with a Schauder basis is arbitrarily distortable, then where do
we have to search in order to find the vectors verifying that $X$ is arbitrarily distortable?

Now, there are various ways of quantifying Banach space properties and my question
is towards understanding who the "difficulty" for finding these vectors can be quantified.

The main tool will be certain families of finite subsets of $mathbb{N}$. These families were discovered (independently) by two groups of researchers: Banach space theorists (Schreier families; see 1 below) and Ramsey theorists (uniform families; see 2 below). In particular, for the discussion below we need for every countable ordinal $xigeq 1$ a family $F_xi$ of finite subsets of $mathbb{N}$ such that:

1. $F_xi$ is regular (i.e. compact, hereditary and spreading; I am sorry for not giving
   the precise definition of these notions but this would make the post too long; but I
   will be happy to answer to any comment).


2. The families are increasing (with respect to $xi$) both in size and complexity. That is, the "order" of $F_xi$ is at least $xi$ and if $zeta<xi$ there there exists $k$ such that all subsets of $F_zeta$ whose minimum is greater than $k$ belong to $F_xi$.


3. For $xi=1$, let us take the Schreier family consisting of all finite subsets of
   $mathbb{N}$ whose size (or cardinality if you prefer) is less than or equal to their minimum.

There many examples of such families, all constructed using transfinite induction.
Some of them have extra important properties. For concreteness (and to simplify things)
let us work with the Schreier families.

Now we come to the following:

Definition: Let $(X,| cdot |)$ be a Banach space with a Schauder basis $(e_n)$ and $xi$ be a countable ordinal with $xigeq 1$. Let us say that $X$ is $xi$-arbitrarily
distortable if for every $r > 1$ there exists an equivalent norm $| cdot |$ on $X$
such that for every normalized block sequence $(v_n)$ of $(e_n)$ there exist a non-empty
set $F$ belonging to the family $F_xi$ and a pair of vectors $x, y$ in $span{v_n: nin F}$ such
that $|x|=|y|=1$ and $|x| / |y| >r$.

In other words, if $X$ is $xi$-arbitrarily distortable, then we have narrow down the
search for the critical set $F$; it has to belong to an a priori given "nice" family
of finite subsets of $mathbb{N}$.

For every Banach space $(X,| cdot |)$ with a Schauder basis $(e_n)$ define

$$AD(X)=min{ xi: X is $xi$-arbitrarily distortable}$$
if $X$ is $zeta$-arbitrarily distortable for some $1leq zeta< omega_1$. Otherwise set $AD(X)=omega_1$.

One can prove the following equivalence:

Let $X$ be a separable Banach space with a Schauder basis.
Then $X$ is arbitrarily distortable if and only if $AD(X)<omega_1$.

This leaves open a number of interesting questions.

Question 1: Is it true that $AD(ell_2)>1$? This is just a restatement of my
previous question.

Question 2: Can we compute $AD(ell_p)$ for every $1 < p < +infty$?

Question 3: Can we find for every countable ordinal $xigeq 1$ an arbitrarily distortable
Banach space $X_xi$ such that $AD(X_xi)>xi$. The answer is yes for $xi=1$; any arbitrarily
distortable asymptotic $ell_1$ space $X$ satisfies $AD(X)>1$.

Notice that an affirmative answer to Question 3 leaves no hope for a "uniform" approach
to distortion on general separable Banach spaces.

---

Some references:

1. D. Alspach and S. Argyros, Complexity of weakly null sequences, Dissertationes Math. 321 (1992), 1-44.




2. P. Pudlak and V. Rodl, Partition theorems for systems of finite subsets of integers,
   Discrete Math. 39 (1982), 67-73.

co.combinatorics - Simple/efficient representation of Stirling numbers of the first kind

Stirling numbers of the second kind can be expressed by means of a simple hypergeometric (considering $n$ fixed) sum

$$S_2(n,k) = frac{1}{k!}sum_{j=0}^{k}(-1)^{k-j}{k choose j} j^n. qquad (1)$$

This can be used for direct calculation of $S_2(n,k)$, without the need to compute any preceding values. But for Stirling numbers of the first kind, one seems to need a nested sum or a recurrence over preceding values, the most direct known representation perhaps being

$$S_1(n,k) = sum_{j=0}^{n-k} (-1)^j {n+j-1choose n-k+j} {2n-k choose n-k-j} S_2(n-k+j,j). qquad (2)$$

Is there a reason to believe that no formula similar to (1) exists for Stirling numbers of the first kind? Does a formula better than (2)+(1) for calculations exist (assume that I have no interest in generating a table of all preceding values)?

sg.symplectic geometry - Has anything precise been written about the Fukaya category and Lagrangian skeletons?

At some point in this past year, some Fukaya people I know got very
excited about the Fukaya categories of symplectic manifolds with "Lagrangian skeletons." As I understand it, a
Lagrangian skeleton is a union of Lagrangian submanifolds which a
symplectic manifold retracts to. One good example would be the
zero-section of a cotangent bundle, but there are others; for example,
the exceptional fiber of the crepant resolution of $mathbb C^2/Gamma$ for $Gamma$ a finite subgroup of $SL(2,mathbb C)$. From the rumors I've heard, apparently there's some connection between the geometry of the skeleton and the Fukaya category of the symplectic manifold; this is understood well in the case of a cotangent bundle from work of Nadler and Nadler-Zaslow

I'm very interested in the Fukaya categories of some manifolds like
this, but the only thing I've actually seen written on the subject is
Paul Seidel's moderately famous picture of Kontsevich carpet-bombing
his research program, which may be amusing, but isn't very
mathematically rigorous. Google searching hasn't turned up much, so I
was wondering if any of you have anything to suggest.

tag removed - Finding Functional form for a given Scaling Condition

Dear all

While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem.

$G(k)$ is a complex valued function, and satisfy the following condition:

$G(kmu) = G(k)^2+ beta$

with $beta,mu$ constant (in my case $beta=frac{2}{9}, mu = frac{4}{3}$)

Is there a way to find the functional form of $G(k)$ which satisfy the condition?

Note that for $beta = 0$, $G(k)=expleft(a k^{log_mu 2}right)$, ($a$ konstant) will satisfy the condition (easily verified), but I have no idea on how to find a solution for non-zero $beta$. I'm a not a math student (I'm studying physics), but I have never seen problems like this before. Is there a way to find analytical expression for $G(k)$? Possible as an expansion?

I can generate a function which has this property on the computer. Writing $G(k)= x(k) + i y(k)$, with $x(k)=x(-k)$ and $y(k)=-y(-k)$ the function should look something like this:

http://dl.dropbox.com/u/483049/xy.pdf

--
jon

lo.logic - "Requires axiom of choice" vs. "explicitly constructible"

I think I'm a bit confused about the relationship between some concepts in mathematical logic, namely constructions that require the axiom of choice and "explicit" results.

For example, let's take the existence of well-orderings on $mathbb{R}$. As we all know after reading this answer by Ori Gurel-Gurevich, this is independent of ZF, so it "requires the axiom of choice." However, the proof of the well-ordering theorem that I (and probably others) have seen using the axiom of choice is nonconstructive: it doesn't produce an explicit well-ordering. By an explicit well-ordering, I simply mean a formal predicate $P(x,y)$ with domain $mathbb{R}timesmathbb{R}$ (i. e., a subset of the domain defined by an explicit set-theoretic formula) along with a proof (in ZFC, say, or some natural extension) of the formal sentence "$P$ defines a well-ordering." Does there exist such a $P$, and does that answer relate to the independence result mentioned above?

More generally, we can consider an existential set-theoretic statement $exists P: F(P)$ where $F$ is some set-theoretic formula. Looking to the previous example, $F(P)$ could be the formal version of "$P$ defines a well-ordering on $mathbb{R}$." (We would probably begin by rewording that as something like "for all $zin P$, $z$ is an ordered pair of real numbers, and for all real numbers $x$ and $y$ with $xneq y$, $((x,y)in P vee (y,x)in P) wedge lnot ((x,y)in P wedge (y,x)in P)$, etc.) On the one hand, such a statement may be a theorem of ZF, or it may be independent of ZF but a theorem of ZFC. On the other hand, we can ask whether there is an explicit set-theoretic formula defining a set $P^*$ and a proof that $F(P^*)$ holds. How are these concepts related:

• the theoremhood of "$exists P: F(P)$" in ZF, or its independence from ZF and theoremhood in ZFC;




• the existence of an explicit $P^*$ (defined by a formula) with $F(P^*)$ being provable.

Are they related at all?

ct.category theory - The simplicial Nerve

About the first question: Yes, the simplicial nerve is an instance of the same general construction which gives you the usual nerve (and e.g. also the Quillen equivalences between simplicial sets and topological spaces, between models for the $mathbb{A}^1$-homotopy category given by simplicial presheaves and sheaves respectively and much more):

A cosimplicial object in a cocomplete category E gives, via Kan extension, rise to an adjunction between $E$ and simplicial sets, where the left adjoint goes from simplicial sets to $E$ and comes from the universal property of the Yoneda embedding (namely that functors from $C$ to $E$, $E$ cocomplete, are the same as colimit perserving functors from $Set^{C^{op}}$ to $E$). This is (part of) Proposition 3.1.5 in Hovey's book on model categories.

Lurie uses the same pattern; it is enough to give a cosimplicial object in simplicial categories, which is done in his definition 1.1.5.1.

homotopy theory - How do you compute the space of lifts of an E-infinity map?

The issue with the underlying monoid seems to complicate everything, in a similar way to how Postnikov decompositions are complicated by the $pi_1$ issue. In that case, the common technique is to fix $G = pi_1$ and consider the Postnikov tower as operating in the category of spaces over $K(G,1)$ rather than in the ordinary category of spaces.

So I don't see a lot of hope immediately for getting the $pi_0$ problem out of the way at the same time; it seems like it colors the whole problem.

Once you've decided on a lifting $pi_0 B to pi_0 X$, though, you can fix the underlying monoid because then you're reduced to studying lifts $B times_{pi_0 Y} pi_0 X to X$ over $pi_0 X$.

If you then fix $M = pi_0 X$ then there's certainly some kind of obstruction theory, but the problem is identifying the obstruction classes as coming from something cohomological that you can actually calculate. It seems to me that one should study the symmetric monoidal category of "spaces over $M$", with product having fibers

$$(X star Y)_m = coprod_{m' m'' = m} (X_{m'} times Y_{m''})$$
(which is some kind of left Kan extension), and try to get some handle on it.

Even when $M = mathbb{N}$ the bookkeeping gets complicated. Then you're studying "graded $E_infty$ spaces" and your obstruction theory will land in something like cohomology with coefficients in the relative homotopy groups of $Y$ over $B$, but you're taking cohomology of the "derived indecomposables" in your $E_infty$ space. The zero'th space of derived indecomposables of an $E_infty$ space $B$ over $mathbb{N}$ is the topological Andr'e-Quillen homology object of $B_0$. Even if $B_0$ is trivial, then the zero'th derived indecomposable space is trivial, the first is $B_1$, and the next is the homotopy cofiber of the squaring map $(B_1 times B_1)_{hmathbb{Z}/2} to B_2$.

Based on this kind of futzing around I am led to believe that your obstructions may possibly occur in the relative topological Andre-Quillen cohomology of $Sigma^infty_+ B$ over $Sigma^infty_+ M$ with coefficients in the relative homotopy of $X$ over $Y$. But the problem seems very difficult for a general monoid $M$.

(Especially evidenced by the fact that Charles Rezk hasn't popped in here with an answer yet.)

rt.representation theory - restriction of a representation of GL(n) to GL(n-1)

Though Peter's answer addresses the finite-dimensional representation theory, I believe that the question asks about the unitary representations on Hilbert spaces, and more general irreps on Banach and Frechet spaces.

This question has been the subject of much recent work by Avraham Aizenbud, Dmitry Gourevitch, Steve Rallis, Gerard Schiffmann, and Eitan Sayag. In particular, Aizenbud and Gourevitch prove the following in their paper "Multiplicity One Theorem for $(GL_{n+1}(R), GL_n(R))$":

Let $F = R$ or $F = C$. Let $pi$ and $tau$ be irreducible admissible smooth Fr'echet representations of $GL_{n+1}(F)$ and $GL_n(F)$, respectively. Then

$$dim left( Hom_{GL_n(F)}(pi, tau) right) leq 1.$$

This paper is on the ArXiv, and now published in Selecta, according to Aizenbud's webpage.

Zhu and Binyong have also proved this, I believe. The result has also been proven for irreducible smooth repreesentations of $GL_{n+1}(F)$ and $GL_n(F)$, when $F$ is a $p$-adic field by Aizenbud-Gourevitch-Rallis-Schiffmann.

Considering the smooth Fr'echet case should suffice for the case of unitary representations on Hilbert spaces, I believe, by considering the subspace of smooth vectors and Garding's theorem. I'd guess it would also work for Banach space representations, but I'm not an expert on these analytic things.

It's important to note that semisimplicity may be lost when one restricts smooth representations in these settings -- so their theorem says something about occurrences of quotients after restriction. It's important to be careful about the meaning of "multiplicity-free" in these situations.

ct.category theory - Can infinite first-order categories be specified other than as categories of models?

I am glad to see that a general question like Is there a relationship between model theory and category theory? receives quite a lot attention and no down-votes for being too general and unspecific. So I feel encouraged to pose some follow-up questions (firstly restricted to first-order model theory).

Preliminaries

I hope the following statements are sufficiently sensible, precise and correct.

• Each first-order theory $T$ with signature $sigma$ unambigously defines a class of (ZF-)models.




• This class of models of $T$ together with the $sigma$-homomorphisms form a category (the category of models of $T$).




• Two first-order theories with two arbitrary signatures may define equivalent categories of models.

Definition: A first-order theory $T$
provides a model of a category $C$ if the
category of models of $T$ is
equivalent to $C$.

• Each category $C$ defines a (possibly empty) set of first-order theories: the set of all $T$ which provide a model of $C$.

Questions

[Remark: I had to work this question over, since it seemed to be ill-posed.]

Old version:
Can infinite concrete categories be specified
other than as categories of (ZF-)models of some (possibly higher-order) theory? Examples?

New version (explicitly restricted to first-order theories):

Given an infinite category of models
of a first-order theory $T$. Can this
category - or one equivalent to it - be
specified/represented/given
independently of any first-order
theory $T$ and its (ZF-)models?

Remark: $T$ of course can be specified/represented/given independently of its models: as a set of formulas.

---

Why is the notion of models of a (concrete)
category so uncommon? (Maybe because the
answer to the first question is "No"?)

---

Is there a genuine model-theoretic notion of two
equivalent theories if these have
two arbitrary signatures?

---

descriptive set theory - Related Open Game in Analytic Determinacy

For this question, please refer to Chapter 33 page 638, Set Theory Millennium Edition, by Thomas Jech.

The proof of analytic games $G_A$ is converted into an open game $G^ast$ on some suitable space. In the game $G^ast$, Player I plays $a_0$, then Player II plays a pair $(b_0, h_0)$, then I plays $a_1$, followed by II playing $(b_1, h_1)$, and so on. Each $h_i$ is an order preserving function from $K_i$ into $kappa$ (refer to the text for the definition of $K_i$). If Player II is able to construct the $h$'s such that $h_{i+1}$ extends $h_i$ for each $iinomega$, then Player II wins. This game $G^ast$, as mentioned, is an open game.

I would like to know specifically what the payoff set is and what the space is (including the topology).

I came up with the following:
The space is $kappa^K$, where $K=bigcup_{ssubseteq x}K_s$, $x=langle a_0, b_0, ldotsrangle$ formed at the end of the run, and the topology is just like that of the Baire space. The payoff set in $G^ast$ is $A^ast={finkappa^K:f;text{is not order preserving}}$. This set is closed by showing that the complement is open (easy). Hence, the game $G^ast$ is an open/closed game.

The above seem plausible, but the issue I have with the set I came up with is

1. it doesn't look like a set that include a "pair" as in $(b_0, h_0)$, for instance, does not appear anywhere, and




2. the set $K$ is particular to the $x=langle a_0, b_0, ldotsrangle$ produced.

I would appreciate if you could let me know if the above is right and provide me a hint as to how to relate to the related notion of a homogeneous tree thereafter. If the above doesn't makes sense, please let me know where it went wrong and a hint as to how to get the right open set in what topological space would be nice.

Thanks!

ac.commutative algebra - Isolated hypersurface singularities, Chow groups and D-branes

Assume $k= mathbf C$ and $W$ homogeneous. Let $X=Proj (k[x_1,cdots,x_n]/(W))$. $X$ is then a smooth hypersuraface in $mathbb P_{n-1}$.

Assume $n=2d$ is even. Corollary 3.10 of the paper you quoted says that $theta=0$ for all pairs iff the homological Chow group $CH^{d-1}_{hom}$ modulo $[h]^{d-1}$ is not torsion (here $[h]$ is the class of the hyperplane section). So your question, in this case, is equivalent to
($l=d-1$):

Examples of smooth hypersurfaces of dimension $2l$ such that $CH^{l}_{hom}/([h]^{l})$ is not torsion ?

(By the way, I think if you phrased your question this way, it probably would become more popular, consider how many geometry-inclined people visit this site! So if you want more and better answers, consider changing the title.)

Now, a cheap way to get examples you want is to take $W= x_1x_{d+1} + cdots + x_dx_{2d}$. Then the cycle defined by $(x_1,...,x_d)$ will not be a multiple of a power of the hyperplane section. Why? Because, I am waving my hand a bit here, if it is then the intersection with the cycle $(x_{d+1},cdots, x_{2d})$ would be positive. But they are disjointed in $X$!

The same trick works for generalized quadrics, i.e. if $W = f_1g_1 +cdots +f_dg_d$.

EDIT: Let me give more details here. In this situation you can easily make $W$ non-homogeneous as you desire. But the trouble is you can't use my argument above as there is no longer a projective variety $X$. But one can get around this. Let $S=k[x_1,cdots,x_{2d}]_{m}$

here $m$ is the irrelevant ideal. Suppose $W = f_1g_1 +cdots +f_dg_d$ and assume that $(f_1,cdots, f_d, g_1,cdots, g_d)$ is a full system of parameters in $S$. Let $R=S/(W)$, $P=(f_1,cdots,f_d)$ and $Q=(g_1,cdots,g_d)$. I claim that $theta^R(R/P,R/Q) neq 0$.

The reason is that $theta^R(R/P,R/Q) = chi^S(S/P,S/Q)$, the Serre's intersection multiplicity (see Hochster's original paper). Because $dim S/P + dim S/Q = d+d =dim S$, we must have $chi^S(S/P,S/Q)>0$ by Positivity, which is known in this case.

More exotic examples should be abound, and I am sure people who know more intersection theory can provide some, once they are aware of what this question is about. I would be interested in hearing more answers along that line.

st.statistics - Distribution function of dependent variable, confidence regions

You need to make your answer more precise. First, the regression should be
y = a + bx + e
where e has some distribution, let's say described by a density g(e)

Now, do you mean:

1) Obtaining the density f(y|x) theoretically?
This one is easy. By the transformation of densities
f(y|x) = 1/(a+bx)*g(y/(a+bx))

2) Estimating the density when you do not know the parameters of the regression model?
This one will depend on what you know and what you don't know, and what data you have at hand, and is a more complex issue, so you need to provide more information.

big list - What are some fundamental "sources" for the appearance of pi in mathematics?

As for the normal distribution, you can characterize it as the unique distribution with the following properties:

Let $X_1, X_2, cdots X_n$ be independent identically distributed normal random variables. Then the joint distribution of the vector $X=(X_1, X_2, cdots X_n)$ is the same as that of $AX$ where $A$ is any orthogonal matrix. So the normal distribution is intimately related to the geometry of real inner product spaces.

The $pi$ comes from the fact that you can integrate such a distribution by first integrating over a sphere and then integrating over $[0,infty]$. Because the distribution is orthogonally invariant, you pick up a constant corresponding to the area of the sphere. For $n=2$ you get the circle, and this is the usual calculation for computing the normalization constant for the normal distribution.

So then the mystery becomes: given that the normal distribution is so closely tied to inner product spaces, why does it show up all the time? The central limit theorem tells us that all that really matters in large scale limits are the first and second moments. The first moment can always be eliminated by re-centering. So all that matters is the second moment. But the second moment comes from the covariance, which is an inner product! (technically, only once you restrict to re-centered random variables, but we are doing that)

I'd venture a guess that most, if not all, appearances of $pi$ in statistics boil down to this fact that covariance is an inner product, and the fact that spheres, which are the norm-level sets for inner product spaces, have areas related to $pi$

physics - Perpetuum Mobile

I think the finite size is a red herring; with ideal sources/sinks and particles it still doesn't "work".

Incorrect approach: 100% of rays from from blue focus go to the red one; 80% of rays from red focus go to the blue one; 100% > 80%; therefore imbalance. Incorrect because we must look at absolute numbers, not percentages, when saying the flow from blue to red must equal the flow from red to blue.

Incorrect approach: Initally both foci emit 100 particles, and during a time period epsilon, 100 particles go from blue to red, but only 80 go from red to blue. True so far, but Incorrect approach because the paths are not all the same length so the travel times are not all the same. When all paths are equally full of particles, the numbers arriving at each focus will be the same.

Correct approach: Short path takes T1 time units to travel; long path takes T2. At equilibrium, during a time period epsilon, 100 particles will leave blue, of which the fraction that hit the tighter-curved mirror will arrive at red after T1, and the fraction that hit the longer-curved mirror will arrive at red after T2. During the same time, 100 particles will leave red; a fraction of those will go to blue in time T1, and a fraction will go back to red in time T2. I'm too lazy to work it all out right now, but that's the correct approach.