nt.number theory - b^(n-1)=-1 mod n

It's clear that b = n-1 with n even gives a solution. But there are many other solutions. Here are the solutions $(b,n)$ not of the form $(2k-1, 2k)$, with n less than or equal to 200, from MAPLE.

L := []: for n from 2 to 200 do 
for b from 1 to n-2 do 
if (b^(n-1) mod n) = n-1 then L := [op(L), [b,n]]; fi:
od: od:
L;
[[3, 28], [19, 28], [23, 52], [43, 52], [17, 66], [29, 66], [35, 66],
[41, 66], [19, 70], [59, 70], [27, 76], [31, 76], [31, 112], [47, 112], 
[99, 124], [119, 124], [49, 130], [69, 130], [11, 148], [27, 148], [87, 154], 
[131, 154], [7, 172], [123, 172], [63, 176], [79, 176], [95, 176], [127, 176],
[23, 186], [29, 186], [77, 186], [89, 186], [29, 190], [59, 190], [69, 190], 
[79, 190], [89, 190], [109, 190], [129, 190], [179, 190], [19, 196], [31, 196]]

For example, $3^{28-1} equiv -1 mod 28$, so the pair [3,28] is on the list.

I can't make sense of this output myself, but maybe someone else can?

Why sin and cos in the Fourier Series?

$1$. Mathematical reason.

There is one reason which makes the basis of complex exponentials look very natural, and the reason is from complex analysis. Let $f(z)$ be a complex analytic function in the complex plane, with period $1$.

Then write the substitution $q = e^{2pi i z}$. This way the analytic function $f$ actually becomes a meromorphic function of $q$ around zero, and $z = i infty$ corresponds to $q = 0$. The Fourier expansion of $f(z)$ is then nothing but the Laurent expansion of $f(q)$ at $q = 0$.

Thus we have made use of a very natural function in complex analysis, the exponential function, to see the periodic function in another domain. And in that domain, the Fourier expansion is nothing but the Laurent expansion, which is a most natural thing to consider in complex analysis.

Here you can make suitable modifications when $f$ is periodic in some domain which is not the whole complex plane. In that case in the $q$-domain, $f$ will be analytic in some circle around $0$, and you can use that to get a Laurent expansion. The modular forms for instance are defined only in the upper-half plane, and what we get here is called the $q$-expansion.

However from the point of view of Real analysis, $L^p$-spaces etc., any other base would do just as fine as the complex exponentials. The complex exponentials are special because of complex analytic reasons.

$2$. Physical reason.

There are historical reasons also. For instance, in electrical engineering or theory of waves, it is very useful to decompose a function into its frequency components and this is the reason for the great importance of Fourier analysis in electrical engineering or in electrical communication theory. The impedance offered by circuits depends on the frequency of the signal that is being fed in, and a circuit consisting of capacitors, inductors etc. react differently to different frequencies, and thus the sine/cosine wave decomposition is very natural from a physical point of view. And it was from this context, and also the theory of heat conduction, that Fourier analysis developed up.

sg.symplectic geometry - Floer's space closed under products?

Floer chooses $epsilon_k$ so that this $epsilon$ space is dense in $L^2$ (this should be equivalent to dense in $C^infty$, since the latter is dense in $L^2$, and Floer's $epsilon$ space sits in $C^infty$).

To show that this subspace is dense in $L^2$, it suffices to show that one can approximate
indicator functions.
For this, he needs the $epsilon_k$ to go to zero very fast. His explicit
construction in Lemma 5.1 (of the "unregularized gradient flow" paper) is to take a fixed cut-off function $beta$,
and approximate the characteristic function of a rectangle. The
approximation to a characteristic function is going to be a product of
terms that behave like $beta(x/delta)$ (with a better approximation
as $delta rightarrow 0$). Thus, the behaviour of the
$epsilon$-norm is going to be roughly:
[
sum_{k=0}^infty epsilon_k delta^{-k} a_k,
]
where $a_k = sup | D^k beta |$. We need this to converge for each $delta > 0$.
Floer takes $epsilon_k = (a_k k^k)^{-1}$. In particular, then, these constants are going to
$0$. Following this argument, it seems we can take the $epsilon_k$ to be on the order of $1/k!$.

Note that if the sequence $epsilon_k$ is not summable, we expect the
space to be very small. In particular, consider this norm on a
compact interval, say $[-pi, pi]$. Then, cos(x) is not in the
space.

The Floer $epsilon$ space forms a Banach algebra if $epsilon_k$ decays faster than $1/k!$.
Then,
[
sum epsilon_k |D^{(k)}(fg)|
le sum_{k=0}^infty sum_{l=0}^{k}
epsilon_k | D^{(l)} f|
|D^{(k-l)} g | binom{k}{l}
= sum_{l=0}^infty |D^{(l)} f| sum_{p=0}^{infty} binom{l+p}{p} epsilon_{p+l} | D^{(p)}g|
]
When $epsilon_{p+l} binom{l+p}{p}
le epsilon_p epsilon_l$, we are then in business.
In particular, this works for Floer's original construction.

reference request - Random noncrossing chords of a circle

Suppose you generate random chords of a circle, with endpoints selected uniformly over the circumference, rejecting any chord that crosses a previously generated chord.
The disk is then partitioned into regions bounded by chords alternating with circular arcs.
For example, here are $n{=}100$ random noncrossing chords, with a region bounded by 5 chords highlighted
(in green).


I am interested in the statistics of the structure of the dual trees for these regions.
Assign each region a node,
and connect two nodes by an edge if they share a chord. In the example above, the highlighted
region's node has degree 5.
Example questions: What is the expected maximum degree of a node for $n$ chords?
Making a max-degree node the root, what is the expected height of the tree?
(In the example above, the height is 21.)
Etc.

Has anyone encountered this model before?
Or a model sufficiently analogous to help establish these statistics?
Thanks for any pointers!

Edit. Many thanks for the wealth of information provided by the community!
I have not yet absorbed all the information in the cited papers,
but so far I have not found this specific question answered (although it is likely implied, perhaps
in the papers they cite):
What is the expected maximum degree of a node as $n rightarrow infty$? What brought me to this
topic in the first place is that I wondered if it might be near 3.

dg.differential geometry - Can all G-connections on a Riemann surface X be induced by maps from X to G

José Figueroa-O'Farrill has already pointed out one necessary condition, namely that your connection must be flat. The remaining condition is that the monodromy should be trivial. In what follows $X$ is any connected smooth manifold, not necessarily a surface, and $G$ is any Lie group.

Let's first consider the analogous situation when $G$ is replaced by $mathbb{R}$. You can think of a one-form $omegain Omega^1(X;mathfrak{g})$ as potentially being the derivative of a map $Xto G$, just as a one-form $etain Omega^1(X;mathbb{R})$ is potentially the derivative of a map $Xto mathbb{R}$. We want to know when these really are the derivative of some map, i.e. when we can integrate these forms. (You mentioned the exponential map, but I think integration is the right metaphor here.)

There is a local obstruction, namely that if $eta$ is to be integrable (meaning $eta=df$ for some $f$) it must be closed, meaning $deta=0$; the Poincaré lemma tells us this is a sufficient condition for $eta$ to be locally integrable. Then there is also a global condition, that the integral of $eta$ around every closed loop must be 0 (unlike $dtheta$ on the circle, which has integral $2pi$); Stokes' theorem tells us this is a necessary condition for $eta$ to be globally integrable. If we have these conditions, recovering the map $f$ from $eta$ is easy; just write $f(p)=int_ast^p eta$, which is well-defined by the above two conditions.

Now let's try to do the same for $mathfrak{g}$-valued one-forms. Start with a connection on the trivial $G$-bundle $Xtimes G$ with connection form $omegain Omega^1(X;mathfrak{g})$. We've talked about the connection being flat, which means that $domega+frac{1}{2}[omega,omega]=0$; but what does that have to do with flatness or integrability? Well, you can show that $domega+frac{1}{2}[omega,omega]$ measures the Lie bracket of two horizontal vector fields, or rather measures the vertical part of the Lie bracket. Thus if this vanishes, the bracket of two horizontal vector fields is horizontal. By the Frobenius integrability theorem, this implies that the horizontal distribution of the connection is integrable; another way to say this is that parallel transport is locally well-defined. Now pick a basepoint $ast$ and restrict your attention to a small neighborhood $U$ of $ast$. Since parallel transport is well-defined on $U$, we get a function $Tcolon Uto G$ by saying that the parallel transport from $ast$ to $u$ (along any path) is multiplication by $T(u)$.

Key point: if you pull back the tautological form on $G$ by this "parallel transport" map $T$, the form you get is the same as your original $omega$!

What this tells us is that if a flat connection on $Xtimes G$ comes from a map $f:Xto G$, then you can recover $f$ by looking at the parallel transport of the connection. (The analogue is that if $eta=f^ast(dx)$ for some $fcolon Xto mathbb{R}$, you can recover $f$ by integrating $eta$, also known as the fundamental theorem of calculus.) Thus flatness, in the form of the Maurer-Cartan equation, is the local obstruction to integrability; here the Frobenius integrability theorem plays the role that the Poincaré lemma does in the real case. To prove the key point is really just a matter of definitions: think about the correspondence between a connection, its connection form, and its parallel transport.

In particular, this tells us that parallel transport must be not just locally well-defined, but globally well-defined (meaning independent of the path), since transport along any path from $ast$ to $p$ is always multiplication by $f(p)in G$. The monodromy of a flat connection is the map $pi_1(X,ast)to G$ which sends a loop to the parallel transport around that loop, and so another way to say "parallel transport is globally well-defined" is that the monodromy is trivial.

This can all be summed up by saying that if $X$ is simply connected, we have an on-the-nose bijection $C^infty(X,G)longleftrightarrow {omegain Omega^1(X;mathfrak{g})vert domega+frac{1}{2}[omega,omega]=0}$. (Here on the left we assume the maps take the basepoint $astin X$ to $1in G$.) If $X$ has fundamental group, we need to add on the right side the additional condition that the monodromy of $omega$ be 0. This is hard to write down just in terms of $omega$, but for the corresponding connection it is just that parallel transport is totally path-independent.

examples - Experimental Mathematics

A new look

Now as the question is five years old and there are certainly more examples of mathematical advances via computer experimentation of various kinds, I propose to consider contributing new answers to the question.

Motivation

I am aware about a few such cases and I think it will be useful to gather such examples together. I am partially motivated by the recent polymath5 which, at this stage, have become an interesting experimental mathematics project. So I am especially interested in examples of successful "mathematical data mining"; and cases which are close in spirit to the experimental nature of polymath5. My experience is that it can be, at times, very difficult to draw useful insights from computer data.

Summary of answers according to categories

(Added Oct. 12, 2015)

To make the question a useful resource (and to allow additional answers), here is a quick summery of the answers according to categories. (Links are to the answers, and occasionally to an external link from the answer itself.)

1) Mathematical conjectures or large body of work arrived at by examining experimental data - Classic

The Prime Number Theorem; Birch and Swinnerton-Dyer conjectures; Shimura-Taniyama-Weil conjecture; Zagier's conjectures on polylogarithms; Mandelbrot set; Gosper Glider Gun (answer), Lorenz attractor; Chebychev's bias (asnwer) ; the Riemann hypothesis; the discovery of the Feigenbaum constant; (related) Feigenbaum-Coullet-Tresser universality and Milnor's Hairiness conjecture (NEW); Solving numerically the so-called Fermi--Pasta--Ulam chain and then of its continuous limit, the Korteweg--de Vries equation

2) Mathematical conjectures or large body of work arrived at by examining experimental data - Current

"Maeda conjecture"; the work of Candès and Tao on compressed sensing; Certain Hankel determinants; Weari-Phelan structure; the connection of multiple zeta values to renormalized Feynman integrals; Thistlethwaite's discovery of links with trivial Jones polynomial; The Monstrous Moonshine; (NEW:) McKay's account on experimentation leading to mysterious "numerology" regarding the monster. (link to the answer); Haiman conjectures on the quotient ring by diagonal invariants

3) Computer-assisted proofs of mathematical theorems

Kepler's conjecture ; a new way to tile the plane with a pentagon: advances regarding bounded gaps between primes following Zhang's proof; Cartwright and Steger's work on fake projective planes; the Seifert-Weber dodecahedral space is not Haken; the four color theorem, the proof of the nonexistence of a projective plane of order 10; Knuth's work on a class of projective planes; The search for Mersenne primes; Rich Schwartz's work; The computations done by the 'Atlas of Lie groups' software of Adams, Vogan, du Cloux and many other; Cohn-Kumar proof for the densest lattice pacing in 24-dim; Kelvin's conjecture;

4) Computer programs that interactively or automatically lead to mathematical conjectures.

Graffiti

5) Various computer programs which allow proving automatically theorems or generating automatically proofs in a specialized field.

Wilf-Zeilberger formalism and software; FLAGTOOLS

6) Computer programs (both general purpose and special purpose) for verification of mathematical proofs.

The verification of a proof of Kepler's conjecture.

7) Large databases and other tools

Sloane's online encyclopedia for integers sequences; the inverse symbolic calculator.

8) Resources:

Journal of experimental mathematics; Herb Wilf's article: Mathematics, an experimental science in the Princeton Companion to Mathematics, genetic programming applications a fairly comprehensive website experimentalmath.info
; discovery and experimentation in number theory; Doron Zeilberger's classes called "experimental mathematics":math.rutgers.edu/~zeilberg/teaching.html;
V.I. Arnol'd's two books on the subject of experimental mathematics in Russian, Experimental mathematics, Fazis, Moscow, 2005, and Experimental observation of mathematical facts, MCCME, Moscow, 2006

Answers with general look on experimental mathematics:

Computer experiments allow new avenues for productive strengthening of a problem (A category of experimental mathematics).

---

Bounty:

There were many excellent answers so let's give the bounty to Gauss...

Related question: Where have you used computer programming in your career as an (applied/pure) mathematician?, What could be some potentially useful mathematical databases?; Results that are easy to prove with a computer, but hard to prove by hand ; What advantage humans have over computers in mathematics?

nt.number theory - S-unit equation and small sets of places

Let $K$ be a number field, and let $S_x$ denote the set of primes of norm at most $x$. Is it possible to find a smaller set of places $T_xsubset S_x$ so that a lot of the solutions of the $S_x$-unit equation $a+b=1$ for $a,bin S_x$ are solutions of the $T_x$-unit equation?

Here's a possible precise statement (although I'd be interested in other formulations as well): Does there exist a constant $0<c<1$, depending on $K$ (but not $x$), so that for each $x$, there is a $T_xsubset S_x$ with $|T_x|lesqrt{|S_x|}$ so that the number of solutions to the $T_x$-unit equation is at least $c$ times the number of solutions of the $S_x$-unit equation?

I'm interested in this mostly by analogy: Bellabas and Gangl have a bound for the set of places of a number field one must check in order to compute $K_2$ of the ring of integers. It would be interesting to know if one could at least get a pretty good approximation for $K_2$ by looking at a much smaller set of places.

nt.number theory - A parametrization of Heronian triangles

Let $a,b,c$ be integers which are the sides of a triangle with integral area, a so called Heronian triangle. This website attributes to Gauss the result that there must then exist integers $m,n,p,q$ such that

$a = mn(p^2+q^2)$

$b = (mp)^2+(nq)^2$

$c = (m+n)(mp^2-nq^2)$

(where I left out a $4pq$ factor designed to make the radius of the circumscribed circle integral as well). It's not hard to see that the triangle defined by these formulas is indeed Heronian, however I could neither prove nor find a reference for the fact that this parametrization is exhaustive.

Can someone do one of these two things?

Thanks!

(Note: I'm communicating this question on behalf of my dad, who is really the person who looked into that but is not easily capable of asking it himself over here. I may be slow to respond on his behalf if questions come up).

tag removed - Is it true that all the "irrational power" functions are almost polynomial ?

Hello all, the $Delta$ operator on functions $mathcal{N} to mathbb{R}$
(where $mathcal N$ denotes $lbrace 1,2, ldots , rbrace$ )defined by
$Delta(f)(n)=f(n+1)-f(n)$ is well-known and
it is not very hard to show by induction that
$f$ is a polynomial of degree $leq k$ iff $Delta^{k+1}(f)$ is identically zero, where
$Delta^{k+1}$ denotes $Delta$ iterated $k+1$ times. Now I say that
a function $f : mathcal{N} to mathbb{R}$ is "almost polynomial" iff
$Delta^{k}(f)$ is a bounded function for some $k$.

My question is : let $lambda >0$ be a non-integer, and let
$f(n)=n^{lambda}$. Is it true that $f$ is almost polynomial ?

botany - Why do plants' leaves become enlarged in low light areas?

I have no idea of the biological mechanisms behind what I'm about to propose. Consider this a hypothesis.

If the purpose of the leave is to collect sunlight for photosynthesis, then the behavior you mention is predictable. I order of trying to maintain a healthy flux of light in dim conditions, the plant adapts by increasing the area of its leaves. Once back in a well-lit environment, the need for larger leaves fades and new leaves are thus smaller, but still catch as much light. Of course, there must be limits to the extent of leaf area variation that one plant can handle.

As for the stem structure, I can only guess it's because plants spend more energy on leaf building than stem maintenance when under such stresses.

fa.functional analysis - Monotone Lipschitz embedding ?

In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the inequality d(u,v) <= || Ku - Kv || <= Ld(u,v) for all u and v in X.
Now, suppose X = l_1 (in this case, L = 2 is best possible). I have the following

Conjecture: Let K: l_1 --> c_0 be a Lipschitz embedding. Then K cannot be monotone w.r.t. the natural duality pairing (.,.) between l_1 and c_0,
i.e., there are some u and v in l_1 such that (u - v, Ku - Kv) < 0.

algebraic groups - Image of a hyperspecial subgroup hyperspecial?

If by "surjective" you mean surjective in the usual sense (for example on $overline{F}$-points) then maybe you have a problem, because $G_1(F)$ may not surject onto $G_2(F)$. So for example $SL(2)$ surjects onto $PGL(2)$ but if $R$ is the integers of $F$ then $SL(2,R)$ is hyperspecial max compact but its image in $PGL(2,F)$ isn't (it's not even maximal, as $PGL(2,R)$ strictly contains the image of $SL(2,R)$).

However if $G_1to G_2$ is, say, a $z$-extension, then (by definition) the kernel is central in $G_1$ and has no $H^1$, so the long exact sequence shows $G_1(F)to G_2(F)$ is surjective. Moreover, if I've got things right, then I think that $G_1$ unramified forces the kernel to be unramified, and if you take a smooth integral model of $G_1$ with $G_1(R)$ equal to the hyperspecial you thought of, then the quotient of this model of $G_1$ by the Zariski closure of the kernel will also be unramified, and the same cohomology argument shows that $G_1(R)$ surjects onto $G_2(R)$, so in this case you win.

What are some of the big open problems in 3-manifold theory?

ADDED (29 May, 2013)

As has been pointed out in the comments, there has been great progress since this answer was first written, and the conjectures below have now been proved, thanks to ground-breaking work of Agol, Kahn--Markovic and Wise. Here's a brief summary of some of the highlights. (Shameless self-promotion: see this survey article for too many further details, including definitions of some of the terms.)

1. Haglund--Wise define the notion of special (non-positively curved) cube complex. If a closed hyperbolic 3-manifold $M$ is homotopy equivalent to a special cube complex then $M$ satisfies L (largeness, defined below).




2. Agol proves that if $M$ is homotopy equivalent to a special cube complex then $M$ also satisfies VFC (the Virtually Fibred Conjecture, also defined below).




3. Kahn--Markovic prove SSC (the Surface Subgroup Conjecture, also defined below), using mixing properties of the geodesic flow. In fact, they construct enough surfaces to show that $M$ is homotopy equivalent to a cube complex.




4. Wise proves (independently of Kahn--Markovic) that if $M$ contains an embedded, geometrically finite surface then $M$ is special.




5. Agol uses a very deep theorem of Wise (the Malnormal Special Quotient Theorem) to prove a conjecture (also of Wise), which states that word-hyperbolic fundamental groups of non-positively curved cube complexes are special. All the properties below follow.

It's quite a story, and many other names have gone unmentioned. There were also very important contributions by Sageev (who's thesis initiated the programme of using cube complexes to attack these problems), Groves--Manning, Bergeron--Wise, Hsu--Wise and another very deep paper of Haglund--Wise. To extend these results to the cusped hyperbolic case you need results of Hruska--Wise and Sageev--Wise. Finally, it turns out that similar results hold for all non-positively curved 3-manifolds, a result established by Liu and Przytycki--Wise.

---

Let $M$ be a finite-volume hyperbolic 3-manifold. (Some of these extend, suitably restated, to larger classes of 3-manifolds. But it follows from Geometrisation that the hyperbolic case is often the most interesting. These are all trivial or trivially false in the elliptic case, for example.)

The Surface Subgroup Conjecture (SSC). $pi_1M$ contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. (Recently proved by Kahn and Markovic.)

The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space with an embedded incompressible subsurface.

Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $widehat{M}$ with $b_1(widehat{M})geq 1$.

Virtually infinite first Betti number (VIFB). $M$ has finite-sheeted covering spaces $widehat{M}_k$ with $b_1(widehat{M}_k)$ arbitrarily large.

Largeness (L). $pi_1(M)$ has a finite-index subgroup that surjects a non-abelian free group.

The Virtually Fibred Conjecture (VFC). $M$ has a finite sheeted cover that is homeomorphic to the mapping torus of a (necessarily pseudo-Anosov) surface automorphism. This is false for graph manifolds. There are fairly easy implications

$LRightarrow VIFB Rightarrow VPFB Rightarrow VHC Rightarrow SSC$.

Also, a fortiori,

$VFCRightarrow VPFB$.

Recently, Daniel Wise announced a proof that $VHCRightarrow VFC$. His proof also shows that, if $M$ has an embedded geometrically finite subsurface, then we get $L$ and other nice properties.

This list is similar to the one that Agol links to in the comments. Also, I suppose it's exactly what Daniel Moskovich meant by 'The Virtually Fibred Conjecture, and related problems'. I thought some people might be interested in a little more detail.

---

Paul Siegel asks in comments: 'Would it be correct to guess that the "virtually _ conjecture" problems can be translated into a question about the large scale geometry of the fundamental group?'

Certainly, it's true that most of these can be translated into an assertion about how (some finite-index subgroup of) $pi_1M$ splits as an amalgamated product, HNN extension or, more generally, as a graph of groups. The equivalence uses the Seifert--van Kampen Theorem in one direction, and something like Proposition 2.3.1 of Culler--Shalen in the other. Rephrased like this, some of the above conjectures turn out as follows.

The Virtually Haken Conjecture (VHC). $M$ has a finite-sheeted covering space $widehat{M}$ such that $pi_1(widehat{M})$ splits.

Virtually positive first Betti number (VPFB). $M$ has a finite-sheeted covering space $widehat{M}$ such that $pi_1(widehat{M})$ splits as an HNN extension.

Largeness (L). $M$ has a finite-sheeted covering space $widehat{M}$ such that $pi_1(widehat{M})$ splits as a graph of groups with underlying graph of negative Euler characteristic.

The Virtually Fibred Conjecture (VFC). $M$ has a finite-sheeted covering space $widehat{M}$ such that $pi_1(widehat{M})$ splits can be written as a semi-direct product

$pi_1(widehat{M}) cong Krtimesmathbb{Z}$

with $K$ finitely generated. (Here we invoke Stallings' theorem that a 3-manifold whose fundamental group has finitely generated commutator subgroup is fibred.)

I don't think I know a way to rephrase $VIFB$ in terms of splittings of $pi_1$.

Often, when people say 'the large scale geometry of $pi_1$' they're talking about properties that are invariant under quasi-isometry. I'm really not sure whether these splitting properties (or, more exactly, 'virtually having these splitting properties') are invariant under quasi-isometry. Perhaps something like the work of Mosher--Sageev--Whyte does the trick?

A good book for history of biology/biotechnology for lay people

I have many friends who are interested in Biology and want to know more about the subject in general (like a history of biology, from Darwin's theory, to DNA structure discovery, to the human genome project). Of course, I cannot suggest to them to read Alberts or Lenninger. Do you know whether such a book exist? I guess that a book that covers most fields of biology cannot be compiled, but even more focused book would do.

Let me try to narrow it down: something like the greatest discoveries in the field of biology (like this article) would be an interesting book to read.

I am not sure how appropriate this question is for SE, but I am sure that I will get the best answer here. Besides, it would be great if lay people can be more excited about biology and contribute to the site growth.

modular forms - Free subquotient of Galois representations coming from Hida theory

$newcommandT{mathbf{T}_{mathfrak{m}}}$
$newcommandQ{mathbf{Q}}$
$newcommandm{mathfrak{m}}$
$newcommandF{mathbf{F}}$
$newcommandFrob{mathrm{Frob}}$
$newcommandrhobar{overline{rho}}$
$newcommandeps{epsilon}$

First, as Professor Emerton mentions, the construction of $L^{+}$ you gave
is not necessarily free over $T$. Thus, I will interpret your question
as asking the following: does there exist an exact sequence:

$$0 rightarrow L^{+} rightarrow (T)^2 rightarrow L^{-} rightarrow 0$$
of $T[G_{Q_p}]$-modules where $L^{+}$ and $L^{-}$ are free
$T$-modules of rank one.

( Edit Perhaps this extra remark might be useful.
Suppose that $L = (T)^2$ admits a free rank one quotient
$L^{-}$. Since $L^{-}$ is free, it admits a section
$L^{-} rightarrow L$, and hence the kernel
$L^{+}$ of $L rightarrow L^{-}$ is also free. Thus
the existence of a free rank one quotient asked
for in the question is equivalent
to the existence of the exact sequence above.)

The answer to this question, in general, is no. The following argument
is implicitly contained in papers of Wiese on the failure of multiplicity
one and weight one forms.

The action of $G_{Q_p}$ on $L^{+}$ is unramified and so acts
via $G_{mathbf{F}_p}$. Thus $Frob_p$ acts on a basis vector
as multiplication by some element of $T$. Since $T$ is determined
by its action on classical eigenforms,
one may identify this element with the Hecke operator $U$. In particular,
$U in T$ (it wasn't clear whether your $T$ included $U$ or not).

The exact sequence remains exact after tensoring with $T/m$,
for dimension reasons. It follows that the sequence is split as a sequence
of $T$-modules. Hence it remains exact after quotienting out by
any ideal of $T$.

Suppose that $rhobar: G_{Q} rightarrow mathrm{GL}_2(F_p)$ is
irreducible and modular (mod-$p$) of weight $1$. Suppose, moreover, that
$rhobar(Frob_p)$ acts by a scalar $lambda$.
Associated to $rhobar$ is a mod-$p$ weight $1$ form
$f = sum a_n q^n in F_p[[q]]$. If $A$ is the Hasse invariant, then
then $Af$ and $f^p$ are both mod-$p$ modular forms of weight $p$. One can check
that all elements of the $F_p$-vector space ${Af,f^p}$ are eigenvalues for all the Hecke operators
$T_l$ for $(l,p) = 1$, but the operator $T$ (and so $U$, which is the
same as $T$ in weight $> 1$) satisfies
$(U - lambda)^2 = 0$ but does not act by a scalar. Since $U$ acts
invertibly on this vector space, it gives rise to a surjective map:
$$T rightarrow F_p[eps]/eps^2,$$
where the image of $T_l$ lands in $F_p$ for all $(l,p) = 1$, but $U$
does not act by a scalar. Let $I$ be the kernel.

The Galois representation on $(T)^2/I simeq (F_p[eps]/eps^2)^2$
is equal to $rhobar oplus rhobar$. This follows from a result of Boston-Lenstra-Ribet,
since $T_l$ is acting by a scalar for each $(l,p) = 1$. It follows, by assumption, that
the action of $G_{Q_p}$ on $L^{+}/I L^{+} simeq T/I$ must also be trivial,
because this is a sub-representation of $rhobar oplus rhobar$. On the other hand, as we have seen, the action of Frobenius on $L^{+}$ and thus $L^{+}/I L^{+} = T/I$ is given
by $U$, which is acting non-trivially $T/I$ by the construction of $I$. This is a contradiction.

Such representations $rhobar$ exist (for example, with $p = 2$, and level $Gamma_0(431)$) as mentioned
in Professor Emerton's answer.

genetics - What is the advantage of circular DNA in bacteria?

To expand a little bit the other answer, I would also add that bacteria can have other (usually circular) DNA segments aside from their main chromosome. These are called plasmids and are double stranded molecules of DNA that can replicate autonomously.

Plasmids often carry genes that allow an organism to survive in certain conditions, for instance they could carry the resistance to an antibiotic, or the gene that encodes for a specific nutrient that may be absent in the environment and so on.

As the other answer says, plasmids can be transferred horizontally between bacteria in a process called bacterial coniugation, and that is made possible by the presence of a specific plasmid, called the F-plasmid in the donor. The F-plasmid encodes, amongst other things, for the F-pilus protein pilin, that allows the formation of the pilus necessary for DNA transfer.

Because of their properties, plasmids are widely used in laboratory as vectors, to transfer genetic material to cells in order to give them specific "abilities" (e.g. you could insert a gene that encodes for a certain receptor normally not expressed by the cells to allow its expression and test its function).

Finally, it is worth remembering that plasmids can also be find in eukariotes (e.g. in yeast).

ag.algebraic geometry - Normal Varieties

I would like to risk an answer that does not use the language of algebraic geometry. For a pair (complex analytic variety $X$; closed analytic subvariety $Y$), $U=Xsetminus Y$,
there exists a triangulation such that $Y$ is a subcomplex (see, for example Triangulations of algebraic sets - Hironaka 1974, can be found with google books). In other words $X$ is a simplicial complex, and $Y$
is a subcomplex. Now, if $X$ is normal its singularities are in real codimension at least $4$. I.e. $X$ is a $PL$ manifold in codimension $4$.

In order to show that the fundamental group of $Xsetminus Y$ surjects onto the fundamental group of $X$, it is sufficient to show that every loop in $X$ can be homotoped into $Xsetminus Y$. Since $Y$ it is contained in the simplicial subcomplex of codimension $2$ it is enough to show that any loop in $X$ can be homotoped so it does not touch any simplex of codim $2$, but this is true for every $PL$ space that is a manifold in codim $2$.

lo.logic - Presburger Arithmetic

Presburger introduced his arithmetic in 1929 the paper was translated into English in 1991. Here is the citation to this paper:

M. Presburger. Ueber die Vollstaendigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. C.R. du I Congr. des Math. des pays Slaves, Warszawa, 1929, pp.92-101

Here is the english translation:

On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation
Mojżesz Presburger; Dale Jabcquette
History and Philosophy of Logic, 1464-5149, Volume 12, Issue 2, 1991, Pages 225 – 233

The pdf here which is mentioned in Jason Dyer's comment to the original question states that in the paper above the system is used to prove its own consistency.

He reduced all statements in his arithmetic to quantifier free statements. To do this he add to extend the system by introducing modular equivalence. The result was a reduction of every statement to the quantifier free form. This led to an algorithm for deciding every statement. There is in fact bounds on the efficiency of the decision algorithm algorithm. It is greater than double exponential and less than triple exponential. For the lower bound see:

M. J. Fischer, M. O. Rabin. Super-Exponential Complexity of Presburger Arithmetic. "Proceedings of the SIAM-AMS Symposium in Applied Mathematics", 1974, vol. 7, pp.27-41

For the upper bound see:

Derek C. Oppen: A 2^2^2^pn Upper Bound on the Complexity of Presburger Arithmetic. J. Comput. Syst. Sci. 16(3): 323-332 (1978)

For there to be inconsistency there would have to be a finite set of inconsistent modular statements. Because of this it is plausible to me that the original paper used the extended system to prove its own consistency.

mathematical writing - When to split/merge papers?

I ran into the following problem. I have some lengthy paper in which I develop some theory to attack some problem. While I was working on this paper, I got some nice result which might interest a bigger audience and can stand by itself (that is, it is interesting out of context of the big work, and has a fairly short proof).

The problem: should I write one long paper or two papers, one long and the other short? If I write just one paper, the short result will strengthen the paper, and it will be, perhaps, more whole. On the other hand, for someone that is only interesting in the short one, it will be better if it appears in a separated paper.

sg.symplectic geometry - Kuranishi structures vs polyfolds

Kuranishi models are a traditional - and beautiful - technique for describing the local structure of moduli spaces cut out by non-linear equations whose linearization is Fredholm. A more elaborate version, "Kuranishi structures", are used by Fukaya-Oh-Ohta-Ono (FOOO) and Akaho-Joyce to handle transversality for moduli spaces of pseudo-holomorphic polygons with Lagrangian boundary conditions, and the compactifications of these spaces. FOOO's book is the result of a decade of dedicated thought by a superb team, but few have assimilated it (I certainly haven't).

Polyfolds, Hofer's "infrastructure project", are designed with the severe demands of symplectic field theory (SFT) foremost in mind. This is a more radical rethink of how to handle transversality, whose aims include absorbing the difficulties of the lack of canonical coordinates when gluing Morse-Floer trajectories. (What difficulties? Try to prove that the moduli space of unparametrized broken gradient flow-lines of an ordinary Morse function is a smooth manifold with corners, and you'll find out.) Several papers into the project, it's still not completely clear how efficiently it will work in applications, especially those outside SFT. I'd hope that polyfolds will help us set up Cohen-Jones-Segal Floer homotopy-types, for instance - but there may still be severe difficulties. I've also never heard a compelling argument that Kuranishi structures are insufficient for SFT.

This paper of Cieliebak-Mohnke suggests an intriguing alternative approach.

My view would be that these mammoths are worth chasing only if you have very clear aims in mind. There are many excellent problems in symplectic topology that don't need such gigantic foundations. If you're interested in Fukaya categories, there's lots to be proved using the definition from Seidel's book, which deals with exact symplectic manifolds. If you want to prove things about contact manifolds, try using symplectic cohomology, a close cousin of SFT requiring less formidable analysis.

soft question - Cocktail party math

On computer role-playing forums, I have seen a lot of general strategy advice regarding difficult group battles ("attack the boss monster first", "start with the highest-damage enemies", etc.). I have decided to see for myself which the optimal tactics is. The answer requires some basic (school-level but nontrivial) mathematics: see http://www.mathlinks.ro/viewtopic.php?t=326811 , scroll down to the remark (the problem turned out to be more or less identical to an American Math. Monthly question which is older than CRPG).

Navigation in mazes.

Is it possible to brute-force a combination lock by repeatedly changing a digit without checking one and the same combination several times? (Yes, by induction, at least if you can cycle every digit all the way from 0, 1, 2, ... to 9 and back to 0. If you cannot move between 9 and 0 without going through the intermediate digits, then no, by a semi-invariant argument. The keyword here is Hamiltonian path. If none exists, the next natural question is how to find a path through every vertex of minimal length...) And as we are talking about Hamiltonian paths, Euler paths can be interesting as well...

Huffman coding. It's in your base compressing your files.

For some reason I never understood, many people not particularly close to mathematics seem to be fascinated by Rubik's cube. As opposed to some other popular riddles like Sudoku, this one becomes more or less trivial once one knows the maths behind it.

Huh, the four-color theorem has not been mentioned yet? This is the best example for the notion of beautiful vs. ugly in mathematics that comes into my mind. The five-color theorem is not difficult and rather nice to prove; the only proofs of the four-color theorem known go the "classify and solve for hundreds of particular cases" way. Mathematics is probably the only science where people care for the difference.

The isoperimetric inequality, with all of its, sorry for the pun, variations (such as the case of a curve in a half-plane with two given ends).

You want to encrypt a message in a way that each of $n$ persons gets a key such that any $k$ of them can, in a joint effort, unambiguously decrypt the it, while any $k-1$ will not have the slightest idea about the message (i. e., every possible message including gibberish will be equiprobable). It's called Shamir's Secret Sharing.

computer science - Travelling Salesman Problem

Yes, as long as your distance does not satisfy the triangle inequality. Here is a series of points which form a shortest route under the Hamming distance. If you plot them on the plane, you will notice that they cross over.

(1,1)
(1,2)
(4,2)
(3,2)
(0,2)
(0,1)

human biology - Does stress physically age our body?

Going by the assumption that stress eventually triggers a flight/fight response, and the subsequent realization that flight/fight puts the body in a system of readiness to use it's available resources (mental, physical).

Does it make sense to think that stress physically ages a person? I am specifically referring to characteristics identified with aging such as bleached hair, telomere shortening, wrinkles around the eyes, etc.

ag.algebraic geometry - projective subvarieties of the moduli space of abelian varieties

Let me comment on Felipe Voloch's answer. The supersingular locus (i.e. the locus of $p$-rank zero) is indeed complete: it is a closed subvariety of $overline A_g$ (choose a toroidal compactification) because the $p$-rank is lower semicontinuous, and it does not meet the boundary because a torus has positive $p$-rank. Moreover it will in fact have the largest possible dimension of a complete subvariety of $A_g$ (in positive characteristic); its dimension is $g(g-1)/2$. Indeed if there exists a complete subvariety of dimension $d$ and $eta$ is an ample divisor class, then $eta^d neq 0$. Now $lambda_1$ (1st Chern class of Hodge bundle) is ample and $lambda_1^{1+g(g-1)/2}$ vanishes. For this see

van der Geer, Gerard. Cycles on the moduli space of abelian varieties. Moduli of curves and abelian varieties, 65--89, Aspects Math., E33, Vieweg, Braunschweig, 1999. MR1722539

In characteristic zero there is no complete subvariety of $A_g$ of this dimension. Hence the largest dimension of a compact subvariety depends on the characteristic!

Keel, Sean; Sadun, Lorenzo. Oort's conjecture for $A_ gotimesBbb C$. J. Amer. Math. Soc. 16 (2003), no. 4, 887--900 (electronic). MR1992828

On special type polynomial inequalities over integers

ADDED: As Mark Sapir and other are pointing out, if you only have $neq$'s, no $=$'s, $<$'s or $>$'s, then there is always a solution. That is to say, if $u_1$, $u_2$, ..., $u_N$ are nonzero polynomials, then there is always a lattice point where all the $u_i$ are nonzero. I assume you are asking the nontrivial question and allowing $<$'s and $>$'s:

---

No. Any set of equations can be turned into a set of special equations. For example, if you have the equation $x^3 y^2 z + x^2 = 7$, just introduce new variables $x_1$, $x_2$, $x_3$, $y_1$, $y_2$ and $z_1$, and write down the special equations $x_1=x_2$, $x_2=x_3$, $y_1=y_2$ and $x_1 x_2 x_3 y_1 y_2 z + x_1 x_2 =7$. This is often called the polarization trick.

So special equations are no simpler than ordinary equations and, as I imagine you know, there is no algorithm to solve Diophantine equations.

I just noticed that you said "inequalities" not equalities. But any Diophantine equation can be rewritten as an inequality: $f(x,y,z)=0$ is the same as $-1 < f(x,y,z) < 1$, and any inequality as an equality: $z geq 0$ is equivalent to $exists (p,q,r,s) : z=p^2+q^2+r^2+s^2$. So this doesn't gain or lose you any generality.

nt.number theory - Why does the Gamma-function complete the Riemann Zeta function?

Gamma function arises when we consecutively differentiate an Appell sequence. An example of Appell polynomials are Bernoulli polynomials. When we differentiate it, the factors combine with themselves:

$$B_n'(x)=nB_{n-1}(x)$$

$$B_n''(x)=n(n-1)B_{n-2}(x)$$

$$B_n'''(x)=n(n-1)(n-2)B_{n-3}(x)$$

They are just another name for Hurwitz Zeta function:

$$B_n(x) = -n zeta(1-n,x)$$

Thus, for $f(s,q)=zeta(s,-q)$

$$fracpartial{partial q}f(s,q)= s f(s+1,q)$$

$$frac{partial^2}{partial q^2}f(s,q)= s(s+1) f(s+2,q)$$

$$frac{partial^3}{partial q^3}f(s,q)= s(s+1)(s+2) f(s+3,q)$$

Since Reihmann zeta is Hurwitz zeta evaluated at $q=1$, the expression you give is apparently consecutive derivative of Hurwitz Zeta, with factor $pi^{-s}$ appearing if we normalize Hurwitz Zeta by stretching it horizontally by factor of pi.

Consecutive derivatives of Hurwitz Zeta in turn are nothing more than just polygamma function.

---

For instance, here is the function $-1/x$:

If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $tan x$. But if we do the same only in one direction, we get "incomplete tangent":

The yellow one is $operatorname{pg}(x)=frac 1pi psi (frac xpi)$, the blue one is $operatorname{cpg}(x)=-frac 1pi psi (1-frac xpi)$. They obey $operatorname{cpg}(x)+operatorname{pg}(x)=-cot(x)$.

Now if we differentiate cpg(x) we get:

$$(operatorname{cpg}(x))^{(s-1)}=pi^{-s}Gamma(s)zeta(s,1-frac xpi)$$

Compare it with yours formula:

$$xi(2s) = pi^{-s}Gammaleft(sright)zeta(2s)$$

l functions - How many L-values determine a modular form?

I think the answer to your first question is "yes." Suppose $L(f,s) = sum_{m} a(m)m^{-s}$ and $L(g,s) = sum_{m} b(m) m^{-s}$, and that $L(f,n) = L(g,n)$ for $n geq n_0$, with $n_0$ large enough that the sums converge absolutely. Then pick an integer $M geq n_0$ and weights $C_M(n)$ so that $sum_{n geq M} C_M(n) m^{-n}$ is $1$ if $m=M$, and $0$ otherwise. One can surely come up with such weights without too much trouble. Then $a(M) = sum_{n geq M} C_M(n) L(f,n) = sum_{n geq M} C_M(n) L(g,n) = b(M)$. It's not too hard to see that if two modular forms eventually have the same Fourier coefficients, then they are the same.

edit: After some further thought, I'm having trouble justifying the existence of those weights. I found a different solution that I'm posting as a separate answer.

ag.algebraic geometry - K3 surfaces with good reduction away from finitely many places

Some thoughts.

There are no such varieties when S = 1. This is a consequence of a theorem of Fontaine, MR1274493 (Schémas propres et lisses sur Z).

I think that one should only expect finitely many such varieties for any fixed S. Let me give an argument that uses every possible conjecture I know. There may be an unconditional proof, but that would probably require knowing something about K3-surfaces.

I first want to claim that the ramification at primes q|S is "bounded" independently of X. The corresponding fact for elliptic curves will be that the power of the conductor for each q|N is bounded by 2 (if p > 3) or (if p = 2 or 3) by some fixed number I can't remember.

The most obvious argument along these lines is to consider the representation on inertia I_q acting on the p-adic etale cohomology groups H^2(X). These correspond to Galois representations with image in GL_22(Z_p). The argument I have in mind for elliptic curves works directly in this case, providing that one has "independence of p" statement for the Weil-Deligne representations at q (quick hint: the image of wild inertia divides the gcd of the orders of GL_22(F_p) over all primes p). This may require the existence of semi-stable models, which one certainly has for elliptic curves, but I don't know for K3-surfaces.

The next step is to use a Langlands-type conjecture. The p-adic representation V on H^2(X) may be reducible, but at least we know that each irreducible chunk will correspond to an irreducible Galois representation of Q into GL_n(Z_p) for some n (at most 22). Each of these, conjecturally, will correspond to a cuspidal automorphic form of fixed weight and level divisible only by q|S. Moreover, from the previous paragraph, the level will be bounded at q|S. Thus there will only be finitely many representations which can occur as H^2(X) for any K3-surface X/Z[1/S]. (Maybe I am assuming here that the Galois representation acting on H^2(X) is semi-simple --- let us do so, since this is a conjecture of Grothendieck and Serre.)

Finally, I want to deduce from any equality H^2(X) = H^2(X') that X is (essentially) X'.
From the Tate conjecture we deduce the existence of correspondences X~~>X' and X'~~>X over Q whose composition induces an isomorphism on H^2(X) --- and now hopefully some knowledge of the geometry of K3 surfaces is enough to show that these sets of "isogenous" K3 surfaces form a finite set.

---

EDIT:

As Buzzard points out, I obscured the fact in the last paragraph that some more arithmetic may be necessary. What I meant to say is that understanding isogeny classes of K3's over Q will first require understanding isogeny classes over C, and hopefully this second task will be the hard part.

As David points out, the Torelli theorem for K3 will surely be relevant here. I think there can be non-isomorphic isogenous K3s, however. If one takes an isogeny of abelian surfaces A->B then one can presumably promote this to an isogeny of the associated Kummer surfaces.

---

EDIT:

Here is another thought. Deligne proves the Weil conjecture for K3 surfaces:

http://www.its.caltech.edu/~clyons/DeligneWeilK3trans.pdf

The philosophy is that there should be an inclusion of motives H^2(X) --> H^1(A) tensor H^1(A) for some abelian variety A (possibly of some huge but uniformly bounded dimension, like 2^19). It may be possible (conjecturally or otherwise) to reduce your question to the analogous statement for A, for which it is known. (Prop 6.5 is relevant here). It may well be possible to show that the variety A is defined over Z[1/2S], for example. I could make this edit more coherent but I'm off to lunch, so treat this as a thought fragment.

ag.algebraic geometry - Does formally etale imply flat?

Formally étale means that the infinitesimal lifting property is uniquely satisfied. If the map is also locally of finite presentation, then it is called étale. One of many characterizations (see EGA 4.5.17) of étale is flat and unramified. So my question is whether the weaker condition of formally étale still implies flatness?

order theory - Name for "lower/upper bounds" of arbitrary relations?

In a pre-order $prec$ (or a category) one can speak of initial objects $0$, or terminal objects $1$, meaning that $0prec x$ for all $x$ --- (or $0rightarrow^! x$ ) --- which also gives the notion of a universal object under several. E.g., among objects preceding both of $b_1,b_2$, with the restricted relation ${(a_1,a_2)|a_1prec a_2 ,a_iprec b_j}$ one can talk again about maximal objects and terminal objects, either of which notions might make a sensible candidate for "greatest lower bound" in this setting.

If you're not assuming the relation is transitive, you might want to take a (possibly graded category) transitive closure, or look at "transitive neighborhoods", or even just immediate neighborhoods as suggesed by Joel David Hamkins.

Of course, this is all quite speculative; I've not done any work where this notion was wanted.

neuroscience - Soma-soma paired neurons

Yep, the soma-soma synapse in that paper appears to be a chemical synapse between two cell bodies. There are no axons or dendrites in that preparation so the synapse must be between the somas. They are using two snail neurons that are far from each other (in different ganglia) in vivo, but which are known to synapse. This in vivo synapse is long-range and so must be mediated by an axon. In the dish, they can encourage the synapse to form even without encouraging axons and dendrites to grow.

Note that this is an artificial in vitro preparation and they are, in some sense, forcing a synapse to occur in an unusual way in a cell culture dish. They do this because it makes it easy to study certain aspects of the signalling mechanisms for generating synapses. In your paper, they appear to be using it as an easy preparation for a proof of principle for the interface between a biological neuron and a non-biological semiconductor substrate.

The paper you cited references this paper that outlines the soma-soma preparation with a pretty picture to make things clear: http://www.jneurosci.org/content/19/21/9306

mg.metric geometry - Metric on one-point compactification

Is there a standard construction of a metric on one-point compactification of a proper metric space?

Comments:

• A metric space is proper if all bounded closed sets are compact.


• Standard means found in literature.

From the answers and comments:

Here is a simplification of the construction given here (thanks to Jonas for ref).
Let $d$ be the original metric. Fix a point $p$ and set $h(x)=1/(1+d(p,x))$.
Then take the metric
$$hat d(x,y)=min{d(x,y),,h(x)+h(y)}, hat d(infty,x)=h(x).$$

A more complicated construction is given here (thanks to LK for ref), some call it "sphericalization".
One takes
$$bar d(x,y)=d(x,y)cdot h(x)cdot h(y), bar d(infty,x)=h(x).$$
The function $bar d$ does not satisfies triangle inequality, but one can show that there is a metric $rho$ such that $tfrac14cdot bar dle rhole bar d$.

nt.number theory - A nice variety without a smooth model

As Minhyong suggests, a curve $C$ (with $C(mathbb{Q}_p)neq0$) which has bad reduction but whose jacobian $J$ has good reduction would do the affair. This works because the cohomology of $C$ is essentially the same as that of $J$, and because an abelian $mathbb{Q}_p$-variety has good reduction if and only if its $l$-adic étale cohomology is unramified for some (and hence for every) prime $lneq p$ (Néron-Ogg-Shafarevich) or its $p$-adic étale cohomology is crystalline (Fontaine-Coleman-Iovita).

I asked Qing Liu for explicit examples. He suggested the curve
$$
y^2=(x^3+1)(x^3+ap^6)qquad (ainmathbb{Z}_p^times)
$$
when $pneq2,3$, and $y^2=(x^3+x+1)(x^3+a3^4x+b3^6)$, with $a,binmathbb{Z}_3^times$, for $p=3$.

He refers to Proposition 10.3.44 in his book for computing the stable reduction of these $C$, and to Bosch-Lütkebohmert-Raynaud,
Néron models, Chapter 9, for showing that $J$ has good reduction.

I "accept" this answer as coming from Minhyong Kim and Qing Liu.

ag.algebraic geometry - Is D-module on flag variety of Lie algebra a scheme?

The de Rham space of a scheme is essentially never a scheme or algebraic space (unless I guess you're Spec of an Artin ring, in which case you'll get a discrete set of points). In particular this applies to the flag variety. I'm not sure which perspective of NC AG you're taking, but certainly if you define the field as the study of Grothendieck categories, or pretriangulated dg categories, etc then D-modules are a very nice (nonproper) noncommutative space. (An interesting comment on this is found at the end of Kontsevich's letter here. Also from the point of view of "function theory" D-modules on a scheme are great, i.e. all functors are given by integral transforms, sheaves on a (fiber) product are tensor product of categories of sheaves, etc, see here.) But I don't think this says anything about the de Rham stack in a classical commutative sense..

ac.commutative algebra - Can I define the polynomial ring A[x] with an isomorphism f: A ---> A[x]?

The question seems to involve a construction of a set-theoretic map, and the indexing (natural numbers?) suggests that A is assumed to have a countable underlying set. That map doesn't even yield a surjection of sets.

I would like to reinterpret the question in the following way: How much structure do we need to forget in order for there to exist an isomorphism $A to A[x]$? YBL pointed out that there is never an A-algebra isomorphism (if A is nonzero) and that there can be a ring-theoretic isomorphism if A is big enough. If A has an infinite underlying set, then there exist isomorphisms on the underlying sets. It is potentially interesting to ask when we get isomorphisms on the underlying additive groups: it is sufficient for A to have a polynomial ring structure, but that is far from necessary: e.g., A could be any field of infinite dimension over its prime field.

Regarding your last question, you can define a polynomial ring using a sequence of embeddings $f_n: a mapsto ax^n$ together with a specified multiplication law. This is a special case of the monoid ring construction. I'm not sure if this was the construction you initially had in mind, but it doesn't yield an isomorphism, since it isn't a single map.

ag.algebraic geometry - Technique to prove basepoint-freeness

Let $X$ be a smooth projective variety over $mathbb{C}$.
And let $L$ be a big and nef line bundle on $X$.
I want to prove $L$ is semi-ample($L^m$ is basepoint-free for some $m > 0$).

The only way I know is using Kawamata basepoint-free theorem:

Theorem. Let $(X, Delta)$ be a proper klt pair with $Delta$ effective.
Let $D$ be a nef Cartier divisor such that $aD-K_X-Delta$ is nef and big for some
$a > 0$. Then $|bD|$ has no basepoints for all $b >> 0$.

Question. What other kinds of techniques to prove semi-ampleness or basepoint-freeness
of given line bundle are?

Maybe I miss some obvious method. Please don't hesitate adding answer although you think your idea on the top of your head is elementary.

Addition : In my situation, $X$ is a moduli space $overline{M}_{0,n}$.
In this case, Kodaira dimension is $-infty$.
More generally, I want to think genus 0 Kontsevich moduli space of stable maps to
projective space, too.
$L$ is given by a linear combination of boundary divisors.
It is well-known that boundary divisors are normal crossing,
and we know many curves on the space such that we can calculate intersection numbers with boundary divisors explicitely.

human biology - Is it purely the nervous system causing vaginal lubrication (arousal)?

My girlfriend was watching some documentary on TLC about a paralyzed woman getting pregnant. I believe that woman still has some feeling, as she spoke about feeling the effects of a bladder inflamation. So supposedly, her brain could fire up her vagina to get ready for sex (in the sense that it gets lubricated and the labia get swollen).

But suppose a woman is paralyzed so they have no nervous connection from the brain to the vagina (or vice versa). Could she still be aroused in the sense of swollen labia, etc? (Presumably via some hormonal system?)

lo.logic - Naturally definable sets of natural numbers (3)

[This shall be the last of a series of questions, see Naturally definable sets of natural numbers (2)]

I cannot explain why I have been so stubborn not to see the most straight-forward definition for naturalness, sorry for the confusion I have caused. (In any case I guess I have learned a lot from previous answers!)

Let $Phi(x)$ be a formula of the first-order language of Peano arithmetic (PA).

Definition: A formula $Phi(x)$ is
natural iff it is provably-equivalent-in-PA to a formula $phi(x)$ in prenex disjunctive normal form with its matrix not containing clauses of which the disjunction is
equivalent [corrected:] provably-equivalent-in-PA to some $p(x) = q(x)$ with
$p(x), q(x)$ polynomials in $x$ with
natural coefficients.

The last condition means "clauses of which the disjunction defines a finite set", thus "adding" an arbirtrary finite set. There must by the way be an analogue condition on the matrix of the prenex conjunctive normal form, which must not contain conjunctive clauses that are equivalent to some $p(x) neq q(x)$, thus "excluding" an arbitrary finite set.

There definitely are natural formulas, e.g. $(exists y) x = 2 cdot y$.

Consider a formula that is not in this form, i.e. its matrix in prenex disjunctive normal form does contain clauses of which the disjunction is equivalent to some $p(x) = q(x)$, e.g. $(exists y) x = 2 cdot y vee x = 17$ Then it is in general not possible to make these clauses disappear by applying logical axioms only. But nevertheless the formula might be natural.

Do you agree

...that the set of natural
formulas is not decidable (but maybe enumerable)?

(Like the set of formulas that define finite sets, see Francois' answer to another question.)

... that nevertheless every formula
which defines an infinite set could be
natural?

Or is there one explicit example of a formula that is provably not natural? I suspect that if (big if) every formula is natural, the formula in the "correct" form (without any clauses equivalent to $p(x)=q(x)$) can be arbitrarily weird.

... that every finite set might be
definable by some natural formula $Phi(x)$ in the form
$lbrace x | Phi(x) wedge bigvee_{k_1 < k_2} n_{k_1} leq x leq n_{k_2} rbrace$

ag.algebraic geometry - Why do we need finiteness conditions for formally étale morphisms?

It follows from a recent answer that even when a ring is formally étale rather than étale, we can check this condition on localizations, and hence stalks. It's not hard to show that we can define a "formally étale" topology on $Aff$. Presumably there's a good reason for requiring finite presentation), but I can't think of a reason why off of the top of my head.

Specifically, we let the covering families be jointly faithfully flat with each morphism formally étale. This is by construction subcanonical.

Questions: Why are finiteness conditions necessary for schemes in general and étale maps in particular.

What kinds of problems will we run into if we do not put finitness conditions on the "formally étale" topology?

If this topology fails in some way, can requiring that covers are finite families of morphisms (quasicompact), or that the morphisms in the cover are themselves flat, or even both? This would give us a topology similar to the fpqc topology, except in that all covering families would be made up of formally étale morphisms. The only difference between this topology and the étale topology is the finite presentation of the morphisms in the covering families. Does this still not work?

ag.algebraic geometry - Methods of showing a map has integral or good reduction

Question

Say we have a map, C->D, of relative curves over a Dedekind scheme, S. What are some of the available methods for showing that this map has good reduction, or integral reduction, at some s∈S? By this I mean: what are some popular conditions that imply this? What are the tricks people usually use?

Clarification

By a map having good reduction I mean that both C_s and D_s are regular integral curves. By integral reduction I mean that both C_s and D_s are integral curves.

You may assume whatever you want, this is part of the question. Assuming, for example, that C->D is generically Galois; or that D is smooth over S; is legitimate. This is pretty open-ended. Hence, community wiki.

ct.category theory - measure spaces as presheaves?

I recently had the idea that maybe measure spaces could be viewed as sheaves, since they attach things, specifically real numbers, to sets...

But at least as far as I can tell, it doesn't quite work - if $(X,Sigma,mu)$ is a measure space and $X$ is also given the topology $Sigma$, then we do get a presheaf $M:O(X)rightarrowmathbb{R}_{geq0}$, where $mathbb{R}_{geq0}$ is the poset of non-negative real numbers given the structure of a category, and $M(U) = mu(U)$, and $M(U subseteq V)$ = the unique map "$geq$" from $mu(V)$ to $mu(U)$, but this is not a sheaf because for an open cover {$U_i$} of an open set $U$, $mu(U)$ is in general not equal to sup $mu(U_i)$ (and sup is the product in $mathbb{R}$).

First off, is the above reasoning correct? I'm still quite a beginner in category theory, but I've seen on Wikipedia something about a sheafification functor - does applying that yield anything meaningful/interesting? Does anyone have good references for measure theory from a category theory perspective, or neat examples of how this perspective is helpful?

co.combinatorics - Mathematical solution for a two-player single-suit trick taking game?

The question on games and mathematics that appeared recently on mathoverflow
(Which popular games are the most mathematical?)
reminded me of a problem I encountered some time ago : starting with the insane
dream of completely solving the game of bridge with a nice mathematical theory,
I ended up considering extremely simplified versions of bridge. One of them was
as follows : there are only 2 players instead of 4, and instead of the usual
deck there are only 2n cards numbered from 1 to 2n. Each player holds half
of the deck, so this is a "complete information" game : each player knows exactly
what is in his opponent's hand. There are no bids, just a sequence of n moves
where each player drops a card ; as in bridge the strongest card wins the trick
and the winner of the game is the player with the largest number of tricks
in the end (take n odd to avoid draws). Also, the winner of the preceding
trick is the first to play (for the very first move the first player is determined
by some rule, random or other ; this is immaterial to the subsequent discussion).

This looks like a very basic kind of game, especially amenable to
mathematization : for example the set of all initial positions is nicely indexed
by the subsets $I$ of $lbrace 1,2, ldots , 2nrbrace$ whose cardinality is $n$
(say $I$ is the set of cards held by the first player). I was
however unable to answer the following questions :

• Is there an algorithm which, given the initial position, finds out which player
  will win if each one plays optimally ? What is the best strategy ?

  
  
  
  • Has this game already been studied by combinatorialists ?
  
  

gr.group theory - nilpotent matrices over polynomial rings

I am looking for an analogue of the Jordan normal form for nilpotent matrices over the
polynomial ring ${mathbb Z}[x_1, dots, x_n]$. More precisely, is there a description for the orbits of action by conjugation of $GL_m({mathbb Z}[x_1, dots, x_n])$ on $M_{m times m}({mathbb Z}[x_1, dots, x_n])$?

evolution - Is there any reason for the variation in mitochondrial DNA size?

Shorter DNA would allows easier synthesis, but more introns could allow for more different transcription factors to influence gene expression.

Human mitochondria probably does not need to be as adaptable as mitochondria in yeast, since the a human cell's environment tends to be more stable. This could provide a viable reason as to why human mitochondria does not need as many introns, but this is only how I imagine things. I have no proof or experience to validate any of this - its just a possiblility.

game theory - Baccarat and the way to win it

Counting cards is possible, but extremely ineffective in baccarat.

In blackjack with n decks, you might model the game as starting out with a house advantage of 0.5%-1% while each card might affect the house advantage by +- 0.5%/m, where m≤n is the number of 52 card decks left in the shoe. So, it's not terribly unlikely that you reach a situation in which the deck is in your favor, although casinos try to shuffle before too many cards are dealt. If you try to monitor the deck's composition and bet more when it is in your favor, then you are counting cards, and you can obtain an advantage. Favorable decks are common enough that you can win at blackjack (until the casino notices and bars you) while varying your bet size by a relatively small amount, e.g., a factor of 4, although this depends on many other specifics such as how far into the decks are dealt before the decks are shuffled.

The problem with baccarat is that card removal has a much lower effect on the house edge. This means you would often have no advantage at any point in the shoe, and you would wait for a vary small advantage. I don't think anyone does it seriously, unlike blackjack.

See The Wizard of Odds on counting in baccarat which contains this comment:

"I hope this section shows that for all practical purposes baccarat is not a countable game. For more information on a similar experiment I would recomment The Theory of Blackjack by Peter A. Griffin. Although the book is mainly devoted to blackjack he has part of a chapter titled 'Can Baccarat Be Beaten?' on pages 216 to 223. Griffin concludes by saying that even in Atlantic City, with a more liberal shuffle point than Las Vegas, the player betting $1000 in positive expectation hands can expect to profit 70 cents an hour."

This assumes you are making no bets, but are keeping perfect count, and then jump in with the occasional $1000 bets on 1 hand out of 500.

Comprehensive reference for synthetic euclidean geometry

(I'm french so sorry for my poor English)

I think the best way to teach Euclidean space to children is to just make it simple:

Two parallel lines will never meet. Like on a map that you can show them.

Then the best way to explain a non-Euclidean space to children is with a Earth globe. You show them the meridians and the longitudes. How the meridians are meeting at the poles even if while watching outside, the Earth seems plane. The globe is the best thing to explain that to a child I think. Just put his finger on the North pole.

How could you expect a child to understand that a triangle doesn't have 180 degrees of corners in a non-Euclidean space ?

Maybe i'm too simplist for you but that's what I think.

co.combinatorics - Why do wedges of spheres often appear in combinatorics?

One way to approach this question quantitatively is suggested by probability. One can put various measures on the space of all simplicial complexes on $n$ vertices. One perhaps fairly natural measure is to take a random graph and then take the clique complex. This doesn't give us all complexes on $n$ vertices but every complex is homeomorphic to the clique complex of some graph, so we are covering everything up to homeomorphism as $n to infty$.

The main point of my paper Topology of random clique complexes is that almost all simplicial complexes arising this way are fairly simple topologically. In particular is shown that for a typical $d$-dimensional clique complex, the homology groups $H_k$ all vanish when $k > lfloor d/2 rfloor$ and when $k< d/4$, and that almost all of whatever homology remains is concentrated in the middle dimension $k=lfloor d/2 rfloor$.

It is currently an open problem to decide whether the homology is vanishing (or merely small) between $k=d/4$ and $k=d/2$. If one could establish this, then one would be well on the way to showing that almost all flag complexes are homotopy to a wedge of spheres; indeed the last thing to do would be to rule out torsion in middle homology with integer coefficients.

I don't have a good feel for whether either of these things is even true, but I do think that this paper gives good anecdotal evidence that most flag complexes are somewhat simple topologically, and is a step in the direction of answering Forman's question. (This particular measure seems especially natural from the point of view of combinatorics, since so many simplicial complexes arise as order complexes of posets, hence are automatically flag complexes.)

UPDATE:

(1) I showed recently that for every $k ge 3$, there is a range of edge probability so that the random clique complex (also called random flag complex) is rationally homotopy equivalent to a wedge of $k$-dimensional spheres. In particular all the rational homology is in middle degree. There is only a very small overlap where there is homology in degree $k$ and in degree $k+1$, but in some sense, most of the time there is only homology in one degree. The conjecture that "rationally homotopy equivalent" can be replaced by "homotopy equivalent" is equivalent to showing that with high probability, homology is torsion free.

(2) On the note of torsion in random homology, in joint work with Hoffman and Paquette, we recently showed that for a slightly different model of random simplicial complex, for most of the range where rational homology is vanishing, integer homology is also vanishing.

There are one or two technical issues in applying the method of (2) in the setting of (1) (namely non-monotonicity of homology), but so far it seems like there is reason to believe that the method will go through eventually.

Together, these two recent results suggest that a random flag complex (for a suitable range of edge probability $p$) is homotopy equivalent to wedges of $d$-dimensional spheres. Random flag complexes seem to me like a very natural model for addressing your question probabilistically, since so many complexes in combinatorics are flag complexes, arising as order complexes of posets, etc.

geometric langlands - A question on group action on categories

The $G((t))$ action and the $Rep(G^vee)$ [or equivalently of $G^vee$ itself after deequivariantization, as Victor explains] are of quite different natures -- the former is a "smooth" action, and the latter an "algebraic" or "analytic" actions (the adjectives smooth and analytic come from analogy with p-adic rep theory).
i.e. there are many kinds of notion of group action, and they are (to me) most conveniently summarized by describing the corresponding notion of group algebra which acts. An algebraic action of a group on a category is an action of the "quasicoherent group algebra" of G, ie the monoidal category of quasicoherent sheaves wrt convolution. (though I'd feel much safer if we said all this in a derived context, makes me uneasy otherwise).
A smooth action is an action of the monoidal category of D-modules on G, the "smooth group algebra" -- analog of smooth functions on a p-adic group. Such an action is the same as an algebraic action, which is infinitesimally trivialized. Such examples are studied in Chapter 7 of Beilinson-Drinfeld's Hecke manuscript and the appendix to the long paper by Gaitsgory-Frenkel, in particular.

PS the "equivariantization" dictionary between categories over BH and categories with H action is a nice simple case of descent --- you describe things over BH as things over a point with descent data, that descent data is given by the map H --> pt,
the two maps H x H---> H, and so on. When you assemble this together (most efficiently using the Barr-Beck theorem) you get the desired dictionary. (Of course if you want to consider categories as forming a 2-category you'd need a 2-categorical version of Barr-Beck, but for most practical purposes I know of you can get by with the current
Lurie [$(infty,$]1-categorical version.