model theory - Cohomology Theories on The Stone Space of Complete n-types

I can not really inform you about this since I don't know, but I can point you to some notes of Angus Macintyre,
http://modular.math.washington.edu/swc/notes/files/03MacintyreNotes.pdf

Here are some excerpts:

"For me personally, the main surprise arising from the discovery of ACFA
was how much there was to be done in terms of a model-theoretic reaction
to the development of etale cohomology and its relatives."

"Again, in a different direction, one begins to see cohomological ideas coming
up all over applied model theory, for example in o-minimality."

I hope that you find this useful.

fa.functional analysis - What function has fourier series the harmonic series?

Qiaochu Yuan's original answer was very helpful, but working through it left me with a question or two. I couldn't put the workings in a comment so I put them here. It turned out that there was a misunderstanding as to exactly what coefficients I wanted. Once that was resolved, his original answer sufficed. However, just in case someone else ponders this question, I'll leave my workings here to save them the effort.

Following the hint from Qiaochu, we start from

$$
sum_{n=0} x^n = (1 - x)^{-1}
$$

and integrate term-by-term to get

$$
sum_{n=1} frac{1}{n} x^n = -log(1 - x)
$$

now we substitute in $x = e^{pi i t}$ to get

$$
sum_{n=1} frac{1}{n} e^{n pi it} = -log(1 - e^{pi i t})
$$

of which we then take the real part:

$$
sum_{n=1} frac{1}{n} cos(n pi t) = -Relog(1 - e^{pi i t})
$$

Excluding the case where $t=0$, we can use the standard branch of the logarithm and so use the identity $log(r e^{itheta}) = log(r) + itheta$ to deduce that we want the logarithm of the absolute value of $1 - e^{pi i t}$. Since squaring is easily taken care of, we are lead to consider:

$$
|1 - e^{pi i t}|^2 = (1 - e^{pi i t})(1 - e^{-pi it}) = 1 - e^{pi i t} - e^{-pi it} + 1 = 2 - 2 cos(pi t)
$$

and thus conclude that

$$
sum_{n=1} frac{1}{n} cos(n pi t) = - frac{1}{2} log 2(1 - cos(pi t))
$$

gt.geometric topology - Osculating conics and cubics and beyond

These highly osculating curves were studied, in particular by V.I. Arnol'd. One of the important refferences will be:

Topological invariants of plane curves and caustics.
Dean Jacqueline B. Lewis Memorial Lectures presented at Rutgers University, New Brunswick, New Jersey. University Lecture Series, 5. American Mathematical Society, Providence, RI, 1994.

More precisely, what was studdied are the points of the curve, where the level of it tangency with (say) conics is higher than expected. I guess these are exactly the points that (using your terminology) separate elliptic part of the curve form hyperbolic.

The key words for these research are Extactic points (therminology proposed by D. Esenbud). Using google scholar you can find a complete text of Arnol'd, called

Remarks on the extatic points of plane curves V.I. Arnold - The Gelfand Mathematical Seminars, 1993-1995.

These article contains some genearlisations of four vertex theorem. http://en.wikipedia.org/wiki/Four-vertex_theorem

One more nice refference is a paper of Tabachnikov and Timorin http://arxiv.org/PS_cache/math/pdf/0602/0602317v2.pdf

at.algebraic topology - Commutativity in K-theory and cohomology

Perhaps the deeper story you want involves the notion of "E-infinity product". The cup product in cohomology, and the sum (and for that matter, the product) in K-theory are commutative (and associative, and unital) not merely up to homotopy, but "up to all possible higher homotopies". You can make this precise by saying that the appropriate binary operation on the representing space (the K(Z,n)'s, or Z x BU), is part of an E-infinity algebra structure on that space.

It seems that most "naturally occuring" sums or products in topology turn out to be E-infinity (addition for any generalized cohomology theory, multiplication for many nice ones such as ordinary cohomology, K-theories, bordism, elliptic cohomology). As you observe, having a strictly commutative operation is very special, and basically forces the representing spaces to be a product of K(A,n)'s, by the Dold-Thom theorem.

To me, the mystery here is that "stricly commutative" is so much more special than "E-infinity commutative". Associative products don't behave this way: "strictly associative" turns out to be no more special than "A-infinity associative", that is, any A-infinity product on a space can be "rigidified" to a strictly associative product on a weakly equivalent space.

Can anyone give me a good example of two interestingly different ordinary cohomology theories?

For any space that has the homotopy type of a CW complex, its cohomology is determined purely formally by the Eilenberg-Steenrod axioms, so a counterexample is necessarily some reasonably nasty space. Here's an example you can see with your bare hands: consider the space $X={1,1/2,1/3,1/4,...,0}$. Now 0th singular cohomology is exactly the group of $mathbb{Z}$-values functions on your space which are constant on path-components, so $H^0(X)=X^mathbb{Z}$ (an uncountable group) naturally for singular cohomology. On the other hand, 0th Cech cohomology computes global sections of the constant $mathbb{Z}$ sheaf, i.e. locally constant $mathbb{Z}$-valued functions on your space. These must be constant in a neighborhood of 0, so the Cech cohomology $H^0(X)$ is actually free of countable rank, generated (for example) by the functions $f_n$ that are $1$ on $1/n$, $-1$ on $1/(n+1)$, and $0$ elsewhere, plus the constant function $1$.

I should add that topologists don't actually care about such examples. The point of the Eilenberg-Steenrod axioms is to show that cohomology of reasonable spaces is determined by purely formal properties, and these formal properties are actually much more useful than any specific definition you could give (the only point of a definition is to show that the formal properties are consistent!). What is of interest is when you remove the dimension axiom to get "extraordinary" cohomology theories, which Oscar talks about in his answer.

at.algebraic topology - Can we make rigorous the 'obvious' characterisation of singular homology?

I think that your theorem is "almost" true if we restrict ourselves to a neighborhood of x in (some quotient of) the k-skeleton. And indeed, this is a way to give a precise meaning to "holes" in singular homology. Let me be more specific.

Let's consider a finite CW-complex. The space is built starting with a finite number of points, and attaching a finite number of cells of various dimensions. The k-skeleton $X_k$ is the union of all cells of dimension less than or equal to k.

The smash $X_k|X_{k-1}$ is obtained from $X_k$ by identifying all points in $X_{k-1}$. If X is a CW-complex, that "smash" is homeomorphic to a bouquet of k-spheres. These spheres are the "holes" we are looking after. From the standard identification
$H_k(X_k|X_{k-1})simeq H_k(X_k,X_{k-1})$ , we see that each sphere gives rise to an element in the relative homology group
$H_k(X_k,X_{k-1})$. There are two ways these elements may fail to give an element in $H_k(X)$.

--> Instead of capturing a k-dimensional "hole", the cell may in fact "fill" a (k-1)-dimensional "hole". That's what happens when, for example, we cap a cylinder with a disk. So, we are only interested in elements in the kernel of the boundary operator $delta_k : H_{k}(X_k,X_{k-1})rightarrow H_{k-1}(X_{k-1})$.

--> The k-dimensional "hole" may be filled by some $k+1$-dimensional cell. So we should quotient $H(X_k,X_{k-1})$ by the image of the operator $H_{k+1}(X_{k+1},X_k)rightarrow H_{k}(X_k)rightarrow H_{k}(X_k,X_{k-1})$. Let us denote that image by $E_k$.

And it works. The group $H_k(X)$ is actually isomorphic to the quotient $ker delta_k / E_k$. So there is a subset of the "holes" in $X_k|X_{k-1}$ that provide a generating family for $H_k(X_k)$. Arguably, the spherical neighborhood you are looking for exists in the smash, not in X, but still, I think it succeeds in making our intuition rigorous. As a reference, I may point to Greenberg "Algebraic topology, an introductory course" (21.8 ff).

big list - Undergraduate Level Math Books

What are some good undergraduate level books, particularly good introductions to (Real and Complex) Analysis, Linear Algebra, Algebra or Differential/Integral Equations (but books in any undergraduate level topic would also be much appreciated)?

EDIT: More topics (Affine, Euclidian, Hyperbolic, Descriptive & Diferential Geometry, Probability and Statistics, Numerical Mathematics, Distributions and Partial Equations, Topology, Algebraic Topology, Mathematical Logic etc)

Please post only one book per answer so that people can easily vote the books up/down and we get a nice sorted list. If possible post a link to the book itself (if it is freely available online) or to its amazon or google books page.

linear algebra - Find a matrix's nullspace from submatrix nullspace

This is probably a basic question, but my linear algebra is weak.

Suppose I want to compute the nullspace of a matrix A using some iterative method (e.g. Lanczos). Suppose further that I know a priori the nullspace of the first n columns of the matrix, i.e., Av = [0 0 0 ... 0 b_n .. B_N], where b_i are nonzero with high probability.

Does starting the iterative method with vector v (instead of a random vector) speed the iterative method (e.g., Lanczos) up at all?

tensor products - The meaning of an intertwiner?

I think that a reformulation of my question is necessary:
An intertwiner $iota:; V_{j_{1}}bigotimes V_{j_{2}}rightarrow V_{j_{3}}$ is defined as:

$forall gin SU(2),;forall u_{i},v_{i},w_{i},...in V_{j_{i}}:;iota((rho_{j_{1}}bigotimesrho_{j_{2}})(g);[v_{1}bigotimes v_{2}])=rho_{j_{3}}(g)[iota(v_{1}bigotimes v_{2})]$

where $rho_{j_{i}}$ is the representation map corresponding to the irrep of spin $j_i$ of $SU(2)$, and $V_{j_i}$ are the invariant spaces upon which acts the $rho_{j_{i}}(g)$ for $gin SU(2)$

I know that the Shur's lemma is: if

$iota:; V_{j}rightarrow V_{k}$

is an interwtwiner, then is it either a scalar (if $j=k$) or zero ($jnot= k$)

Now, what I want to know, is if $V_{j_{1}}bigotimes V_{j_{2}} = dotsoplus V_{j_{2}} oplus dots$ take an example $V_{j_{1}}bigotimes V_{j_{2}} = V_{j_{4}} oplus V_{j_{2}} oplus V_{j_{5}}$ I can write:

$forall gin SU(2),;forall u_{i},v_{i},w_{i},...in V_{j_{i}}:;iota((rho_{j_{4}}oplusrho_{j_{3}}oplusrho_{j_{5}})(g);[v_{1}bigotimes v_{2}])=rho_{j_{3}}(g)[iota(v_{1}bigotimes v_{2})]$

in this case how to prove that $iota$ is a scalar? (by Shur's lemma)

Historical transition from classical homotopy to modern homotopy theory.......

I am surely not a historian of topology, but I might try a few words.

That the usual literature concerning model categories is quite far away from traditional homotopy as presented in Whitehead's classic, is no wonder. Indeed, model categories are abstracted from homotopy theory, but not really that of the classical flavour. Quillen's lecture notes are not without reason entitled 'Homotopical Algebra'. As discussed in its introduction, its main object is to present an abstract framework where one can consider simplicial objects in categories of relevance for algebra. This leads to a theory of "non-additive derived functors", e.g. Andre-Quillen homology.

In particular, the example of the model structure on topological spaces inducing the classical homotopy category is not presented in Quillen's book - only the one using Serre fibrations, generalized CW-complexes and weak homotopy equivalences. The model structure with Hurewicz fibrations/cofibrations and homotopy equivalences had to wait until Strom's The homotopy category is a homotopy category . As a consequence, the first absorbers of the theory of model categories were more simplicial minded guys. See for example Bousfield and Kan's Homotopy limits, completions and localizations.

One reason, why the notion of a model category is today so omnipresent in algebraic topology is that they provided a very good framework to discuss the homotopy theory of spectra and it was important to work both simplicially and in topological spaces. But the model structure on topological spaces used here was again the Quillen model structure.

I think, it is only in the last years that topologists are caring more again to reunion classical homotopy theory and model categories. One important work for this is Cole's Mixing model structures . Here, a model structure on topological spaces is constructed, where the weak equivalences are again the weak homotopy equivalences, but fibrations are now the Hurewicz fibrations. This leads to a theory, where the cofibrant objects are all spaces homotopy equivalent to a CW-complex. This model structure interacts rather well with more classical homotopy theory (using Hurewicz cofibrations and so on) as is seen e.g. here or in (section 8 of) this, which is also used in the five-author paper Units of ring spectra and Thom spectra. The reason, why the latter needs the connection to more classical homotopy theory is that the theory of $E_infty$-spaces stems from classical homotopy theory and is simultaneously deeply linked to modern stable homotopy theory.

ag.algebraic geometry - Langlands Dual Groups

As Peter mentioned, reductive groups are determined by their root data, and the Langlands dual is given by switching weights with coweights, and roots with coroots.

There is a "construction" of a group from a root datum in SGA III Exp 25 (in vol 3). It starts by reducing to the simply connected semisimple case (meaning there are ways of going from this case to the general case). The weights give you a torus T, and the positive/negative roots give you unipotent groups U+ and U-. You form a scheme Omega = U- x T x U+, and create G by gluing a few disjoint copies of Omega together, and writing down a composition law. This yields a split group over an arbitrary base scheme.

gt.geometric topology - Rotation part of short geodesics in hyperbolic mapping tori

This should follow from Minsky's work on a priori bounds for surface groups, which is used in the proof of the ending lamination conjecture.
The punctured torus case is simpler and more explicit (see Theorem 4.1 and equations 4.4 and 4.5).

Addendum: Once I thought about it for a bit, I think it follows from much more elementary
considerations (in fact, I'm pretty sure someone explained this to me before, but I forgot the argument). Let $Sigma$ be a surface. Suppose one has a very short geodesic $gammasubset M$, where $Mcong Sigmatimes mathbb{R}$ is a hyperbolic manifold, then Otal's argument proves it is unknotted (this was actually known to Thurston, and generalized to multiple components by Otal). In fact, one may find a pleated surface $f:Sigma to M$ so that $gamma$ is a closed geodesic on the image of this surface. Then the Margulis tube $V$ of $gamma$ is of very large radius, and therefore its boundary $partial V$ is very close to being a horosphere (i.e., its principle curvatures are very nearly $=1$) and is isometric to a Euclidean torus. The boundary slope $gamma'subset partial V$ of the surface $Sigma$ is a Euclidean geodesic of bounded length - this follows from an area estimate of a pleated annulus $A subset Sigma$ such that $f(A)$ cobounds $gamma$ and $gamma''$, where $gamma''sim gamma'subset partial V$, which has $Area(A) approx gamma''$ by a Gauss-Bonnet argument (if $V$ were a horocusp, then this would be an equality). But
$$length(gamma')leq length(gamma'')approx Area(A) leq Area(f^{-1}(Sigma)) = -2pi chi(Sigma).$$ The meridian $musubset partial V$ is a curve intersecting $gamma'$ once. We may assume that $gamma',musubset partial V$ are chosen to be Euclidean geodesics. Then $partial V backslash (gamma'cup mu)$ is a Euclidean parallelogram, with one pair of sides of bounded length corresponding to $gamma'$. Since $V$ has very large radius, $mu$ must be extremely long. The rotational part corresponds to the fraction of the offset between the two sides of the parallelogram corresponding to $mu$.
But this implies that the rotational part of $gamma$ is less than $$2pi length(gamma')/length(mu),$$ which is very small, and approaches zero as $length(gamma)to 0$.

mg.metric geometry - Can the circle be characterized by the following property?

If a figure had an axis of symmetry in three non-parallel
but non-concurrent axes, then composing these suitably would
give a translative symmetry, which is impossible if the figure
is bounded. So all the axes of symmetry of your putative curve
are concurrent through a point $O$ which we shall call a centre.
Then all rotations about the centre $O$ are symmetries. The only
simple closed curves with this property are circles centred at $O$.

algebraic number theory - Monic polynomial from absolute value information

Here's a solution using lattice reduction:

1) Find degree $d$ polynomials $p_i(x)$ such that $p_i(x_j) = |M(x_i)| delta_{i j}$.

2) Let $c_i$ be the coefficient of $x^d$ in $p_i(x)$, and $c$ the $d+1$ long column vector whose coordinates are $c_i$.

3) Find a matrix $U in SL_{d+1}(mathbb{Z})$ such that $U c = e$, where $e$ is the $d+1$ long column vector with a 1 in the first coordinate and zeros elsewhere. Added later:

not quite. Let $D$ be a common denominator of all the elements of $c$, and form a $d+1 times d+1$ matrix, $A$, whose first column is $cD$ and the lower $d times d$ block is the identity matrix (with the rest of the top row 0). Find the Hermite normal form: $U in SL_{d+1}(mathbb{Z})$, $U A = H$. The first column of $H$ will be zeros below the first entry, which will be a positive integer $r$. In order for there to be a solution it is necessary that $r | D$. Form a new matrix $U'$ by multiplying the top row of $U$ by $D/r$.

4) The answer (see below) is a vector in the $mathbb{Z}$-lattice generated by the bottom $d$ rows of $U'$ which is close to the top row.

Namely, form the matrix $W$ by adjoining a $d+1 times d+1$ identity matrix to the right of $c$. Since only the coefficient of $x^d$ matters in the answer, we can see that an answer will be given by some integer linear combination of the rows of $W$ which has $pm 1$ in the last $d$ coordinates. The squared Euclidean length of that vector will be $d+1$, which is quite short. There are a number of algorithms for finding a closest vector (in theory for general lattices it's a hard problem, but in practice in a lattice like this it's not too hard). For a nice account of how to do it look in http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.81.8089 starting around page 14.

The idea here is to prepend a column to $U'$ which is a unit vector with 1 in the first position and 0's everywhere else. Now multiply the rest of the matrix (all columns but the first) by a large scaling factor, $s$. This will make sure that the first row will show up in the linear combination forming the shortest basis vector.

Lattice reduction will supply a short vector in the lattice which we know that our answer is. We then read off the coefficients of the $p_i$ in the last $d+1$ coordinates.

I've programmed this, and tested it on random polynomials of degree 20, and it successfully finds the $pm 1$ combination leading to a monic polynomial.

In Tony Scholl's example of three different polynomials having the same data, the lattice generated has a lot of short vectors, so in that case one needs to enumerate short vectors to pick out the answers.

nt.number theory - The norm of a non-Galois extension of local fields

Background and Motivation

Local Class Field Theory says that abelian extensions of a finite extension $K/mathbb{Q}_p$ are parametrized by the open subgroups of finite index in $K^times$. The correspondence takes an abelian extension $L/K$ and sends it to $N_{L/K}(L^times)$, and this correspondence is bijective.

If one starts instead with a galois extension $L/K$ that isn't abelian, one can then ask "What abelian extension does $N_{L/K}(L^times)$ correspond to?" The answer is the maximal abelian extension of $K$ contained in $L$.

---

The hypothesis of being galois isn't necessary in the statement of the non-abelian theorem: both the question and the answer still make sense. I am thus asking

Assume that $L/K$ as above. Is the abelain extension of $K$ corresponding to $N_{L/K}(L^times)$ the maximal abelian extension of $K$ contained in $L$?

A couple examples to illustrate this problem (including the example that I was told would sink this):

If $p > 2$, then consider $L = mathbb{Q}_p(sqrt[p]{p})$. The norm subgroup that I am anticipating is all of $mathbb{Q}_p^times$. Moving into the galois closure and using the theorem, one gets that $N_{L/mathbb{Q}_p}(L^times)$ contains $p^mathbb{Z} times (1 + pmathbb{Z}_p)$. Moreover, the norm of a number $a in mathbb{Z}_p$ will just be $a^p$, which is congruent to $a$ mod $p$, so the norm will also hit something congruent to any given root of unity in $mathbb{Q}_p$, and that was all that I was missing from the earlier note.

One can also, using the same idea (moving into the galois closure and getting a lot of information from that, and then using the explicit structure of the field that one started with) show that this works for $K(sqrt[n]p)$ with $(n, p^{f(K/mathbb{Q}_p)}) = (n,p) = 1$ as well.

lie algebras - Invariant forms

Not a full answer, but you can get plenty of examples by taking $V$ to be the coadjoint representation of a Lie algebra with nonzero Killing form. Then $omega in Lambda^3mathfrak{g}^*$, defined by
$$omega(X,Y,Z) = mathrm{Tr}~mathrm{ad}([X,Y])mathrm{ad}(Z),$$
is invariant and nonzero.

Similarly, any metric Lie algebra with $V$ the adjoint representation also works. The invariant 3-form is then
$$omega(X,Y,Z) = langle [X,Y], Zrangle,$$
with $langle-,-rangle$ the ad-invariant inner product.

ds.dynamical systems - Spectrum of a generic integral matrix.

Yes, a generic integer matrix has no more than two eigenvalues of the same norm. More precisely, I will show that matrices with more than two eigenvalues of the same norm lie on a algebraic hypersurface in $mathrm{Mat}_{n times n}(mathbb{R})$. Hence, the number of such matrices with integer entries of size $leq N$ is $O(N^{n^2-1})$.

Let $P$ be the vector space of monic, degree $n$ real polynomials. Since the map "characteristic polynomial", from $mathrm{Mat}_{n times n}(mathbb{R})$ to $P$ is a surjective polynomial map, the preimage of any algebraic hypersurface is algebraic.
Thus, it is enough to show that, in $P$, the polynomials with more than two roots of the same norm lie on a hypersurface. Here are two proofs, one conceptual and one constructive.

Conceptual: Map $mathbb{R}^3 times mathbb{R}^{n-4} to P$ by
$$phi: (a,b,r) times (c_1, c_2, ldots, c_{n-4}) mapsto (t^2 + at +r)(t^2 + bt +r) (t^{n-4} + c_1 t^{n-5} + cdots + c_{n-4}).$$

The polynomials of interest lie in the image of $phi$. Since the domain of $phi$ has dimension $n-1$, the Zariski closure of this image must have dimension $leq n-1$, and thus must lie in a hyperplane.

Constructive: Let $r_1$, $r_2$, ..., $r_n$ be the roots of $f$. Let
$$F := prod_{i,j,k,l mbox{distinct}} (r_i r_j - r_k r_l).$$
Note that $F$ is zero for any polynomial in $mathbb{R}[t]$ with three roots of the same norm. Since $F$ is symmetric, it can be written as a polynomial in the coefficients of $f$. This gives a nontrivial polynomial condition which is obeyed by those $f$ which have roots of the sort which interest you.

big picture - Intuitions/connections/examples for "eigen-*"

@vonjd

As explained by SandeepJ, "eigen..." is related to the spectrum of something. In particular, when one calculates the eigenvalues and their corresponding eigenvectors from A.x = lambda*x, adding the same vector x times a constant to both sides of the equation (say tau, i.e., one adds tau*x), only the eigenvalues get shifted by an amount equivalent to tau, but not the eigenvectors. The corollary that one can get from this statement, is that "eigen..." is about locating some characteristic functions or operators, which get unchanged under the shift of a constant, or something similar. We can state it in another way: in the process of studying "eigen..." we want to find some functions on the spectrum of some sort of functional, which remain invariant under certain types of transformation. This last phrase may sound pretty vague, but the thing is that "eigen..." is not something for which we can always define a straightforward algorithm.

The place which I liked the most for answering this question of yours is "Numerical Recipes in Fortran. The Art of Scientific Computing" (William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery)

pr.probability - If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

Yes, here's a nice and beautiful argument!

First you should draw a picture of axes a and b. You're asked to select uniformly a point in the square [0,1]x[0,1]. Now because of the symmetry (sic!) it's equivalent to choosing the points a and b uniformly in the triangle cut from the square by b > a.

So you're actually uniformly selecting a point inside triangle defined by lines a>=0, b<=1, 'b>=a'.

Now let's find the conditions to be able to make a triangle of short sticks. We should have a + (1-b) > b-a, b-a + (1-b) > a and b > 1 - b which indeed, as you say, boils down to

b > 1/2,  a < 1/2,  b-a < 1/2  

It remains to note that those lines create inside the big triangle a small triangle which is similar to big but with all lengths 1/2 of the big, so this small triangle has area of exactly 1/4 of original!

ag.algebraic geometry - Hironaka desingularisation theorem -- new proofs in literature?

For a long time, Hironaka's paper was probably the canonical reference. New proofs started to crop up in the late 90's, and truly changed the perception of the problem. The motivation was to make Hironaka's proof more constructive, and ultimately, make it tractable by computer algebra. This was ultimately done by Schicho and Bodnar in Maple (desing project). (Of course, the complexity is atrocious, but it still allows to run example beyond what one could reasonably do by hand.)

Two main groups worked on it in the late 90's, and they have been referenced already in the prior answers: Bierstone-Milman on the one hand, and Encinas-Villamayor and maybe a couple of others on the other hand (spanning many papers). There was a year-long seminar at Purdue in 2000-2001 led by Kenji Matsuki and Andrei Gabrielov to work through the Encinas-Villamayor proof. This was an opportunity to clarify quite a few things in the construction, and Matsuki posted his notes (128 p.) on Arxiv.

Ultimately, this lead Jaroslaw Wlodarczyk to offer the shortest proof of desingularization (by far, 24 pages vs. 100+ for everyone else) .

Jaroslaw Wlodarczyk
Simple Hironaka resolution in characteristic zero.
J. Amer. Math. Soc. 18 (2005), no. 4, 779--822
Arxiv

I hope this helps! (Sorry for the delay in my answer, I only joined MO last week-end.)

(Added later) How could I forget? There is also a nice expository paper that covers all these developments. It should be very useful for someone trying to understand how the various works fit together (does predate Wlodarczyk's paper though).

H. Hauser
The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand).
Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 3, 323--403

soft question - Betting markets for unsolved problems?

"I've heard that betting markets generally have a good track record of aggregating the views of people and pointing to reasonably correct results, as long as there's enough participation and people have something real at stake"

In my opinion, there is no reason to believe that prediction markets will be useful to aggregate information which is not there. For most mathematical and scientific questions people have no real information that aggregating together will say something useful.

It is not clear also how good are financial markets in predicting their own future behavior. (But it is a complicated question how to formally as it.)

It would be useful to look skeptically also into these claims about the good track record of prediction markets that the asker heard about.

Probably the question is more suitable to a "meta discussion".

Further discussion can be found in this post over Shtetl optimized following this one.

ag.algebraic geometry - What are the higher $mathrm{Ext}^i(A,mathbf{G}_m)$'s, where $A$ is an abelian scheme?

It seems that here you are talking about the local exts, i.e. about the sheaves $underline{Ext}^{i}(A,G_m)$ on $S$. The question is rather tricky actually. The problem is that before we try to answer it, we should first specify what we mean by $underline{Ext}^{i}(A,G_m)$. We could mean exts in the category of sheaves of abelian groups in the flat topology on $S$, or we could mean exts in the category of commutative group schemes. The latter is not an abelian category so you have to do something before you can define exts. It is possible to do this though. The standard lore is to use Yoneda exts. This is carried out in detail in the LNM 15 book by Oort. Among other things Oort checks that if $S$ is the spectrum of an algebraically closed field, then ext sheaves are all zero for $i geq 2$.

Over general base schemes the situation is more delicate. First of all there are examples of Larry Breen showing that the ext sheaves in the category of sheaves of abelian groups are strictly larger than the ext sheaves in the category of commutative group schemes. In his thesis

Breen, Lawrence
Extensions of abelian sheaves and Eilenberg-MacLane algebras.
Invent. Math. 9 1969/1970 15--44.

Breen also showed that over a regular noetherian base schemes the global ext groups (in either category) are torsion if $i geq 2$. Later in

Breen, Lawrence
Un théorème d'annulation pour certains $E{rm xt}sp{i}$ de faisceaux abéliens.
Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 3, 339--352.

he strengthened his result to show that the higher exts sheaves are always zero for $1 < i < 2p-1$, where $p$ is a prime which is smaller than the (positive) residue characteristic of any closed point in $S$.

gn.general topology - How can one characterise compactness-by-experiment?

There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that fit the concept of doing experiments on an object to find out about it.

Let me explain the idea in a little more detail. In some branches of mathematics, we try not to study an object itself too closely. Category theory could be viewed as the extreme example of this, but it also finds a home in homotopy theory, cohomology theory, differential topology and geometry, and no doubt in others as well. The idea being that one has a family of "nice" objects and uses them to "probe" or "coprobe" the object in question. More prosaically, one tries to figure out the shape of the object in question either by throwing mud at it (to see what sticks) or by taking photographs of it. Less prosaically, we study $X$ by looking at morphisms $g colon Y to X$ or $f colon X to Y$ (where $Y$ runs over our family of "nice" objects).

For some property of our object, we can then ask "Can we detect it by doing these experiments?", or, more generally, "Is there something a bit like it that I can detect using experiments?" (the belief that these are the same question could go on the "false mathematical beliefs" question!).

Let's home in on the case I'm interested in: compactness. If we were probing our space by looking at maps from $mathbb{N}$ (aka sequences), then we would come up with the notion of "sequentially compact". If we were probing our space by looking at maps to $mathbb{R}$ (aka functionals), then we would come up with the notion of "pseudocompactness" (no, I'd never heard of it either before I looked it up on Wikipedia a couple of days ago!).

So, to my question: are there other examples of variations on the theme of compactness that fit this pattern? I'm particularly interested in the case of maps from $mathbb{R}$ (path-compactness, perchance?) but anything of this flavour would be helpful.

I'm specifically looking for stuff that's known. If there's nothing that's known then I'm interested in trying to figure out what it should look like, but that's not a good MO question so I'm not asking it. Anyone interested in this wider question is welcome to join in a discussion on it over at the nForum.

nt.number theory - Proof that pi is transcendental that doesn't use the infinitude of primes

The infinitude of primes (more precisely, the existence of arbitrarily large primes) might actually be necessary to prove the transcendence of $pi$. As I explained in an earlier answer, there are structures which satisfy many axioms of arithmetic but fail to prove the unboundedness of primes or the existence of irrational numbers. Shepherdson presented a simple method for constructing such models, I will present such a model where $pi$ is rational!

The Shepherdson integers $S$ consist of all Puiseux polynomials of the form
$$a = a_0 + a_1T^{q_1} + cdots + a_kT^{q_k}$$
where $0 < q_1 < cdots < q_k$ are rationals, $a_0 in mathbb{Z}$, and $a_1,dots,a_k in mathbb{R}$. This is a discrete ordered domain, where $a < b$ iff the most significant term of $b-a$ is positive; this corresponds to making $T$ infinitely large. This ring $S$ satisfies open induction axioms
$$phi(0) land forall x(phi(x) to phi(x+1)) to forall x(x geq 0 to phi(x))$$
where $phi(x)$ is a quantifier free formula (possibly with parameters). So the ring $S$ satisfies the same basic axioms as $mathbb{Z}$, but only a very limited amount of induction. In the field of fractions of $S$, $pi$ is equal to the ratio $pi T/T$. In other words, $pi$ is a rational number!

Is $pi T/T$ really $pi$? The integers form a subring of $S$, and if $p,q in mathbb{Z}$ then $p/q < pi T/T$ in $S$ if and only if $p/q < pi$ in $mathbb{R}$. So $pi T/T$ defines the same Dedekind cut as $pi$ does, which is a very accurate description of $pi$. Indeed, any proof of the transcendence of $pi$ must ultimately be based on the comparison of $pi$ and its powers with certain rational numbers, which $pi T/T$ will accomplish just as well as the real number $pi$. However, the usual definitions of $pi$ are not easily formalizable in this basic theory, so there is much room for debate here and I wouldn't claim that $pi T/T$ satisfies all reasonable definitions of $pi$. Shepherdson only presented this argument for real algebraic numbers like $sqrt{2}$, which have a finitary description in this theory and leave little room for debate. In any case, the conclusion to draw from this is that basic arithmetic with open induction does not suffice to prove that $pi$, or any other real number, is irrational (never mind transcendental).

What about primes? In the ring $S$, the only primes are the ones from $mathbb{Z}$. Although there are infinitely many primes in $S$, it is not true that there are arbitrarily large primes. For example, there are no primes larger than $T$. Thus $S$ is a model where the unboundedness of primes fails and so does the irrationality of $pi$. This only shows that basic arithmetic with open induction does not suffice to prove either result. A possible line of attack to show that the unboundedness of primes is necessary to prove the transcendence of $pi$ would be to show that the minimum amount of induction necessary to prove that $pi$ is transcendental also suffices to prove the unboundedness of primes. Unfortunately, I do not know how much induction is necessary to prove the transcendence of $pi$. (And the minimum amount of induction necessary to prove the unboundedness of primes is still an open problem.)

---

Well, here is a partial answer, which is a bit of a bummer. There is another Shepherdson domain $S_0$ similar to the above where $pi$ is transcendental over $S_0$ and $S_0$ does not have arbitrarily large primes. This shows that the transcendence of $pi$ does not imply the unboundedness of primes over basic arithmetic with open induction. The ring $S_0$ is the subring of $S$ where the coefficients of the Puiseux polynomial are restricted to be algebraic numbers. The unboundedness of primes fails in $S_0$ because the real algebraic numbers form a real closed field just like $mathbb{R}$. The number $pi$ is transcendental over $S_0$ because it is transcendental over the field of real algebraic numbers.

This is not entirely surprising since open induction is a very weak base theory and the Shepherdson type rings are very pathological. To constrain such pathologies Van Den Dries suggested requiring that the domain is integrally closed in its field of fractions; he called such domains normal but I don't know if this is standard terminology. Neither $S$ nor $S_0$ are normal. More convincing examples would be normal discrete ordered domains. The methods of Macintyre and Marker (Primes and their residue rings in models of open induction, MR1001418) suggest that normal analogues of $S$ and $S_0$ might exist.

The conclusion that I draw from this is that open induction is probably too weak a base theory to study this question. Stronger base theories run into the difficulty that it is still not known just how little induction is necessary to prove the unboundedness of primes. The next reasonable candidate is bounded-quantifier induction (IΔ₀), which is not known to imply the unboundedness of primes. Using the Euler product $pi^2/6 = prod_p (1-p^{-2})^{-1}$ looks promising, but so far I can only make sense of this product in IΔ₀ + Exp which is known to prove the unboundedness of primes.

differential equations - Point boundary problems

I think you may have gotten things backwards.

The point of differential equations is to describe macroscopic (global) phenomena via microscopic (local) physical laws, as differential operators are strictly local objects. Solving a differential equation one often finds families of solutions, which can live in various different function spaces of different regularity. The study of well-posedness of a differential equation often becomes the study of under what conditions can we obtain the existence of a unique solution. It is often found that the "degrees of freedoms" in the families of solutions can be restricted by prescribing boundary data of sufficient regularity. This is the case of elliptic operators and leads to the Dirichlet problem.

Now, the amount of data to be prescribed at the "boundary" is not the same for every problem. For elliptic type problems one only need to give a Dirichlet or Neumann type condition, but in general giving both conditions may lead to non-existence of a solution (over-determined problem). But for a different type of boundary and a different type of equation (say hyperbolic/wave equation in the initial value problem formulation), it is necessary to prescribe both the "Dirichlet" and the "Neumann" conditions for the question to be well-posed.

What you are asking is sort of an opposite problem: you are asking that given the value of a function on some subset of, say, the plane, what differential equations are well-posed for this data. This problem has too many solutions. Just to give a few

• As I mentioned in the comments, the equation $partial^2f = 0$ (vanishing Hessian)


• Or, let $v,w$ be arbitrary unit vectors not parallel to the sides of the triangle, then you can set $v(f) = w(w(w(f))) = 0$. The solution is constant in the $v$ direction, and along integral lines of $w$ must be quadratic, which is determined by three constants.


• Or, let $v$ be the unit vector going from point 1 to point 2, and let $w$ be the unit vector that connects point 3 to the line formed by point 1 and point 2 perpendicularly. Let $a$ be a number that is not a multiple of the distance between points 1 and 2, and $b$ be a number that is not a multiple of the distance between point 3 and the line, then take $v(v(f)) = - a^2 f$ and $w(w(f)) = - b^2 f$. The general solution is a trigonometric function depending on one translation coordinate and one scaling parameter.

But if you ask that also the function represents the steady state of a solution to the classical heat equation (in other words a solution to the Laplace equation), then the answer is no: you can extend the temperature profile on the boundary of your triangle arbitrarily from the three fixed data points you gave. For every extension (say differentiable) there is a corresponding solution to the Laplace equation. In other words, there are many, many steady-states whose temperature are as given at those three points on the triangle. So the map from your data to admissible steady-state temperature distributions is necessarily non-unique. So unless you prescribe additional conditions to pick out which of the many possible solutions you want, it is in general impossible (by definition) to write down a well-posed differential equation doing what you want it to do.

Edit: I just saw your answer to Qiaochu's comment

Yes, the devil is in the details on how you insert the constraint though. The Dirichlet problem is well posed. The three-point version isn't. By counting dimensions your constraint needs to be strong enough to mod-out a infinite dimensional set. For example, my first example of an equation $partial^2 f = 0$, is one possibility. A solution to that equation most certainly solves the Laplace equation. It is equivalent to extending the data to be linear along the boundary of the triangle, and solving the Laplace equation. It also happens to be the one that also minimizes the $H^2$ norm among all solutions to the Laplace equation. Is it a meaningful one? I dunno, what do you think? But it certainly is optimal when considering one metric.

In any case, any conditions you can impose that leads to a unique solution most certainly will be equivalent to one that leads to a unique set of compatible boundaries. Then you can impose differential conditions (ask that $f$ solves a second order ODE along each segment of the boundary) or algebraic conditions. Unless you have a physical justification, optimality really is in the eye of the beholder.

rt.representation theory - How to make commutative algebraic groups strongly dualizable?

Let's use the notation of [A=>B] for Hom(A, B). Take a 1-dimensional algebraic torus Gm and higher-dimensional torus T and let's live in the category of commutative algebraic groups over k.

Out of four expressions like [Gm=> [Gm=>T]] etc. half give back T (*), others the dual torus TV, in the sense that X*(T) := [Gm=> T] = [TV=> Gm] =: X*(TV).

The equality (*) can be proven by using the following formula with B = Gm

   (**)              A otimes [B=>B] ==== [[A=>B] => B].

Question: Is there another example of commutative algebraic group or a similar generalized object B, for which the identity (**) is true or true in some generalized sense?

One thing I specifically have in mind is that if we could write [X => Y] = X*otimes Y whenever X and Y are groups, as if they were vector spaces, the formula would hold for all A and B. So, what's a category that is related to algebraic groups but which posesses this property?

gn.general topology - Shape of long sequences in C(ω_1)

Apologies for the vague title - I couldn't come up with a single sentence that summarised this problem well. If you can, please edit or suggest a better one!

This question is also rather specific and contains lots of annoying technical detail. I must admit to not really expecting an answer unless there's an obvious solution I'm missing (which is very possible - I feel like any solution is either going to be obvious or very deep), but some pointers in plausible sounding directions would be greatly appreciated. I suspect the answer will depend on the combinatorics of $omega_1$, which I know relatively little about.

Let $V$ be a normed space. For $A subseteq V$, define $r(A) = inf { r : exists V, A subseteq B(v, r) }$. Define a bad sequence in $V$ to be a sequence ${ v_alpha : alpha < omega_1 }$ with the properties that:

$forall beta, r({ v_alpha : alpha < beta }) leq 1$

$inf_beta r({ v_alpha : alpha geq beta }) > 1$

An example of a space with a bad sequence is $c_0(omega_1)$ (the set of all bounded real-valued sequences of length $omega_1$ such that ${ alpha : |x_alpha| > 0 }$ is countable). The sequence $2 * 1_{{alpha}}$ is bad. The radius of any tail is $2$ because the center must be eventually 0. The radius of the initial segments is $leq 1$ because the segment up to $alpha$ is contained in the closed ball of radius 1 around $1_{[0, alpha]}$, which is in $c_0(omega_1)$ because $alpha < omega_1$.

I have two (three depending on how you count it) major examples of spaces which have no bad sequences:

• Any separable space: you can choose centers to lie in the countable dense set, so one center must work as a radius for the initial segment for unboundedly many and thus for all $alpha$.


• Any space which has what I'm imaginatively calling the chain-radius condition: The union of a chain of sets of radius $leq r$ has radius $leq r$. This includes:
  
  • Any reflexive space: If $U_alpha$ forms a chain, the sets $F_alpha = bigcap_{v in U_alpha} overline{B}(v, r + epsilon)$ form non-empty closed and bounded convex sets with the finite intersection property, so compactness in the weak topology implies they have non-empty intersection. Any element of the intersection contains the union of the chain in $overline{B}(c, r + epsilon)$
  
  
  • any space with the property that $textrm{diam}(A) = 2 r(A)$ (in particular the $l^infty$ space on any set) because it's clear that unions of chains of diameter $leq 2r$ have diameter $leq 2r$.
  
  

So... that's all the backstory for this question. Given that, my actual question is very simple: Does $C(omega_1)$ contain a bad sequence?

I feel like the answer "must" be no. In particular note that the projection of any sequence onto the first $alpha$ entries is not bad (because it's a sequence in a separable space) and that if you drop the restriction for continuity the answer is immediately yes. So it sits right between two classes of examples where there are no bad sequences, and I feel that one really should be able to take advantage of that. But on the other hand, functions in $C(omega_1)$ are eventually constant, so maybe you can take advantage of that to construct some sets with arbitrary bad tails.

For bonus kudos, I'd love to know for what compact Hausdorff spaces $K$, $C(K)$ contains a bad sequence.

mp.mathematical physics - Examples where physical heuristics led to incorrect answers?

I'll describe below a controversy in statistical mechanics in the 1980's: the case of the lower critical dimension of the Ising model with an applied random magnetic field.

Background

Let me give a little background, though you might want to read Terry Tao's discussion of basic statistical mechanics instead. The Ising model is a statistical mechanical model of "spins" on a hypercubic lattice. The energy functional is: $E=sum_{langle ijrangle}frac{1}{2}(1-S_iS_j)-sum h_iS_i$ where the first sum is taken over nearest neighbor pairs on the lattice and the second is taken over all sites, and $S_i$ is a $pm1$ valued variable on each site called the spin and $h_i$ is the real-valued "externally applied magnetic field" applied to each site. Each possible configuration of spins on the lattice is assigned a probability proportional to its Boltzmann weight $e^{-beta E}$ where $beta>0$ is a parameter that is interpreted physically as the inverse temperature $T$.

Given such a model, one question is to determine the "phase behavior", or what are the typical properties of the ensemble of configuration at a given $beta$, and how does this change with $beta$.

Considering at the moment just the Ising model with $h_i=0$, one might expect that for large $beta$, the typical configuration will tend to have lower energy, and hence have all its spins aligned to either all $+1$ or all $-1$. At small $beta$, all the Boltzmann factors tend to 1 and the typical configuration will have random spins. This rough argument is just meant to guide the intuition that there might be a phase transition between "mostly aligned" configurations to "mostly random" configurations at some special value of $beta$.

As it turns out, what happens is highly dependent on the dimensionality of the lattice.

The lower critical dimension $d_L$ of a model is the dimension below which no phase transitions can occur because even as $betarightarrowinfty$, there is not enough of an energy gain from ordering to create a phase with long-range correlations. In the ordinary Ising model (with all $h_i=0$), the lower critical dimension is 1, and hence at any finite $beta$, the average $langle S_iS_{j}rangle$ over configurations weighted with the Boltzmann distribution will approach zero (exponentially fast, even) as the distance between sites $i$ and $j$ approaches $infty$. For two dimensions and above, it can be shown that above a certain $beta_c$ (depending on dimension) this average will be finite in that long-distance limit.

Controversy

In the 1980's there was a controversy in the physics literature over the value of $d_L$ for the Random Field Ising model, a model where the $h_i$ are independent Gaussian random variables with zero mean and constant variance $epsilon^2$.

I'm not in a position to describe the history accurately, but I believe that there were physical arguments by Imry and Ma originally that $d_Lleq 2$, which were disputed when an amazing connection between random systems in $d$ dimensions and their pure counterparts in $d-2$ dimensions was found, known as the "Parisi-Sourlas correspondence". My understanding of Parisi-Sourlas is that it is based on a hidden supersymmetry in some series representation of the model which yields order-by-order agreement in the "epsilon expansions" of the two systems. Their argument was also made rigorous by Klein, Landau and Perez (MR). Based on this, since the Ising model has $d_L=1$, the RFIM was argued to have $d_L=3$ by various authors, though this was never a consensus view.

This controversy was settled by work of John Imbrie (MR) and later work of Bricmont and Kupianen (MR) building off his results that proved rigorously that $d_Lleq2$ in this system. Apparently terms like $e^{-1/epsilon}$ become important and the epsilon expansion breaks down in low dimensions, though I'm not sure if this has been made precise, and even today the RFIM is far from being completely understood.

Links between Riemann surfaces and algebraic geometry

This relationship is a very beautiful one.

Imagine a Riemann surface. There are different ways to introduce it, but since you gave kind of a reference point, let's just define it as a projective variety in the complex projective plane. Now the people call it a surface because it looks two-dimensional from a real point of view. You can also draw a picture of Riemann surface covering a sphere by projection onto a coordinate.

Now what could be an object of study of algebraic geometry? Why, certainly it should be some geometric object defined by algebraic means. Among the different ways to start learning algebraic geometry let's say we selected the abstract definition of an algebraic curve. To recap, this a a geometry locally defined by algebraic equations in some space so that the resulting manifold is one-dimensional.

These algebraic curves can be studied purely abstractly. You can, e.g., define algebraic forms on these, and prove various theorems relating to their geometry.

But the beautiful fact is that those are two sides of the same medal. That's right:

Every Riemann surface is a complex algebraic curve and every compact complex algebraic curve can be embedded into a projective plane and drawn as the Riemann surface.

There are lots of gems in this short statement. For example, as I said there is a way to count the algebraic forms in terms of inner geometry of algebraic curve. This gives some number, which could be 0, 1, 2, etc. On the other hand, if you draw a Riemann surface, you notice that it can be studied in topology and then it has the invariant called the number of handles which could also be 0 (sphere), 1 (torus), 2, etc. It turns out this is exactly the same thing though defined in a completely different way by a completely different branch of mathematics.

The whole algebraic geometry is, so to say, our attempt to make ourselves comfortable about this amazing connection between things we calculate (algebra) and things we draw (geometry).

fa.functional analysis - L_p norm balls for 12?

The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same shape. The L2 ball is shaped like a circle.

Hypothesis: For all p in the interval (1,2), there is q>2 such that the q-ball and the p-ball are similar. One further hypothesis is that this occurs precisely when p,q are Hölder conjugates.

I wasn't sure how to tag this.

gt.geometric topology - Kirby calculus and local moves

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and only if the links can be related by a sequence of Kirby moves and isotopies. This is pretty similar to Reidemeister's theorem, which says that two link diagrams correspond to isotopic links if and only if they can be related by a sequence of plane isotopies and Reidemeister moves.

Note however that Kirby moves, as opposed to the Reidemeister moves, are not local: the second Kirby move involves changing the diagram in the neighborhood of a whole component of the link. In "On Kirby's calculus", Topology 18, 1-15, 1979 Fenn and Rourke gave an alternative version of Kirby's calculus. In their approach there is a countable family of allowed transformations, each of which looks as follows: replace a $pm 1$ framed circle around $ngeq 0$ parallel strands with the twisted strands (clockwise or counterclockwise, depending on the framing of the circle) and no circle. Note that this time the parts of the diagrams that one is allowed to change look very similar (it's only the number of strands that varies), but still there are countably many of them.

I would like to ask if this is the best one can do. In other words, can there be a finite set of local moves for the Kirby calculus? To be more precise, is there a finite collection $A_1,ldots A_N,B_1,ldots B_N$ of framed tangle diagrams in the 2-disk such that any two framed link diagrams that give homeomorphic manifolds are related by a sequence of isotopies and moves of the form "if the intersection of the diagram with a disk is isotopic to $A_i$, then replace it with $B_i$"?

I vaguely remember having heard that the answer to this question is no, but I do not remember the details.

ag.algebraic geometry - Examples of rational families of abelian varieties.

David, I don't know if you are still interested in this, it's been over a year. I just stumbled upon your question in the depths of MO. I often found that Weil restrictions of elliptic curves give nice families of examples on which you can test things.

E.g. take a family of elliptic curves $y^2=x^3+t$ and Weil restrict it from ${mathbb Q}(i)$ to ${mathbb Q}$. Writing $x=x_1+x_2i$ and similarly for $y$ and $t$, expanding the equation and breaking it into real and imaginary parts, you get a family of 2-dimensional abelian varieties over ${mathbb Q}(t_1,t_2)$ given by two equations in a 4-dimensional space,
$$
y_1^2-y_2^2 = x_1^3 - 3x_1 x_2^2 + t_1, qquad
2y_1y_2 = x_1^2x_2 - 3x_2^3 + t_2.
$$
Alternatively, you fix the elliptic curve, but you let the extension vary with $t$ (e.g. ${mathbb Q}(t^{1/3})$), or both, and you also get interesting families.

The really nice thing is that as opposed to Jacobians, Weil restrictions are trivial to write down in terms of equations. Over the algebraic closure they are isogenous isomorphic to products of elliptic curves (making them boring), but for arithmetic applications they are interesting. There is a small extension of this construction, when you do not base change the elliptic curve but you "tensor it with a ${mathbb Z}-$module with a Galois action", which is not necessarily a permutation module. This is explained in Milne's paper "On the arithmetic of abelian varieties" (Invent. Math. 1972) section 2, and it is useful if you want to write down non-principally polarised examples.

pr.probability - Cover time of weighted graphs

A search for "algebraic connectivity" of a graph may be helpful, as well as the extensive literature on rapid mixing.

The problems mainly occur when the graph is nearly disconnected because different component-like sets have too few edges between them, or edges with weights too close to zero (which is the weight of a non-edge). If you make certain edges exponentially small, the cover time also becomes exponential.

Your adjacency matrix has Perron root and spectral radius $W$, and the usual bounds on mixing or covering time are in terms of the eigenvalue of second-highest magnitude, as a fraction of $W$. (Bipartite graphs are a special case, where $-W$ is also an eigenvalue.) The limiting distribution, in this case uniform, is the $W$ eigenvector, and is the only all-positive eigenvector. Convergence to uniform is exponential, but with base the ratio
of $W$ to other eigenvalues, by the spectral decomposition theorem. Sometimes you can actually get a clue to the worst component-like pieces from the positive and negative parts of large eigenvectors.

nt.number theory - Can select many disjoint pairs with prescribed differences from Z_n?

Suppose we have a sequence $d_i<2n$ for $i=1,ldots,n$ and we want to select $n$ disjoint pairs from $Z_p$, $x_i,y_i$ such that $x_i-y_i=d_i mod p$. Then how big $p$ has to be compared to $n$ to do this? I am primary interested on an upper bound on $p$. Is it true that there is always a $ple (1+epsilon)2n+O(1)$?

My comments. It is trivial that $pge 2n$ because all the numbers $x_i,y_i$ must be different and $d_1=1, d_2=2$ shows that this is not always enough. I also guess that it helps if $p$ is a prime, maybe the smallest prime bigger than $2n$ works which would answer the question.

order theory - Kuratowski closure-complement problem for other mathematical objects?

Here's a paper that might be of interest:

D. Peleg, A generalized closure and complement phenomenon, Discrete Math., v.50 (1984) pp.285-293.

Other than what's found in the above paper I do not know of any general theory or framework specifically aimed at organizing results similar to the Kuratowski closure-complement problem, i.e., those which involve starting with a seed object (or objects) and repeatedly applying operations to generate further objects of the same type in a given space.

Here's a general sub-question I thought of recently, that might be interesting to study:

"What's the minimum possible cardinality of a seed set that generates the maximum number of sets via the given operations?"

A few years ago I proposed a challenging Monthly problem (11059) that essentially asks this question for the operations of closure, complement, and union in a topological space. It does turn out there's a space containing a singleton that generates infinitely many sets under the three operations, but it's a bit tricky to find. I haven't looked into the question yet for other operations. As far as I know it hasn't been discussed yet in the literature (apart from the specific case addressed by my problem proposal).

pr.probability - If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

Yes, here's a nice and beautiful argument!

First you should draw a picture of axes a and b. You're asked to select uniformly a point in the square [0,1]x[0,1]. Now because of the symmetry (sic!) it's equivalent to choosing the points a and b uniformly in the triangle cut from the square by b > a.

So you're actually uniformly selecting a point inside triangle defined by lines a>=0, b<=1, 'b>=a'.

Now let's find the conditions to be able to make a triangle of short sticks. We should have a + (1-b) > b-a, b-a + (1-b) > a and b > 1 - b which indeed, as you say, boils down to

b > 1/2,  a < 1/2,  b-a < 1/2  

It remains to note that those lines create inside the big triangle a small triangle which is similar to big but with all lengths 1/2 of the big, so this small triangle has area of exactly 1/4 of original!

nt.number theory - Strongest known version of Baker's theorem

There is a big difference between linear forms in many logarithms and in two (or three) logarithms. The first case is covered in the archimedean case by the work of E. Matveev; Matveev's original works are hard even to specialists but there is a very nice survey [Yu. Nesterenko, Linear forms in logarithms of rational numbers, in Diophantine approximation (Cetraro, 2000), 53--106, Lecture Notes in Math., 1819, Springer, Berlin, 2003. MR2009829 (2004i:11082)]. The $p$-adic case was mostly done by Kunrui Yu. The best estimate for the case of two logarithms, which is of importance because of Tijdeman's application to Catalan's equation, is given in [M. Laurent, M. Mignotte et Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), no. 2, 285--321. MR1366574 (96h:11073)]. The latest news in the last direction (also in relation to Catalan's) are reviewed in [M. Mignotte, Linear forms in two and three logarithms and interpolation determinants. Diophantine equations, 151--166, Tata Inst. Fund. Res. Stud. Math., 20, Tata Inst. Fund. Res., Mumbai, 2008. MR1500224 (2010h:11119)]

dg.differential geometry - How to construct a vector fields with isolated zeros?

Your question isn't very well defined. A manifold on its own is not an object where constructions come by easily. But there is a generic way to construct vector fields with isolated zeros. Any vector field can be approximated by one with isolated zeros. This is a consequence of Sard's theorem. So start off with the zero vector field and choose any small random perturbation of that, and there you go.

If you want a more constructive answer you'll have to assume a more constructive situation. Like say if your manifold is triangulated, or has a handle decomposition, or a morse function.

Chapman describes the Morse situation so I'll give the triangulation situation.

The vector field has these properties:

There is a critical point at the barycentre of every cell in the triangulation. The vertices are repellors. The barycentres of the top-dimensional simplices are the attractors. A 1-simplex is a (1,n-1)-index critical point -- meaning there's two orbits approaching (along the 1-simplex) and an n-2-dimensional family of reverse orbits attracting. Etc. A j-simplex barycentre has a j-1-dimensional family of attracting orbits, and an n-j-1-dimensional family of reverse orbits attracting.

That isn't quite explicit as one needs an explicit smoothing of the triangulation to put this all together. But it gives you the idea.

st.statistics - Distribution of the sum of the $m$ smallest values in a sample of size $n$

This is following on from Douglas Zare's answer. While not an answer in its own right,it was getting too long for a comment. Briefly, we can hit the question with brute force and look for a generating function for the desired expected values.

So, put
$$f_j(y) = sum_{k=0}^{j-1} {n choose k} y^k (1-y)^{n-k}$$
so that using the notation of Douglas' answer, $g_j(x)=f_j(F(x))$, and the expectation of the $j$th order statistic is
$$ E_j := int_0^infty f_j(F(x)) dx $$

Put $G(y,z) = sum_{j=1}^n f_j(y) z^j$ where $z$ is a formal variable. By linearity we have
$$ sum_{j=1}^n E_j z^j = int_0^infty G(F(x),z) dx $$

We can try to write $G$ as a rational function in $y$ and $z$.
Expanding out and interchanging the order of summation gives
$$ eqalign{
G(y,z) & = sum_{j=1}^nsum_{k=0}^{j-1} {nchoose k} y^k (1-y)^{n-k} z^j \\
& = sum_{k=0}^{n-1} sum_{j=k+1}^n z^j {nchoose k} y^k (1-y)^{n-k} \\
& = sum_{k=0}^{n-1} left( sum_{j=1}^{n-k} z^j right) cdot z^k{nchoose k} y^k (1-y)^{n-k} \\
}
$$

Now
$$ eqalign{
sum_{j=1}^{M-1}jz^j
= z frac{d}{dz}sum_{j=0}^{M-1} z^j
& = z frac{d}{dz}left[frac{1-z^M}{1-z}right] \\
& = frac{z-z^{M+1}}{(1-z)^2} - frac{Mz^M}{1-z}
& = frac{z-z^M}{(1-z)^2} - frac{(M-1)z^M}{1-z}
} $$
so substituting this back in we get
$$ eqalign{
G(y,z)
& = sum_{k=0}^{n-1} left[ frac{z-z^{n-k+1}}{(1-z)^2} - frac{(n-k)z^{n-k+1}}{1-z} right] cdot z^k{nchoose k} y^k (1-y)^{n-k} \\
& = sum_{k=0}^{n-1} frac{z}{(1-z)^2}{nchoose k} (yz)^k (1-y)^{n-k} \\
& - sum_{k=0}^{n-1} frac{z^{n+1}}{(1-z)^2} {nchoose k} y^k (1-y)^{n-k} \\
& - sum_{k=0}^{n-1} frac{nz^{n+1}}{1-z} { {n-1} choose k } y^k(1-y)^{n-k} \\
& = frac{z}{(1-z)^2} left[ (1-y+yz)^n - (yz)^n right]
- frac{z^{n+1}}{(1-z)^2} (1-y^n)
- frac{nz^{n+1}}{1-z}
} $$

I guess that in theory one could plug this back in to obtain a "formula" for the generating function $sum_{j=1}^n E_j z^j$, but I can't see how that formula might then simplify to something calculable, unless $F$ has a rather special form.

algebraic groups - theorem of Borel and Tits

This is too long for a comment, but like others who have commented I don't expect to find anything like a "complete proof" of the Borel-Tits theorem written in English. Borel and Tits have each written at times in French, English, German, but their
serious joint work has been in French and is not especially hard to follow. The actual mathematics is difficult, however, requiring a lengthy technical treatment in their 1973 Annals paper on abstract homomorphisms. (Even so, they stopped short of allowing anisotropic groups even while suspecting there were would be some good results in that case; perhaps later work by people like Gopal Prasad sheds light there.)

The underlying question originates with work of Dieudonne, O'Meara, and others on the automorphisms of various classical groups over fields (then more general rings), including compact groups. On the other hand, Steinberg gave in 1960 a unified treatment of the automorphisms of finite Chevalley groups, which I imitated for infinite Chevalley groups in 1967 using more from algebraic groups. What Borel and Tits did was far more comprehensive and sophisticated than any of these concrete investigations. There was a short survey given in French, but also an earlier 1968 preview in English by both authors (not indexed in MathSciNet) which gives a short overview of the method of proof:

On "abstract" homomorphisms of simple algebraic groups, pp. 75-82, Proc. of the Bombay Colloquium in Algebraic Geometry, 1968. (This was published in book form for Tata Institute).

As the time lag between 1968 and 1973 suggests, the full proofs took a lot of work but achieved near-definitive results in a reasonably unified framework.

oa.operator algebras - Is there an i.c.c. nonamenable simple group that is inner amenable?

Hey Jon

So my initial thought would be no.

First, in full generality every group is virtually inner-amenable. Meaning that for any group $G$, the group $G times mathbb{Z}/2mathbb{Z}$ is inner amenable. In fact, any non-icc group is inner amenable just by taking the mean to be the counting measure on a finite conjugacy class, and 0 elsewhere.

Even if we restrict to icc groups then, for any icc group $G$, $Gtimes S_infty$ (or just choose the second group to be anything inner amenable) is still inner amenable.

And because the group is formed as a direct product there is not any way for the generators of $S_infty$ to sort of "slow down" the growth in the $G$ factor.

Now a final way to maybe make something out of this is to ask

"If $G$ is inner amenable and along with all of its quotients, then is there a growth contsraint."

This will get rid of the examples above. Amenable groups fall into this class, and I would be willing to bet that there are others as well (if anyone knows examples that would be nice) but I can't think of any on the spot.

AS for this class.... I have no idea.

co.combinatorics - association schemes, infinite schemes, semi-schemes, quasi-schemes

A (probably not very brilliant) partial answer to the first question.

The following definition of infinite association scheme semi-rings makes sense: A set $A_i,iinmathcal I$ (with $mathcal I$ not necessarily finite) of infinite matrices (indexed by $mathbb N$ or $mathbb Z$) with coefficients in ${0,1}$ containing the infinite identity matrix such that $sum_{iin mathcal I}A_i=J$ where $J$ denotes the infinite all $1$ matrix and
$A_iA_j=sum_{kinmathcal I}gamma_{i,j}^kA_k$ with $gamma_{i,j}^k$ in
$mathbb Ncuplbraceinftyrbrace$ and where the last sum is finite.
We require moreover the equalities $gamma_{i,j}^k=gamma_{j,i}^k$.
All operations are then well-defined on
the semiring $sum_{iinmathcal I}lambda_i A_i$ of finite sums with coefficients in
$mathbb R_{geq 0}cup{infty}$ and can even be extended to infinite sums (this is
useful since the Hadamard product identity, $J$ is an infinite sum).
Negative coefficients should be avoided.

I am not convinced of the interest of such a structure.

na.numerical analysis - trapezoidal rule error approximation. What if f''(x)/12n^2 doesn't work?

My first guess would be to use the Euler-Maclaurin summation formula (Wikipedia article). This proves, amongst other things, that the error goes down exponentially if the integrand is a periodic function on [0,1].

Added: After thinking about it a bit more, I'm wondering about some things. Firstly, the formula given in the question is not the trapezoidal rule (as promised in the title and suggested by the result for the error), but it is the rectangle rule which is only first order. Secondly, if the integrand has poles in [0,1] (that is, if $P_n(cos(pi x))=0$ for some $xin[0,1]$), then the error estimate becomes meaningless; in this case you probably need different techniques like complex analysis to prove anything. A final remark: perhaps you can use the elementary techniques explained in: Weideman, "Numerical integration of periodic functions: a few examples", Amer. Math. Monthly 109 (2002), no. 1, 21-36 (MathSciNet).

I think I need some more background in order to have further help. In particular, do you know anything about the polynomials $P_n$, and what kind of result do you hope to get?

mg.metric geometry - Compute the Centroid of a 3D Planar Polygon

In response to JBL's comment, I offer this answer merely to close out this topic. It has been effectively answered in the comments:
Simply project to xy and to xz and compute the centroid there. (One tiny wrinkle not addressed is if the polygon lies in a plane perpendicular to xy or to xz. But then simply chose the coordinate planes in which it does not lie.)

On the advice of Andrew Stacey, I am designating this answer "community wiki," and hope that someone will vote it up so it will no longer be bumped to the top of the active list by the MO background process.

soft question - Fundamental Examples

It is not unusual that a single example or a very few shape an entire mathematical discipline. Can you give examples for such examples? (One example, or few, per post, please)

I'd love to learn about further basic or central examples and I think such examples serve as good invitations to various areas. (Which is why a bounty was offered.)

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Related MO questions: What-are-your-favorite-instructional-counterexamples,
Cannonical examples of algebraic structures, Counterexamples-in-algebra, individual-mathematical-objects-whose-study-amounts-to-a-subdiscipline, most-intricate-and-most-beautiful-structures-in-mathematics, counterexamples-in-algebraic-topology, algebraic-geometry-examples, what-could-be-some-potentially-useful-mathematical-databases, what-is-your-favorite-strange-function; Examples of eventual counterexamples ;

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To make this question and the various examples a more useful source there is a designated answer to point out connections between the various examples we collected.

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In order to make it a more useful source, I list all the answers in categories, and added (for most) a date and (for 2/5) a link to the answer which often offers more details. (~year means approximate year, *year means a year when an older example becomes central in view of some discovery, year? means that I am not sure if this is the correct year and ? means that I do not know the date. Please edit and correct.) Of course, if you see some important example missing, add it!

Logic and foundations: $aleph_omega$ (~1890), Russell's paradox (1901),
Halting problem (1936), Goedel constructible universe L (1938), McKinsey formula in modal logic (~1941), 3SAT (*1970), The theory of Algebraically closed fields (ACF) (?),

Physics: Brachistochrone problem (1696), Ising model (1925), The harmonic oscillator,(?) Dirac's delta function (1927), Heisenberg model of 1-D chain of spin 1/2 atoms, (~1928), Feynman path integral (1948),

Real and Complex Analysis: Harmonic series (14th Cen.) {and Riemann zeta function (1859)}, the Gamma function (1720), li(x), The elliptic integral that launched Riemann surfaces (*1854?), Chebyshev polynomials (?1854) punctured open set in C^n (Hartog's theorem *1906 ?)

Partial differential equations: Laplace equation (1773), the heat equation, wave equation, Navier-Stokes equation (1822),KdV equations (1877),

Functional analysis: Unilateral shift, The spaces $ell_p$, $L_p$ and $C(k)$, Tsirelson spaces (1974), Cuntz algebra,

Algebra: Polynomials (ancient?), Z (ancient?) and Z/6Z (Middle Ages?), symmetric and alternating groups (*1832), Gaussian integers ($Z[sqrt -1]$) (1832), $Z[sqrt(-5)]$,$su_3$ ($su_2)$, full matrix ring over a ring, $operatorname{SL}_2(mathbb{Z})$ and SU(2), quaternions (1843), p-adic numbers (1897), Young tableaux (1900) and Schur polynomials, cyclotomic fields, Hopf algebras (1941) Fischer-Griess monster (1973), Heisenberg group, ADE-classification (and Dynkin diagrams), Prufer p-groups,

Number Theory: conics and pythagorean triples (ancient), Fermat equation (1637), Riemann zeta function (1859) elliptic curves, transcendental numbers, Fermat hypersurfaces,

Probability: Normal distribution (1733), Brownian motion (1827), The percolation model (1957), The Gaussian Orthogonal Ensemble, the Gaussian Unitary Ensemble, and the Gaussian Symplectic Ensemble, SLE (1999),

Dynamics: Logistic map (1845?), Smale's horseshoe map(1960). Mandelbrot set (1978/80) (Julia set), cat map, (Anosov diffeomorphism)

Geometry: Platonic solids (ancient), the Euclidean ball (ancient), The configuration of 27 lines on a cubic surface, The configurations of Desargues and Pappus, construction of regular heptadecagon (*1796), Hyperbolic geometry (1830), Reuleaux triangle (19th century), Fano plane (early 20th century ??), cyclic polytopes (1902), Delaunay triangulation (1934) Leech lattice (1965), Penrose tiling (1974), noncommutative torus, cone of positive semidefinite matrices, the associahedron (1961)

Topology: Spheres, Figure-eight knot (ancient), trefoil knot (ancient?) (Borromean rings (ancient?)), the torus (ancient?), Mobius strip (1858), Cantor set (1883), Projective spaces (complex, real, quanterionic..), Poincare dodecahedral sphere (1904), Homotopy group of spheres, Alexander polynomial (1923), Hopf fibration (1931), The standard embedding of the torus in R^3 (*1934 in Morse theory), pseudo-arcs (1948), Discrete metric spaces, Sorgenfrey line, Complex projective space, the cotangent bundle (?), The Grassmannian variety,homotopy group of spheres (*1951), Milnor exotic spheres (1965)

Graph theory: The seven bridges of Koenigsberg (1735), Petersen Graph (1886), two edge-colorings of K_6 (Ramsey's theorem 1930), K_33 and K_5 (Kuratowski's theorem 1930), Tutte graph (1946), Margulis's expanders (1973) and Ramanujan graphs (1986),

Combinatorics: tic-tac-toe (ancient Egypt(?)) (The game of nim (ancient China(?))), Pascal's triangle (China and Europe 17th), Catalan numbers (18th century), (Fibonacci sequence (12th century; probably ancient), Kirkman's schoolgirl problem (1850), surreal numbers (1969), alternating sign matrices (1982)

Algorithms and Computer Science: Newton Raphson method (17th century), Turing machine (1937), RSA (1977), universal quantum computer (1985)

Social Science: Prisoner's dilemma (1950) (and also the chicken game, chain store game, and centipede game), the model of exchange economy, second price auction (1961)

Statistics: the Lady Tasting Tea (?1920), Agricultural Field Experiments (Randomized Block Design, Analysis of Variance) (?1920), Neyman-Pearson lemma (?1930), Decision Theory (?1940), the Likelihood Function (?1920), Bootstrapping (?1975)

ag.algebraic geometry - A comprehensive overview of finite fields

I've read numerous introductions to finite fields, but I feel like my intuition about them is fairly lacking. Considering that finite fields are the the most "inert" objects in algebraic geometry, I think I could use a serious surge of perspective.

What I would like to read now is a comprehensive overview that tells me "everything I need to know" about how finite fields and their algebraic closures work, algebraically. I don't mind working out the proofs on my own if they are terse or absent; I'm just looking for quality and quantity of results. Hopefully some intense reading will help steep out some of my insecurities about characteristic p.

Can anyone recommend a single source for such an overview?

Thanks!

How to figure out the type of the bifurcation in a dynamical system?

Suppose we have a dynamical system

$dot{x} = f(x,r)$

in which x is a state variable and r is a bifurcation parameter. How to figure out which kind of bifurcation(s) (e.g. saddle-node, transcritical, pitchfork, hopf and etc) the system undergoes?

Edit 1: consider the space as 1D or 2D.

ct.category theory - What is the motivation for maps of adjunctions?

In Mac Lane, there is a definition of an arrow between adjunctions
called a map of adjunctions. In detail, if a functor $F:Xto A$ is left
adjoint to $G:Ato X$ and similarly $F':X'to A'$ is left adjoint to
$G':A'to X'$, then a map from the first adjunction to the second is a
pair of functors $K:Ato A'$ and $L:Xto X'$ such that $KF=F'L$,
$LG=G'K$, and $Leta=eta'L$, where $eta$
and $eta'$ are the units of the first and second adjunction. (The
last condition makes sense because of the first two conditions; also,
there are equivalent conditions in terms of the co-units, or in terms
of the natural bijections of hom-sets).

As far as I can see, after the definition, maps of adjunctions do not
appear anywhere in Mac Lane. Googling, I found this definition also
in the unapologetic mathematician,
again with the motivation of being an arrow between adjunctions.

But what is the motivation for defining arrows between adjunctions
in the first place? I find it hard to believe that the only
motivation to define such arrows is, well, to define such arrows...

So my question is: What is the motivation for defining a map of
adjunctions? Where are such maps used?

Besides the unapologetic mathematician, the only places on the web
where I found the term ''map of adjunctions'' were sporadic papers,
from which I was not able to get an answer to my question (perhaps
''map of adjunctions'' is non-standard terminology and I should have
searched with a different name?).

I came to think about this when reading Emerton's first answer
to a question about completions of metric spaces.
In that question, $X$ is metric spaces with isometric embeddings, $A$
is complete metric spaces with isometric embeddings, $X'$ is metric
spaces with uniformly continuous maps, $A'$ is complete metric
spaces with uniformly continuous maps, and $G$ and $G'$ are the
inclusions. Now, if I understand the implications of Emerton's answer
correctly, then it
is possible to choose left adjoints $F$ and $F'$ to $G$ and $G'$ such
that the (non-full) inclusions $Ato A'$ and $Xto X'$ form a map of
adjunctions. This made me think whether the fact that we have a map
of adjunctions has any added value. Then I realized that I do not
even know what was the motivation for those maps in the first place.

[EDIT: Corrected a typo pointed out by Theo Johnson-Freyd (thanks!)]