nt.number theory - Why no abelian varieties over Z?

Comments by Anweshi

The essential point is what Emerton mentioned, ie the analogy with Minkowski's theorem on number fields with ramification. The basic principle is that "arithmetic is geometry". Number rings are in a sense zero dimensional objects, elliptic curves one dimensional objects and abelian varieties correspond to higher dimensions. So we have Minkowski's theorem. And we ask, can we extend it to higher dimensions? Tate, after setting up the theory correctly as in his famous survey article on the arithmetic of elliptic curves, proved it rather trivially for elliptic curves(as Emerton mentions). Now the task is for abelian varieties.

Fontaine comes along, and proves that it is indeed the case. But the proof turns out to be much more complicated than expected. He built a whole lot of "Fontaine theory" around this. It goes into $p$-adic Hodge theory, $p$-adic Galois representations etc. He worked on it for some 15 years in isolation, it is said. The first major success of his theory was this theorem, and later it gained popularity. Now it is a major stream of research in arithmetic geometry.

References:

• Neukirch, Algebraic number theory, for the general philosophy that "arithmetic is geometry".


• Notes of Robert Coleman's course on Fontaine's theory of the mysterious functor


• The Bourbaki expose of Bearnadette Perrin-Riou. Fonctions L p-adiques des représentations p-adiques, Astérisque 229, (1995).


• Tate, The Arithmetic of Elliptic Curves, Survey Article, Inventiones.

It could be also worthwhile to have a look at the articles on finite flat group schemes in the volume Arithmetic Geometry of Cornell and Silverman, and in the volume Modular forms and Fermat's Last Theorem by Cornell, Silverman and Stevens. This is all intimately connected with them, as Emerton mentions. In fact, you can find a particular viewpoint by Fontaine on Finite Flat group Schemes.

There could be also be a simpler motivic explanation of this, without getting into the intricacies of Fontaine theory. The reason I think so is the following. I have heard the answer that there is no elliptic curve over $F_1$ because from the zeta functions the motives turn out to be mixed Tate. But, on the other hand, my own "proof" of this fact was that if there were an elliptic curve or abelian variety over $F_1$, it would be extensible to $Spec Z$ and there by Fontaine's theorem the only abelian scheme is the trivial one. Ever since I have wondered, whether it is possible to substitute Fontaine's theory arguments with motivic ones.

Emerton clarified to me in this connection: From a number theorist's point of view, p-adic Hodge theory is one of the key ingredients in the theory of motives, so these arguments are motivic, in a certain sense. (Perhaps one can say that p-adic Hodge theory encodes arithmetic properties of motives in a way analogously to the way that Hodge theory encodes geometric and analytic properties.)

Thus, by Emerton's answer, Fontaine theory seems to be thus a deeper part of motives. However, this "no abelian variety over Z" theorem of Fontaine was the first major application of Fontaine's theory. I imagined, if any results of Fontaine's theory were to be replaced by usual motivic arguments, then this ought to be the first candidate.

Before stopping, I must mention the intimate connection all this has with Iwasawa theory. Fontaine's theory is very much tangled with it, as could be seen in the expose of Perrin-Riou. However the more knowledgeable people should clarify on this.

This might be an apt place to mention the conference in honor of Fontaine. He is about to retire, after his great achievements.

Comment by Ilya

I think this should be indeed related to motives. (update: I think others provided some good references.)

Comments by Emerton

(1) There were earlier applications of Fontaine's results on finite flat group schemes; e.g. they played a role in Mazur's proof of boundedness of torsion of elliptic curves over $mathbb Q$. I say this just to emphasize that Fontaine's theory didn't really develop in isolation.
His theory is deep and technical, and it took people time to absorb it. But the theory of finite flat group schemes and $p$-divisible groups has a long history intertwined with arithmetic: there are results going back to Oda, Raynaud, and Tate; Fontaine generalized these; they were used by Mazur in his work, and by Faltings; Fontaine generalized further to $p$-adic Hodge theory (a theory whose existence was in part conjectured earlier by Grothendieck, motivated by, among other things, the work of Tate); ... . One shouldn't think of these ideas as being esoteric (despite the ``black magic'' label); they are and always have been at the forefront of the interaction between geometry and arithmetic, in one guise or another. (As another illustration, Fontaine's theory also closely ties in with earlier themes in the work of Dwork.)

(2) I'm not sure that there is any particular kind of usual motivic argument. The phrase motive conjures up a lot of different images in different peoples minds, but one way to think of what motivic means is that it is the study of geometry via structures on cohomology. From this point of view, $p$-adic Hodge theory is certainly a natural and important tool.

Here are some papers that give illustrations of $p$-adic Hodge theoretic reasonsing in what might be regarded as a motivic context:

Grothendieck, Un theoreme sur les homomorphismes de schemas abeliens, a wonderful paper.
Although the results are essentially recovered and generalized by Delignes work in his Hodge II paper, it gives a fantastic illustration of how $p$-adic Hodge theoretic methods can be used to deduce geometric theorems.

Kisin and Wortmann, A note on Artin motives

Kisin and Lehrer, Eigenvalues of Frobenius and Hodge numbers

James Borger, Lambda-rings and the field with one element

These three are chosen to illustrate how $p$-adic Hodge theory arguments can be used to make geometric/motivic deductions. The paper of Borger is also an attempt in part to provide foundations for the theory of schemes over the field of one element, and illustrates how $p$-adic Hodge theory plays a serious role in their study.

Maulik, Poonen, Voisin, Neron-Severi groups under specialization, a terrific paper, which
illustrates the possibility of using either $p$-adic Hodge-theoretic arguments or classical Hodge-theoretic arguments to make geometric deductions. (This is the same kind of complementarity as in Grothendieck's paper above compared to Deligne's Hodge II.)

Comments by Anweshi.

@Emerton, or anybody else: If there is something which does not make sense in my foray into "motivic" pictures, or something else which does not make sense, please feel free to erase and edit in whatever way you wish.

a further question by Thomas:

The great references given above let me ask about the current status of the many conjectures and open questions in Illusie's survey, e.g. finiteness theorems, crystalline coefficients, geometric semistability,... ?

• ilya's comment: I think it would be very useful if somebody posted a question along the lines of what Thomas suggests, especially filling in some background from Illusie's paper (I would do it, but I don't have the paper itself).

** Anweshi's comment:** Fontaine's theory uses a great deal of crystalline cohomology. For instance please see Robert Coleman's notes referred above.

lo.logic - randomness in nature

The field of statistical physics exists for this question. Basically when you have a nonequilibrium state that is complicated (e.g., has high entropy, Kolmogorov complexity, or whatever you like) and some kind of hyperbolic dynamics, the process of averaging leads to effective parabolicity. Thus you have things like the heat equation emerging from the effectively deterministic but complex Newtonian (quantum effects really aren't responsible for anything but perhaps the averaging scale, which is extremely small) microdynamics of particle collisions.

ca.analysis and odes - Applications of Measure, Integration and Banach Spaces to Combinatorics

Here's a nice application of measure theory, precisely, of the the theory of orthogonal polynomials, to a classic problem of counting derangements.

Problem: How many anagrams with no fixed letters of a given word are there?

For instance, for a word made of only two different letters, say $n$ letters $A$ and $m$ letters $B$, the answer is, of course, 1 or 0 according whether $n = m$ or not, for the only way to form an anagram without fixed letters is, exchanging all the $A$ with $B$, and this is possible if and only if $n=m$.

In the general case, for a word with $n_1$ letters $X_1$, $n_2$ letters $X_2$, ..., $n_r$ letters $X_r$, you will find (after the proper use of the inclusion-exclusion formula) that the answer has the form of a sum of products, that looks very much like the expansion of a product of sums, yet it is not. It is not, exactly because of the presence of terms $k!$, that would formally make a true expansion of a product of sums, if only they where replaced by corresponding terms $x^k$. This suggests to express them with the Eulerian integral $k!=int_0^infty x^ke^{-x}dx$, with the effect that the said expression becomes an integral (with the weight $e^{-x}$) of a true product of sums: precisely,

$$int_0^infty P_{n_1} (x) P_{n_2}(x)cdots P_{n_r}(x), e^{-x}, dx,$$

with a certain sequence of polynomials $P_n$, where $P_n$, has degree $n$. But the above answer for the case $r=2$ gives an orthogonality relation, whence the $P_n$, are the Laguerre polynomials, (up to a sign that is easily decided). Note that in the case with no repeated letters, all $n_i=1$, one finds again the more popular enumeration of permutations without fixed points.

Disclaimer: I partially copied this from wikipedia; it's me who wrote it there. The above is my personal amateur's solution, and possibly differs slightly from the vulgata. An on-line reference, with generalizations of the problem, is e.g.

Weighted derangements and Laguerre polynomials, D.Foata and D.Zeilberger, SIAM J. Discrete Math. 1 (1988) 425-433.

dg.differential geometry - Smooth manifolds that don't admit a partition of unity

The answer to your stated question ("Does anyone study non-paracompact manifolds?") is certainly yes. Here are a few papers which do just this:

Gauld, David.
Manifolds at and beyond the limit of metrisability. (English summary) Proceedings of the Kirbyfest (Berkeley, CA, 1998), 125--133 (electronic),
Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999.

http://www.emis.de/journals/GT/ftp/main/m2/m2-7.pdf

Among other things, Gauld references that there are two paracompact and two nonparacompact 1-manifolds, and $aleph_0$ paracompact and $2^{aleph_1}$ non-paracompact 2-manifolds. (That's a lot!)

Foliations and foliated vector bundles, 1969 MIT lecture notes

http://www.foliations.org/surveys/FoliationLectNotes_Milnor.pdf

Milnor entertains non-paracompact manifolds. In particular he constructs a (necessarily non-paracompact) surface with uncountable fundamental group. Milnor also says: "The main object of this exercise is to imbue the reader with suitable respect for non-paracompact manifolds."

Balogh, Zoltan; Gruenhage, Gary.
Two more perfectly normal non-metrizable manifolds. (English summary)
Topology Appl. 151 (2005), no. 1-3, 260--272.

The existence of perfectly normal, non-metrizable (hence non-paracompact) manifolds is shown to depend upon one's set-theoretic assumptions.

And so forth. I could find 10 more papers without much effort. I'm not sure I could find 100. (A MathSciNet search with "manifold" and ("nonmetrizable" or "non-paracompact") in the anywhere fields doesn't return many hits.) So some serious mathematicians take non-paracompact manifolds seriously enough to write some papers about them. On the other hand, although one could use more complimentary language than "extremely eccentric", your lecturer's take on non-paracompact manifolds seems to be an accurate reflection of how most geometric topologists feel: they seem mostly to be used as a source of counterexamples and to be of interest to general and set-theoretic topologists.

ds.dynamical systems - Light rays bouncing around inside a sphere in d-dimensions

Suppose $S=mathbb{S}^d$ is a unit sphere in $(d-1)$ dimensional space, with $d=3$ of special interest.
The surface of $S$ is a perfect (internal) mirror.
You stand at point $x$ (not the sphere center $c$) inside $S$ and emit a single laser light ray in direction $u$.
What happens? I believe that the light ray will remain within the plane containing the three points
${ x, x+u, c }$.

Now suppose instead that from $x$ you shine a flashlight, a cone with angular extent $pm epsilon$.
Does this fill the sphere with constant-density energy for any $epsilon > 0$? Are there are no dark points within $S$?

A somewhat related question is: What would the flashlight-holder see from $x$?
What would the visual image be, say in a graphics ray-tracing system (in $d$ dimensions!)?

I've asked enough questions for one MO posting, but ellipsoids in $mathbb{R}^d$ are the
obvious extension. Are they integrable or chaotic?

lo.logic - Algebraic structures at hypernatural parameters

You can build your group directly as an ultraproduct without fussing about the particular language. Namely, let U be any nonprincipal ultrafilter on ω, and let S be the ultraproduct Π_n S_n/U. That is, one defines f ≡ g in the product Π S_n if and only if { n | f(n) = g(n) } ∈ U. This is an equivalence relation, and the ultraproduct is the collection of equivalence classes [f], where the algebraic structure is well-defined coordinate-wise. Los's theorem then states that S will satisfy a statement about [f] in the language of groups if and only if the set of n for which f(n) has the property in S_n has the property is in U. For example, S will have elements of infinite order, since we can let f(n) select an element of S_n of order n, and then observe that for any fixed k, the set of n for which f(n) has order at least k is co-finite and hence in U. As Gerald observed, S is not itself a full symmetric group, since one will never be able to define the standard/nonstandard cut.

But let me describe a more general way to accomplish what you want, for all kinds of structures simultaneously. There is little reason when taking ultraproducts to restrict to any particular structure. Rather, one should simply consider the ultrapower of the entire set-theoretic universe V. This results in a new set-theoretic universe W = V^ω/U that satisfies all the truths about any [f] that are true of f(n) in V on a set in U. In particular, S is the just [〈 S_n | n ∈ ω 〉] in the universe W. (What this shows is that ultraproducts of any set structures are elements of the ultrapower of the universe, and in this sense, ultraproducts are a special case of ultrapowers, although one usually hears the converse.) With this construction, we are not limited to the language of group theory when discussing properties of your group S, and we can freely refer to properties involving subgroups, group extensions and whatever else is expressible in set theory. If almost every S_n (that is, on a set in U) has some property expressible in set theory, then S will have this property in W. Thus, the group S is a "finite" permutation group inside W on the nonstandard natural number N represented by the identity function id(n) = n. That is, W thinks that S is just S_N, where N = [id]. So any property that you can prove about all S_n, will be true of S in W. Some but not all of these properties are absolute between W and V, and it is often interesting to compare the differences.

What this shows is that there IS a sense in which your group S is a full group of permutations on a set, because it is the group of all permutations on N inside W. That is, it is a full symmetric group inside the alternative set-theoretic universe W, rather than in the original universe V. In particular, Gerald's counterexample permutation does not exist in W, since W also cannot define the standard/nonstandard cut.

nt.number theory - Do finite places of a number field also correspond to embeddings?

The Archimedean places of a number field K do not quite correspond to the embeddings of K into $mathbb{C}$: there are exactly $d = [K:mathbb{Q}]$ of the latter, whereas there are
$r_1 + r_2$ Archimedean places, where:

if $K = mathbb{Q}[t]/(P(t))$, then $r_1$ is the number of real roots of $P$ and $r_2$ is the number of complex-conjugate pairs of complex roots of $P$. In other words, $r_1$ is the number of degree $1$ irreducible factors and $r_2$ is the number of degree $2$ irreducible factors of $P(t) in mathbb{R}[t]$.

There is a perfect analogue of this description for the non-Archimedean places. Namely, the places of $K$ lying over the $p$-adic place on $mathbb{Q}$ correspond to the irreducible factors of $mathbb{Q}_p[t]/(P(t))$; or equivalently, to the prime ideals in the finite-dimensional $mathbb{Q}_p$-algebra $K otimes_{mathbb{Q}} mathbb{Q}_p$.

More generally: if $L = K[t]/P(t)/K$ is a finite degree field extension and $v$ is a place of $K$ (possibly Archimedean), then the places of $L$ extending $v$ correspond to the prime ideals in $L otimes_K K_v$ or, if you like, to the distinct irreducible factors of $P(t)$ in $K_v$, where $K_v$ is the completion of $K$ with respect to $v$.

See e.g. Section 9.9 of Jacobson's Basic Algebra II.

By coincidence, this is exactly the result I'm currently working towards in a course I'm teaching at UGA. I'll post my lecture notes when they are finished. (But I predict they will bear a strong resemblance to the treatment in Jacobson's book.)

rt.representation theory - Fun applications of representations of finite groups

Maybe this is not an "application", but I certainly found it fun when learning about representations of the symmetric group. Combinatorially, there is a clear correspondence between transpose-invariant partitions (ie partition diagrams that are symmetric along the main diagonal) and partitions involving distinct odd integers. (eg (3,2,1) corresponds to (5,1). )

Here is a sketch of the representation theory behind the result: the Specht modules from transpose-invariant partitions are precisely the irreducible representations of S_n that decompose into 2 irreducible representations when restricted to A_n. We view irreducible characters and conjugacy-class-indicator-functions as two bases on the vector space of A_n class functions, and deduce that the number of these reducible-on-restriction representations is equal to the number of S_n conjugacy classes which split as two A_n conjugacy classes. An S_n conjugacy class splits into two A_n conjugacy classes precisely when it doesn't commute with any odd cycles, which is to say all factors of its cycle decomposition have distinct odd length.

I've seen other instances where representation theory "explains" a combinatorial coincidence (eg q-dimension formula of various Lie algebras), so I think of this example as "typical" of the connection between representation theory and combinatorics.

EDIT: The background comes from pages 18-25 of the lecture notes here: http://www.dpmms.cam.ac.uk/~ae284/characters.html , and this particular statement is exercise 1 of sheet 3. I have what I believe is a complete solution to the exercise, but I'm having a little trouble pdf-ing it, hopefully it should be on my homepage (under "writings") by Tuesday (along with my own exposition of the background, which may expand on those official notes a little).

lo.logic - Is theory with domain of interpretation in second order objects a First Order Theory?

Thank everybody for answering my previous questions: first, and second.

Here I would like to ask about some important thing which I do not understand clearly.

1. Is it necessary for theory to have given interpretation in some
   universe by definition or not, it is not necessary for set of axiom to be a theory?

The meaning of question above is: is this a correct way of defining theory if we do not give domain of interpretation for it? What obstacles may arise from that? For example: As far as I know standard category theory do not qualify clearly which is its domain of interpretation, then, may we say that it is "defined theory" from a formal point of view? I have heard that there are some efforts to define category theory by means of metacategories which probably has as one of its aims to give clear definition of domain of interpretation.

So there another question arises for which I will try to give some introduction by example.

I may think about theory which when interpreted in set universe defined by some formulas, relations whatever may be consistent and have models, whilst in other universes do not. Is this a case? For example in answer about possible definition of domain of discourse for first order theories here Joel David Hamkins wrote: "If V is the universe of all sets, we can define certain classes in V, such as { x | φ(x) }, where φ is any property."

What if $phi(x)$ is not property from first order logic?

For example we may consider sentence $S= {x | phi (x) }$ as meaning:"for every theory x when schema of axioms of induction is present in axiom set". It is clearly not first order sequence, so maybe model which is interpreted in such universe is not first order model even if axiom of theory forms first order set? The only difference here is in a way we introduce a domain of interpretation so maybe this level of freedom is too much?

Usual definition of domain of interpretation which I have found in wikipedia here uses set "structure" but not state clearly that domain of discourse has to be first order set from the beginning! It is something which is not clear for me here. Should I believe that some "interpretations" of theory may be inconsistent whilst other may be consistent when we drop interpretation part as a definition and we leave it free?

1. If we construct theory in first order logic ( for example as in my second question), and then we will try to use it on domain of objects defined by means of higher order logic, do we end with higher order theory, or rather first order theory which is saying something about second order logic objects? Or maybe the answer is" "it depends" and there should be stated additional requirements?

I have hope it is no obviously wrong question...

---

Note 1. As I do not want to be found very speculative, I will point to blog of Terence Tao here where You may find some information about nonfirstorderisability and specially this sentence:

which is part of the fundamental
theorem of linear algebra, does not
seem to be expressible as stated in
first order set theory

So there are pretty useful practical statement, not very abstract, in normal mathematics which cannot be expressed in first order theory language.

---

Note 2. In wikipedia here I found the following remark:

"MK can be confused with second-order
ZFC, ZFC with second-order logic
(representing second-order objects in
set rather than predicate language) as
its background logic. The language of
second-order ZFC is similar to that of
MK (although a set and a class having
the same extension can no longer be
identified), and their syntactical
resources for practical proof are
almost identical (and are identical if
MK includes the strong form of
Limitation of Size). But the semantics
of second-order ZFC are quite
different from those of MK. For
example, if MK is consistent then it
has a countable first-order model,
while second-order ZFC has no
countable models."

Is there a canonical Hopf structure on the center of a universal enveloping algebra?

Let $mathfrak g$ be a finite-dimensional Lie algebra over $mathbb C$. Define $mathcal Z(mathfrak g)$ to be the center of the universal enveloping algebra $mathcal Umathfrak g$, and define $(mathcal Smathfrak g)^{mathfrak g}$ to be the ring of invariant elements of the symmetric algebra $mathcal Smathfrak g$ under the induced adjoint action of $mathfrak g$. (Clearly $mathcal Z(mathfrak g) = (mathcal Umathfrak g)^{mathfrak g}$ via the adjoint action.) The Duflo isomorphism is an isomorphism of algebras $mathcal Z(mathfrak g) cong (mathcal Smathfrak g)^{mathfrak g}$. At the level of vector spaces, the trick is to realize that the PBW map $mathcal Umathfrak g to mathcal S mathfrak g$ is an isomorphism of $mathfrak g$-modules. For the isomorphism of algebras in the semisimple case, see for example my unedited notes on the class by V. Serganova.

(I read here that this isomorphism can be realized as a composition of the PBW vector-space isomorphism $mathcal Umathfrak g to mathcal Smathfrak g$ with the map $mathcal Smathfrak g to mathcal Smathfrak g$ given by $x mapsto sinh(x/2)/(x/2)$. But it's not at all obvious that this composition is even well-defined or linear when restricted to $mathcal Z(mathfrak g)$. I should mention that $mathcal Z$ is not a functor, I think. The PBW isomorphism is non-canonical, although a canonical version can be given via the symmetrization map, and I guess on the center it is canonical.)

When $mathfrak g$ is semisimple of rank $n$, at least, one can further show that $(mathcal Smathfrak g)^{mathfrak g} cong mathbb C[x_1,dots,x_n]$, although you have some choice about how to make this isomorphism. Thus, at least when $mathfrak g$ is semisimple, $mathcal Z(mathfrak g)$ is a polynomial ring.

But any polynomial ring can be given a Hopf structure. By choosing an isomorphism with $mathbb C[x_1,dots,x_n]$, we can take the Hopf structure generated by $Delta: x_i mapsto x_i otimes 1 + 1 otimes x_i$. In fact, this structure doesn't depend quite on the full choice of isomorphism. A Hopf structure on a commutative algebra $R$ is by definition the same as an algebraic group structure on $text{Spec}(R)$. But $text{Spec}(mathbb C[x_1,dots,x_n])$ is $n$-dimensional affine space — the algebra isomorphisms of $mathbb C[x_1,dots,x_n])$ are precisely the affine maps — so picking a commutative group structure is the same as picking an origin. (For certain values of $n$ there are also non-commutative group structures on affine $n$-space, and so non-cocommutative Hopf structures on the polynomial ring. For example, the group of upper-triangular matrices with $1$s on the diagonal is affine.)

My question is whether this Hopf structure can be picked out canonically.

Question: If $mathfrak g$ is a finite-dimensional Lie algebra over $mathbb C$, can the center $mathcal Z(mathfrak g)$ of the universal enveloping algebra be given a canonical (cocommutative) Hopf algebra structure? If no, how much extra structure on $mathfrak g$ is needed?

Here by "canonical" I of course don't mean that there is a unique one, so you may make choices once and for all. But there should be some definition/construction that does not require the user to make any choices to implement it. By "extra structure" I mean either extra structure (an invariant metric, for example) or extra properties (semisimplicity, for example).

My suspicion is that the answer is "yes" for a metric Lie algebra, which is a Lie algebra $mathfrak g$ along with a choice of an invariant nondegenerate metric, i.e. a chosen isomorphism of $mathfrak g$-modules $mathfrak g cong mathfrak g^*$. Metric Lie algebras include the semisimples and the abelians, and certain extensions of these (in fact, I believe that there is a structure theorem that any metric Lie algebra is a metric extension of semisimples and abelians, but don't quote me), but generally there are many choices of metric (e.g. any metric on an abelian Lie algebra $mathfrak a$ is invariant, so there are $mathfrak{gl}(dim mathfrak a)$ many choices).

The motivation for my question is this: by studying Vassiliev invariant and/or perturbative Chern-Simons theory, Bar Natan and others have defined a certain commutative and cocommutative Hopf algebra $A$ of "diagrams". Any choice of metric Lie algebra $mathfrak g$ determines an algebra homomorphism $A to mathcal Z(mathfrak g)$. I would like to know if this can be made into a Hopf algebra homomorphism.

fa.functional analysis - Characterizing the harmonic oscillator creation and annihilation operators in a rotationally invariant way

Let me try to answer this question. I apologise if my notation is slightly different, since I will work in some more generality, since the equivariance properties of the creation and annihilation operators are actually more transparent, I believe, relative to the the general linear group instead of the orthogonal group. Also the fact that this is the harmonic oscillator is a red herring. In the case of the harmonic oscillator, we have to introduce more structure, reducing the group of symmetries.
It is also in this case, where the grading to be discussed below coincides (up to a choice of scale) with the energy of the system.

---

Let $E$ be an $n$-dimensional real vector space and let $E^*$ denote its dual. Then on $H = E oplus E^* oplus mathbb{R}K$ one defines a Lie algebra by the following relations:
$$ [x,y]= 0 = [alpha,beta] qquad [x,alpha] = alpha(x) K = - [alpha,x] qquad [K,*]=0$$
for all $x,y in E$ and $alpha,beta in E^*$. This is called the Heisenberg Lie algebra of $E$, denoted $mathfrak{h}$.

The automorphism group of $mathfrak{h}$ is the group $operatorname{Sp}(Eoplus E^*)$ of linear transformations of $Eoplus E^*$ which preserve the symplectic inner product defined by the dual pairing:
$$omegaleft( (x,alpha), (y,beta) right) = -alpha(y) + beta(x).$$

Let $mathfrak{a} < mathfrak{h}$ denote the abelian subalgebra with underlying vector space $E oplus mathbb{R}K$. One can induce a $mathfrak{h}$-module from an irreducible (one-dimensional) $mathfrak{a}$-module as follows. Let $W_k$ denote the one-dimensional vector space on which $E$ acts trivially and $K$ acts by multiplication with a constant $k$. Then letting $U$ be the universal enveloping algebra functor, we have that
$$ V_k = Umathfrak{h} otimes_{Umathfrak{a}} W_k$$
is an $mathfrak{h}$-module. The Poincaré-Birkhoff-Witt theorem implies that $V_k$ is isomorphic as a vector space to the symmetric algebra of $E^*$, which we may (as we are over $mathbb{R}$) identify with polynomial functions on $E$.

The subgroup of $operatorname{Sp}(Eoplus E^*)$ which acts on $V_k$ is the general linear group $operatorname{GL}(E)$ and hence $V_k$ becomes a $operatorname{GL}(E)$-module. In fact, $V_k$ is graded (by the grading in the symmetric algebra of $E^*$ or equivalently the degree of the polynomial):
$$V_k = bigoplus_{pgeq 0} V_k^{(p)}$$
and each $V_k^{(p)}$ is a finite-dimensional $operatorname{GL}(E)$-module isomorphic to $operatorname{Sym}^p E^*$.

Every vector $x in E$ defines an annhilation operator: $A(x): V_k^{(p)} to V_k^{(p-1)}$ via the contraction map
$$E otimes operatorname{Sym}^p E^* to operatorname{Sym}^{p-1} E^*$$
whereas every $alpha in E^*$ defines a creation operator: $C(alpha): V_k^{(p)} to V_k^{(p+1)}$ by the natural symmetrization map
$$E^* otimes operatorname{Sym}^p E^* to operatorname{Sym}^{p+1} E^*.$$

Both of these maps are $operatorname{GL}(E)$-equivariant, and this is perhaps the most invariant statement I can think of concerning the creation and annihilation operators.

ct.category theory - Does the product (by an object) in an abelian category ever have a right adjoint?

In an additive category the functor F(–) = – × A = – ⊕ A cannot have a right adjoint unless A = 0. If F had a right adjoint then it would preserve coproducts and in particular A = F(0) = F(0 ⊕ 0) = F(0) ⊕ F(0) = A ⊕ A via the fold map. This means Hom(A, K) = Hom(A, K) × Hom(A, K) for every K, but Hom(A, K) is nonempty (we have zero maps) so Hom(A, K) = • and thus A = 0 by Yoneda.

complex geometry - Dolbeault Cohomology of $mathbb{P}^1$

I wrote a blog post about almost exactly this question. I'll give a summary here:

Since $H^{1,1}(X)$ is one dimensional, I could answer your question by giving anythng with the correct integral. However, I'll try to give you the kind of cocycle which actually comes out of the proof of Dolbeaut-Cech equality.

Your cocycle isn't $dz/z$ but, rather, $dz/z$ with a specific choice of open cover of $X$. Lets say your choice is $U_1 cup U_2$, where $U_1 = { z : z neq infty }$ and $U_2 = { z : z neq 0 }$. Refine your cover to $V_1 cup V_2$, where $V_1 = { z : |z| > r }$ and $V_2 = { z : |z| < r^{-1} }$ for some $r < 1$.

Let $theta_1$ and $theta_2$ be $1$ forms on $V_1$ and $V_2$ such that $theta_1|_{V_1} - theta_2|_{V_2} = dz/z$. Then $overline{partial} theta_1$ and $overline{partial} theta_2$ have equal restrictions to $V_1 cap V_2$. The $(1,1)$-form you are looking for is their common value, which I'll call $omega$.

---

Let's first do a fake solution. A real solution would look like a $C^{infty}$ smearing out of this one.

We'll work in the degenerate case $r=1$, so we are only gluing along a circle, not an annulus. We'll take $theta_1 = (1/2) overline{z} dz$ and $theta_2 = -(1/2) dz / (overline{z} z^2)$. Notice that both $theta_1$ and $theta_2$ restrict to $dz/z$ on the unit circle, but are constructed to extend smoothly to the appropriate discs.

So $overline{partial} theta_1 = (1/2) d overline{z} d z$ and $overline{partial} theta_2 = (1/2) dz d overline{z} / (overline{z}^2 z^2)$. Our $omega$ is formed by gluing these two differential forms together.

---

A genuine smooth solution would be like this, but would interpolate smoothly between these two. If you push forward in a brute force manner, you'll get something with bump functions in it.

If you are more clever, you may discover the solution
$$theta_1 = frac{dz}{z} left( 1- frac{1}{1+z overline{z}} right)$$
and
$$theta_2 = - frac{dz}{z} left( frac{1}{1+z overline{z}} right).$$

You should check that $theta_1|_{V_1} - theta_2|_{V_2} = dz/z$ and that $theta_i$ is smooth and well-defined on $U_i$.

Then
$$overline{partial} theta_1 = overline{partial} theta_2 = frac{dz doverline{z}} {(1+z overline{z})^2}.$$

This is, as Scott guessed, the Fubini-Study form.

nt.number theory - Cusp forms and L^2

This is a long answer because the question asks quite a lot of things. I agree that Gelbart's book, although inspirational, is hard for someone without a strong analytic background. The Boulder and Corvallis proceedings are full of articles which are worth studying if you want to get an understanding of automorphic theory. It is hard to summarise but here's a (necessarily oversimplified) sketch of the "big picture" you may be looking for.

First, two important things to get straight:

1) "Cuspidal" is not the same as "vanishing at the cusps". Sticking to the upper half-place for the moment, a function is cuspidal if at each cusp the 0-th Fourier coefficient vanishes. For a holomorphic function the Fourier coefficients are constant, so cuspidal and vanishing at cusps are the same thing. But for those functions $E_phi$ the 0-th Fourier coefficient at $infty$ is some function $c_0(y)$ vanishing in a neighbourhood of infinity, but certainly non-zero in general. In fact, $fin L^2(Gammabackslashmathfrak{H})$ is cuspidal iff it is orthogonal to all the $E_phi$ (compute the Petersson product to see this), so ${E_phi}$ generates a complement to the cusp forms in $L^2$. Arithmetically, though, they aren't interesting.

2) Automorphic forms and $L^2(Gammabackslash G)$ are different beasts entirely. An automorphic form is a smooth function on $Gammabackslash G$ satisfying various properties (moderate growth, K-finite, and killed by an ideal of finite codimension in the centre of the universal enveloping algebra). It does not have to be in $L^2$. For example, holomorphic Eisenstein series are automorphic forms which are not in $L^2$.

Hecke theory: for $SL_2(mathbb{Z})$ you can just mimic the classical definitions in this more general setting, but for groups other than $GL_2/mathbb{Q}$ you really need to look at the adelic setting. Here an automorphic form (for a reductive group $G/mathbb{Q}$) is a smooth function on $G(mathbb{Q})backslash G(mathbb{A})$, satisfying a bunch of properties (see Borel and Jacquet's article in Corvallis for a precise definition, indeed for just about everything here). The finite adelic group $G(mathbb{A}_f)$ acts by right translation on the space of automorphic forms, and this is the right generalisation of Hecke operators. The Lie group $G(mathbb{R})$ does not act on the space of automorphic forms (K-finiteness is not preserved) but there is a suitable algebra of invariant differential operators (the Hecke algebra of $G(mathbb{R})$) which does.

Cuspidal automorphic forms - in the adelic setting - are those $F$ for which the function
$$
F_N(g) = int_{N(mathbb{Q})backslash N(mathbb{A})} F(hg) dh
$$
vanishes for suitable unipotent subgroups $N$. For $GL_2$ these are just the unipotent radicals of Borel subgroups (defined over $mathbb{Q}$), and since these are all conjugate it's enough to verify the vanishing for the upper triangular unipotent subgroup.

A classical cusp newform $f$ of weight $k$ then gives rise to a cuspidal automorphic form $F$ on $GL_2/mathbb{Q}$. There are two different things one can now do with $F$:

1) The translates of $F$ under $GL_2(mathbf{A}_f)$ generate an irreducible representation $V_f$, which encodes the action of the Hecke operators (and more). Elements of this representation are none other than oldforms associated to $f$. More precisely, $V_f$ is an infinite tensor product of representations $V_p$ of $GL_2(mathbb{Q}_p)$. If $p$ doesn't divide the level of $f$, then $V_p$ tells you the Hecke $T_p$ and $R_p$ eigenvalues. At bad primes, it contains much more delicate information than classical Atkin-Lehner theory does (one reason to use the adelic approach even for $GL_2$).

2) The translates of $F$ under the Hecke algebra at infinity, on the other hand, generate an irreducible representation $V_infty$ of a particular type (discrete series with parameter given by $k$), inside which $F$ is characterised as the lowest weight vector (this is the group-theoretic interpretation of the holomorphy of $f$).

The space of all translates of $F$ (i.e. by the finite adelic group and the Hecke algebra at infinity) is just $V_inftyotimes V_f$. It is an example of an automorphic representation. (In general, an automorphic representation is any irreducible subquotient of the spaces of automorphic forms under these actions.)

The map $Fmapsto F_N$ allows one to describe the quotient (automorphic forms)/(cusp forms) by induced representations from parabolic subgroups. Although explicit, this quotient is quite a complicated representation - in particular, it is very far from being semisimple, even for $GL_2$.

So, if you are looking at this from a number-theorist's perspective, why care about $L^2$ ? The reason is the trace formula, which needs to be formulated in the setting of Hilbert spaces. There is essentially no difference between an $L^2$-form which is cuspidal and an automorphic cusp form, so the trace formula, appropriately wielded, can tell you a lot about cusp forms. It is a powerful and indispensable tool. But to apply it you need also to look at the rest of the $L^2$, in which lives the continuous spectrum, accounted for by real-analytic Eisenstein series at $Re(s)=1/2$. Here there is a lot of analysis but you can usually find a friendly expert to help you out.

PS: Some people like to define a cuspidal automorphic representation as an irreducible subspace of $L^2_{mathrm{cusp}}$. This has some advantages: (a) it's concise, and (b) at infinity one is working with genuine (unitary) representations of the real Lie group, rather than $(mathfrak{g},K)$-modules (equivalently, representations of the real Hecke algebra). But it only gives the correct answer for cuspidal representations.

soft question - Has mathoverflow yet led to mathematical breakthroughs?

Hey Tim. I'm not so sure whether my model of how "proving a theorem" works is the same as yours. But let me give some kind of an example of something and you can take it or leave it. I'm writing a paper with Toby Gee at the minute, and we're both number theorists, and the arguments in the paper are "robust" but the details need checking. We're now at the point where we're writing up technical calculations and these technical calculations are mostly in the area of representation theory of reductive algebraic groups, an area which I think it's fair to say that neither Toby nor I would call ourselves experts in. So we have this overall "robust" argument, and a write-up that exists but occasionally says "lemma: (statement in representation theory); proof: TO BE ADDED". We 'know' these lemmas are true because they fit into our overall picture, but occasionally when I write one of these things up I can't go from my intuition to a rigorous proof. Here's an example of an occasion when I got stuck:

This question of mine.

Ben Webster made a crucial remark that enabled me to finish the argument, so that lemma went from "must be true but proof not yet written or even discovered by authors" to "lemma proved".

So if I were interpreting your question in a particularly anal way, one might argue that had this lemma been the last of the lemmas we need to write up the proof of, then Ben's contribution might be "just what I needed to complete my research project". Unfortuately there are several more to go :-).

Having said all that, it's not clear to me that MO was "crucial" to solving the problem. I could have worked more on the problem until I'd done it myself. I could have asked one of the representation theorists in my own department. I could have left it and hoped that my co-author sorted it out. All of these would have been viable approaches. Why did I ask at MO? Simply because I am sick of writing this paper and asking at MO was by no means the only way of solving the problem, but I had high hopes that it would be the quickest.

Kevin

ag.algebraic geometry - Lifting abelian varieties in (the closed fiber of) a fixed Neron model

Suppose that $R$ is a dvr with field of fractions $K$ and residue field $k$ and that $A_K$ is an abelian variety over $K$ with Neron model $A$ over $R$. Then the closed fiber $A_k$ is a smooth commutative group scheme over $k$. Suppose now that $B_k$ is an abelian variety over $k$ that is a quotient of $A_k$.

Does there exist an abelian scheme $B$ over $R$ and a morphism
$Arightarrow B$ of smooth commutative $R$-groups whose base change to $k$ is the quotient mapping $A_k rightarrow B_k$?

The answer is NO, I'm fairly sure, as the generic fiber of $A$ could be simple with
$A_k$ an extension of an abelian variety by a torus. Let us
therefore assume that $B_k$ can be lifted up to isogeny, i.e. that there exists $C$
an abelian scheme over $R$ such that 1) $C_k$ is isogenous to $B_k$ and
2) There exists a map of smooth groups $Arightarrow C$ over $R$ whose base change to $k$
is the quotient map $A_krightarrow B_k$ followed by the isogeny $B_krightarrow C_k$. With this added assumption, is the answer to the question above still NO?

I'm inclined to think that this is the case, but can't immediately convince myself of this.

---

Reformulation

Consider the following theorem of Chevalley (see 9.2/1 of the book "Neron Models" by Bosch, Lutkebohmert and Raynaud):

Theorem: Let $k$ be a perfect field and $G$ a smooth and connected algebraic $k$-group. Then there exists a smallest connected linear subgroup $L$ of $G$ such that the quotient $G/L$ is an abelian variety. Furthermore, $L$ is smooth and of formation compatible with extension of $k$.

Definition: We write $av(G)$ for $G/L$ as in the Theorem.

Now fix a dvr $R$ of mixed characteristic $(0,p)$ with fraction field $K$ and residue field $k$. Let $A_K$ be an abelian variety over $K$. There exists an abelian variety
quotient $B_K$ of $A_K$, unique up to isogeny, with the following properties:

1. $B_K$ has good reduction


2. Any abelian variety quotient $A_Krightarrow C_K$ of $A_K$ having good reduction
   factors through $A_Krightarrow B_K$.

If we impose the additional assumption that the kernel of $A_Krightarrow B_K$ is connected (i.e. an abelian sub-variety of $A_K$), then $B_K$ is uniquely determined. We call this $B_K$ the maximal good reduction quotient of $A_K$.

The surjection $A_Krightarrow B_K$ induces a mapping $Arightarrow B$ on Neron models over $R$ and hence a mapping on identity components of closed fibers $A^0_k rightarrow B_k$ which yields a homomorphism of abelian varieties
$$varphi:av(A^0_k)rightarrow B_k.$$

Question: Is the kernel of $varphi$ an abelian sub-variety of $av(A^0_k)$?

mp.mathematical physics - A reading list for topological quantum field theory?

I think it might be worth pointing out that there are two kinds of topological quantum field theory, (Albert) Schwarz-type theories and Witten-type theories. In Schwarz type theories (like Chern-Simons theory and BF-theory), you have an action which is explicitly independent of the metric and you expect that the correlation functions computed by the path integral will also be independent of the metric. In Witten-type theories (Donaldson theory, Gromov-Witten theory), metric independence is a little bit more subtle. In these theories, you do have to choose a metric to get started. But you have some extra structure that allows you to compute some quantities which are metric independent.

(Slightly) more precisely: In a Witten-type theory, you have some operator Q which squares to zero, which you think of as a differential. (Witten type theories are also called cohomological field theories.) You also have an operator T, taking values in (2,0)-tensors, which a) is Q-exact ( T = [Q,G] for some G), and b) generates changes in the metric g. The latter means that if we compute the expectation value < epsilon(T)A > as a function of g, we find that it's equal to the expectation value of A computed with respect to g + epsilon. Here epsilon is a "small" (0,2) tensor we pair with T to get a scalar. In these theories, you can show that the correlation functions of operators which are Q-exact must vanish, which implies that small deformations of g don't change the correlation functions of Q-closed operators A. If you choose A so that its expectation value behaves like a function on the space of metrics, this tells you it's constant on the space of metrics. If you choose some fancier A so that the correlation functions behave like differential forms on the space of metrics, cohomological complications can arise.

Most of the references here are for Schwarz-type theories. For a physics treatment of Witten-type theories, it's worth looking at Witten's "Introduction to cohomological field theory". There's also a long set of lecture notes by Cordes, Moore, & Ramgoolam. The mathematical treatments of the idea are less complete. Hopkins, Lurie, & Costello's stuff is about the most comprehensive, but it's pretty far removed from actions and path integrals. For a starter, you might enjoy Teleman's classification of 2d semi-simple "families topological field theories".

at.algebraic topology - Details for the action of the braid group B_3 on modular forms

You can think of the space of positively oriented covolume-one bases of $mathbb{R}^2$ as a torsor under $SL_2(mathbb{R})$, i.e., it is a manifold with a simply transitive action of the group. If you choose a preferred basepoint, such as $(mathbf{i},mathbf{j})$, you get an identification with the group. You can think of elements of the universal cover as positively oriented area-one bases equipped with a homotopy class of paths in $mathbb{R}^2 - {0}$ from $mathbf{i}$ to the first element of the basis. There is a reasonably straightforward composition law that involves multiplying matrices and composing paths.

Gannon's description of lifting to $SL_2(mathbb{R})$ implies the lifts of even weight modular forms to $widetilde{SL_2(mathbb{R})}$ are invariant under the action of $B_3$. In particular, classical modular forms are rather boring from the perspective of the braid group. In order to detect the central extension, you need to consider modular forms of fractional weight. When the weight is not an integer, you don't get an action of $SO(2)$, but instead, an action of the universal cover $mathbb{R}$. The resulting action of $B_3$ is necessarily nontrivial, since the restriction to the center is by a nontrivial character of $mathbb{Z}$. I don't know many explicit constructions of fractional weight forms, other than half-integer weight forms like $eta$ and theta functions, and vector-valued forms constructed from them. However, you can generate a family of examples by choosing powers of the cusp form $Delta$, which admits a logarithm since it is globally regular and nonvanishing.

My understanding of the explicit relationship to configurations of points and elliptic curves is the following: Given a path of triples of distinct points $(a_1(t),a_2(t),a_3(t))$, we get a path on the space of elliptic curves of the form $y^2 = (x-a_1(t))(x-a_2(t))(x-a_3(t))$, but this will throw away an action of real translations and dilations (irrelevant) together with the central extension and the circle action (important). If we just look at the isomorphism types (i.e., the $j$-invariants) of the curves, we get a path through the quotient of the upper half plane by $SL_2(mathbb{Z})$. We need to choose a discrete structure to remove the quotient by the center, and a one dimensional continuous structure to promote our space to three dimensions. To retain the angular information that we lost by passing to elliptic curve isomorphism, we fix a tangent direction at infinity to remove rotational symmetry. This tangent direction is manifested when we choose our discrete structure: a homotopy class of nonintersecting paths from the three points to infinity, because we demand that the paths asymptotically approach infinity in that direction. The elliptic curve is a double cover of the complex projective line, ramified at the three points and infinity. We can choose once and for all a uniform convention for lifting the three paths to primitive homology cycles, such that any pair generates $H_1$, and one cycle is the sum of the other two, so those two form a preferred basis. To get the parametrization of $widetilde{SL_2(mathbb{R})}$ from the first paragraph, we choose a preferred basepoint configuration of three points with paths from infinity and asymptotic direction, and consider a triple $(a_1(t),a_2(t),a_3(t))$ that starts at the basepoint. By uniqueness of homotopy lifting, we get a family of tangent vectors at infinity together with a family of elliptic curves with oriented bases of homology. By rescaling the bases in $mathbb{R}^2$ to have unit covolume, we get the parametrization in the first paragraph.

co.combinatorics - Equivalence relations on permutations and pattern avoidance

I'm working on the interaction between equivalence relations on permutations and pattern avoidance. I've only considered Knuth equivalence and cyclic shifts until now and I'm looking for other equivalence relations to test some conjectures on. So my question is:

What interesting equivalence relations on permutations are there?

Background/Motivation

By a permutation of $n$ I mean a bijection $lbrace 1,2,dots,nrbrace to lbrace 1,2,dots,n rbrace$. The set of all permutations of $n$ will be denoted $S_n$. I'll use the one-line notation for permutations, e.g., $132$ means the permutation $1to1$, $2to3$, $3to2$.

A pattern will also be a permutation, but we are interested in how patterns occur in permutations. E.g., the pattern $132$ occurs in the permutation $215314$ as the letters
$2$, $5$, $3$, because these have the same relative order as the pattern. Note that the letters do not have to be adjacent in the permutation. When a pattern does not occur in a permutation we say that the permutation avoids that pattern.

Now, fix an equivalence relation on $S_n$. For any pattern $p$ we define two subsets of $S_n$:

$X_n(p) = lbrace sigma in S_n phantom{i}|phantom{i} sigma
text{ and every equivalent permutation avoids } p rbrace$

$Y_n(p) = lbrace sigma in S_n phantom{i}|phantom{i} sigma
text{ avoids } p text{ and every equivalent pattern}rbrace$

What I am mostly interested in is how these two sets are related.

Example: Assume our equivalence relation is cyclic shifts, meaning that two permutations (or patterns) $p$, $q$ are equivalent if we can write $p = r_1 * r_2$ and $q = r_2 * r_1$ where $*$ means concatenation. E.g., $1234$ is equivalent to $3412$. Here the two sets $X_n(p)$ and $Y_n(p)$ are equal for any $p$. (This is also true if we generalize our patterns to bivincular patterns, whose definition I'll omit here)
[Note considering permutations up to cyclic shifts is equivalent to looking at circular permutations.]

Example: If our equivalence relation is Knuth equivalence, meaning that two permutations (or patterns) are equivalent if they have the same insertion tableaux under the RSK-correspondence, then the two sets are not always the same. They are the same for any pattern of length $3$, but $X_4(1324) neq Y_4(1324)$.

When the two sets are equal then counting $Y_n(p)$ can sometimes be done more easily by looking at $X_n(p)$. When the relation is Knuth equivalence one can use the hook-length formula for instance.

So I would very happy if someone could tell me about other interesting equivalence relations on permutations so I could see how $X_n(p)$ and $Y_n(p)$ are related in other cases.

at.algebraic topology - circle action on sphere

I do not know, in which category your actions live. In the following answer I want to consider isometric actions.

Another classification problem might be:

Classify all periodic one parameter subgroups of $SO(n)$ up to conjugacy, which is the same problem as the classification of all such actions up to isometry.

There is a normal form for orthogonal matrices over the reals, which says, that every such matrix is conjugated to a block matrix consisting of $(2,2)$ rotation matrices and $(1,1)$ diagonal entries, which are $pm 1$.

My claim would be that every isometric action decomposes (after congugation) uniquely as a induced action on each of the components of the decomposition

$S^n = S^1 * S^1 * ldots * S^1 * S^0 * ldots * S^0$
($*$ should denote the join here). The isometric actions of $S^1$ on $S^1$ can be classified by the integers. The only possibilities are $(t,x)mapsto mt+x$ for $minmathbb{Z}$.

I think the upper normal forms for matrices should imply this. But I am not sure.

inequalities - Is the Jaccard distance a distance?

Here is an elementary proof of the Steinhaus transform (from which said metricity follows as a special case, as noted in Suresh's answer).

Lemma. Let $p,q,r > 0$ such that $p le q$. Then, $frac{p}{q} le frac{p+r}{q+r}.$

Corollary. Let $d(x,y)$ be a metric. Then, for arbitrary (but fixed) $a$,
begin{equation*}
delta(x,y) := frac{2d(x,y)}{d(x,a)+d(y,a)+d(x,y)},
end{equation*}
is a metric.

Proof. Only the triangle inequality for $delta$ is nontrivial. Let $p=d(x,y)$, $q=d(x,y)+d(x,a)+d(y,a)$, and $r=d(x,z)+d(y,z)-d(x,y)$. Applying the lemma, we obtain
begin{eqnarray*}
delta(x,y) &=& frac{2d(x,y)}{d(x,a)+d(y,a)+d(x,y)} le frac{2d(x,z)+2d(y,z)}{d(x,a)+d(y,a)+d(x,z)+d(y,z)}\
&=& frac{2d(x,z)}{d(x,a)+d(z,a)+d(x,z)+d(y,z)+d(y,a)-d(z,a)} + frac{2d(y,z)}{d(y,a)+d(z,a)+d(y,z)+d(x,z)+d(x,a)-d(z,a)}\
&le& delta(x,z)+delta(y,z),
end{eqnarray*}
where the last inequality again uses triangle inequality for $d$.

lo.logic - Is there a known natural model of Peano Arithmetic where Goodstein's theorem fails?

(I've previously asked this question on the sister site here, but got no responses).

Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It turns out that this theorem is unprovable in Peano Arithmetic ($PA$) but is provable in $ZFC$.

I'd like to "discuss" this (both the proof in $ZFC$ as well as the proof that it's impossible in $PA$) in an hour long lecture to a group of grad students (with no assumed background, handwaving is not only allowed, but encouraged). Because of previous talks I've given, I think it will not take too long to cover/remind them of the basics of a semester course in first order logic (e.g., the compactness and completeness theorem, etc).

The problem is that the proofs of unprovability I've found (the same as those linked in the Wikipedia article) are rather too difficult for this setting. In a nutshell, I'm looking for the easiest known proof.

For example, I would love a proof of unprovability which works by exhibiting a model of $PA$ in which Goodstein's theorem fails. Such models neccesarily exist by the completeness theorem, since "$PA$ + Goodstein's theorem is false" is consistent.

Has anyone proven the independence of Goodstein's theorem by exhibiting a model of $PA$ where Goodstein's theorem has failed?

In the interest of getting as simple a proof as possible, I'd love to see a proof which uses the compactness and completeness ideas - something like showing there is a set $Sigma = {phi_n}$ of explicit first order sentences(in a slightly larger language, say) such that

1) for any finite $Sigma_0subseteq PAcup Sigma$, the standard model $mathbb{N}$ models $Sigma_0$ and

2) The theory $PAcup Sigma$ proves Goodstein's theorem is false.

Is such a proof known? More generally, is there a known proof of the unprovability of Goodstein's theorem which is accessible to someone with only a semester or 2 of logic classes?

Thank you and please feel free to retag as necessary.

math philosophy - synthetic differential geometry and other alternative theories

There are models of differential geometry in which the intermediate value theorem is not true but every function is smooth. In fact I have a book sitting on my desk called "Models for Smooth Infinitesimal Analysis" by Ieke Moerdijk and Gonzalo E. Reyes in which the actual construction of such models is carried out. I'm quite new to this entire subject and I only stumbled upon it because I was trying to find something like non-standard analysis for differential geometry.

Already I'm liking the more natural formulations for differentials and tangent vectors in the new setting although I can see that true mastery of all the intricacies will require more background in category theory like Grothendieck topologies. So my questions are a bit philosophical. Suppose some big conjecture is refuted in one of these models but proven to be true in the classical setting then what exactly would that mean for classical differential geometry? Is such a state of affairs possible or am I missing something that rules out such a possibility like a metatheorem that says anything that can be proven in the new models can be proven in the usual classical model? More specifically what is the exact relationship between the new models and the classical one? Could one even make any non-trivial comparisons? References for such discussions are welcome. I'm asking the question here because I suspect there might be some experts familiar with synthetic differential geometry that will be able to illuminate the connection to the classical theory.

Edit: Found a very lively and interesting discussion by John Baez, Andrew Stacy, Urs Schreiber, Tom Leinster and many others on n-category cafe called Comparative Smootheology although I couldn't make out the exact relation to SDG.

at.algebraic topology - de Rham Cohomology of surfaces

For an orientable 2-dimensional surface, it's best to view it as a Riemann surface. Any standard reference on Riemann surfaces (in my day, this was Gunning's Lectures on Riemann surfaces) will work out the deRham cohomology, as well as its decomposition into Dolbeault cohomology. And then if you look in, say, Griffiths and Harris, you can see how to compute the deRham cohomology of complex projective spaces in any dimension.

For standard manifolds that are quotients of compact Lie groups, I believe you can compute deRham cohomology using averaging. I don't know where to find this, though. I vaguely recall learning this from notes by Bott.

ct.category theory - Do normal categories have pullbacks?

Background

I am reading Tom Blyth's book Categories as I am thinking of using it as a guide for a fourth-year project I'll be supervising this academic year. The books seems the right length and level for the kind of project we require of our final year single honours students in Edinburgh.

In the fourth chapter of the book, a normal category is defined as one with zero objects, in which every morphism has a kernel and a cokernel, and in which every monomorphism is a kernel. This last condition can be rephrased as saying that monomorphisms are normal.

My confusion stems from Theorem 4.6 in the book which states that a normal category has pullbacks. The proof in the book seems to use that in the diagram
$$begin{matrix}
& B cr
& downarrow cr
A rightarrow & C cr
end{matrix}
$$
whose limit is the desired pullback, the morphism $A to C$ is a kernel. This seems to me an unwarranted assumption, since it would seem to imply, in particular, that generic morphisms are monic.

Alas, I have not been able to find an independent proof of the theorem and I am starting to suspect that this may not be true. Googling seems not to be of much help. For one thing one has to wade through a surprising number of hits which have nothing to do with category theory.

Since much of the rest of the chapter seems to depend on this result, I am a little stuck. I have emailed the author, who is an emeritus professor across the firth in St Andrews, but so far no reply. So I thought I would try it here in MO, since I'm sure to get an authoritative answer to my

Two questions

1. Is the result true? And if so,




2. Can someone point me to a proof?

Thanks in advance!

fa.functional analysis - When is A : C(X) --> C(Y) a composition operator?

I'll risk making this a post, not a commment.

I think the real numbers $mathbb R$ are a hemicompact $k$-space. Certainly $mathbb R = bigcup_n [-n,n]$ and if $Ksubseteqmathbb R$ is compact, then it's bounded, hence in some $[-n,n]$. It's a k-space, for if $Ksubseteqmathbb R$ has closed intersection with all compacts, then by looking at sequences, it's easy to see that $K$ is closed.

But $mathbb R$ is not compact, so I guess you really mean to look at $C^b(mathbb R)$, the algebra/space of all bounded continuous functions. Is that right? If not, then it's a whole new ball game (as $C(mathbb R)$ the space of all continuous functions is not a Banach space).

But if so, then $C^b(mathbb R)$ has character space $betamathbb R$, and we can apply Jonas's construction: just pick a point $winbetamathbb Rsetminus mathbb R$ and evaluate there. This gives an algebra homomorphism $C^b(mathbb R)rightarrowmathbb C$ which is not a composition operator.

Edit: Yes, the original question was about all continuous functions on X, not just the bounded ones. My mistake...

co.combinatorics - Is there an English translation of Kuratowski's theorem on forbidden minors of planar graphs?

In case you are asking for the original paper "Sur le problème des courbes gauches en Topologie" by Kuratowski where he first proves his characterization of planar graphs, then a translation by J.Jaworowski can be found in "Graph Theory, Łagów", 1981, M. Borowiecki, J. W. Kennedy and M. M. Sysło. It is the proceedings of a conference held in Łagów, dedicated to the memory of K.Kuratowski.

books - Erratum for Cassels-Froehlich

Hendrik Lenstra says:

Below my 51 errata that I didn't see on your list or in William Stein's mail
yet. Most are of a typographical nature, but some have mathematical
substance. I did at the present occasion not verify the correctness of those.

And: I did not do any proofreading of my list either!! I trust you will apply
your own sound judgment.

Good luck!!

And best regards,

Hendrik

Errata for Cassels-Fr"ohlich
copied by Hendrik Lenstra from his own copy
Jan 13, 2010

Page 3, Proposition 1. This Proposition is misstated, and the proof has the
wrong reference: Chapter II, section 10 has no such result, but Chapter II,
section 5 does. The Theorem in the latter section is the correct formulation:
it is not the extension of the valuation, but the completion that one wants to
be unique. More or less coincidentally, the Proposition is correct as stated
(exercise!), but that statement is neither used (by anybody) nor proved (in the
book).

Page 45, line 5: for "=n" read "=n+1".

Page 52, part (3) of the first definition: for "K" read "V".

Page 54, line -5: replace roman "A" by italic "A" (twice).

Page 73, line 6: replace "vica" by "vice".

Page 75, line 1: replace "(19.9)" by "(19.10)".

Page 78, first line of display (A.19): replace "b_{ij}" by "b_{1j}".

Page 78, line -8 (display (A.24)): replace the third subscript "P" by "R".

Page 78, line -6: replace the third subscript "P" by "R".

Page 79, line -8: remove the three commas within the parentheses.

Page 98, the lower "delta" in the diagram should have a "hat" (the upper
one has one, though it is barely visible in my copy).

Page 123, last line before section 2.5: replace roman "C" by italic "C".

Page 129, line 10: replace "]" by "])".

Page 130, line 1: replace "2.7" by "2.8".

Page 130, line 14: replace "2.5" by "1.5".

Page 131, last line before Corollary 1: replace "2.7" by "2.8".

Page 131, line -10: replace the second "K_{nr}" by "K_{nr}^*".

Page 135, line 6 of Lemma 4: replace the second "M)" by "M))".

Page 139, line 14 (the first display): put ")" before the final ".".

Page 140, line 3 of section 2.3: replace "H^2(G,Z)" by "H^1(G,Q/Z)" (with
Q, Z boldface), since the isomorphism delta hasn't been applied yet!

Page 140, line -8: replace "s" by "s_alpha".

Page 140, line -2: replace "Prop. 2" by "Prop. 1". Also, the proof is
confused. One defines s'alpha to be (alpha,L'/K), and the fact that
salpha maps to s'_alpha under the natural map G^{ab} -> (G/H)^{ab}
FOLLOWS from the equality of character values rather than playing a role
in the proof of that equality.

Page 141, first line after the first diagram: replace the last "K" by "K'".

Page 143, line -3: replace "Lubin" by "Lubin-".

Page 147, first line after Definition: put ")" at the end.

Page 150, proof of Proposition 1: (c) is not proved that way.

Page 150, line -10: for "left-and" read "left- and".

Page 151, line 13: replace the last "[a]" by "[b]".

Page 154, line 18: replace "r_pi" by "r_pi(omega)".

Page 154, line 19: in my copy, there is the scrawled complaint "why is K_pi
from sec. 3.6 equal to K_pi from section 2.8?", and a three line additional
argument, which reads as follows: "Adopt the definition of K_pi as in sec.3.6
(or Theorem 3(b)). Then r(pi) is trivial on K_pi (def. of r), and so is
vartheta(pi) by Cor. to Prop. 6. Also r(pi) and vartheta(pi) are F on
K_{nr}. Hence r and vartheta agree on pi, hence on all of K^*. (Hence also
K_pi=Kpi !)

Page 154, line 2 of section 3.8: replace "2.3" by "2.7".

Page 154, line -8: replace "I_K" by "I'_K".

Page 155, line -11: replace roman "G" by italic "G".

Page 156, line 3: replace "3.3" by "3.4".

Page 156, line 10: replace "beta_j" by "beta^j".

Page 157, line 9: replace "intertia" by "inertia".

Page 158, line -4: replace "|" by "/".

Page 168, line 5: replace roman "F" by italic "F".

Page 168, line -16: replace roman "C" by italic "C".

Page 170, line -18: replace "U^S an arbitrarily small" by "U^S contained in
an arbitrarily small" (because U^S is generally not open).

Page 175, line 2 after the diagram: for "N_{M/K}" read "N_{M/K}J_M".

Page 179, line 12: put ")" before the second "=".

Page 183, line 1: there is no "Proposition 2". Probably "Proposition 2.3" is
meant.

Page 183, display (7): replace the second "K" by "K^*".

Page 192, line -11: replace "infiinte" by "infinite".

Page 211, line -12 (counting the footnote as -1): for "does or does not"
read "does not or does". (This is what I scrawled, I did not verify it at the
present occasion!)

Page 211, line -7: for "seq" read "seq.".

Page 236, line 5: for "2.5", read "1.2, Prop. 1".

Page 353, line 4: for "(lambda,b)_v", read "(b,lambda)_v".

Page 360, last line of Exercise 5.1: for "4.3", read "4.4".

Page 366, under "Tchebotarev, N.,", also list "165," and "227,".

----------------------THE---END-------------------------------------------------

recreational mathematics - length of 'digital' recurring expansion of rational number

Here's a question of a 'recreational' nature.

A similar question has already been posed for radix 10 in particular:
Integer division: the length of the repetitive sequence after the decimal point .
My question has been partly answered there; however, I'd be interested in anything further that can be added.

Suppose you have a rational number $n/d$ in simplest terms and a radix $r ge 2$. Are there any results on the cycle length $l$ of the recurring part, and less importantly on the length of the prefix (numerals between the decimal point and the start of the recurring section)?

One result is that $l|d-1$.

I wonder whether there are simple results for various cases:

• $r$ is a prime $p$ (such as $2$)


• $r$ is a prime power $p^k$ (such as $16=2^4$)


• $r$ is square-free (such as $10 = 2 cdot 5$), i.e. $r$ is a product of distinct primes, i.e $p^2|r$ for no prime $p$

Some equivalent statements about primitive algebras

Lam (A first course in noncommutative rings, 2ed) does it for (unital) rings $R$ in Lemma 11.28 (page 186):

If such a $B$ exists, we may assume (after an application of Zorn's Lemma) that it is a maximal left ideal. The annihilator of the simple left $R$-module $R/B$ is an ideal in $B$, and so it must be zero. This shows that $R$ is left primitive. Conversely, if $R$ is left primitive, there exists a faithful simple left $R$-module, which we may take to be $R/B$ for some (maximal) left ideal $B subsetneq R$. A nonzero ideal $C$ cannot lie in $B$ (for otherwise $C$ annihilates $R/B$) and so must be comaximal with $B$.

Here, the statement equivalent to the Axiom of Choice is Zorn's Lemma; it says that:

Every partially ordered set in which every chain (i.e. totally ordered subset) has an upper bound contains at least one maximal element.

In this proof we get to see one of its most common uses: it assures that any unital ring has a maximal ideal (see the Wikipedia page for more information).

ho.history overview - Biographic Data/Stories about André Néron

It seems that Colliot-Thélène was asked this question three times: by me, by Olivier, and by Kevin Buzzard. Here is the reply that he gave to Kevin and forwarded to me:

Le patron d'André Néron était Albert Châtelet.

Néron eut un autre étudiant en thèse, Gérard Ligozat,
qui après plusieurs travaux sur les courbes modulaires
quitta le département de mathématiques.

André Néron parlait le langage de la géométrie algébrique
de Weil à une époque où l'école de Grothendieck était devenu
dominante. Les jeunes fringants allaient naturellement voir
du côté des schémas.

La notion de patron et d'élève dans les années 1970
en France était souple. Après une excellente scolarité,
on obtenait un poste quasi-permanent soit à l'Université
soit au CNRS sans avoir publié une ligne dans une revue.
Ensuite on faisait une thèse si on en avait envie. Voir
son patron une fois par an était souvent suffisant.

Et le système, sur la durée, a marché aussi bien
que dans notre époque de publish ou perish.

André Néron mourut d'un cancer en 1985.

Kevin explicitly gave Colliot-Thélène the option of responding in French, which seems appropriate given the subject matter.

---

English translation by François G. Dorais:

André Néron's advisor was Albert Châtelet.

Néron had one more thesis student, Gérard Ligozat, who left the Department of Mathematics after much work on modular curves.

André Néron spoke the language of Algebraic Geometry in the style of Weil at a time when the Grothendieck school had become dominant. The dashing youth naturally leaned toward [Grothendieck's] schemes.

The notion of advisor and student was flexible in France during the 1970's. After excellent scholarly work, one could obtain a quasi-permanent job either at the University or at the CNRS, without having published a single line in a journal. Then, if desired, one could do a thesis. Seeing one's advisor once per year was often enough.

And this system, while it lasted, worked just as well as the publish or perish [system] of our times.

André Néron died of cancer in 1985.

(I took a few very minor liberties for readability, but the translation is mostly literal.)

l functions - What is a path in K-theory space?

I know nothing about Alexander polynomials but let me try to answer the Iwasawa theory part. As is well known, in classical Iwasawa theory one considers cyclotomic $mathbb{Z}_p$ extension $F{infty}$ of $F$. We take the $p$-part of the ideal class group $A_n$ of the intermediate extension $F_n$ of $F$ of degree $p^n$. The inverse limit of $A_n$ with respect to norm maps, say $A$, has an action of $G= Gal(F_{infty}/F)$. Since $A$ is pro-$p$, it becomes a $mathbb{Z}_p[G]$-module. However, it is not finitely generated over this group ring (and for various other reasons) one considers the completion $mathbb{Z}_p[[G]]$ of $mathbb{Z}_p[G]$. Since $A$ is compact, it becomes a $mathbb{Z}_p[[G]]$-module. As $G cong mathbb{Z}_p$, the ring $mathbb{Z}_p[[G]] cong mathbb{Z}_p[[T]]$, the power series ring in variable $T$. There is a nice structure theory for finitely generated modules over $mathbb{Z}_p[[T]]$. The module $A$ is a torsion $mathbb{Z}_p[[G]]$-module (i.e.$Frac(mathbb{Z}_p[[G]]) otimes A = 0$). For such modules one can define the characteristic ideal using the structure theory. Iwasawa's main conjecture asserts that there is a canonical generator for this ideal called the $p$-adic $L$-function.

In generalised Iwasawa theory (more precisely, to formulate the generalised main conjecture à la Kato), one wants to consider extensions whose Galois groups are not necessarily $mathbb{Z}_p$ (but most formulations of the main conjecture still require that the cyclotomic $mathbb{Z}_p$-extension of the base field be in the extension). For the completed $p$-adic groups rings of such Galois groups, the structure theory completely breaks down even if the Galois group is abelian.

However, one can still show that $A$ is a torsion Iwasawa module (which again just means that $Frac(mathbb{Z}_p[[G]]) otimes A = 0$. Note that it is always possible to invert all non-zero divisors in a ring even in the non-commutative setting). Hence the class of $A$ in the group $K_0(mathbb{Z}_p[[G]])$ is zero. Strictly speaking, here I must assume that $G$ has no $p$-torsion so that I can take a finite projective resolution of $A$, or I must work with complexes whose cohomologies are closely related to $A$. But I will sinfully ignore this technicality here. Now, since the class of $A$ in $K_0(mathbb{Z}_p[[G]])$ is zero, there is a path from $A$ to the trivial module 0 in the $K$-theory space. In Iwasawa theory this is most commonly written as

There exists an isomorphism $Det_{mathbb{Z}_p[[G]]}(A)$ $to$ $Det(0)$.

This isomorphism replaces the characteristic ideal used in the classical Iwasawa theory. The $p$-adic $L$-function then is a special isomorphism of this kind. (Well one has to be careful about the uniqueness statement in the noncommutative setting but it is a reasonably canonical isomorphism). Hence the main conjecture now just asserts existence of such a $p$-adic $L$-function.

Thus the $p$-adic $L$-function may be thought of as a canonical path in the $K$-theory space joining the image of Selmer module (or better- a Selmer complex), such as the ideal class group in the above example, and the image of the trivial module.

I hope this answer helps until Minhyong sheds more light on his remarks and relations between $p$-adic $L$-functions and the Alexander polynomials.

[EDIT: Sep. 30th] To answer Daniel's questions below-
1) Take the projective resolution of A to define its class in $K_0$. In any case the $K$-theory of the category of finitely generated modules is the same as the $K$-theory of finite generated projective modules. 2) I do not know if there are any sensible/canonical multiplicative sets at which to localise $mathbb{Z}_p[G]$. The modules considered in Iwasawa theory are usually compact (or co-compact depending on whether you take inverse limit with respect to norms or direct limit with respect to inclusion of fields in a tower) and so the action of the ring $mathbb{Z}_p[G]$ extends to an action of the completion $mathbb{Z}_p[[G]]$ by which we mean $varprojlim mathbb{Z}_p[G/U]$, where $U$ runs through open normal subgroups of $G$. 3) We do not usually get a loop in the $K$-theory space of $mathbb{Z}_p[[G]]$. However, modules which come up from arithmetic are usually torsion as $mathbb{Z}_p[[G]]$-modules i.e. every element in the module is annihilated by a non-zero divisor or in other words if $X$ is the module then $Frac(mathbb{Z}_p[[G]]) otimes X=0$. (it is usually very hard to prove that the modules arising are in fact torsion). But once we know that they are torsion we get a loop in the $K$-theory space of $Frac(mathbb{Z}_p[[G]])$. If we know that for some multiplicatively closed subset $S$ of $mathbb{Z}_p[[T]]$ annihilates $X$ i.e. $(mathbb{Z}_p[[G]]_S) otimes X=0$, then we already get a loop in the $K$-theory space of $mathbb{Z}_p[[G]]$. In noncommutative Iwasawa theory it is often necessary to work with a multiplicative set strictly smaller than all non-zero divisors.

arithmetic geometry - When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?

[Added: Just to set the scene, this answer is discussing the $p$-adic etale cohomology of a variety $X$ over a finite extension of $mathbb Q_p$ (or, a little more generally,
a finite extension of the fraction field of the Witt vectors of a perfect field of char. $p$).]

Crystalline implies semi-stable implies de Rham implies Hodge-Tate. The cohomology is crystalline if $X$ is proper with a smooth model, is semi-stable if $X$ is proper with a semi-stable model, and is de Rham always. (In the proper case, the last statement is a theorem of Tsuji; in the general case, of Kisin. The crystalline and semi-stable cases are due, in various degrees of generality, to Fontaine--Messing, Hyodo--Kato, Faltings, and Tsuji.)

Added: It might help to note that de Rham and potentially semi-stable coincide (as discussed in the comments below, Berger reduced this to a result in the theory of $p$-adic differential equations, which was then proved independently by Andre, Kedlaya, and Mebkhout). Also, unramified is a very strong condition in this setting ($p$-adic etale cohomology of varieties over $p$-adic fields), which essentially never
holds unless we are looking just at $H^0$, and if the geometrically connected components
of $X$ are defined over an unramified extension.

linear algebra - Order of "one minus automorphism"

Your first condition implies that $r$ is the order of $p$ modulo $q$.

If the order of $1-t$ is $q$, then $p$ divides
$$ prod_{i=1}^{p-1} prod_{j=1}^{p-1} (1 - zeta^i - zeta^j) quad (*)$$
where $zeta$ is a primitive $p$-th root of unity.

If these two conditions are satisfied, we can find $t$ with the required properties.

Proofs:

First, suppose that $t$ obeys your first condition. Let $f$ be the characteristic polynomial of $t$. If $f$ factored as $gh$, and $v$ were in the kernel of $g(t)$, then $mathrm{Span} (t^i v)$ has dimension at most $deg g$, contradicting your hypothesis.

So $f$ is irreducible. In particular, $f$ divides $x^{p^r}-x$ and so $q | p^r-1$. Also, if $f$ divided $x^{p^s} - x$ for some $s<r$, then $f$ has a nontrivial factor of degree $leq s$, contradicting that $f$ is irreducible. So $r$ is the order of $p$ modulo $q$.

Now, suppose that $1-t$ has order $q$. Let $Phi_q(x) = (x^q-1)/(x-1)$. Then $f(x) | Phi_q(x)$ and $f(1-x) | Phi_q(x)$ as well. So $f$ divides $mathrm{GCD}(Phi_q(x), Phi_q(1-x))$. In particular, the resultant of $Phi_q(x)$ and $Phi_q(1-x)$ is zero modulo $p$. The product $(*)$ is precisely that resultant.

Finally, we must reverse the argument. Suppose that $p$ divides $(*)$. So there is an irreducible polynomial $f$ dividing $Phi_q(x)$ and $Phi_q(1-x)$. Let $s = deg f$. Let $V$ be the field $mathbb{F}_{p^s}$ and let $t$ be the action of a root of $f$ on $V$. Then $t$ obeys condition 1. (If not, $mathrm{Span} (t^i v)$ is a $t$-stable subspace of $V$, contradicting that $f$ is irreducible.) By the first paragraph of the proof, this shows that $s$ is the order of $p$ modulo $q$, so $s=r$ and we have constructed the desired $t$.

rt.representation theory - extending cusp forms

I believe the answer should be yes, by some version of the following sketch of an argument:

(Note: by restriction of scalars, I regard all groups as being defined over $mathbb Q$,
and I write ${mathbb A}$ for the adeles of $mathbb Q$.)

We are given $V_{pi} subset Cusp(G(F)backslash G({mathbb A}_F)).$

Let $tilde{C}$ denote the maximal $mathbb Q$-split torus in the centre of $tilde{G}$
(this is just a copy of $mathbb G_m$), and write $C = tilde{C}cap G$. (I guess
this is just $pm 1$?)

Now $C(mathbb A)$ acts on $V_{pi}$ through some character $chi$ of $(mathbb A)/C(mathbb Q)$. Choose an extension
$tilde{chi}$ of $chi$ to a character of $tilde{C}(mathbb A)/tilde{C}(mathbb Q)$,
and regard $V_{pi}$ as a representation of $tilde{C} G$ by have $tilde{C}$ act
through $tilde{chi}$. Since $tilde{C} G$ is normal and Zariksi open in $tilde{G}$,
we should be able to further extend the $tilde{C} G(mathbb A)$-action on $V_{pi}$
to an action of $tilde{G}(mathbb Q)tilde{C}G(mathbb A).$

Now if we consider $Ind_{tilde{G}(mathbb Q)tilde{C} G(mathbb A)}^{tilde{G}(mathbb A)} V_{pi},$ we should be able to
find a cupsidal representation $V_{tilde{pi}}$ of the form you want (with $tilde{C}(A)$ acting via $tilde{chi}$).

The intuition is that automorphic forms on $G(mathbb A)$ are $Ind_{G(mathbb Q)}^{G(mathbb A)} 1,$
and similarly for $tilde{G}$. We will consider variants of this formula that takes into account central characters, and think about how to compare them for $G$ and $tilde{G}$.

Inside the automorphic forms, we have those where $C(mathbb A)$ acts by $chi$; this we can
write as $Ind_{G(mathbb Q)C(mathbb A)}^{G(mathbb A)} chi$, and then rewrite as
$Ind_{tilde{G}(mathbb Q)tilde{C}(mathbb A)}^{tilde{G}(mathbb Q)tilde{C}G(mathbb A)} tilde{chi}.$ This is where $V_{pi}$ lives, once we extend it to a repreresentation
of $tilde{G}(mathbb Q)tilde{C}G(mathbb A)$ as above.

Now the automorphic forms on $tilde{G}(mathbb A)$ with central character $tilde{chi}$
are $Ind_{tilde{G}(mathbb Q)tilde{C}(mathbb A)}^{tilde{G}(mathbb A)} tilde{chi},$
which we can rewrite as
$Ind_{tilde{G}(mathbb Q) tilde{C}G(mathbb A)}^{tilde{G}(mathbb A)}
Ind_{tilde{G}(mathbb Q)tilde{C}(mathbb A)}^{tilde{G}(mathbb Q)tilde{C}G(mathbb A)}
tilde{chi}.$ This thus contains $Ind_{tilde{G}(mathbb Q)tilde{C}G(mathbb A)}^{tilde{G}(mathbb A)}V_{pi}$ inside it, and so an irreducible constituent of the latter
should be a $V_{tilde{pi}}$ whose restriction (as a space of functions) to $G(mathbb A)$ contains $V_{pi}$.

What I have just discussed is the analogue for $G$ and $tilde{G}$ of the relation between automorphic forms on $SL_2$ and $GL_2$ discussed e.g. in Langlands--Labesse. Hopefully I haven't blundered!

set theory - Higher-rank Borel sets

The Borel hierarchy is, of course, strictly increasing at
every step, so there will be sets of every countable
ordinal rank. Furthermore, all of these sets are relatively
concrete, obtained as countable unions of sets having lower
rank. But you asked for natural examples, so let me give a
few.

1) The collection of true statements of number theory, in
the ring of integers Z. This is well known to have
complexity Sigma^0_omega, which is a large step up from the
complexity you mentioned. This is, of course, a light-face
notion. The corresponding bold-face notion would be truth in arbitrary countable structures. That is, the
set of pairs (A,phi), where A is a first-order structure
and phi is true in A. This has complexity Sigma^0_omega for
the same reason as (1). (I regard this as a highly natural example, but Kechris' remark was about natural examples specifically from topology and analysis.)

2) The set of countable graphs with finitely many connected
components. This has complexity Sigma^0_3, since you must
say: there is a finite set of vertices, for all other
vertices, there is a path to one of them.

There are numerous other statements about finiteness that
also arise this way.

3) The set of (reals coding) finitely generated groups.
This is Sigma^0_3, since you say: there is a finite set in
the group, such that for all other elements of the group,
there is a generating word.

4) finitely-generated other structures, etc. etc.

5) The set of pairs (A,p), such that A is a real and p is a
Turing machine program with oracle A that accepts all but
finitely many input strings. Sigma^0_3, since "There is a
finite set of strings, for all other strings, there is a
computation showing them to be accepted."

ap.analysis of pdes - Partial $L^2$ control on (part of) the Hessian of a harmonic function.

Okay so I thought about this some more and I believe it is just a (really) straightforward application of the Poincare/Wirtinger inequality and the obvious way you would solve $u_{xy}=0$ classically.

Lets work on the square $S=(0,2pi)_xtimes(0,2pi)_y$ as it is simpler to work there and doesn't change much.
We assume $u$ is $C^infty_0(S)$ (i.e. smooth and of compact support) for the time being but do not need it to be harmonic.

Set $f(s)=frac{1}{2pi}int_0^{2pi} u_x(s, t)dt$ so $int_0^{2pi} u_{x}(s, t)-f(s) dt=0$ for all $s$.

By Wirtinger's inequality we have
$int_0^{2pi} (u_x(s,t)-f(s))^2 dt leq int_0^{2pi} u_{xy}^2(s,t) dt$.

Now pick a $F$ so that $F'(x)=f(x)$.
Set $G(t)=frac{1}{2pi}int_0^{2pi} u(s,t)-F(s) ds$ so $int_0^{2pi} u(s,t)-F(s)-G(t) ds=0$ for all $t$. Since $frac{d}{ds}left( u(s,t)-F(s)-G(t)right)=u_x(s,t)-f(s)$ we have by Wirtinger's inequality that:

$int_0^{2pi} (u(s,t)-F(s)-G(t))^2 ds leq int_0^{2pi} (u_x(s,t)-f(s))^2 $ for all $t$.
Then by Fubini we have that

$int_{S} (u(x,y)-F(x)-G(y))^2 leq int_S u^2_{xy}$

Everything is valid if $u$ is in $H^2(S)$. Notice the $F,G$ are given explicitly in terms of $u$.
If $u$ is harmonic with the normalizing properties at $0$ then the $F=ax^2, G=-ay^2$.

At least I think this works...

knot theory - Intutive interpretation about Linking forms

Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with
$H_{*}(M;Q)=H_{*}(S^3;Q)$

As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's Lecture notes in Algebraic toplogy book, we have a $Q/Z$ valued linking form, $lambdacolon H_{1}(M;Z)times H_{1}(M;Z)to Q/Z$ defined by the adjoint to following isomorphism.

(Actually we can define linking form in more general setting, e.g.) odd dimensional manifold without the restriction such as rational homology sphere condition. Because I want to just intuitive idea about linking form, I restricted the case)

$H_1(M;Z)cong H^2(M;Z)cong H^1(M;Q/Z)=Hom(H_1(M;Z),Q/Z)$

First isomorphism : Poincare duality,

Second isomorphism : Inverse of Bockstein homomorphism $delta$ induced form $0rightarrow Z rightarrow Qrightarrow Q/Zrightarrow 0$. More precisely, induced long exact sequence is that $ldotsrightarrow H^1(M;Q)rightarrow H^1(M;Q/Z)rightarrow H^2(M;Z)rightarrow H^2(M;Q)$ and here $H^1(M;Q)$ and $H^2(M;Q)$ vanishes. Therefore, the long exact sequence shows that $H^1(M;Q/Z)$ and $H^2(M;Z)$ are naturally isomorphic (if $M$ is rational homology 3-sphere) and we call that homomorphism as Bockstein homomorphism.

Third isomorphism : Universal Coefficient theorem

In short, $lambdacolon H_{1}(M;Z)times H_{1}(M;Z)to Q/Z$ is defined by $lambda(x,y)=<tilde{x}cupdelta^{-1}(tilde{y}),[M]>$, where $tilde{x},tilde{y}$ are Poincare dual to $x,y$ and $deltacolon H^1(M;Q/Z)to H^2(M;Z)$ is a Bockstein homomorphism as defined above. Cup products are defined in $H^1(M;Z)times H^2(M;Q/Z) to H^3(M;Q/Z)$ induced from the Bilinear map $Z times Q/Z to Q/Z$.

I understand this linking form only algebraic viewpoint. Therefore, I can play with this form only by using dilluminating algebraic topology.

I'm struggle to find geometric interpretation but I have no idea to express the Q/Z valued in terems of geometric language which seems to be highly algebraic. (Feeling like Injective, divisible, Ext or something linke that )

Are there any intuitve and geometric (clean) interpretation about this linking form? (e.g.)such as Alexander duality, like the argument that .........we can find a dual basis which represents meridian......)

ag.algebraic geometry - any linear algebraic group rational?

This is really just Pete Clark's answer -- the new bit is to note that
the Levi decomposition isn't needed.

Let G be a (reduced, connected) linear algebraic group over an
alg. closed k, and let R be the unipotent radical of G. Choose a Borel
group B of G with unipotent radical U < B (so R < U).

There is a dense B-orbit ("big cell") V in G/B which is a rational
variety.

Since U and R are (split) unipotent, [Springer, LAG 14.2.6] shows that there
is a section $s:U/R to U$ to the natural projection $U to U/R$.

If $f:G to G/B$ is the quotient mapping, using $s$ you can find a "local
section" of $f$ over the big cell V.

This show that $f$ is a locally trivial B-bundle (in the Zariski
topology) and in particular $f^{-1}(V)$ is an open subvariety of G
isomorphic to the rational variety V x B.

ct.category theory - Yoneda embedding target

Various categories of graphs are presheaf categories.

The category of directed graphs is (equivalent to) presheaves on $C$, where $C$ is a category with two objects, call them $V$ and $E$, and two parallel morphisms $s, t : V to E$. If you have never seen this example, you should compute for yourself that a functor $G : C^{op} to text{Set}$ is the same thing as a directed graph. You may find this "Guided tour of the topos of graphs" illuminating.

Other categories of graphs are (almost) presheaf categories. For example, take the monoid $M$ of all endomaps $lbrace 0,1rbrace to lbrace0,1rbrace$. This is a four-element monoid whose elements are the identity $id$, two constant maps $0$ and $1$, and the "twist" map $t$. View $M$ as a category (one object, four morphisms). The presheaves on $M$ are what is sometimes called "reflexive" graphs. Since this is not apparent at first sight, let me spell it out a bit. Consider a fuctor $F : M^{op} to text{Set}$, which is the same thing as a set $S$ with a right action of $M$. The corresponding graph $G$ has as its vertices the set $V = lbrace x in S mid x cdot 0 = xrbrace$ of elements fixed by the action of the constant map $0$ (exercise: the points fixed by the constant map $0$ are the same as the points fixed by the constant map $1$). The edges of $G$ are the elements of $S$. An edge $e in S$ has as its source the vertex $e cdot 0$ and the target $e cdot 1$. But since we also have the action of the twist map $t$, the situation is symmetric: to every edge $e$ going from $e cdot 0$ to $e cdot 1$ there corresponds the opposite edge $e cdot t$ going from $(e cdot t) cdot 0 = e cdot 1$ to $(e cdot t) cdot 1 = e cdot 0$. So we are talking about symmetric graphs. Our graphs may be degenerate in the sense that an edge $e$ could be its own opposite (and then it is also a loop since $e cdot 1 = e cdot 0$). The graphs are reflexive because a homomorphism between them is allowed to "squish" edges to vertices, which is another exercise in computing natural transformations.

All of this and more (perhaps too much) can be found in:

Categories of spaces may not be generalized spaces as exemplified by directed graphs, F. William Lawvere, Revista Colombiana de Matematicas, XX (1986) 179-186. (Republished in: Reprints in Theory and Applications of Categories, No. 9 (2005) pp. 1-7)