ho.history overview - Newton and Newton polygon

If memory serves correct the history of Newton's polygon and Puiseaux series has some subtleties, so be a bit wary of secondary historical sources. Histories of mathematics are bursting at the seams with romanticized legends, so it is always best to consult primary sources if you wish to know the real history. The following note from Chrystal's Algebra may serve as a helpful entry into the primary literature.

Historical Note. - As has already been remarked, the fundamental idea of
the reversion of series, and of the
expansion of the roots of algebraical
or other equations in power-series
originated with Newton. His famous"
Parallelogram" is first mentioned in
the second letter to Oldenburg; but is
more fully explained in the
Geometria Analytica (see Horsley's edition of Newton's Works, t. i., p.
398). The method was well understood
by Newton's followers, Stirling and
Taylor; but seems to have been lost
sight of in England after their time.
It was much used (in a modified form
of De Gua's) by Cramer in his
well-known Analyse dea Lignes
Courbea Algebriques (1750). Lagrange
gave a complete analytical form to
Newton's method in his "Memoire sur
l'Usage des Fractions Continues,"
Nouv. Mem. d. l'Ac. roy. d. Sciences d. Berlin (1776). (See OEuvres de
Lagrange, t. iv.)

Notwithstanding its great utility, the
method was everywhere all but
forgotten in the early part of this
century, as has been pointed out by De
Morgan in an interesting account of
it given in the Cambridge
Philosophical Transactions, vol.ix.
(1855).

The idea of demonstrating, a priori,
the possibility of expansions such as
the reversion-formulae of S.18
originated with Cauchy; and to him, in
effect, are due the methods employed
in SS.18 and 19. See his memoirs on
the Integration of Partial
Differential Equations, on the
Calculus of Limits, and on the Nature
and Properties of the Roots of an
Equation which contains a Variable
Parameter,
Exercices d'Analyse et de Physique Mathematique, t. i. (1840), p. 327;
t. ii. (1841), pp. 41, 109. The form
of the demonstrations given in SS. 18,
19 has been borrowed partly from
Thomae, El. Theorie der Analytischen
Functionen einer Complexen
Veranderlichen (Halle, 1880), p. 107;
partly from Stolz,
Allgemeine Arithmetik, I. Th. (Leipzig, 1885), p. 296.

The Parallelogram of Newton was used
for the theoretical purpose of
establishing the expansibility of the
branches of an algebraic function by
Puiseaux in his Classical Memoir on
the Algebraic Functions (Liouv. Math.
Jour., 1850). Puiseaux and Briot and
Bouquet (Theorie des Fonctions
Elliptiques (1875), p. 19) use
Cauchy's Theorem regarding the number
of the roots of an algebraic equation
in a given contour; and thus infer the
continuity of the roots. The
demonstration given in S.21 depends
upon the proof, a priori, of the
possibility of an expansion in a
power-series; and in this respect
follows the original idea of Newton.

The reader who desires to pursue the
subject further may consult Durege,
Elemente der Theorie der Functionen einer Complexen Veranderlichen
Grosse, for a good introduction to
this great branch of modern
function-theory.

The applications are very numerous,
for example, to the finding of
curvatures and curves of closest
contact, and to curve-tracing
generally. A number of beautiful
examples will be found in that
much-to.be-recommended text-book,
Frost's Curve Tracing. -- G. Chrystal: Algebra, Part II, p.370

soft question - Theorems for nothing (and the proofs for free)

I'd say the Tutte-Berge formula, which is a wonderful result that tells you (almost) everything you want to know about matchings in graphs. Although there are many proofs of this theorem, there is a beautiful proof for free using matroids. Strictly speaking, there is a proof for free of Gallai's Lemma (from which Tutte-Berge follows easily).

Gallai's Lemma.

Let $G$ be a connected graph such that $nu(G-x)=nu(G)$, for all $x in V(G)$. Then $|V(G)|$ is odd and def$(G)=1.$

Remark: $nu(G)$ is the size of a maximum matching of $G$, and def$(G)$ denotes the number of vertices of $G$ not covered by a maximum matching.

Proof for free.
In any matroid $M$ define the relation $x sim y$ to mean $r(x)=r(y)=1$ and $r({x,y})=1$ or if $x=y$. (Here, $r$ is the rank function of $M$). We say that $x sim^* y$ if and only if $x sim y$ in the dual of $M$. It is trivial to check that $sim$ (and hence also $sim^*$) defines an equivalence relation on the ground set of $M$.

Now let $G$ satisfy the hypothesis of Gallai's Lemma and let $M(G)$ be the matching matroid of $G$. By hypothesis, $M(G)$ does not contain any co-loops. Therefore, if $x$ and $y$ are adjacent vertices we clearly have $x sim^* y$. But since $G$ is connected, this implies that $V(G)$ consists of a single $sim^*$ equivalence class. In particular, $V(G)$ has co-rank 1, and so def$(G)$=1, as required.

Edit. For completeness, I decided to include the derivation of Tutte-Berge from Gallai's lemma. Choose $X subset V(G)$ maximal such that def$(G-X) -|X|=$ def$(G)$. By maximality, every component of $G-X$ satisfies the hypothesis in Gallai's lemma. Applying Gallai's lemma to each component, we see that $X$ gives us equality in the Tutte-Berge formula.

Looking for an undergraduate research problem in algebraic geometry or algebraic number theory

I'm looking for a small research problem an undergraduate would be capable of after taking just an abstract algebra course, introductory algebraic geometry (at level of Miles Reid's book and Ideals, Varieties & Algorithms), and a course in number theory. Is there a website that would have a decent listing, or possibly a book one can recommend that may have small open research problems?

Normal operators and it's spectrum in C*-algebras

If $A$ is a C*-algebra and $n$ is a normal element of $A$, then we have: (By Gelfand duality for example.)

$operatorname{spec}( |N| ) = | operatorname{spec}(N) | := left{ | lambda | ; lambda in operatorname{spec}(N) right}$

where we define: $|n|:=(n^*n)^{1/2}$. My question is, does the converse also hold?

That is if $ain A$ and for $r>0$:

$left{ r e^{it} : t in [0, 2pi[ right} cap operatorname{spec}(a)$ is not empty if and only if $rin operatorname{spec}(|a|)$

implies that $a$ is normal. (Possibly some exceptions made for the zero-element) Or bluntly speaking if the mapping $a$ to $a^*a$ does not create any "new" (or removes any "old") elements in the spectrum then $a$ is normal.

For example if $e$ is an idempotent in $A$, then $e$ is a projection if and only if $||e||=1$. Hence if $e$ is a non-projection idempotent we have $left{0,1right} = operatorname{spec}(e) subsetneq operatorname{spec}(|e|)$, since $||e||>1$ and by the spectral radius $operatorname{spec}(e^*e)$ contains an element strictly bigger that one.

Clearly if p is a projection, then p=|p|.

ag.algebraic geometry - Analogues of the Weierstrass p function for higher genus compact Riemann surfaces

I think Hunter and Greg's answers make it hard to see the forest for the trees. Let X be a compact Riem. surface of genus >= g. Let Y be the universal cover of X equipped with the complex structure pulled back from X. As a complex manifold, Y is isomorphic to the upper half plane, and the deck transformations form a subgroup Gamma of PSL_2(R). There will be characters chi of Gamma for which there are nonzero functions f on Y such that f(gz) = chi(g) f(z). For chi ample enough (not defined here), we will be able to choose functions (f_1, f_2, f_3) such that z --> (f_1(z) : f_2(z) : f_3(z)) gives an immersion X --> P^2. All of this works in any genus.

The technical issue is that this map is an immersion, not an injection, meaning that the image can pass through itself. One can either decide to live with this, or work with maps to P^3 instead.

Most books that I have seen don't lift all the way to the universal cover of X. Instead, they take the covering of X which corresponds to the commutator subgroup of pi_1(X). This can be motivated in a particular nice way in terms of the Jacobian. This is a complex manifold with the topological structure of a 2g dimensional torus. There is a map X --> J, so that the map pi_1(X) --> pi_1(J) is precisely the map from pi_1(X) to its abelianization. People then work with the universal cover of J, and the preimage of X inside it. This has three advantages: the universal cover of J is C^g, not the upper half plane; the group of Deck transformations is Z^{2g}, not the fundamental group of a surface, and the action on C^g is by traslations, not Mobius transformations. The functions which transform by characters, in this setting, are called Theta functions*, and they are given by explicit Fourier series.

*This is a slight lie. Theta functions come from a certain central extension of the group of Deck transformations. It is certain ratios of Theta functions that will transform by characters as sketched above. The P function itself, for example, is a ratio of four Theta functions. In the higher genus case, in my limited reading, I haven't seen names for these ratios, only for the Theta functions.

ag.algebraic geometry - Equivalence of derived categories which is not Fourier-Mukai

Schlichting gave an example of two categories of singularities which are derived equivalent but whose K-groups are not isomorphic. Dugger and Shipley (arXiv:0710.3070) expanded on this example and noted that it gives two dga's which are derived equivalent but not by an integral transform.

Otherwise, Lunts and Orlov's results on uniqueness of enhancements give a large class of triangulated categories for which one might lift exact functors to dg-functors and apply Toen's result.

ag.algebraic geometry - Higher vanishing cycles

My apologies if this is too much or too little; leave a comment and I can try and correct it. He's talking about a specific issue in homotopy theory that we'd like a better understanding of.

The stable homotopy category (implicitly localized at a prime p) has a stratification into "chromatic" layers, which correspond to a connection to formal group laws. We geometrically think of the stable homotopy category as some kind of category of sheaves on a moduli stack X which has a sequence of open substacks X(n) - these are the "E(n)-local categories", and there are Bousfield localization functors taking a general element M to its E(n)-localization L_E(n) M, which you can think of as restricting to the open substack. (A general Bousfield localization will take some notion of "equivalence" and construct a universal new category where those equivalences become isomorphisms, but in an appropriately derived way.)

The difference X(n) X(n-1) between two adjacent layers is a closed substack of X(n), which in our language is the "K(n)-local category". There is also a Bousfield localization functor that takes an element M to its K(n)-localization L_K(n) M. Bousfield localization is pretty general machinery and in the previous "open" situation it acted as restriction; in this "closed" situation it acts as a completion along the closed substack.

We have some general understanding of the K(n)-local categories. They act a lot like some kind of quotient stack of some Lubin-Tate space classifying deformations of a height n formal group law by the group scheme of automorphisms of said formal group law, which is the n'th Morava stabilizer group S_n. Geometrically we think about it as a point with a fairly large automorphism group (even though this is, of course, the wrong way to think about things). These are places where you can get dirty and do specific computations and examine one chromatic layer at a time.

There are two remaining pieces of data we need, then, to understand M itself from its localizations L_K(n) M: we need to understand how they are patched together into the E(n)-localizations, and we need to understand the limit of the L_E(n)M. The latter is a "chromatic convergence" question and not immediately relevant to the point under discussion.

In general there is a "patching" diagram, which is roughly something like the data you'd usually associate with a recollement. (My favorite reference for data in this kind of situation is Mazur's "Notes on etale cohomology of number fields".) We have a (homotopy) pullback diagram

LE(n) M  ->  LE(n-1) M
     |             |
     V             V
LK(n) M  ->  LE(n-1) LK(n) M

that tells us that a general E(n)-local object is reconstructed from a K(n)-local object (something concentrated on the closed stack), an E(n-1)-local object (concentrated on the open stack), and patching data (a map from the object on the open stack to the restriction of the complete object to the open stack). This roughly follows because the K(n)-localization of any E(n-1)-local object is trivial.

The functor that takes an object concentrated near the closed substack and restricts it (in a derived way) to the open substack is what Morava considers. Here, in the language of Bousfield localization, it is E(n-1)-localization applied to K(n)-local objects. What he seems to be proposing is that this general Bousfield localization setup should be one way of thinking about the vanishing cycles functors (and I concur with his dislike for the "vanishing" terminology) in which we can, in a fully derived way, view sheaves on a large stack as coming from patching data on an open-closed pair.

Just to close the loop, what we don't really understand at all in this picture is what this "trans-chromatic-layer" stuff really does. We have, for example, two stabilizer groups connected to formal group laws of adjacent heights, and we don't really understand what the specialization functor is really doing in this case.

hyperbolic geometry - Poincaré disk model: is this locus a known curve?

In the Klein model, one may see that this is also a circle. Consider a line segment with one point on the center of the disk. One side of the triangle goes through the center. Then orthogonal lines to a line through the center are also orthogonal in the hyperbolic metric, e.g. since they are preserved by reflection. So one sees that a circle is traced out which goes through the origin. If you'd rather center the curve at the origin, then it will be an ellipse, since hyperbolic isometries of the Klein model are projective transformations.

To convert to the Poincare model, take a hemisphere sitting over the disk, and project vertically. The projection of the circle is given by the intersection of a cylinder over the circle with the upper hemisphere. This upper hemisphere is conformally equivalent to the Poincare model, e.g. by inversion through a sphere centered at the south pole of the lower hemisphere. I haven't computed the curve this traces out though.

ca.analysis and odes - Sequence that converge if they have an accumulation point

The following version of the mean ergodic theorem is taken from the book of Krengel, "ergodic theorems".

Let T be a bounded linear operator in a Banach space X. The Birkhoff averages are denoted by $A_n = {1over n} Sigma_{k=0}^{n-1} T^k$.
Assume that the sequence of operator norms $||A_n||$ is bounded independently of $n$. Then for any x and y in B, the following is equivalent :

-- y is a weak cluster point of the sequence $(A_nx)$,

-- y is the weak limit of the sequence $(A_nx)$,

-- y is the strong limit of the sequence $(A_nx)$.

(note that we talk about cluster points instead of converging subsequences because we didn't assume B separable. Hence the weak topology is not necessarily metrizable.)

This theorem implies e.g. the ergodic theorem for Markov operators on $C(K)$ (sequential compactness follows from Azrela-Ascoli), or the ergodic theorem for power bounded operators defined on reflexive Banach spaces (sequential compactness follows from Eberlein-Smulian).

There is a whole set of theorems in ergodic theory along these lines. Let me mention the convergence of the one sided ergodic Hilbert transform, discussed in Cohen and Cuny (see Th 3.2) as another example.

ca.analysis and odes - Are these two notions of Lipschitz hypersurface equivalent?

Let $S$ be a subset of $mathbb{R}^n$. I would like to call $S$

1. a Lipschitz(1) hypersurface if for every $xin S$ there is a hyperplane $H$ so that the orthogonal projection onto $H$ is a bi-Lipschitz map from a neighbourhood of $x$ in $S$ onto an open subset of $H$, and




2. a Lipschitz(2) hypersurface if for every $xin S$ there is a bi-Lipschitz map $psi$ from $Btimes(-1,1)$ onto a neighbourhood of $x$ in $mathbb{R}^n$ so that $psi^{-1}(S)=Btimes{0}$, where $B$ is an open subset of $mathbb{R}^{n-1}$.

It seems clear enough (*) that Lipschitz(1) implies Lipschitz(2). But is the converse true? And if not, what is a simple counterexample?

I have come across the notion of regions with Lipschitz boundaries in a number of papers on boundary value problems for PDEs. But every such paper seems to take the notion of Lipschitz-ness for granted.

(*) If $S$ is Lipschitz(1), then after a rotation of the axes, it locally looks like the graph of a Lipschitz function $gammacolonmathbb{R}^{n-1}tomathbb{R}$. Put $psi(x,t)=(x,gamma(x)-t)$ to obtain the Lipschitz(2) property.

dg.differential geometry - Cauchy-Crofton formula for curvature

I doubt it. Let's first dissect your Cauchy-Crofton formula. Up to minor technical assumptions, what we have is that
$$mbox{length}(gamma) = int_{mathbb{R}^2} delta_gamma(x) d^2x$$
where $delta_gamma(x)d^2x$ represents the measure concentrated on $gamma$. Locally you can think of its as the pull back of the Dirac delta via the charaterization function of $gamma$ as a submanifold.

Next we do a change of coordinates. Observe that (not being too precise here) $mathbb{R}^2 = TS^1 times mathbb{R} / S^1$ (basically you can rotate a standard coordinate system; in other words, the set of all lines on the plane can be identified with $TS^1$, and we take its Cartesian product with $mathbb{R}$ to measure along each of the lines). So let $pi$ be the canonical projection map from $TS^1times mathbb{R}$ to $mathbb{R}^2$. Then your integral can be re-written as
$$ mbox{length}(gamma) = frac{1}{|S^1|}int_{TS^1times mathbb{R}} pi^*delta_gamma(y) dy$$
where $pi^*delta_gamma(y) dy$ is the pull back measure. Now you integrate out the fibers $mathbb{R}$ first and by simple geometry you see that
$$ int_{TS^1timesmathbb{R}}pi^*delta_gamma(y) dy = int_{TS^1} n(s) ds $$
where $n(s)$ is the number of times the line indexed by the coordinate $s$ intersects $gamma$. So up to some normalization constants we recover the Cauchy-Crofton formula.

Now, if you want to get an integral of the curvature $k$ along the curve, you can write it as
$$ mbox{curvature integral} = int_{mathbb{R}^2} |k(x)|^p delta_gamma(x) d^2x$$
the procedure of lifting to the space of lines is no problem, so you can again get an integral over $TS^1times mathbb{R}$. The problem is that I can't see any obvious way of integrating out the additional factor of $mathbb{R}$. In the Cauchy-Crofton case, each time the line hits your curve it picks up a unit bundle of mass. In the curvature case, you pick up some number which depends on the curvature at the point of intersection.

Using circles won't help either. You can consider a foliation (or almost a foliation) of $mathbb{R}^2$ by some family of curves, each of which can be identified with some curve $m$. Then you can do the same thing as above: let $M$ be the parameter space of this foliation (the foliation can be moved under rigid motion transformations to get a new foliation) then $mathbb{R}^2$ can be identified as $Mtimes m / G$ by $G$ being some subgroup of the Euclidean symmetries. So formally your integral can be re-written as
$$ frac{1}{|G|}int_{Mtimes m} mbox{something} dy $$
but to simplify down to just an integral over the parameter space $M$, you need to integrate out along the fibre $m$, and to get a simple expression at the end you almost certainly need that the family of curves $m$ in your foliation must be very special. Unless $m$ is adapted to $gamma$ such that the integral along $m$ of $|k|^pdelta_gamma$ can be easily evaluated, you have no hope of arriving at a simple integral expression.

co.combinatorics - Finding integer points on an N-d convex hull

Because the facets of your convex hull are themselves polytopes (of one lower dimension—$d{=}21$ in your case), it seems your question is equivalent to asking how to count lattice points in a polytope. One paper on this topic is "The Many Aspects of Counting Lattice Points in Polytopes" by J. A. DeLoera. Section 4 is entitled, "Actually Counting: Modern Algorithms and Software." A key reference in that section is to a paper by Barvinok entitled, "Polynomial time algorithm for counting integral points in
polyhedra when the dimension is fixed" (Math of Operations Research, Vol. 19, 769–779, 1994), whose title (polynomial time) seems to provide an answer of sorts to your question.

lo.logic - Proofs that require the existence of large finite numbers

This isn't addressed to logicians, but it may be of interest. I happen to know of an example in PDE that was necessary in proving the well-posedness of radial solutions of the Nonlinear Schrodinger Equation:

$$i u_{t}+Delta u=|u|^{4}u$$

for which J. Bourgain was awarded his Fields Medal for treating. (J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, JAMS 12 (1999), 145-171).

In one of the many many critical steps required in this proof, a bound on energy is required. A team (J. COLLIANDER, M. KEEL, G. STAFFILANI, H. TAKAOKA, and T. TAO) have now treated the non-radial case and make explicit the large ordinals used for bounding the energy. I quote from page 36 of their paper "Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R^3":

"If one then runs the induction of energy argument in a direct way (rather than arguing by contradiction as we do here), this leads to very rapidly growing (but still finite) bound for M(E) for each E, which can only be expressed in terms of multiply iterated towers of exponentials (the Ackermann hierarchy). More precisely, if we use X ↑ Y to denote exponentiation X^Y,
X↑↑Y :=X↑(X↑...↑X) to denote the tower formed by exponentiating Y copies of X,
X↑↑↑Y :=X↑↑(X↑↑...↑↑X)
to denote the double tower formed by tower-exponentiating Y copies of X, and so forth, then we have computed our final bound for M(E) for large E to essentially be
M(E) ≤ C ↑↑↑↑↑↑↑↑ (CE^C).
This rather Bunyanesque bound is mainly due to the large number of times we invoke the induction hypothesis Lemma 4.1, and is presumably not best possible."

http://arxiv.org/abs/math/0402129

ag.algebraic geometry - no lines/conics on a degree 4/5 surface?

I will work over $mathbb C$. Although I have not checked, the example below should work for characteristic different from $3$.

To exhibit a degree $d$ projective surface $S subset mathbb P^3$ not containing any line you can consider surfaces of the form $t^d = f(x,y,z)$ where $f$ is homogeneous polynomial of degree $d$.

Let $C subset mathbb P^2$ be the curve determined by the polynomial $f$ and $pi: S to mathbb P^2$ be the linear projection from the point $p=[0:0:0:1]. $ If $ell$ is a line contained in $S$ then $pi(ell)$ is a line tangent to $C$ at a total inflection point $q$, i.e. the contact between $C$ and the line $pi(ell)$ at $q$ is of order $d$. For details see Section 6 of Counting lines on surfaces by Boissière and Sarti.

This reduces the problem of finding a surface without lines to the one of finding an
algebraic curve without total inflection points. For quartic curves we have not to look much, as Klein determined all the inflections of his famous quartic
$$
x^3y + y^3 z+ z^3 x = 0.
$$
All its $24$ inflection points are simple, see for instance Jeremy Gray's paper in
The Eightfold Way: The Beauty of Klein's Quartic Curve. Thus the surface
$$
t^4 - x^3 y - y^3 z - z^3 x =0
$$
has no invariant lines.

It should be possible to pursue this argument further to determine the sought examples.

Edit: The quartic surface above ( as any quartic of the form ${ t^4- f(x,y,z)=0 }$ ) has many conics, as the pre-image of a bitangent line (there are $28$) is the union of two conics. On the other hand, the surface
$$
t^5 - x^4y - y^4 z - z^4 x =0
$$
seems to be a good candidate for a quintic without lines nor conics.

soft question - Mathematics as a hobby

I try to learn and understand as many facts as I can. Of course, many people would like to benefit from the opposite, that is, digging into a certain branch as deep as they can.

I try to do the opposite, which I see as my main advantage, as the opposite to professional mathematicians. This is because they have their own careers and has their professional criteria to fulfill (writing articles in journals, gaining citation points, etc.).

As an amateur I am not obliged to do so, and this is a great freedom. If you want to be creative, you may try to dig here and there, and probably you will be lucky to find certain problems which are not penetrated, or you may find just something interesting enough (for example, your own point of view on a well-known area, maybe you find a surprising connection and, even if it is well known, it is funny to discover it once more, etc.) to write it somewhere, maybe on a blog.

In summary: I read as much as I can, I learn as much as I can, and I ask as much as I can.

As regards to low-level entry (you need of course to be genius to discover it, but nothing more;-), an example is Feigenbaum's famous discovery about chaos, etc. As far as I know, he used only a programmable calculator to discover it. He was just inquisitive, nothing more, nothing less.

ac.commutative algebra - Can a quotient ring R/J ever be flat over R?

1) If A is arbitrary and I is an ideal of finite type such that A/*I* is a flat A-module, then V(I) is open and closed. In fact, A/*I* is a finitely presented *A*-algebra and thus Spec(A/I)->Spec(A) is a flat monomorphism of finite presentation, hence an étale monomorphism, i.e., an open immersion (cf. EGA IV 17.9.1).

2) If A is a noetherian ring then A/*I* is flat if and only if V(I) is open and closed (every ideal is of finite type).

3) If A is not noetherian but has a finite number of minimal prime ideals (i.e., the spectrum has a finite number of irreducible components), then it still holds that A/*I* is flat iff Spec(A/I)->Spec(A) is open and closed. Indeed, there is a result due to Lazard [Laz, Cor. 5.9] which states that the flatness of A/*I* implies that I is of finite type in this case.

4) If A has an infinite number of minimal prime ideals, then it can happen that a flat closed immersion is not open. For example, let A be an absolutely flat ring with an infinite number of points (e.g. let A be the product of an infinite number of fields). Then A is zero-dimensional and every local ring is a field. However, there are non-open points (otherwise Spec(A) would be discrete and hence not quasi-compact). The inclusion of any such non-open point is a closed non-open immersion which is flat.

The example in 4) is totally disconnected, but there is also a connected example:

5) There exists a connected affine scheme Spec(A), with an infinite number of irreducible components, and an ideal I such that A/*I* is flat but V(I) is not open. This follows from [Laz, 7.2 and 5.4].

[Laz] Disconnexités des spectres d'anneaux et des préschémas (Bull SMF 95, 1967)

Edit: Corrected proof of 1). An open closed immersion is not necessarily an open immersion! (e.g. X_red->X is a closed immersion which is open but not an open immersion.)

Edit: Raynaud-Gruson only shows that flat+finite type => finite presentation when the spectrum has a finite number of associated points. Lazard proves that it is enough that the spectrum has a finite number of irreducible components. Added example 5).

gr.group theory - Periodic Automorphism Towers

As a follow up to my previous answer, I have computed all the automorphism series for groups of order up to 24 within the limits of what I could do with GAP. I will share some of what I found, throughout I will use the notation $(n,d)$ for the group of order $n$ with GAP library id $d$. For large groups not contained in the library, I will use $<r,n>$ to denote a group with minimum generating set of size $r$ and of order $n$, if I know only a bound for $r$, I will use $<leq r,n>$ to denote that fact. We will say that a group $G$ stabilizes (at $k$) if $Aut^{(k-1)}(G) notsimeq Aut^{(k)}(G) simeq Aut^{(k+1)}(G)$. We will call a group $G$ stable if $G simeq Aut(G)$.

I have verified that all groups of order up to 24 stabilize except for the following groups:

$(16,5)$, $(16,6)$, $(16,7)$, $(16,8)$, $(16,9)$, $(16,10)$, $(16,11)$, $(24,4)$, $(24,6)$, $(24,7)$, $(24,9)$, and $(24,10)$.

Of note is that many of these groups have the same automorphism group, hence their series is identical from $k=1$ onwards. Specifically, $(16,5)$, $(16,6)$, $(16,8)$, $(24,9)$, and $(24,10)$ all have $(16,11)$ as their automorphism group. The groups $(16,7)$, $(16,9)$ have the same automorphism group and so do the groups $(24,4)$, $(24,6)$, and $(24,7)$.

The following is a list of the groups I know to be stable, it is complete only up to order 24, and I give a Structure Description of the ones of order up to 24:

$(1,1) simeq mathbb{Z}_1$, $(6,1) simeq S_3$, $(8,3) simeq D_8$, $(12,4) simeq D_{12}$, $(20,3) simeq mathbb{Z}_5 rtimes mathbb{Z}_4$, $(24,12) simeq S_4$.

These are still in the library: $(40,12)$, $(42,1)$, $(48,48)$, $(54,6)$, $(110,1)$, $(144,183)$, $(336,208)$, $(384,5678)$, $(432,734)$, $(1152,157849)$.

These are too big to be in the library: $<2,40320> simeq S_8$, $<4,442368> simeq Aut^{(6)}((16,3))$.

These results give me two (general) ideas on how to attack this problem: One, analyze those groups for which I haven't been able to determine stabilization to see if I can find anything they have in common and use it show the conjecture is false. Two, Analyze the stable groups to see what causes them to be stable and use that knowledge to (somehow) show that every group must eventually stabilize. Both will likely require a detailed analysis of how $Aut(G)$ arises from $G$, to see how $G$ controls the properties of $Aut(G)$.

Edit:
I now have a complete list of stable groups of order up to 511, their GAP structure descriptions already reveal some very interesting patterns:

$(1,1)simeqmathbb{Z}_{1}$

$(6,1)simeq S_{3}$

$(8,3)simeq D_{8}$

$(12,4)simeq D_{12}$

$(20,3)simeqmathbb{Z}_{5}rtimesmathbb{Z}_{4}$

$(24,12)simeq S_{4}$

$(40,12)simeqmathbb{Z}_{2}times(mathbb{Z}_{5}rtimesmathbb{Z}_{4})$

$(42,1)simeq(mathbb{Z}_{7}rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2}$

$(48,48)simeqmathbb{Z}_{2}times S_{4}$

$(54,6)simeq(mathbb{Z}_{9}rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2}$

$(84,7)simeqmathbb{Z}_{2}times((mathbb{Z}_{7}rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2})$

$(108,26)simeqmathbb{Z}_{2}times((mathbb{Z}_{9}rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2})$

$(110,1)simeq(mathbb{Z}_{11}rtimesmathbb{Z}_{5})rtimesmathbb{Z}_{2}$

$(120,34)simeq S_{5}$

$(120,36)simeq S_{3}times(mathbb{Z}_{5}rtimesmathbb{Z}_{4})$

$(144,182)simeq((mathbb{Z}_{3}timesmathbb{Z}_{3})rtimesmathbb{Z}_{8})rtimesmathbb{Z}_{2}$

$(144,183)simeq S_{3}times S_{4}$

$(156,7)simeq(mathbb{Z}_{13}rtimesmathbb{Z}_{4})rtimesmathbb{Z}_{3}$

$(168,43)simeq((mathbb{Z}_{2}timesmathbb{Z}_{2}timesmathbb{Z}_{2})rtimesmathbb{Z}_{7})rtimesmathbb{Z}_{3}$

$(216,90)simeq(((mathbb{Z}_{2}timesmathbb{Z}_{2})rtimesmathbb{Z}_{9})rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2}$

$(220,7)simeqmathbb{Z}_{2}times((mathbb{Z}_{11}rtimesmathbb{Z}_{5})rtimesmathbb{Z}_{2})$

$(240,189)simeqmathbb{Z}_{2}times S_{5}$

$(252,26)simeq S_{3}times(mathbb{Z}_{7}rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2}$

$(272,50)simeqmathbb{Z}_{17}rtimesmathbb{Z}_{16}$

$(312,45)simeqmathbb{Z}_{2}times(mathbb{Z}_{13}rtimesmathbb{Z}_{4})rtimesmathbb{Z}_{3}$

$(320,1635)simeq((mathbb{Z}_2timesmathbb{Z}_2timesmathbb{Z}_2timesmathbb{Z}_2)rtimesmathbb{Z}_5)rtimesmathbb{Z}_4$

$(324,118)simeq S_{3}times(mathbb{Z}_9rtimesmathbb{Z}_3)rtimesmathbb{Z}_2)$

$(336,208)simeq PSL(3,2)rtimesmathbb{Z}_2$

$(342,7)simeq (mathbb{Z}_{19}rtimesmathbb{Z}_{9})rtimesmathbb{Z}_2$

$(384,5677)simeq((((mathbb{Z}_{4}timesmathbb{Z}_{4})rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2})rtimesmathbb{Z}_{2})rtimesmathbb{Z}_{2}$

$(384,5678)simeq((((mathbb{Z}_{2}timesmathbb{Z}_{2}timesmathbb{Z}_{2}timesmathbb{Z}_{2})rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2})rtimesmathbb{Z}_{2})rtimesmathbb{Z}_{2}$

$(432,520)simeq(((mathbb{Z}_{3}timesmathbb{Z}_{3})rtimesmathbb{Z}_{3})rtimes Q_{8})rtimesmathbb{Z}_{2}$

$(432,523)simeq(((mathbb{Z}_{6}timesmathbb{Z}_{6})rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2})rtimesmathbb{Z}_{2}$

$(432,533)simeqmathbb{Z}_{2}times((((mathbb{Z}_{2}timesmathbb{Z}_{2})rtimesmathbb{Z}_{9})rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2})$

$(432,734)simeq(((mathbb{Z}_{3}timesmathbb{Z}_{3})rtimes Q_{8})rtimesmathbb{Z}_{3})rtimesmathbb{Z}_{2}$

$(480,1189)simeq(mathbb{Z}_{5}rtimesmathbb{Z}_{4})times S_{4}$

$(486,31)simeq(mathbb{Z}_{27}rtimesmathbb{Z}_{9})rtimesmathbb{Z}_{2}$

$(500,18)simeq(mathbb{Z}_{25}rtimesmathbb{Z}_{5})rtimesmathbb{Z}_{4}$

$(506,1)simeq(mathbb{Z}_{23}rtimesmathbb{Z}_{11})rtimesmathbb{Z}_{2}$

at.algebraic topology - Why are Delta-generated spaces locally presentable?

EDIT (12/18/15)
The below argument using countably-generated spaces achieves the estimate that $Delta$-generated spaces are locally $(2^{2^{aleph_0}})^+$-presentable. An improvement (probably optimal in light of Zhen Lin's comment) to local $(2^{aleph_0})^+$-presentability can be obtained by using sequential spaces instead of contably-generated ones, and an axiomatization of sequential spaces by Gutierres and Hoffman. The same ideas are applicable: by axiomatizing spaces in terms of convergence, it becomes easy to compute sufficiently-filtered colimits.

I'll keep the below argument here, though, because it easily generalizes to show that the $mathcal{A}$-generated spaces are locally presentable for any small full subcategory $mathcal{A} subset mathsf{Top}$.

---

Here's a sketch of a more direct proof that $Delta$-generated spaces (call this category $Delta-mathrm{Gen}$) are locally presentable. It's essentially a "compiling-out" of Fajstrup and Rosický's proof. The best estimate I'm able to extract for the accessibility rank is $(2^{2^{aleph_0}})^+$, though from Zhen Lin's comment one should probably expect the true accessibility rank to be $(2^{aleph_0})^+$.

First we show that the category $aleph_0-mathrm{Gen}$ of countably generated spaces -- those spaces which are to the countable topological spaces as $Delta$-generated spaces are to simplices -- is locally presentable. Since $Delta-mathrm{Gen}$ is a full subcategory of $aleph_0-mathrm{Gen}$ which is closed under colimits, with a dense generator given by the simplices, it is also locally presentable.

(That argument might sound like it requires Vopenka's principle, but it doesn't: it just uses the characterization of locally presentable categories as those cocomplete categories with a dense generator [this hypothesis can be weakened to: strong generator] of presentable objects. If $mathcal{K}$ is locally presentable then every object there is a cardinal $lambda$ such that the object is $lambda$-presentable [since for every object there is a $lambda$ such that it is a $lambda$-small colimit of canonical generators, and the $lambda$-presentable objects are closed under $lambda$-small colimits]. If $mathcal{L}$ is a full subcategory closed under colimits, then all the objects of $mathcal{L}$ are also presentable, so if $mathcal{L}$ has a dense generator, it is locally presentable. Explicitly in this case, the simplices are continuum-sized colimits of countable spaces, so they are presentable. If the countable spaces were $(2^{aleph_0})^+$-presentable, the simplices would be too; as it is though, I can only show that the countable spaces are $(2^{2^{aleph_0}})^+$-presentable, so that's the best estimate I have for the simplices, too.)

The reason for bringing $aleph_0-mathrm{Gen}$ into the picture is that in $aleph_0-mathrm{Gen}$, it's easy to describe the topology on a colimit $X = varinjlim X_i$ when the colimit is sufficiently filtered. Namely, $X$ has the topology where

a countably-supported ultrafilter $mathcal{F} in beta_omega X$
converges to a point $x in X$ if and only if "$mathcal{F}$ already
converges to $x$ at some stage of the colimit", i.e. iff there exists
an $X_i$ and an $x_i in X_i$ mapping to $x$ and a countably-supported
ultrafilter $mathcal{F}_i in beta_omega X_i$ which pushes forward
to $mathcal{F}$, such that $mathcal{F}_i$ converges to $x_i$ in
$X_i$.

Here we use the notion of ultrafilter convergence: an ultrafilter $mathcal{F} in beta X$ is said to converge to a point $x in X$ iff every neighborhood of $x$ is an element of $mathcal{F}$. A countably-supported ultrafilter $mathcal{F} in beta_omega X$ is just an ultrafilter which contains a countable subset of $X$.

It's obvious from this description of a sufficiently-filtered colimit that spaces of sufficiently small cardinality are presentable, because a function between (countably-generated) topological spaces is continuous iff it sends convergent (countably-supported) ultrafilters to convergent ultrafilters. Since the countable spaces are dense in $aleph_0-mathrm{Gen}$, it follows that $aleph_0-mathrm{Gen}$ is locally presentable.

The subtlety, of course, comes in verifying that this description of ultrafilter convergence in a sufficiently-filtered colimit actually arises from a topology (and that this topology is countably-generated). Barr showed that a relation $R subseteq beta X times X$ defined for all ultrafilters arises from a topological space if and only if $R$ is a lax algebra for the ultrafilter monad $beta$, giving a (concrete) equivalence of categories between topological spaces and lax $beta$-algebras. By replacing $R$ with a relation $R subseteq beta_omega X times X$ in this definition, we get a "lax-algebraic" description of $aleph_0-mathrm{Gen}$, which we can use to compute sufficiently-filtered colimits as above.

---

To be precise about the ultrafilter description of $aleph_0-mathrm{Gen}$, let me first review the ultrafilter description of general topological space. Consider a relation $ beta X overset{pi_1}{leftarrow} R overset{pi_2}{to} X$, and write $mathcal{F} rightsquigarrow x$ if $(mathcal{F},x) in R$, i.e. $R= {(mathcal{F},x) mid mathcal{F} rightsquigarrow x}$. Then $R$ is the convergence relation for a topology on $X$ if and only if the following conditions hold:

1. For every $x in X$, $mathrm{prin}(x) rightsquigarrow x$, where $mathrm{prin}(x)$ is the principal ultrafilter at $x$.




2. If $mathcal{G}$ is an ultrafilter on the set $R$ itself, and if $(pi_2)_*(mathcal{G}) rightsquigarrow x$, then $sum (pi_1)_*(mathcal{G}) rightsquigarrow x$.

Here $()_*$ is the pushforward of ultrafilters, $f_*(mathcal{F}) = {A mid f^{-1}(A) in mathcal{F}}$ and $sum: beta beta X to beta X$ is the sum of ultrafilters $sum mathcal{H} = {A mid hat{A} in mathcal{H}}$, where $hat{A} = {mathcal{F} mid A in mathcal{F}}$.

Analogously, consider a relation $ beta_omega X overset{pi_1}{leftarrow} R overset{pi_2}{to} X$, with the notation $mathcal{F} rightsquigarrow x$ as before. Then $R$ is the convergence relation (restricted to countably-supported ultrafilters) for a countably-generated topology on $X$ if and only if the following conditions hold:

1. For every $x in X$, $mathrm{prin}(x) rightsquigarrow x$, where $mathrm{prin}(x)$ is the principal ultrafilter at $x$.




2. If $mathcal{G}$ is an ultrafilter on the set $R$ itself, and if $(pi_2)_*(mathcal{G}) in beta_omega X$ and $(pi_2)_*(mathcal{G}) rightsquigarrow x$, and if $sum (pi_1)_*(mathcal{G}) in beta_omega X$, then $sum (pi_1)_*(mathcal{G}) rightsquigarrow x$.

Note that in (2), $mathcal{G}$ is not required to be countably supported, nor is $(pi_1)_*(mathcal{G})$. So a countably-generated space is not apparently the same thing as a lax algebra for the monad $beta_omega$ of countably-supported ultrafilters -- it needs to satisfy a stronger associativity condition (2) which refers back to the full ultrafilter monad $beta$. There ought to be some general 2-categorical or equipment-theoretic description of the relationship between these two monads and of this sort of "hybrid" lax algebra for them, but I haven't worked out what it should be.

I doubt that $Delta-mathrm{Gen}$ can be described directly in terms of a submonad of the ultrafilter monad $beta$ -- this is the reason for bringing countably-generated spaces into the story.

career - How does one handle two-body job searches?

It's important to know that if you want to discuss this with the chair of a department where you're interviewing, or have an offer, you will have to bring it up yourself; it's illegal (at least in the U.S., or at least in the states I'm familiar with) for a university to ask you about your marital status. You'll find that most departments are eager to help to the extent they can, and that there's wide variation in what that extent is.

Update: Based on the comments below, let me add that some chairs don't know what the law is, or don't care; so it's clearly not universally true that you won't be asked about your spouse.

Maybe I'll also add that, if I remember right, I never brought up my spouse until I had an offer. The interview stage is when you're trying to convince them that you should work there; the post-offer stage is the reverse. The problem with bringing up spousal issues at the interview stage is that the department may start thinking "We're never going to find a job for her husband in post-Inca ethnography, so maybe we should make an offer to somebody we're more likely to get." I think this would be somewhat unethical, but then again so is illegally asking you about your marital status, and it's sometimes done.

So the difficult question: what to do if you don't want to talk about your two-body problem at the interview, and you're asked about it? On general "keep the tone light" grounds I would advise against saying "You are breaking the law." If you said, "I'd rather keep this just about me for the moment," I would be fine with it; but for the sake of honesty I should say I'd expect some chairs would find it weird.

Perhaps the best thing would be to concede that you have a spouse, and to say something about what field they're in, but to play down any sense of a two-body "problem." If the spouse is an academic (which I think is the situation in the posted question), I think it is totally OK to create the impression that a TT job for your spouse would be a big draw, but not a necessity. This allows the math department to get the ball rolling on an attempt to do a double hire, but reduces the risk that the department will find out there's no job for your spouse and give up on you, too.

Note that the advice above only applies if you're (illegally) asked about your spouse in the interview. If you choose to bring it up yourself, I think you should be totally upfront about what will be required to hire you. If you are deadset against considering anything other than a double TT offer (e.g. if you are already situated in a place where you have one TT and one non-TT offer) then you should go ahead and bring this up to avoid wasting anyone's time.

ag.algebraic geometry - How many people fully understand the proof of Fermat's Last Theorem?

Dear Michael,

The methods introduced by Wiles, and by Taylor and Wiles, in the two papers that proved FLT, as well as the methods introduced by Ribet in his earlier paper reducing FLT to Shimura--Taniyama, are at the heart of much modern work in algebraic number theory and automorphic forms, such as (in addition to the proofs of Shimura--Taniyama and FLT) the proofs of Serre's conjectures and the Sato--Tate conjecture.

Conferences/workshops in these fields typically attract on the order of magnitude of 100 or so particants, which gives you some sense of the number of students/researchers thinking about these questions: its in the tens or hundreds, but probably not in the thousands. Of course, not all these people know all the details, but the people at the top of the field surely do. (Of course, there is a question of what "understand" means exactly. I don't know how many people have both carefully studied all the details of the trace formula arguments that underly Jacquet--Langlands, Langlands--Tunnell, and base-change, and also carefully studied the details of the p-adic Hodge theory and other arithmetic geometry that is used in the arguments. But certainly the top people do understand the significance of these techinques, and are fluent in their use and application, and understand both the overall structure and strategy, as well the technical details, of the proof of FLT itself (and of various more recent related results).

Finally, let me note that the best evidence for the final claim of the previous paragraph is that this is currently an extremely vibrant area of research, which has progressed at a rapid clip over the last ten years or so. (The reason for this being that people have not only assimilated the arguments of Wiles/Taylor--Wiles but have improved upon them and pushed them further.)

examples - Do good math jokes exist?

Kurd Lasswitz, mathematician, writer, inventor of science fiction in Germany, wrote this "nth part of Faust" for the Breslau Mathematical Society 1882:

"Personen:
Prost, Stud. math. in höheren Semestern, steht vor dem Staats-Examen,
Mephisto, Dx (sprich De-ix), Differentialgeisterkönig, ein Fuchs.
Ort Breslau. Zeit: Nach dem Abendessen. (Rechts ein Sofa, auf dem Tische zwischen allerlei Büchern ein Bierseidel und Bierflaschen, links eine Tafel auf einem Gestell, Kreide und Schwamm. Auf der Tafel ist eine die gesamt Fläche einnehmende ungeheuerliche Differentialgleichung aufgeschrieben).

Prost am Tische, mit den Büchern beschäftigt. Er stärkt sich.

Prost

Habe nun, ach, Geometrie, Analysis und Algebra
und leider auch Zahlentheorie studiert,
und wie, das weiß man ja!
Da steh' ich nun als Kandidat
und finde zur Arbeit keinen Rat.
Ließe mich gern Herr Doktor lästern;
zieh' ich doch schon seit zwölf Semestern
herauf, herab und quer und krumm
meine Zeichen auf dem Papiere herum,
und seh', daß wir nichts integrieren können.
Es ist wahrhaftig zum Kopfeinrennen.

Zwar bin ich nicht so hirnverbrannt,
daß ich mich quälte als Pedant,
wenn ich 'ne Reihe potenziere,
zu seh'n, ob sie auch konvergiere,
... "

linear algebra - An "existence contra partition of unity" statement for integer matrices?

I believe that what you say is true. I'll sketch an argument.

Let f:Zⁿ ---> Z²ⁿ be the map of free Z-modules given by the matrices B₁, B₂ put in column (i.e. the direct sum of the morphisms given by B₁ and B₂). Now we rephrase conditions (1) and (2) in a slightly more abstract way:

• (1) fails to hold if, and only if there exists p:Z²ⁿ ---> Zⁿ such that, together with f, fit in a short exact sequence

  
  
  

  0 ---> Zⁿ ---> Z²ⁿ ---> Zⁿ ---> 0 (*)

Indeed, the failure of (1) means that any v in Qⁿ such f(v) in Z²ⁿ must be integral (i.e. v in Zⁿ). In particular, this implies that f is injective. Moreover, take w in Z²ⁿ representing a nonzero torsion element in the cokernel of f. As w represents a torsion element, Nw belongs to the image of f for some big enough positive integer N, so there is v in Zⁿ such that f(v) = Nw. But now f(1/N v) = w, and this means, by the failure of (1), that 1/N v is integral, so w is in the image of f and the cokernel of f has no torsion. As a finitely generated torsion-free Z-module is free, we get an exact sequence like (*) above. This argument can easily be reversed, to show the equivalence between the existence of this exact sequence and the failure of (1).

• (2) holds if, and only if there exists a morphism of Z-modules r:Z²ⁿ ----> Zⁿ such that rf = id.

Let r be represented by a matrix (A₁,A₂). Then gf has matrix A₁B₁ + A₂B₂, and gf = id if, and only if (2) holds.

Now, the proof of what you asked for is easy. (1) fails if, and only if we can form the exact sequence (*), but such an exact sequence is always split because Z^n is projective, so we can form such exact sequence if, and only if there exists a splitting r:Z²ⁿ ----> Zⁿ, which is exactly condition (2).

at.algebraic topology - Analogue to covering space for higher homotopy groups?

Just like there is a universal cover of every space, there is a natural $n$-connected space $X_n$ that maps to any space $X$. To construct this space, you can add cells of dimension $n+2$ and higher to $X$ to get a space $Y$ together with a map $X to Y$ which is an isomorphism on $pi_i$ for $i leq n$, but such that $pi_i(Y)=0$ for $i>n$. The homotopy fiber $X_n to X$ of this map is then the "$n$-connected cover" of $X$; $X_n$ is $n$-connected but has the same homotopy groups as $X$ above $n$, as can easily be seen from the long exact sequence of the fibration. Details of this, as well as a proof of uniqueness of the $n$-connected cover, are in Hatcher starting on page 410.

More generally, if you started with an $(n-1)$-connected space, you could both kill the homotopy groups of $X$ above $n$ and kill a subgroup of $pi_n(X)$, and then the homotopy fiber would be an "$n$-cover" of $X$ corresponding to that subgroup of $pi_n(X)$.

arithmetic geometry - Galois representation attached to elliptic curves

Since your representation $overline{rho}$ is defined over $mathbb F_p$, you can't do things like the Hasse bounds, since
the traces $a_{ell}$ of Frobenius elements at unramified primes are just integers mod $p$,
and so don't have a well-defined absolute value.

One thing you can do is check the determinant; this should be the mod $p$ cyclotomic character if $overline{rho}$ is to come from an elliptic curve. In general (or more precisely, if $p$ is at least 7), that condition is not sufficient (although it is sufficient if $p = 2,3$ or 5);
see the various results discussed in this paper of Frank Calegari,
for example. In particular, the proof of Theorem 3.3 in that paper should
give you a feel for what can happen in the mod $p$ Galois representation attached to
weight 2 modular forms that are not defined over $mathbb Q$, while the proof of Theorem 3.4
should give you a sense of the ramification constraints on a mod $p$ representation imposed
by coming from an elliptic curve.

enumerative combinatorics - Convex Polyhedra

Dear Ali,

Well there are various tools which are useful to the study of convex polytopes. The following list is perhaps not complete and it certainly should not be frightening. (I dont know very well various of these tools.)

1) Basic tools of linear algebra and convexity.

The notions of supporting hyperplanes, seperation theorems, Caratheodory, Helly and Radon theorem etc.

2) Combinatorics

Some of the study of convex polytopes translates geometric questions to purely combinatorial questions. So familiarity with combinatorial notions and techniques is useful.

3) Graph theory

As Joe mentioned the study of polytopes in 3 dimensions is closely related to the study of planar graphs. There are few other connections to graph theory so it is useful to be familiar with some graph theory.

4) Gale duality

The notion of Gale duality is a linear-algebra concept which privides an important technique in the study of convex polytopes.

5) Some basic algebraic topology

Euler's theorem and its higher dimensional analogues is of central imoprtance and this theorem is closely related to algebraic topology. Another example: there is a result by Perles that the [d/2]-dimensional skeleton of a simplicial polytope determines the entire combinatorial structure. (See this paper by Jerome Dancis.) The proof is based on an elementary topological argument. Borsuk-Ulam theorem also has various nice applications for the study of polytopes.

6) Some functional analysis

There is a result by Figiel, Lindenstrauss, and Milman that a centrally symmetric convex polytope in d dimension satisfies $$log f_0(P) cdot log f_{d-1}(P) ge gamma d$$ for some absolute positive constant $gamma$. The proof is based on a certain variation of Dvoretzky theorem and I am not aware of an alternative approach.

7) Some commutative algebra

Several notions and results from commutative algebra plays a role in the study of convex polytopes and related objects. Especially important is the notion of Cohen Macaulay rings and results about these rings.

8) Toric varieties

Understanding the topology of certain verieties called "toric varieties" turned out to be quite important for the study of convex polytopes.

All these items refer to general polytopes. There is also a (related) reach study of polytopes arising in combinatorial optimization. Here is a link to a paper entitled "Polyhedral combinatorics an annotated bibliography" by Karen Aardal and Robert Weismantel.

references

Here are some relevant references: Ziegler's book: Lectures on Convex Polytopes, and the second edition of Grunbaum's book "Convex polytopes" will give a very nice introduction to topics 1) - 4). The connection with commutative algebra and some comments and references to the connection with toric varieties (topics 7 and 8) can be found in (chapters 2 and 3 of) the second edition of Stanley's book "Combinatorics and Commutative Algebra". Relations with algebraic topology and with functional analysis can be found in various papers. This wikipedia article can also be useful.

pr.probability - Shuffling decks of cards where not all cards are distinguishable

Suppose a deck of cards consists of $a_1+a_2+cdots+a_k$ cards of $k$ types, where there are $a_i$ indistinguishable cards of each type. How many shuffles does it take, on average, to randomize the deck? For example, $a_i=4$ and $k=13$ gives a standard deck of playing cards; $a_i=4$ for $1le ile9$, $i_{10}=16$, $k=10$ gives the cards for blackjack.

In general I would expect that this would be an easier task than shuffling a deck where all cards are indistinguishable. A standard deck has about 226 bits of entropy, while the same deck without suits has only 166 bits of entropy.

---

Consider a standard deck of 52 playing cards. Bayer & Diaconis (famously) showed that, under a certain model, it takes 7 riffle shuffles to sufficiently randomize the deck. Salem applies a different methodology and gets 6 idealized riffle shuffles. Mann uses a much stricter measurement and determines 11.7 as the expected stopping time for the riffle unshuffle (and thus the riffle shuffle) to randomize the deck.

In particular, Mann gives a formula:
$$E(mathbf{T})=sum_{k=0}^inftyleft[1-{2^kchoose n}frac{n!}{2^{nk}}right]$$
which lends itself to generalization nicely.

I'm partial to Mann's method, but my question applies broadly.

---

[1] David Aldous and Persi Diaconis, "Shuffling cards and stopping times", The American Mathematical Monthly 93:5 (1986), pp. 333-348.

[2] Dave Bayer and Persi Diaconis, "Trailing the dovetail shuffle to its lair", The Annals of Applied Probability 2:2 (1992), pp. 294-313. JSTOR: 2959752

[3] Brad Mann, How many times should you shuffle a deck of cards?

[4] Michael P. Salem, How many shuffles are necessary to randomize a standard deck of cards?

at.algebraic topology - Are fundamental groups of aspherical manifolds Hopfian?

A group $G$ is Hopfian if every epimorphism $Gto G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth manifolds Hopfian?

Perhaps the manifold structure is irrelevant and makes examples harder to construct, so here is another variant that may be more sensible. Let $X$ be a finite CW-complex which is $K(pi,1)$. If it helps, assume that its top homology is nontrivial. Is $pi=pi_1(X)$ Hopfian?

Motivation. Long ago I proved a theorem which is completely useless but sounds very nice: if a manifold $M$ has certain homotopy property, then the Riemannian volume, as a function of a Riemannian metric on $M$, is lower semi-continuous in the Gromov-Hausdorff topology. (And before you laugh at this conclusion, let me mention that it fails for $M=S^3$.)

The required homotopy property is the following: every continuous map $f:Mto M$ which induces an epimorphism of the fundamental group has nonzero (geometric) degree.

This does not sound that nice, and I tried to prove that some known classes of manifolds satisfy it. My best hope was that all essential (as in Gromov's "Filling Riemannian manifolds") manifolds do. I could not neither prove nor disprove this and the best approximation was that having a nonzero-degree map $Mto T^n$ or $Mto RP^n$ is sufficient. I never returned to the problem again but it is still interesting to me.

An affirmative answer to the title question would solve the problem for aspherical manifolds. A negative one would not, and in this case the next question may help (although it is probably stupid because I know nothing about the area):

Question 2. Let $G$ be a finitely presented group and $f:Gto G$ an epimorphism. It it true that $f$ induces epimorphism in (co)homology (over $mathbb Z$, $mathbb Q$ or $mathbb Z/2$)?

ac.commutative algebra - An example where GCD depends on the domain

First some notation. Given a domain $R$ and $x,a,b in R$, I write $x=gcd(a,b)_R$ to mean that $x$ is one gcd of $a$ and $b$ in $R$.

I want to find an example of an GCD-domain $R$, a subdomain $S subseteq R$, and two elements $a, b in S$ such that there isn't any $x in S$ such that $x=gcd(a,b)_R$ and $x=gcd(a,b)_S$. Notice that it is not enough to find one element $x in S$ such that $x=gcd(a,b)_R$ but $x neq gcd(a,b)_S$.

I can prove that this is impossible in as little as a Bezout domain, but I cannot prove that this is impossible in a mere GCD-domain. I do not know that many examples of GCD-domains which are not Bezout domains in the first place.

ETA: As suggested below, I also wanted $S$ to be a GCD-domain.

graph theory - Can one make Erdős's Ramsey lower bound explicit?

As was mentioned in the previous answers, the answer is no. Or more accurately I'd say that the answer is currently no, but possibly yes.

Also, consider the related question of constructing a bipartite graph with parts of size $2^n$, which contains no $K_{k,k}$ and whose complement contains no $K_{k,k}$ where $k = O(n)$. Such an explicit construction will have as far as I can tell huge impact on derandomization of randomized algorithms, among other topics in theoretical computer science. See e.g. this paper, where such an explicit construction is given for $k = 2^{n^{o(1)}}$.

You might also be interested in the following accompanying paper (seems like I cannot post it, being a new user; you can google it though, its title is "Pseudorandomness and Combinatorial Constructions") to Luca Trvisan's talk at ICM '06. This may contain more connections between explicit constructions of combinatorial objects and applications in theoretical computer science.

measure theory - Are there sigma-algebras of cardinality $kappa>2^{aleph_0}$ with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $sigma$-algebras. The only proof that I know is via a contradiction argument which yields no estimate on the minimum cardinality of an infinite $sigma$-algebra.

Given an a set $X$ of infinite cardinality $kappa$, the $sigma$-algebra of all co-countable subsets of $X$ is of cardinality $2^kappa$ $kappa^{aleph_0}$. This example doesn't tell me whether there are $sigma$-algebras of cardinality below $2^{aleph_0}$, if I don't assume the Continuum Hypothesis.

My question is as the title says: Are there $sigma$-algebras of every uncountable cardinality?

Edit: The combined answer with Stephen, Matthew proves that the cardinality of a $sigma$-algebra is necessarily at least $2^{aleph_0}$. Further, for each cardinality $kappage 2^{aleph_0}$ with uncountable cofinality, the $sigma$-algebra of countable (or cocountable) subsets of a set $X$ with cardinality $kappa$, is of cardinality $kappa$.

What is left is whether for $kappage 2^{aleph_0}$ with $cf(kappa)=aleph_0$ are there $sigma$-algebras of cardinality $kappa$. (I changed the title to reflect this.)

Thanks Stephen, Matthew, Apollo, for the combined work!

tiling - Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?

The Gosper islands are a fundamental domain for the translation action of the Eisenstein integers $mathbb{Z}[frac{1+sqrt{-3}}{2}]$, since the shapes can be constructed by deforming a Voronoi decomposition of the plane with respect to that lattice. It is therefore reasonable to look for functions in the field generated by the Weierstrass $wp$ function for the Eisenstein integers and its first derivative $wp'$, since this field comprises all of the meromorphic functions that are periodic with respect to the lattice.

It is not clear what selection rule you want to apply to favor one function over another. Elliptic functions do not have canonical fundamental domains, and one has to choose extra data (e.g., a basis of the lattice, and a pair of paths in the homotopy class representing the basis) to write down boundaries in the usual theory.

I suppose you may want to find a function $f(z)$ such that the boundary configuration is equal to ${ z : |f(z)| = 1 }$, but I am somewhat doubtful that such a function exists, simply by degree considerations.

rt.representation theory - Orbits of a symplectic group on its Lie algebra in the finite field case

It won't tell you everything about the orbits, but the decomposition of $mathfrak{sp}_{2n}textbf{F}_p$ as an $text{Sp}_{2n}textbf{F}_p$-representation is known. This can be found in Hogeweij, "Almost-classical Lie algebras. I." Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 441-452, but it is hard to extract the answer from that paper, so I'll briefly give the argument.

If $p$ is odd, then $mathfrak{sp}_{2n}textbf{F}_p$ is irreducible with highest weight $2omega_1$, where $omega_1,ldots,omega_n$ are the fundamental weights for $text{Sp}_{2n}textbf{F}_p$.

If $p=2$, we proceed as follows. Let $Happrox textbf{F}_p^{2n}$ be the standard representation of $text{Sp}_{2n}textbf{F}_p$. Note that as a subspace of $mathfrak{gl}_{2n}textbf{F}_papprox H^*otimes H$, the condition defining $mathfrak{sp}_{2n}textbf{F}_p$ describes exactly the subspace $text{Sym}^2 H$ inside $Hotimes Happrox H^*otimes H$, so we are looking at orbits of $text{Sp}_{2n}textbf{F}_p$ on $text{Sym}^2 H$.

In characteristic 2 we have an embedding of $H$ into $text{Sym}^2 H$ by $xmapsto xcdot x$, which is linear since $(x+y)^2 = x^2+y^2$ (in general this twists by Frobenius but we are over $textbf{F}_2$). Since $xcdot y=ycdot x=-ycdot x$, the quotient $text{Sym}^2 H/H$ is isomorphic to $bigwedge^2 H$. Now $bigwedge^2 H$ has two invariant subrepresentations. One is trivial, spanned by the vector $omega=a_1wedge b_1+cdots+a_nwedge b_n$. The other is the kernel $K$ of the contraction $bigwedge^2 Hto textbf{F}_2$, defined by $a_iwedge b_imapsto 1$, $a_iwedge a_jmapsto 0$, $b_iwedge b_jmapsto 0$, and $a_iwedge b_jmapsto 0$. Note that under this contraction $omega$ is taken to $nin textbf{F}_2$; thus $omega$ is contained in $K$ iff $n$ is even. Finally, $K$ is irreducible when $n$ is odd, and $K/langleomegarangle$ is irreducible when $n$ is even.

If I'm not mistaken, this means the invariant subrepresentations are thus just $H$, $langleomegarangle$, $Hoplus langleomegarangle$, and $H+K$ (the kernel of the contraction $text{Sym}^2 Hto textbf{F}_2$).

ag.algebraic geometry - Does reduced+Noetherian space imply Noetherian scheme

In the special case of affine schemes, there is an exercise on Hartshorne saying that when Spec A is a Noetherian topological space A may not be a Noetherian ring. While it is easy to find an example for that when A has nilpotent elements, e.g. $A=k[x_1,...,x_i,...]/(X_1, x_2^2,...,x_i^i,...).$ It is not clear to me that whether we could still find a counter example when A is a domain. In the general case, I am asking for a reduced scheme, if the underlying topological space is Noetherian, is the scheme necessarily a Noetherian scheme?

My guess would be no, consider the direct limit of the series of localizations,

$$underline{lim} k[x^{1over 2^n}]_{(x^{1over 2^n})}$$

each process within the limit is a one dimension scheme, and I think the limit is also a one dimension scheme. (In general, will dimension necessarily be held constant, non increasing or non decreasing in a limit process? or none of the above?). However, it is not hard to see the limit ring is not a Noetherian ring.

mathematics education - Theorems in Euclidean geometry with attractive proofs using more advanced methods

A nice example is Pascal's theorem for the circle:

If a hexagon is inscribed in a circle then the intersections of opposite
sides are collinear.

Plücker gave an elegant proof of Pascal's theorem as a consequence of
Bézout's theorem. View the unions of alternate sides of the hexagon as
cubic curves

$l_{135}=0$ and $l_{246}=0$.

They meet in 9 points, 6 of which are the vertices on the circle $c=0$. But we
can choose constants $a,b$ so that the cubic

$al_{135}+bl_{246}=0$

passes through any point. Taking this point on the circle, the circle and the cubic have at least 7 points in common. By Bézout's theorem, the curves have a common component, necessarily the circle $c=0$, since $c$ is irreducible.

Hence $al_{135}+bl_{246}=cp$, for some polynomial $p$, which must be
linear. Since $al_{135}+bl_{246}=0$ contains all 9 points common to
$l_{135}=0$ and $l_{246}=0$, while $c=0$ contains only 6, the remaining 3
(intersections of opposite sides of the hexagon) must lie on the line $p=0$.

gr.group theory - Does $mathrm{Aut}(mathrm{Aut}(...mathrm{Aut}(G)...))$ stabilize?

I don't know about non-stabilizing, but rigidity provides many examples that stabilize quickly.

1) Let π be the fundamental group of a finite volume hyperbolic manifold M of dimension ≥ 3 with no symmetries (that is, no nontrivial self-isometries). Negative curvature implies that π is centerless, so the map π -> Aut(π) is injective. Mostow-Prasad rigidity says that Out(π) = Isom(M), so the lack of isometries implies that Out(π) is trivial and Aut(π) = π. [This works verbatim for lattices in higher-rank semi-simple Lie groups subject to appropriate conditions.]

2) Let π=F_d be a free group of rank 2≤d<∞. Then Aut(F_n) is a much larger group; however, Dyer-Formanek showed that Out(Aut(F_n)) is trivial. Thus since Aut(F_n) is clearly centerless, we have Aut(Aut(F_n)) = Aut(F_n).

3) Interpolating between these two examples, if π=π₁(S_g) is the fundamental group of a surface of genus g≥2, then Aut(π) is the so-called "punctured mapping class group" Mod_g,*, which is much bigger than π. Ivanov proved that Out(Mod_g,*) is trivial, and since Mod_g,* is again centerless, we have Aut(Aut(π₁(S_g))) = Aut(π₁(S_g)).

In each of these cases, rigidity in fact gives stronger statements: Let H and H' be finite index subgroups of G = Aut(F_n) or Mod_g,*. (This class of groups can be widened enormously, these are just some examples.) Then any isomorphism from H to H' comes from conjugation by an element of G, by Farb-Handel and Ivanov respectively. In particular, Aut(H) is the normalizer of H in G. Rigidity gives the same conclusion for H = π₁(M) as in the first example and G = Isom(Hⁿ) [which is roughly SO(n,1)]. It seems that by carefully controlling the normalizers, you could use this to construct examples that stabilize only after n steps, for arbitrary large n.

---

Edit: I find the examples of D₈ and D_∞ unsatisfying because even though Inn(D) is a proper subgroup of Aut(D), we still have Aut(D) isomorphic to D. Here is a general recipe for building similarly liminal examples. Let G be an infinite group with no 2-torsion so that Aut(G) = G and H¹(G;Z/2Z) = Z/2Z. (Edited: For example, by rigidity, any hyperbolic knot complement with no isometries has these properties; by Thurston, most knot complements are hyperbolic.) The condition on the 2-torsion implies that for any automorphism G x Z/2Z -> G x Z/2Z, the composition

G -> G x Z/2Z -> G x Z/2Z -> G

is an isomorphism. From this we see that Aut(G x Z/2Z) / G = H¹(G;Z/2Z) = Z/2Z. By examination the extension is trivial, and thus Aut(G x Z/2Z) = G x Z/2Z. However, the image Inn(G x Z/2Z) is the proper subgroup G.

Comments: looking back, this feels very close to your original example of R x Z/2Z. Interesting that it's (seemingly) much harder to find group-theoretic conditions to force the behavior the way you want, while topologically it's easy.

Also, if you instead take G with H¹(G;Z/2Z) having larger dimension, say H¹(G;Z/2Z) = (Z/2Z)², this blows up quickly. You get Aut(G x Z/2Z) = G x (Z/2Z)², but then Aut(Aut(G x Z/2Z)) is the semidirect product of H¹(G;Z/2Z²) = (Z/2Z)⁴ with Aut(G) x Aut(Z/2Z²) = G x GL(2,2). Already the next step seems very hard to figure out. However, if you had enough control over the finite quotients of G, perhaps you could show that the linear parts of these groups don't get "entangled" with the rest, so that the automorphism groups would act like a product of G x (Z/2Z)ⁿ with something else, with n going to infinity. If so, this could yield an example where the isomorphism types of the groups never stabilize.

Does there exist a sequence of groups whose representation theory is described by plane partitions?

Not if you want the direct analogue of the branching rule to hold: namely, if V is the representation of G_n corresponding to a plane partition A of n, then the restriction of V to G_n-1 is the direct sum of one copy of the representation corresponding to each plane partition of n-1 contained in A. That would allow you to compute the dimension of the representation corresponding to A as the number of paths in the containment poset of plane partitions from the empty partition to A. Some computation then shows that the order of G₃ would be 1+4+4+1+4+1=15, but there's only one group of order 15, the abelian one, which doesn't work.

You could imagine some variations of the branching rule, though, such as "if B is obtained from A by replacing k by k-1 then the irrep corresponding to A contains k copies of the irrep corresponding to B", and maybe something like that would work.

metamathematics - Complete mathematics

You probably intended to restrict the question to effectively axiomatizable theories. Otherwise, for example, the first-order theory of the standard model of arithmetic is a complete theory, as is the theory of the standard model of ZFC.

Gödel's incompleteness theorem establishes some limitations on which effective theories can be complete. It shows that no effective, complete, consistent theory can interpret even weak theories of arithmetic such as Robinson arithmetic. However, there are many mathematically interesting theories that do not interpret the natural numbers.

Examples of complete, consistent, effectively axiomatizable theories include:

• For any prime $p$, the theory of algebraically closed fields of characteristic $p$


• The theory of real closed ordered fields, mentioned by Ricky Demer


• The theory of dense linear orderings without endpoints


• Many axiomatizations of Euclidean geometry

soft question - Value of "of course" in the mathematical literature

Hello,

I agree with some of the comments above: "of course" is useful to point out that some step is trivial (e.g. direct consequence of the definition), as opposed to the rest of non-trivial parts of the proof. Sometimes, "of course" is useful just as an stylistic resource in the writing, to introduce and connect a sentence to the previous one. But it can be very frustrating for the reader if this step is non-trivial, even though the author claims it is.

I was curious about this question and decided to find some examples in the "mathematical literature", as the original poster suggested. I looked through "A Course in Arithmetic", by J-P. Serre (which many consider a very good writer of mathematics) and the expression "of course" appears exactly twice. In both cases, "of course" appears in a parenthetical remark:

1) (p.35) Corollary. - For two nondegenerate quadratic forms over $mathbb{F}_q$ to be equivalent it is necessary and sufficient that they have same rank and same discriminant.
(Of course the discriminant is viewed as an element of the quotient group $mathbb{F}_q^ast/mathbb{F}_q^{ast 2}$.)

2) (p.73) Let $A$ be a subset of $P$ [$P$ is the set of prime numbers]. One says that $A$ has for density a real number $k$ when the ratio
$$ left(sum_{pin A}frac{1}{p^s} right)/ left(log frac{1}{s-1}right)$$
tends to $k$ when $sto 1$. (Of course, one then has $0 leq k leq 1$.)

In example (1), the way the corollary is stated, a remark is needed - but (i) it is clear from the context that this is what the author means, and (ii) it is typical in this context to consider discriminants only up to squares. Here I see this "of course" as a reminder of (ii).

Example (2) is trickier, as it is not immediately obvious that the limit of the expression as $sto 1$ is between $0$ and $1$. But I do not interpret this "of course" as a "clearly" in this case, but rather a sort-of "do not worry, if you go back and check Cor 2 in p. 70, you can convince yourself that $0leq k leq 1$, and it makes sense to call this number a density".

Álvaro

PS: In "A Course in Arithmetic", the word "clearly" appears many times, while "obviously" was never used in the entire book.

mp.mathematical physics - The bosonic and fermionic parts of the orthosymplectic super Lie-Algebra

The way I would understand it that $osp(6,2|4)$ is the group of linear tansformations
of a real super vector space with a non-degenerate symmetric inner product. The even (bosonic) vector space has dimension 8 and the inner product is symmetric with signature
$(6,2)$ the odd (fermionic) part has dimension 4 and a symplectic form.

The even part is then the product of the groups of these two vector spaces, namely $o(6,2)$ and $sp(4)$. There is an isomophism of rank two Lie algebras $sp(4)cong so(5)$; to see this note that the spin representation has dimension 4 and has an invariant symplectic form.

I realise you have $so(5,1)$ where I have $so(6,2)$. I don't know what is going on here but $so(6,2)$ is the group of conformal transformations of $R^{5,1}$.

soft question - Isn't a graph to be considered isomorphic to its complement, actually?

It is a strange question, but maybe a useful answer can make it a bit better.

Certainly for many purposes a graph will look totally different from its complement. For instance, a graph and its complement have completely different spectra, diameter, perfect matchings, etc. So that side of the question is kind-of lame, I agree.

On the other hand, for some purposes a graph is much the same as its complement. One obvious case is when you are interested in the automorphism group of a graph, or in the computational problem of graph isomorphism. Then you might as well think of a graph as a bicoloring of the edges of a complete graph. It then has a natural extra automorphism given by switching the two colors. More generally a colored graph with $n$ colors is equivalent, for graph isomorphism and graph automorphism questions, to a complete graph with $n+1$ colors. This viewpoint is more useful than you might first think, because a natural partial algorithm and preparatory step in the graph isomorphism and automorphism problems is to recolor every vertex by its valence, then color every edge by the colors of its vertices, etc., until the recoloring process stabilizes. Many graphs can be completely identified this way in practice. Recognizing the equivalence between a graph and its complement makes it easier to understand what these algorithms are really doing.

Even some specific graphs, such as the Higman-Sims graph, are mainly used for their automorphisms and similar purposes, and it might be better to think of them as colorings of a complete graph.

A much deeper example is the perfect graph theorem of Lovasz. The theorem is that a graph is perfect if only if its complement is perfect. For perfect graphs, taking the graph complement is closely related to the dual or polar polytope of a convex polytope.

linear algebra - Connected subset of matrices ?

The answer is always yes. Indeed the set is path-connected.

Let $C(f)$ denote the companion matrix associated to the monic
polynomial $f$. Every matrix $A$ is similar to a matrix in rational
canonical form:
$$B=C(f_1)oplus C(f_1 f_2)opluscdotsoplus C(f_1 f_2,cdots f_k)$$
where here $oplus$ denotes diagonal sum. Then $m$ is the degree of
$f_1 f_2cdots f_k$. Starting with $B$ deform each $f_i$ into a power
of $x$. We get a path from $B$ to
$$B'=C(x^{a_1})oplus C(x^{a_2})opluscdotsoplus C(x^{a_k})$$
inside $E_m$. There's a path from $B'$ in $E_m$ given by
$$(1-t)C(x^{a_1})oplus (1-t)C(x^{a_2})opluscdotsoplus C(x^{a_k})$$
ending at
$$B_m=Ooplus C(x^m).$$
Thus there is a path in $E_m$ from $A$ to $UB_mU^{-1}$
where $U$ is a nonsingular matrix. If $det(U)ne0$ then there is
a path in $GL_n(mathbf{R})$ from $U$ to $I$ and so a path in $E_m$
from $A$ to $B_m$. If $m< n$ then there is a matrix $V$ of negative determinant
with $VB_m V^{-1}=B_m$ so that we may take $U$ to have positive determinant.

The only case that remains is when $m=n$. In this case $E_m$ contains
diagonal matrices with distinct entries, and each of these commutes
with a matrix of negative determinant.

lo.logic - What is the reverse mathematics of first-order logic and propositional logic?

First of all, there is a difference between 'strength' and 'expressiveness'. This is not always unambiguous used in articles and literature.

With 'strength' is usually meant the possibility of a system to proof certain sentences. When comparing two systems for strength, one can limit the comparison to a certain subset of sentences.

Considering your question, I do think that you mean 'expressiveness'.

If you have first order logic + induction scheme + definitions for addition and multiplication, you can express any problem of discrete mathematics.

You need to build a 'pairing' construction and a 'transitive reflexive closure' construction. With those two, you can do anything. With some tricks with addition and multiplication, you can construct both.

You can also opt to start with a pairing operator and closure, and construct addition and multiplication from that.

If you have higher order logic, you don't need the addition and multiplication, because higher order logic allows the construction of pairing and closure in a different way.

So, I think the answer to your question is that you can codify all logics (with finite sentences) with First Order PA (or the question is not clear).

When talking about strength. First order PA + addition + multiplication, can't prove the consistency of itself. You need second order logic, with the ability to have second order formulas as induction hypothesis to prove consistency of first order logic + PA.

Lucas

ag.algebraic geometry - Barsotti--Weil formula over separably closed fields

Let $S$ be a noetherian scheme and let $A$ be an abelian scheme on $S$ with dual $A^vee$. The generalised Barsotti Weil formula states that there is a canonical and functorial (in $S$ and $A$) isomorphism of groups

$$A^vee(S) cong mathrm{Ext}^1_S(A,mathbb G_m)$$

the classical version being the special case where $S$ is the spectrum of an algebraically closed field. There are at least two references for this in the literture: The standard one is Oort's LNM on commutative group schemes, where he proves that the Barsotti--Weil formula over $S$ holds if it holds over all residue fields of $S$. He then says that the formula is known to hold over any field and quotes Serre's Groupes algébriques et corps de classes, VII.16, théorème 6. The other one is by Milne, exactly along the same lines and also quoting Serre.

My problem with this is that Serre makes right at the beginning of chapter VII in loc. cit. the assumption that the ground field is algebraically closed.

Hence, as long as all residue fields of $S$ are perfect Oort's proof is fine (functoriality in $S$, so one can go down with Galois). For the general case one should/must check that Serre's arguments also work over separably closed fieds. A first careless walkthrough didn't reveal any serious problem with that, so probably everything is just fine. Since "probably just fine" is not good enough:

Here is my request: A proof of the Barsotti--Weil formula over separably closed fields, preferably a reference, but I will appreciate any text or note where this is carefully checked.

By the way: Even if one is only interested in abelian varieties over, say, a finite field, it is important to have Barsotti--Weil over all schemes of finite type over that field, the reason is that one often wants to fppf--sheafify the Barsotti--Weil isomorphism so to get an isomorpihsm of abelian varieties $A^vee cong underline{mathrm{Ext}}^1(A,mathbb G_m)$ over the ground field.

ds.dynamical systems - Homomorphisms from R to $Diffeo^+(R)$, or "fractional iterations"

This has some relation with this question, but it is obviously different.

In general, a homeomorphism of $mathbb{R}$ which preserves the orientation may have or not fixed points.

If it has no fixed points, then it is conjugated to a translation and thus, one can easily construct such $phi_f$.

The other case is not much more difficult, since one can consider the set of fixed points of $f$ (that is, such that $f(x)=x$) which is closed and then do the trick in the complement of that set and leave fixed the set of fixed points for every $t$ (when defining $phi_f(t)$). Notice that in either case, there is in general no unique way to do this.

It is also interesting that the diffeomorphism case is quite different, in particular, one can easyly construct a diffeomorphism $f:mathbb{R}to mathbb{R}$ such that there is no diffeomorphism $g$ such that $gcirc g =f$. This can be seen in the paper provided by Helge (in fact it has to do with distortion and the fact that if you take one contracting point, there are restrictions to construct, for example a square root, see Section 1 of this paper).

ADDED RELATED REFERENCE: In this paper, Palis gives a not so difficult proof that $C^1$-generic diffeomorphisms (which belong to a $G_delta$-dense subset of $Diff^1(M)$) of a compact manifold, the diffeomorphisms are not the time one map of a flow.

soft question - Which pair of mathematicians has the most joint papers?

I was searching on MathSciNet recently for a certain paper by two mathematicians. As I often do, I just typed in the names of the two authors, figuring that would give me a short enough list. My strategy was rather dramatically unsuccessful in this case: the two mathematicians I listed have written 80 papers together!

So this motivates my (rather frivolous, to be sure) question: which pair of mathematicians has the most joint papers?

A good meta-question would be: can MathSciNet search for this sort of thing automatically? The best technique I could come up with was to think of one mathematician that was both prolific and collaborative, go to their "profile" page on MathSciNet (a relatively new feature), where their most frequent collaborators are listed, alphabetically, but with the wordle-esque feature that the font size is proportional to the number of joint papers.

Trying this, to my surpise I couldn't beat the 80 joint papers I've already found. Erdos' most frequent collaborator was Sarkozy: 62 papers (and conversely Sarkozy's most frequent collaborator was Erdos). Ronald Graham's most frequent collaborator is Fan Chung: 76 papers (and conversely).

I would also be interested to hear about triples, quadruples and so forth, down to the point where there is no small set of winners.

---

Addendum: All right, multiple people seem to want to know. The 80 collaboration pair I stumbled upon is Blair Spearman and Kenneth Williams.

gn.general topology - Explanation for E_8's torsion

This doesn't directly address your question, but it does give you a way of thinking about torsion in the cohomology of Lie groups in general.

(This is all coming from Borel and Serre's Sur certains sous-groupes des groupes de Lie, which can be found in Commentarii mathematici Helvetici Volume 27, 1953)

As you mentioned above, every compact lie group is rationally a product of odd spheres. But how many odd spheres? Turns out, if G is compact and rank k, then it is rationally a product of k spheres (of various dimensions).

There is an analogous result for torsion. That is, one can define the 2-group of G to be any subgroup which is isomorphic to $(mathbb{Z}/2mathbb{Z})^n$ or some n. One defines the 2-rank of a group as the maximal $n$ of any 2-group in G. (On can show that for connected $G$, the 2-rank is bounded by twice the rank, and is thus finite).

Just to point out something that really threw me when I first learned of these - while the rank is an invariant of the algebra (i.e., all Lie groups with the same algebra have the same rank), the 2-rank of a Group is NOT an invariant of the algebra. For example, the 2-rank of SU(2) is 1 (in fact, -Id is the UNIQUE element of SU(2) of order 2), while the 2-rank of SO(3) is 2 (generated by diag(-1,-1,1) and diag(-1,1,-1) ). The 2-rank of O(3) is 3 (generated by diag(-1,1,1), diag(1,-1,1), and diag(1,1,-1) ).

Now, given $Tsubseteq G$, the maximal torus, it's clear that simply by taking the maximal 2-group in T, that the 2-rank of G is AT LEAST the rank of G. When is it strictly bigger? Precisely when the group G contains 2-torsion.

The analogous result for p-groups and p-torsion (p any prime) also holds.

In short, to understand the existence of the 5-torsion in $E_{8}$, one need only understand why there is a subgroup isomorphic to $(mathbb{Z}/5mathbb{Z})^nsubseteq E_8$ for some $ngeq 9$.

Gaussian primes, quaternion primes, ... octonions?

Is there a notion of an octonion prime?
A Gaussian integer $a + bi$ with $a$ and $b$ nonzero is prime if $a^2 + b^2$ is prime.
A quaternion $a + bi + cj + dk$ is prime iff $a^2 + b^2 +c^2 + d^2$ is prime.
I know there is an eight-square identity that underlies the octonions.
Is there a parallel statement, something like: an octonion is prime if its norm is prime?

I ask out of curiosity and ignorance.

Addendum.
The Conway-Smith book Bruce recommended is a great source on my question.
As there are several candidates for what constitutes an integral octonion,
the situation is complicated. But a short answer is that unique factorization
fails to hold, and so there is no clean notion of an octonion prime.
C.-S. select out and concentrate on what they dub the octavian integers, which,
as Bruce mentions, geometrically form the $E^8$ lattice.
Here is one pleasing result (p.113): If $alpha beta = alpha' beta'$, where
$alpha, alpha', beta, beta'$ are nonzero octavian integers, then the angle between $alpha$ and $alpha'$ is
the same as the angle between $beta$ and $beta'$.

---

A non-serious postscript:
Isn't it curious that
$mathbb{N}$,
$mathbb{C}$,
$mathbb{H}$,
$mathbb{O}$
correspond to N, C, H, O, the four atomic elements that comprise all proteins and much of organic
life? Water-space: $mathbb{H}^2 times mathbb{O}$, methane-space: $mathbb{C} times mathbb{H}^4$, ...

books - What are some good group theory references?

As was mentioned Rotman's book is a very good basic book in group theory with lots of exercises.

For finite group theory Isaacs has a relatively new book. I didn't read much from the book, but the little I did, was very nice. Generally, Isaacs is a very good teacher and a writer.

Old fashion references for finite group theory are Huppert's books (the second and third with Blackburn) and Suzuki's books. They are out of print, old fashion and the first of Huppert’s book is in German. But they are encyclopaedic, useful, and popular.

Robinson’s book is a good book especially for infinite group theory, an area which is hard to find in other books.

In my corner of group theory, DDMS, Analytic pro-p groups is standard if you are interested in linear pro-p group, Wilson’s Profinite groups is more general profinite groups theory, and there is also Ribes and Zelesski which I am not familiar with, but I think is more geometric in nature.

A book worth mentioning in my view is Subgroup Growth by Lubotzky and Segal. It contains a lot of group theory and touches on many topics. So by reading it, it is possible to get a good overview of the all area.

triangulated categories - Is there a constructive description of type in the p-local stable homotopy category?

The title pretty much sums it up - but let me give a little bit of background first.

In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) the Morava K-theories are a non-negative integer indexed family of spectra (morally they are residue objects or "homological fibre functors") which classify the thick subcategories of SH^{(fin)_p the p-local stable homotopy category of finite spectra. This classification assigns to each p-local finite spectrum a type (namely the smallest thick subcategory it occurs in).

As far as I know the only definitions of type are directly in this way via the Morava K-theories or in terms of periodic self maps which are still really in terms of Morava K-theory.

Is there a more "constructive" definition of type? For instance can one determine the type of a p-local finite spectrum in terms of how bad the obstruction to it generating the whole of SH^{(fin}}_p is? Really I would be happy with any answer which was somewhat more constructive or to find out that the question is open/ridiculous.

Now let me explain somewhat the motivation for this question and how it came about. One can view the Morava K-theories (as I mentioned above) as residue objects/homological fibre functors (the term homological fibre functor has less algebraic geometry bias and sounds cooler) in the sense that their behaviour is analogous to residue fields of points in the derived category of quasi-coherent modules on a scheme (to be safe one should really take the derived category of O_X-modules with quasi-coherent cohomology) and with kappa-modules in modular representation theory. Namely they all give tensor functors to some flavour of graded vector space category which classify thick subcategories.

The mod n Moore spectra are `Koszul objects' which we want to view as analogues of the usual Koszul complexes on a scheme and of Carlson modules in modular rep theory. In other words a Koszul object is a cone on some (possibly graded and maybe also twisted) element of the endomorphism ring of the tensor unit of our category.

Now one can associate some geometry to a biexact tensor product (by biexact tensor product I mean symmetric monoidal structure which is exact in each variable, there are no decency assumptions or extra axioms regarding compatibility with the triangulation required) on an essentially small triangulated category. It is possible to cook up a locally ringed space associated to such a category with tensor product (this is work of Paul Balmer). In the two algebraic cases, derived categories of schemes and stable categories in modular rep theory, one gets (with some mild hypothesis in the algebraic geometry case) back the scheme or recovers the projective support variety. This comes down in some sense to the fact that the Koszul objects determine the topology, or equivalently that they determine the thick subcategories in some sense.

This fails for the stable homotopy category of finite spectra (both globally and p-locally). The mod n Moore spectra are not enough - one needs the Morava K-theories. The locally ringed space one gets is not a scheme (nor an algebraic space). One can then ask if there is some "global" reason that this happens (even though it is not at all a surprise) other than the fact that the Morava K-theories are just there generating subcategories p-locally. This is motivated partially by trying to understand the failure of a certain comparison map to be injective (which I didn't mention - it would be interesting to have a good criterion for its injectivity and I currently only have quite hard to check ones) and to try to get some feel for what properties can cause the associated locally ringed space to fail to be algebraic (one really needs more examples computed for this and I haven't found the time yet unfortunately).

So basically I feel that if there were some definition of type that made clear the failure to be able to reduce the type by taking triangles and suspensions (or something other than just the residue objects being there) it might be quite enlightening. In particular, in general one wouldn't expect to produce an algebraic gadget from a topological triangulated category (in the sense of Schwede ). That is why I mentioned the extension problem as if topologicalness provided some global obstruction to Koszul objects being enough that would be very interesting.

That got very long - I hope it is interesting/useful.