ag.algebraic geometry - sheaves of representations on galois groups, can there be interesting cohomology?

Consider a field $K$ (of characteristic 0, say) and its absolute galois group $G_K^{ab} = Gal(overline{K}/K)$, given the Krull topology: $U_E(sigma) = sigma Gal(overline{K}/E)$ form a basis of the topology, ranging over $sigma in G_K^{ab}$ and $E/K$ finite galois.

Fix a group $G$ and denote by $R_E$ its representation ring over $E$, and by $R_E^sigma subset R_E$ the elements of $R_E$ fixed by $sigma$.

We can construct a sheaf $mathcal{F}$ on $G_K^{ab}$ by setting $mathcal{F}(U_E(sigma)) = R_E^sigma$. It is a simple exercise to verify the axioms.

One might hope that the sheaf cohomology of $mathcal{F}$ encodes information about the splitting behaviour of representations of G over various ground fields, but this is not the case: $G_K^{ab}$ is known to be totally disconnected, hausdorff and compact. It is a theorem [1, 5.1] that $H^r(G_K^{ab}, mathcal{F}) = 0$ for $r > 0$. Furthermore the $U_E(sigma)$ are actually clopen, so most useful subsets I can think of are also compact, hence their cohomology is equally uninteresting.

Is there a way to produce a useful cohomology along these lines?

Here "useful" essentially means "non-trivial", and "along these lines" basically "involving the galois action on $R_E$ for various $E$".

[1] http://www.jstor.org/stable/2035693

zoology - Why is there a difference in the rotation of the tail fin in fish compared to marine mammals?

While fish tend to move from side to side (lateral undulation) for which a vertical tail makes sense, the land ancestors of marine mammals had their limbs under them and so their spines were already adapted to up and down movement (dorsoventral undulation). When these animals moved to marine environments, they continued up and down movement in their swimming, for which a horizontal tail makes sense.

(The wikipedia article on fins gives some more detail, and links to this webpage on Berkeley.edu. A paper by Thewissen et al. suggests that for cetaceans, dorsoventral undulation as a swimming strategy came first, and the horizontal tail evolved later.

---

More detail:

In a third example beyond fishes and marine mammals, the icthyosaurs and other aquatic reptiles developed vertical tails, even though like marine mammals they evolved from four-footed land animals. This may be because the legs/spines/gaits of land reptiles differ from those of land mammals, so that the earliest icthyosaurs swam with lateral undulation, as reflected in their spinal modifications.

This blog post by Brian Switek gives a really superb run-down on the issue, with figures and citations. I'll quote this part which deals with the dorsoventral undulation theory:

...mammals and their relatives were carrying their legs underneath their bodies and not out to the sides since the late Permian, and so the motion of their spine adapted to move in an up-and-down motion rather than side-to-side like many living reptiles and amphibians. Thus Pakicetus[the pre-cetacean land mammal] would not have evolved a tail for side-to-side motion like icthyosaurs or sharks because they would have had to entirely change the way their spinal column was set up first.

Switek goes on to to talk about exceptions in some marine mammals:

At this point some of you might raise the point that living pinnipeds like seals and sea lions move in a side-to-side motion underwater. That may be true on a superficial level, but pinnipeds primarily use their modified limbs (hindlimbs in seals and forelimbs in sea lions) to move through the water; they aren’t relying on propulsion from a large fluke or caudal fin providing most of the propulsion with the front fins/limbs providing lift and allowing for change in direction. This diversity of strategies in living marine mammals suggests differing situations encountered by differing ancestors with their own suites of characteristics, but in the case of whales it seems that their ancestors were best fitted to move by undulating their spinal column and using their limbs to provide some extra propulsion/direction.

V-filtration of D-modules associated to a monomial

Hi

In Mixed Hodge modules Saito computes the Verdier specialisation of a D-modules with respect to a monomial $g = x_1^{m_1}ldots x_n^{m_n}$. This is a very nice result as I find such explicit computations are quite rare in the litterature but I'm having problems understanding his proof.

Let $X$ be a polydisc, $D_i = {x_i = 0}$, $D_I = bigcap_{iin I} D_i$. Consider $M$ a regular holonomic algebraic quasi-unipotent right D-module with characteristic variety contained in the union of the conormal bundles to the $D_I$. For $nu = (nu_1,ldots,nu_n)$, set
$$
M^nu := langle uin M ~|~ u(x_ipartial_i - nu_i)^p = 0~~ textrm{for $pgg 0$} rangle
$$
We have $bigoplus_{nu in mathbb{Q}^n} M^nu = M$.

Let $g:Xto S$, $S$ an open disk, given by $g=x_1^{m_1}ldots x_n^{m_n} = x^m$ and $i_g:X to Xtimes S$ its graph. The D-module $tilde{M} = (i_g)_* M$ is $M[partial_t]$ with the action of $D_{Xtimes S}$ given by:
$$
(uotimes partial_t^j)x_i = ux_iotimes partial_t^j, quad
(uotimes partial_t^j)partial_i = upartial_iotimes partial_t^j - u(partial_ig)otimes otimes partial_t^{j+1}
$$
$$
(uotimes partial_t^j)t = ugotimes partial_t^j + ju otimes partial_t^{j-1} , quad
(uotimes partial_t^j)partial_t = ux_iotimes partial_t^{j+1}
$$

Theorem (3.4 p. 280): The V-filtration of $tilde{M}$ along $t = 0$ is generated over $D_X$ by
$$
M^nu otimes 1 ~~textrm{with $nu_i leq m_ialpha$} quad textrm{if $alpha< 0$}
$$
$$
M^nu otimes partial_t^j ~~textrm{with $nu_i leq m_i(alpha-j)$} quad textrm{in general}
$$

It is enough to check that the filtration defined in the theorem satifies the properties of the V-filtration. Saito says:

"We have
$$
(uotimes partial_t^j)x_ipartial_i = (uotimes partial_t^j)(N_i + nu_i - m_i(s-j) )
qquad forall u in M^nu
$$
where $s = tpartial_t$ and $(uotimes partial_t^j)N_i = u(x_ipartial_i-nu_i)otimes partial_t^j$ if $uin M^nu$. Thus we get that $s-alpha$ is nilpotent on $Gr^V_alpha tilde{M}$ and $V_alpha tilde{M}$ ar $V_0D_{Xtimes S}$-submodules."

I don't understand this last statement. $Gr^V_alpha tilde{M}$ is generated over $D_X$ by the $u otimes partial_t^j in V_alpha tilde{M}$ with at least one $i$ so that $nu_i = m_i(alpha-j)$. For such $nu$, the above equation reads
$$
(uotimes partial_t^j)x_ipartial_i = (uotimes partial_t^j)(N_i - m_i(s-alpha))
$$
Why would this imply that $s-alpha$ is nilpotent?

Edit: I think I have found the answer. Consider $uotimes partial_t^j$ with $nu_i leq m_i(alpha-j)$ forall $i$. Set $I := langle i ~|~ nu_i = m_i(alpha-j),~ m_ineq 0rangle$. Then
$$
(uotimes partial_t^j)(prod_{iin I} x_ipartial_i) = (u prod_{iin I}x_i otimes partial_t^j) (prod_{iin I}partial_i)
$$
is in $V_{<alpha} tilde{M}$. So, by applying the relation above, we have
$$
(uotimes partial_t^j)prod_{iin I} (N_i - m_i(s-alpha)) =
(uotimes partial_t^j)(prod_{iin I} x_ipartial_i) = (u prod_{iin I}x_i otimes partial_t^j) (prod_{iin I}partial_i) = 0
$$
in $Gr^V_alpha tilde{M}$. So
$$
(uotimes partial_t^j)(s-alpha)^{|I|} = (uotimes partial_t^j)(sum_{k=0}^{k=|I|-1} sigma_k(N_i/m_i)(s-alpha)^k) )
$$
The operator on the right hand side is a polynomial in the $N_i$'s with coefficient in $Q[s-alpha]$ and no constant term. So if $p_i$ is the nilpotency order of $N_i$ operatating on $uotimes partial_t^j$ and we elevate the relation to the power $sum p_i$ we find that $(uotimes partial_t^j)(s-alpha)^{|I|(sum p_i)} = 0$. So $s-alpha$ is indeed nilpotent.

riemannian geometry - Hermitian symmetric spaces vs Hermitian homogeneous spaces

Here is a geometric answer to (2), or more precisely a slight and classical reinterpretation. First, note that Hermitian symmetric spaces fit into the more general concept of riemannian symmetric space; this can helps you find references.

Denote your space by $X$ (assumed to be Riemannian, or Hermitian if you prefer) and take a point $p$. Consider the geodesic symmetry around $p$: it maps a point $q$ to the point $sigma_p(q)$ so that $sigma_p(q), p, q$ lie on a constant speed geodesic, at times $-1,0,1$ respectively. This map is at least defined locally. Then $X$ is symmetric if and only if for all $pin X$, $sigma_p$ is globally defined and an isometry (ask it to be also holomorphic if your are in the Hermitian setting, but I guess it will automatically be so since it is conjugate to $-mathrm{Id}$ by the exponential map). Of course, $sigma_p$ is your involution.

This condition automatically implies that the isometry group of $X$ acts transitively (because any $q$ is mapped to any $q'$ by the map $sigma_p$ where $p$ is a midpoint of $[q,q']$).

It also gives you an involution $theta$ on the Lie algebra $mathfrak{g}$ of the isometry group of $X$. Now remark that $theta$ is a linear endomorphism, and $theta^2=1$ implies that
$mathfrak{g}$ decomposes into two components, the eigenspace associated to the eigenvalue $1$ of $theta$ and the one associated to $-1$. They are usually denoted by $mathfrak{h}$ and $mathfrak{p}$; the latter identifies with the tangent space to $X$ at $p$. This is called Cartan decomposition and is fundamental to the study of these spaces.

The most common reference is Helgason's Differential geometry, Lie groups, and symmetric spaces but I find it quite difficult to read. In the nonpositively curved case, I found Eberlein's Geometry of nonpositively curved manifolds useful.

Concerning the motivation, it seems that the interest in symmetric spaces is that they provide examples between constant curvature spaces, which are very constrained, and homogeneous spaces that are very numerous.

Last, a lead concerning (1). You shall find in Berger's A panoramic view of Riemannian geometry, section 15.8.1 page 719 a formula enabling one to compute the curvature of a homogeneous space. I guess it can be used to show the existence of many such spaces, even in the Hermitian world, that are negatively curved but not symmetric.

rt.representation theory - Role of nontrivial component groups in Springer Correspondence?

Set-up for classical Springer Correspondence:

$G$ = reductive group (usually assumed to be semisimple of adjoint type) over $mathbb{C}$, with Borel subgroup and
maximal torus $B supset T$, Weyl group $W=N_G(T)/T$.

Fix a unipotent $u in G$ with component group $A(u) = C_G(u)/C_G(u)^circ$.

$mathcal{B} = G/B$ (flag variety), containing Springer fiber $mathcal{B}_u$
= fixed points under $u$, $d=dim mathcal{B}_u$ (= half codimension
of class of $u$ in unipotent variety)

Then $W times A(u)$ acts on cohomology ($ell$-adic or classical)
$H^*(mathcal{B}_u)$ with top cohomology in degree $2d$.

(Springer) Each irreducible representation of $W$ occurs, for some pair
$(u,phi)$ with $u$ unipotent and$phi$ an irreducible character of $A(u)$, as an isotypic component of the $W times A(u)$ representation
on the top cohomology. Here all pairs $(u,1)$ occur.

Assume $A(u) neq 1$ (possible except in type A).

(1) Must some pair $(u,phi)$ with $phi neq 1$ occur?

(2) Is the representation of $A(u)$ on $H^*(mathcal{B}_u)$
always a permutation permutation?

The answers to both questions seem to be yes, but I don't know
any uniform approach using Springer theory. For example,
(1) can be checked using case-by-case study of simple
types, but is there a general reason for it? For (2) there is
a sophisticated indirect argument using work of Bezrukavnikov,
Mirkovic, Rumynin. All of this ties in naturally with some
unsolved problems about representations of related Lie algebras
in characteristic $p$.

ag.algebraic geometry - Proper definition of a moduli problem

Since not many people have had anything to say, I thought I might make a few remarks. But beware that this is all what I've passively picked up over the years---it's not the result of an actual study of things.

I think the right definition of a moduli problem is a fibered category $p:Eto B$. This should be thought of as a family of categories parametrized by $B$, just like you think of a map $X to S$ of spaces as being a family of spaces parametrized by $S$. Here is an example: $E$ is the category whose objects are maps of schemes $Xto S$ making $X$ a family of elliptic curves over $S$ (i.e. an abelian scheme of relative dimension 1), and whose maps are the (hopefully) evident cartesian squares; $B$ is the category of schemes, and the functor $Eto B$ sends an object $Xto S$ to $S$. The fibered structure is given by pull back: given $Xto S$ in $E$ and a map $S'to S$ in $B$, we get the object $Xtimes_S S'to S'$ (which maps under $p$ to $S'$). If we let $E_S$ denote the fiber of this fibered category over an object $S$, then in this example, $E_S$ is the category of families of elliptic curves parametrized by $S$. Thus the fibered category encodes the data of all possible families of elliptic curves and how they behave under base change. So I hope my point that the fibered category is the moduli problem seems reasonable.

You then say the fibered category is representable if there is an object $U$ of $E$ such that for any $S$ in $B$, the pull back functor from the discrete category consisting of the set $mathrm{Hom}(S,p(U))$ to the category $E_S$ is an equivalence. This is pretty unlikely. For instance, it implies that each category $E_S$ is discrete---all maps are isomorphisms and all automorphisms are the identity. This is certainly not the case with the elliptic curve example. Every elliptic curve has a nontrivial automorphism given by the inverse map with respect to the group structure.

Another version of representability of a fibered category is the following. Let $F:Btomathrm{Sets}$ denote the functor which sends an object $S$ to the set of isomorphism classes of objects of $E_S$. Then the moduli problem is (weakly?) representable if the functor $F$ is representable. This definition is surely weaker in general than the one above, but it is often equivalent in the examples that people look at in algebraic geometry. For example, the two are probably equivalent if each $E_S$ is a discrete category.

If the moduli problem is not representable, then you get into other issues, such as whether you have effective descent with respect to some topology on $B$ (i.e. $E$ is a "sheaf of categories" over $B$), and if so, stack-theoretic issues, such as whether $E$ can be represented by a category object in $E$, and if so, whether it's a groupoid object.

soft question - Favorite popular math book

Title: What is mathematics?

Author: Herbert Robbins and Richard Courant

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

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

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