human biology - What are the clotting factors' effect on avascular necrosis development?

First of all we should distinguish between the physiological clotting factors that are parts of the normal clotting pathways, and those that might affect clotting pathways but are not observed under healthy conditions.

TPA (tissue plasminogen activation), PAI-1 (plasminogen activator inhibitor) and prothrombin are normal clotting factors that are essential for blood clotting. Any deficiency (congenital or acquired) might cause certain pathologic conditions, including the osteoarthritis and avascular bone necrosis.

Lupus anticoagulant, however, is an antibody against cell membrane that is common for Lupus. This antibody can intervene into the process of clotting and increase the prothrombine time (the common lab method to test blood clotting properties), hence the name "anticoagulant". But this effect is just a sort of "side-effect" of this antibody, that leads to certain symptoms in case of auto-immune attack against different target organs.

So, after this introduction I can rephrase your first question as: "Does the deficiency of clotting factors TPA and PAI-1 lead to degenerative arthritis in the same way that lupus anticoagulant and prothrombin deficiency might?"

The part of the answer is already in this question: yes, the deficiency of these factors might and under certain conditions, leads to the development of osteoarhritis and avascular necrosis.

I must confess, I found no evidence whether the underlying mechanism is different or not. Papers just prove that these factors are discovered in the affected joins, and see the correlation in mice models between the factors and the articifially induced arthritis. But I found no study which would try to address the mechanisms of these actions or to show, whether these are different.

ap.analysis of pdes - Why don't existence and uniqueness for the Boltzmann equation imply the same for Navier-Stokes?

Okay, after figuring out which paper you were trying to link to in the third link, I decided that it is better to just give an answer rather then a bunch of comments. So... there are several issues at large in your question. I hope I can address at least some of them.

The "big picture" problem you are implicitly getting at is the Hilbert problem of hydrodynamical limit of the Boltzmann equations: that intuitively the ensemble behaviour at the large, as model by a fluid as a vector field on a continuum, should be derivable from the individual behaviour of particles, as described by kinetic theory. Very loosely tied to this is the problem of global existence and regularity of Navier-Stokes.

If your goal is to solve the Navier-Stokes problem using the hydrodynamic limit, then you need to show that (a) there are globally unique classical solutions to the the Boltzmann equations and (b) that they converge in a suitably regular norm, in some rescaling limit, to a solution of Navier-Stokes. Neither step is anywhere close to being done.

As far as I know, there are no large data, globally unique, classical solutions to the Boltzmann equation. Period. If we drop some of the conditions, then yes: for small data (perturbation of Maxwellian), the recent work of Gressman and Strain (0912.0888) and Ukai et al (0912.1426) solve the problem for long-range interactions (so not all collision kernels are available). If you drop the criterion of global, there are quite a bit of old literature on local solutions, and if you drop the criterion of unique and classical, you have the DiPerna-Lions solutions (which also imposes an angular-cutoff condition that is not completely physical).

The work of Golse and Saint-Raymond that you linked to establishes the following: that the weak solution of DiPerna-Lions weakly converges to the well-known weak solutions of Leray for the Navier-Stokes problem. While this, in some sense, solve the problem of Hilbert, it is rather hopeless for a scheme trying to show global properties of Navier-Stokes: the class of Leray solutions are non-unique.

As I see it, to go down this route, you'd need to (i) prove an analogue of DiPerna-Lions, or to get around it completely differently, and arrive at global classical and unique solutions for Boltzmann. This is a difficult problem, but I was told that a lot of very good people are working on it. Then you'd need (ii) also to prove an analogue of Golse-Saint-Raymond in a stronger topology, or you can use Golse-Saint-Raymond to first obtain a weak-limit that is a Leray solution, and then show somehow that regularity is preserved under this limiting process. This second step is also rather formidable.

I hope this somewhat answers your question.

ho.history overview - Where can I find the text of Weyl's Fields Medal speech for Serre?

I thought about asking this question a while ago, but decided against it. But now I see a question about Eichler's "modular forms" quote, so while I guess it's probably still, um, questionable, what the hey.

So when Serre won the Fields Medal in 1954, Hermann Weyl (I guess) presented the award and described Serre's work. The Wikipedia article on Serre describes it thus: "...Weyl praised Serre in seemingly extravagant terms, and also made the point that the award was for the first time awarded to an algebraist."

If you're still not sure what I'm talking about, this is the speech where Weyl says something like "Never before have I seen such a rapid or bright ascension of a star in the mathematical sky as yours," if you've heard that quote.

Anyway, I've been trying to find a full version of Weyl's remarks (just out of curiosity), but to no avail. I would guess it would probably be in the congress proceedings somewhere, but I don't really have a copy on me. Anyone know where else I could find it?

palaeontology - Descendants of dinosaurs

No. No other group found in the fossil record after the K-T boundary (the extinction "event") descends from dinosaurs.

It is likely that the extinction event was not itself instantaneous so if you wanted to be extremely picky you could argue that small numbers of individuals survived the K-T boundary but, apart from birds, none of these survivors went on to form long lasting lineages.

Non-avian dinosaurs that survived the extinction event are referred to as "palaeocene dinosaurs", there are some links on Wikipedia that might interest you.

You can see a diagram linking vertebrate groups here, similar diagrams can be found in most evolution texts.

differential equations - What happens to the solutions of a fourth-order boundary-value problem as you turn off the fourth-order coefficient?

Background

Lagrangian mechanics on $mathbb R^n$ is usually defined by picking a Lagrangian function $L: {rm T}mathbb R^n to mathbb R$, where ${rm T}mathbb R^n = mathbb R^{2n}$ is the tangent bundle of the configuration space $mathbb R^n$. Such a function determines the Euler-Lagrange equations:
$$ frac{d}{dt}left[ frac{partial L}{partial v^i}bigl( dotgamma(t), gamma(t)bigr) right] - frac{partial L}{partial q^i} bigl( dotgamma(t), gamma(t)bigr) = 0$$
Here $(v^i,q^i)$ for $i=1,dots,n$ are the standard coordinates on ${rm T}mathbb R^n$, $gamma: [0,T] to mathbb R^n$ is a smooth function, and $dotgamma^i(t) = frac{dgamma^i}{dt}$. Suppose that the matrix $frac{partial^2 L}{partial v^ipartial v^j}(v,q)$ is invertible for any $(v,q) in {rm T}mathbb R^n$. Then the Euler-Lagrange equations are a nondegenerate second-order differential equation on $mathbb R^n$. I am interested in the boundary-value problem for $L$. Namely, fix $T > 0$ and $q_1,q_2 in mathbb R^n$; the BVP asks to find the set $C(q_1,q_2,T)$ of all paths $gamma: [0,T] to mathbb R^n$ with $gamma(0) = q_1$ and $gamma(t) = q_2$. Generically, this is a discrete set.

My question

Suppose that if instead of the Euler-Lagrange equations above, I pick some small parameter $epsilon$ and consider the differential equation
$$ frac{d}{dt}left[ frac{partial L}{partial v^i}bigl( dotgamma(t), gamma(t)bigr) right] - frac{partial L}{partial q^i} bigl( dotgamma(t), gamma(t)bigr) = epsilongamma^{(4)}(t)^i $$
where $gamma^{(4)}(t)^i$ is the $i$th component of the fourth derivative of $gamma$ with respect to $t$ (the "jounce", a word I just learned from Wikipedia). For $epsilon neq 0$, the EL equations are a nondegenerate fourth-order differential equation, and so generically solutions to the boundary-value-problem above form a two-dimensional family. To restrict to a discrete set, we should fix more boundary values. Pick $(v_1,q_1), (v_2,q_2) in {rm T}mathbb R^n$ and $T > 0$, and define $C_epsilon(v_1,q_1,v_2,q_2,T)$ to be the set of solutions $gamma$ to the $epsilon$-dependent EL equations with $(dotgamma(0),gamma(0)) = (v_1,_1)$ and $(dotgamma(T),gamma(T)) = (v_2,q_2)$.

My question is: as $epsilon to 0$, in what sense do we have $C_epsilon(v_1,q_1,v_2,q_2,T) to C(q_1,q_2,T)$?

mathematics education - How seriously should a graduate student take teaching evaluations?

Positions not associated with teaching (such as industrial or government labs) will very rarely care about your teaching. When I applied for these, I didn't even bother to list my teaching on the cv.

Teaching positions (such as at a community college) will probably care about it a lot more, since they want some proof that you can teach well. But I can't say much since I don't have experience with these.

Research universities are somewhere in-between. In general, their main priority is the quality of your research. So for a standard tenure-track faculty positions, they will likely focus on selecting an interesting (research-wise) colleague rather than the best teacher.

Of course, research universities need to teach too, and they do feel the pressure to teach well. Also, "research university" is not a uniform designation; different universities will have different priorities which may include more or less emphasis on teaching.

Generally, teaching works like this at a research university. The department (math, in your case) needs to teach some courses. These are service courses to other departments (such as "calculus 1 for biology students") and internal courses (e.g. "graduate group theory"). These need to be taught adequately. If the service courses are not taught well, other departments will complain and your dean will not like it. If the internal courses are not taught well, then your colleagues will have underprepared students to deal with, and they will not like it. So people will want to know that you can teach adequately. Generally, at a research university, I would take "adequately" to mean that you will not leave the students grossly underprepared. Whether they love your teaching or not is less of an issue. So, as long as you have some teaching experience, I would say you are OK.

Now, you don't have to list ALL teaching evaluations on your cv. If the evaluations are great, mention them. If not, you can omit them and just list the course. For example:

TEACHING

Fall 2008: Calculus 1

Spring 2009: Algebra (received 4.5 / 5 evaluation)

Fall 2009: Linear algebra

...

Also, I don't think good evaluations will affect your candidacy negatively. It's true that some people might interpret interest in teaching as lack of interest in research, but I don't think good evaluations are enough for that. If you teach a lot, if you publish papers on teaching, go to teaching conferences, etc. -- in that case, yes, people might be suspicious of whether you are interested in research at all (especially if you don't have an equally active research program). But I don't think that just having good evaluations will do you any harm.

bioinformatics - Main methods used to predict functional annotations in GO

I believe the most common source of electronic annotations comes from analysis of peptide sequences. A collection of InterPro to GO mappings were created manually and can generate GO annotations. DNA binding domains of transcription factors would be given "DNA binding" GO annotations say.

This method has its flaws - if the domain detected has evolved away from the function used for the Interpro to GO listing, there is a potential for error in this method.

ag.algebraic geometry - What does the nilpotent cone represent?

Notation

Let $mathfrak g$ be a the Lie algebra of an algebraic group $Gsubseteq GL(V)$ over a(n algebraically closed) field $k$ (I'm actually thinking $G=GL_n$, so $mathfrak g=mathfrak{gl}_n$). Then any element $X$ of $mathfrak g$ can be uniquely written as the sum of a semi-simple (diagonalizable) element $X_s$ and a nilpotent element $X_n$ of $mathfrak g$, where $X_s$ and $X_n$ are polynomials in $X$. The nilpotent cone $mathcal N$ is the subset of nilpotent elements of $mathfrak g$ (elements $X$ such that $X=X_n$).

---

People often talk about the nilpotent cone as having the structure of a subvariety of $mathfrak g$, regarded as an affine space, but usually don't say what the scheme structure really is. To really understand a scheme, I'd like to know what its functor of points is. That is, I don't just want to know what a nilpotent matrix is, I want to know what a family of nilpotent matrices is (i.e. what a map from an arbitrary scheme $T$ to $mathcal N$ is). Since any scheme is covered by affine schemes, it's enough to understand what an $A$-valued point (a map $mathrm{Spec}(A)to mathcal N$) is for any $k$-algebra $A$. So my question is

What functor should $mathcal N$ represent?

A guess

Well, an $A$-point of $mathfrak g$ is "an element of $mathfrak g$ with entries in $A$" (again, I'm really thinking $mathfrak g=mathfrak{gl}_n$, so just think "a matrix with entries in $A$"), so I would expect that such an $A$-point happens to be in $mathcal N$ exactly when the given matrix is nilpotent. That is, $mathcal N(mathrm{Spec}(A))={Xin mathfrak{g}(mathrm{Spec}(A))| X^N=0$ for some $N}$.

However, this is wrong. That functor isn't even an algebraic space, even for the nilpotent cone of $mathfrak{gl}_1$. If it were, the identity map on it would correspond to a nilpotent regular function $f$ (a nilpotent $1times 1$ matrix), and this would be the universal nilpotent regular function; every other nilpotent regular function anywhere else would be a pullback of this one. But whatever the degree of nilpotence of this function (say $f^{17}=0$), there are some nilpotent regular functions which cannot be a pullback of it (something with nilpotence degree bigger than 17). If this version of the nilpotent cone were representable, you can show that the $mathfrak{gl}_1$ version would be too.

Another guess

I think the answer might be that an $A$ point of $mathcal N$ is a matrix ($A$ point of $mathfrak g$) so that all the coefficients of the characteristic polynomial vanish. This is a scheme and it has the right field-valued points, but why should this be the nilpotent cone? What is the meaning of having all coefficients of the characteristic polynomial vanish for a matrix with entries in $A$?

human biology - Can an adult without genetic lactase persistence still develop a tolerance for dairy foods?

Here are my thoughts on the topic.

Dairy good is not only milk, but also the following milk products: sour milk products (like yougurt, kefir, katik, buttermilk, etc.), cheese, etc. These products can contain less lactose than in the milk solids (due to fermentation during processing). It is also common for some of these products to contain living lactase-active bacteria, that can digest lactose.

Regular intake of these products can lead to certain changes in our small intestine microflora, so that the lacking lactase activity is substituted by the bacteria and thereby the tolerance grows. Maybe this is the intend of the Chinese government? Maybe they encourage the intake of sour milk products and not solid milk?

It is known, for example, that children with congenital/primary enzymatic deficiency develop the tolerance at the age of 6-10 years so that no special diet is needed anymore. This is true at least for such common enzymatic deficiencies like phenylketonuria and coeliacia.

In Russia, where I got my medical degree, it is common to use bacteria in the treatment of many types of primary intolerance combining this treatment with dietary support.

gr.group theory - Symmetric Groups and Poisson Processes

This isn't a problem I've looked at before, but I've been thinking about it since reading your post, and there does seem to be an interesting limit. The following looks like it should all work out, but I haven't gone through all the details yet.

Embed the set {1,...,n} into the unit interval I=[0,1] by θ(i) ≡ i/n. Then look at θ(A) for the fixed points A ⊂ {1,..,n} of a random permutation. Increasing n to infinity, the distribution of θ(A) should converge weakly to the Poisson point process on I with intensity being the standard Lebesgue measure.

That is only the fixed points though. You can look at the limiting distribution of the set of all points contained in orbits of bounded size (≤ m, say), and at the action of the random permutation on this set. This should have a well defined limit, and you can then take the limit m → ∞ to to give a random countable subset of I and a random permutation on this, such that all orbits are finite.

Let I_n={1/n,2/n,...,n/n} ⊂ I, and consider a random permutation π of this. The probability that any point P ∈ I_n is an element of an orbit of size r is (1-1/n)(1-2/n)...(1-(r-1)/n)(1/n). The expected number of points lying in such orbits is (1-1/n)...(1-(r-1)/n) and the expected number of orbits of size r is (1-1/n)...(1-(r-1)/n)/r, which tends to 1/r as n → ∞.

Now represent orbits of size r of the permutation π by sequences of distinct elements of the interval I, (x₁,...,x_r), where π(x_i) = x_i+1 and π(x_r) = x₁. We'll identify cyclic permutations of such sequences, as they represent the same orbit and the same action of π on the orbit. So (x₁,...,x_r) = (x_r,x₁,...,x_r-1). Let I^(r) be the space of such sequences up to identification of cyclic permutations.

Then, every permutation π of I_n gives a set of fixed points A₁ ⊂ I⁽¹⁾=I, orbits of size 2, A₂ ⊂ I⁽²⁾, orbits of size 3, A₃ ⊂ I⁽³⁾, etc.
Then, I propose that A₁, A₂, A₃,... will jointly converge to the following limit; (α₁,α₂,α₃,...) are independent Poisson random measures on I⁽¹⁾,I⁽²⁾,I⁽³⁾,... respectively such that the intensity measure of α_r is uniform on I^(r) with total weight 1/r.
Taking the union of all these orbits gives a random countable subset of I and a random permutation of this.

I used the unit interval as the space in which to embed the finite sets {1,...,n} but, really, any finite measure space (E,ℰ,μ) with no atoms will do. Just embed the finite subsets uniformly over the space (i.e., at random).

ag.algebraic geometry - Dualizing sheaf of reducible variety?

With regards part 2.

Let's assume that you have two components $X_1$ and $X_2$ (or even unions of components) such that $X_1 cup X_2 = X$=. Let $I_1$ and $I_2$ denote the ideal sheaves of $X_1$ and $X_2$ in $X$.

Set $Z$ to be the scheme $X_1 cap X_2$, in other words, the ideal sheaf of $Z$ is $I_1 + I_2$.

It is easy to see you have a short exact sequence
$$0 to I_1 cap I_2 to I_1 oplus I_2 to (I_1 + I_2) to 0$$
where the third map sends $(a,b)$ to $a-b$.

The nine-lemma should imply that you have a short exact sequence

$$0 to O_X to O_{X_1} oplus O_{X_2} to O_Z to 0$$

If you Hom this sequence into the dualizing complex of $X$, you get a triangle
$$omega_Z^. to omega_{X_1}^. oplus omega_{X_2}^. to omega_{X}^. to omega_Z^.[1]$$

You can then take cohomology and, depending on how things intersect (and what you understand about the intersection), possibly answer your question.

If $X_1$ and $X_2$ are hypersurfaces with no common components (which should imply everything in sight is Cohen-Macualay) then these dualizing complexes are all just sheaves (with various shifts), and you just get a short exact sequence
$$0 to omega_{X_1} oplus omega_{X_2} to omega_{X} to omega_{Z} to 0$$

Technically speaking, I should also probably push all these sheaves forward onto $X$ via inclusion maps.

Introductory text on Riemannian geometry

I would differenciate between some books. If you simply want to learn how to calculate things, for instance because you are interested in physics, I'd recommend to you the book "Geometry, Topology and Physics" by Nakahara which has a very good part on Riemannina goemetry. However, it may contain more than you actually want to know.
The book with which I have really understood Riemannian Geometry was Jürgen Jost's "Riemmanian Geometry and Geometric Analysis". However, it might require some prerequisites that you haven't got so far.
Alternatively, you could read Do Carmo's book. It is very concisely written and features some nice examples.
I would not advise you to read the Gallot-Hutin because some of the proofs are simply only outlined. For a person that begins learning Riemannian geometry this could be very discouraging. However, I admit that the great advantage of this book is the number of exercises.

Something I want to add because I really want to advertize one book: Riemmanian geometry by Takahashi sakai. If you once have got the basics, this books takes you further. A quantum leap further. But you really need the basics. I would use this book for a second course in Riemmanian Geometry, assuming the student's familiarity with differentiable manfiolds and fiber bundles and a first course in Riemannina Geometry, such as for instance material covered in Jost's book in the chapters 1-4.

evolution - Genetic Diversity and Adaptation

Great question! A lot of things affect how quickly a population or species can adapt to a new environment, including population size, mutation rate, generation time, standing genetic diversity, and selective pressure.

The diversity of life encompasses practically all combinations of those variables. A bacterial population might very well contain enough diversity to allow a portion of the population to overcome a rapid change. In fact, applying antibiotics to a bacterial population and counting the survivors is a common measure of mutation rates.

On the other hand, organisms with small populations and long generation times will be much less likely to overcome a rapid environmental change. This is why there is so much concern over anthropogenic changes to the environment, including climate change.

It's hard to provide a definitive answer, since "rapid change" is a relative term, and the difficulty of adapting isn't known. Some striking adaptations can be caused by a single base pair mutation, such as in beetles and Monarch butterflies that are insensitive to toxic plant compounds.

It's important to point out that natural selection is based on relative fitness. Therefore, an adaptive mutation will spread because it's likely that carriers will be more fit than all of their neighbors. This does not necessarily mean that individuals without the mutation will die or fail to reproduce, only that those with the mutation will do it better.

Likewise, adaptation is not necessarily caused by the environment changing and wiping out all but a few lucky mutants like in the bacteria example. Instead, the environmental change might make things more difficult, but as long as the population can persist, then mutations will continue to enter the population which could confer a selective advantage against the change. So, no, a population does not (cannot) carry all possible adaptations. A population cannot adapt to an environment it hasn't encountered.

Finally, it's worth pointing out that a species can expand its range through migration. A new, unsuitable environment can act as a migration sink (that is, migrants make it there but fail to establish) for a nearby population. If this happens for long enough, some of the migrants may have a mutation allowing them to establish in the new environment.

reference request - Inverse limit in metric geometry

Question. Did you ever see inverse limits to be used (or even seriousely considered) anywhere in metric geometry (but NOT in topology)?

The definition of inverse limit for metric spaces is given below. (It is usual inverse limit in the category with class of objects formed by metric spaces and class of morphisms formed by short maps.)

Definition.
Consider an inverse system of metric spaces $X_n$ and short maps $phi_{m,n}:X_mto X_n$ for $mge n$;
i.e.,(1) $phi_{m,n}circ phi_{k,m}=phi_{k,n}$ for any triple $kge mge n$ and (2) for any $n$, the map $phi_{n,n}$ is identity map of $X_n$.

A metric space $X$ is called inverse limit of the system $(phi_{m,n}, X_n)$ if its underlying space consists of all sequences $x_nin X_n$ such that $phi_{m,n}(x_m)=x_n$ for all $mge n$ and for any two such sequences $(x_n)$ and $(y_n)$ the distance is defined by

$$ | (x_n) (y_n)| = lim_{ntoinfty} | x_n y_n | .$$

Why: I have a theorem, with little cheating you can stated it this way: The class of metric spaces which admit path-isometries to Euclidean $d$-spaces coincides with class of inverse limits of $d$-polyhedral spaces.
In the paper I write: it seems to be the first case when inverse limits help to solve a natural problem in metric geometry. But I can not be 100% sure, and if I'm wrong I still have time to change this sentence.

lo.logic - When forcing with a poset, why do we order the poset in the order that we do?

The reason is that in the corresponding Boolean algebra, 0 is less than 1. That is, stronger conditions correspond to lower Boolean values in the Boolean algebra. The trivial condition (which is often the empty function in the cases you mention), corresponds to the element 1 in the Boolean alebra.

We definitely want to regard lower elements of the Boolean algebra as stronger, since they have more implications in the Boolean algebra sense. (After all, 0 = false is surely the strongest assumption you could make, right?)

Meanwhile, you can be comforted by the fact that Shelah and many researchers surrounding him (and a few others) use the alternative forcing-upwards notation. Nevertheless, the forcing-downwards notation is otherwise nearly universal.

This difference in culture sometimes causes some funny problems when authors from opposing camps collaborate. Sometimes a compromise is struck to never officially to use the order explicitly, and to write "stronger than" or "weaker than" in words, rather than take sides. Another alternative is the use the forcing turnstyle symbol itself as the order, but this solution suffers from the fact that it only works when the order is separative.

---

Edit. I looked at Cohen's PNAS 1963 article, and in that article, he does not use the forcing-downwards notation at all. Rather, he uses the containment symbol $supset$ explicitly. Thus, the assumption in the question that Cohen did indeed use the downward-oriented relation may be unwarranted. (Perhaps this view is a little softened by the observation that he consistently uses $supset$ rather than $subset$.)

Here is my theory. In logic and set theory there has been a long-standing tradition of consistently using the relation ≤ in preference to ≥, presumably to avoid the problems associated with mixing up the greater-than less-than order. Perhaps this goes back to Cantor? Now, in the case of forcing, it is usually the case that you have a condition P already, and you want to ask whether there is Q stronger than P with a certain property (one rarely asks for weaker conditions this way). Thus, if you have the downward-oriented relation, you can economically say "there is Q ≤ P such that..." This is just how Cohen's text reads, since he says "there is Q supset P such that ...". Generalizing Cohen's containment order to an arbitrary partial order, one then wants to interpret containment as ≤. And then the further support for this convention arrives with the fact that it agrees with the Boolean algebra order a few years later, so it became standard (except for the Shelah school and a few others).

co.combinatorics - Help with a double sum, please

Let's define
$$
beta_n doteq sum_{ile (n-1)/2 } binom{n-(i+1)}{i} (-1)^i frac{1}{ (2i+1) 2^{2i+1} }.
$$
The following problem is equivalent to proving that $S=0$:
prove that the sequence $beta_n$ satisfies the recursion
$$
beta_{n+1} = frac{2n+1}{2n+2} beta_n +frac{1}{(n+1) 2^{n+1}}.
$$
Similar with $S=0$, numerical computations suggest that this statement is true. Unfortunately, I didn't see a straightforward way to prove it.

Below is one way to think about the problem, which led to the above reformulation.

The connection between the above problem and $S=0$.

Using the notation developed in the previous answer, let's define
$$
F(m,n) = sum_{k=0}^m (-1)^k binom{m}{k} binom{2(n+k)}{n+k} frac{1}{2^{2(n+k)}} sum_{l=1}^{k+n}
frac{2^l}{l binom{2l}{l} },
$$
and
$$
f(n)= F(0,n)= binom{2n}{n} frac{1}{2^{2n}}
sum_{l=1}^n frac{2^l}{l binom{2l}{l} }.
$$
The statement $S=0$ is the same as $F(m,m)= 0$. Note that $F$ satisfies
$$
F(m,n) = frac{1}{2} F(m-1,n) - frac{1}{2}F(m-1,n+1) ~~~~~~text{(r1)}
$$
Define the difference operator $D(x_1,x_2) = (x_1 - x_2)/2.$ (r1) in terms of $D$
is
$$
F(m,n) = D( F(m-1,n), F(m-1,n+1) ).
$$
Define $D^k$ by iterating $D$:
$$
D^n(x_1,x_2,x_3,ldots,x_{n+1}) = D( D^{n-1}(x_1,x_2,x_3,ldots,x_{n}), D^{n-1}(x_2,x_3,ldots,x_{n+1} ))
$$
Iterating (r1) gives

$$
F(m,n) = D^m( f(n),f(n+1),f(n+2), f(n+3),cdots,f(n+m)).
$$

In particular:
$$
F(m,m) = D^m( f(m),f(m+1),f(m+2), f(m+3),cdots,f(m+m)).
$$

Define
${mathcal D}:{mathbb R}^inftyrightarrow {mathbb R}^infty$ as follows:
the $i^{th}$ component of ${mathcal D}(x_{1}^infty)$ is
$$D^n(x_n,x_{n+1},x_{n+2},ldots,x_{2n}).$$

We can restate our original problem as follows:
show that $(f(1),f(2),f(3),...,f(n),...)$ is in the kernel
of ${mathcal D}$.

Because we are looking for a zero of this operator,
the $1/2$ in the definition of $D$ is not important; thus let us assume that $D(x_1,x_2)$ is simply
$x_1 -x_2$.

Note that
$D^{n}(f(n),f(n+1),...,f(2n)) =0$
is the same as
$$
D^{n-1}(f(n),f(n+1),f(n+2),...,f(2n-1)) = D^{n-1}(f(n+1),f(n+2),f(n+3),...,f(2n)).
$$
A numerical computation reveals that these discrete derivatives equal $frac{1}{(2n-1)2^{2n-1}}$.
One can go back from these values
to an element
of the kernel of
${mathcal D}$
by inverting each $D$ in the above display.
A bit of computation in this direction yields
the vector $beta$ in the first display.
By its construction $beta$ is in the kernel of ${mathcal D}$.
Thus if one can prove that $f$ equals $beta$ then we are done.

Finally, using its definition, we see that $f$ satisfies:
$$
f(n+1) = frac{2n+1}{2n+2} f(n) + frac{1}{(n+1)2^{n+1}}, ~~~ f(1) = 1/2.
$$
These relations determine $f$ and thus we can take them as $f$'s definition.
Thus to verify $f=beta$ it is enough to show that $beta$ satisfies this recursion.

nt.number theory - Values of the j-function

One crude but effective method is to compute all the h conjugates a_j numerically to high precision, from which you can find the polynomial Π(x-a_j) they are the roots of using the fact that it has integral coefficients, (h=class number, and the values a_j are the values of j at the imaginary quadratic integers with the same discriminant.)

Alternatively see the paper On singular moduli by Gross and Zagier, which gives an explicit expression for the values of j as products of many small algebraic integers.

ag.algebraic geometry - intersection cohomology when the resolution is not semi-small

When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection cohomology?
Someone was telling me something about this with shifting the IC or something, but I cannot remember the precise statement.

ct.category theory - What is Yoneda's Lemma a generalization of?

I like to view Yoneda's lemma as a generalization of the description of Galois coverings in topology. To any functor $F: C to Set$ we can associate its category of elements $El(F)$. Its objects are pairs $(x,a)$, $ain C$, $xin F(a)$. A morphism $f:(x,a)to (y,b)$ is a morphism $f_*: a to b$, such that $F(f_*)(x) = y$. Such a category is equipped with a natural projection $Q_F : El(F) to C$, sending $(x,a)$ to $ain C$ and a morphism in $El(F)$ to the underlying morphism in $C$. Then it is easy to see that a natural transformation $mu: (p;cdot) to F(cdot)$ is just the same as a morphism of fibrations over $C$ $$int mu: El(p;cdot) simeq p/C to El(F)$$

This is an example of Grothendieck's construction, applied to set-valued functors. It is itself a categorical version of the correspondence between sheaves of sets and their etale spaces in algebraic geometry.

Consider for example $Nat[(p;cdot);(p;cdot)]$. By Yoneda's lemma it equals to $Hom_C(p;p)$. This is exactly the fibre of $p/C$ over $C$ under the Grothendieck's construction for $(p;cdot)$. The whole automorphism of $(p;cdot)$ is thus determined by the image of $1:pto p$. This reminds that an automorphism of Galois covering is uniquely defined by choosing the image of one element in the fibre, thus $$Aut(Mstackrel{p}{to} N) = p^{-1}(x),;xin N$$

A morphism of Galois coverings $f:Xto Y$ with $X$ connected is likewise uniquely determined (if it exists) by choosing some element of a fibre of $Y$. If $X$ is contractible, then a morphism always exists. This means that slice categories $p/C to C$ are actually similar to contractible fibrations. I don't know how far the analogy goes, but via the classifying space functor slice categories really map to contractible spaces, because they have initial objects.

human biology - Relationship between our microbiome and personalized nutrition

does the microbiome affect food metabolism?

Most definitely (and not surprisingly). The Arumugam paper [1] notes that

The drivers of [enterotype 1] seem to derive energy primarily from carbohydrates and proteins through fermentation, … because genes encoding enzymes involved in the degradation of these substrates (galactosidases, hexosaminidases, proteases) along with glycolysis and pentose phosphate pathways are enriched in this enterotype […]

Enterotype 2 … is enriched in Prevotella … and the co-occurring Desulfovibrio, which can act in synergy to degrade mucin glycoproteins present in the mucosal layer of the gut […]

Enterotype 3 is […] enriched in membrane transporters, mostly of sugars, indicating the efficient binding of mucin and its subsequent hydrolysis as well as uptake of the resulting simple sugars by these genera. […]

The enriched genera indicate that enterotypes use different routes to generate energy from fermentable substrates available in the colon, reminiscent of a potential specialization in ecological niches or guilds. In addition to the conversion of complex carbohydrates into absorbable substrates, the gut microbiota is also beneficial to the human host by producing vitamins. Although all the vitamin metabolism pathways are represented in all samples, enterotypes 1 and 2 were enriched in biosynthesis of different vitamins […]

[All emphasis mine.]

is the food that we eat affecting the microbiome?

Yes, just as certainly. I don’t have a publication handy but it should be obvious that our food influences our gut microbiome – in the extreme case, it can kill it (consider antibiotics side effects).

[1] Manimozhiyan Arumuga, Jeroen Raes & al., Enterotypes of the human gut microbiome, Nature 473, 174–180, May 2011.

big list - What is your favorite "strange" function?

Just a simple construction to illustrate Nate Eldredge's answer about functions with dense graphs. Pick any $mathbb{R}$-vector space E with a norm. On E, choose a non-continuous linear form $L: E to mathbb{R}$; now this can only be done if $dim(E)=infty$, of course.

Then, pick y such that $L(y)=1$, and let $T: E to E$ be defined by $Tx=x-L(x)y$. Then obviously T maps E onto the kernel of L; it is not difficult to prove that $ker (L)$ must be dense in E for any non-continuous L (the two conditions are even equivalent), and thus the graph of T must be dense in $E times E$.

ca.analysis and odes - How to find a solution to a particular Bottcher equation

Functional equations of the form $f(g(x))=(f(x))^p$, where $g(x)$ is known, is called Bottcher equation. Generally, we have only asymptotic formula for the solution $f(x)$ under certain conditions. In one of my study, I come across the following problem,

How to find a nontrivial (i.e., nonconstant) function $f(x)in C^infty(0 ,+infty)$ satisfying the following functional equation $$f(frac{2x^3+a}{3x^2})=f(x)^2,$$ where $a>0$ is a constant.

Or can you tell me finding a explicit form for $f(x)$ is impossible?

ac.commutative algebra - Ring-theoretic characterization of open affines?

Theorem 1: Let $R$ be an integral domain with field of fractions $K$, and $R to A$ a homomorphism. Then $Spec(A) to Spec(R)$ is an open immersion if and only if $A=0$ or $R to K$ factors through $R to A$ (i.e. $A$ is birational over $R$) and $A$ is flat and of finite type over $R$.

Proof: Assume $Spec(A) to Spec(R)$ is an open immersion and $A neq 0$. It is known that open immersions are flat and of finite type. Thus the same is true vor $R to A$. Now $R to K$ is injective, thus also $A to A otimes_R K$. In particular, $A otimes_R K neq 0$. Open immersions are stable under base change, so that $Spec(A otimes_R K) to Spec(K)$ is an open immersion. But since $Spec(K)$ has only one element and $Spec(A otimes_R K)$ is non-empty, it has to be an isomorphism, i.e. $K to A otimes_R K$ is an isomorphism. Now $R to A to A otimes_R K cong K$ is the desired factorization.

Of course, the converse is not as trivial. It is proven in the paper

Susumu Oda, On finitely generated birational flat extensions of integral domains
Annales mathématiques Blaise Pascal, 11 no. 1 (2004), p. 35-40

It is available online. In the section "Added in Proof." you can find some theorems concerning the general case without integral domains. In particular, it is remarked that in E.G.A. it is shown that

Theorem 2: $Spec(A) to Spec(R)$ is an open immersion if and only if $R to A$ is flat, of finite presentation and an epimorphism in the category of rings.

More generally, in EGA IV, 17.9.1 it is proven that a morphism of schemes is an open immersion if and only if it is flat, a (categorical) monomorphism and locally of finite presentation.

There are several descriptions of epimorphisms of rings (they don't have to be surjective), see this MO-question.

pr.probability - Skewing the distribution of random values over a range

Let's clean this up. First, assume a = 0 (if not, just add it on at the end). Second, don't deal with closed intervals (0 <= n <= M, 0 <= n' <= b) but with half-open intervals (0 <= n < M, 0 <= n' < b). This second simplification is exactly the 'bit of tricky math' in the comment, which just replaces M by M+1 and b by b+1.

So now we have a random number 0 <= n < M, and we want to transform it into a random number 0 <= n' < b, where b <= M. This is an interesting question (if not necessarily the one that the OP asked). The solution in the OP's code fragment is to take n' = nb/M. Assuming that b doesn't divide M exactly (for then there is no problem), n' is not uniformly distributed: it takes a number of values (in fact, M mod b of them) with probability ⌈M/b⌉/M, and the remaining values with probability ⌊M/b⌋/M. Exactly the same is true if we take n'= n mod b; there is no statistical difference.

If M/b is big enough, then ⌈M/b⌉/M and ⌊M/b⌋/M are nearly equal; and this may be good enough for your application. If not, then the simplest solution is the following:
Loop:
Generate random 0 <= n < M
if (n < M - (M mod b)) return n mod b
goto Loop

The problem is that this algorithm is not guaranteed to halt! But you can get a closer and closer approximation to uniformity by setting a bigger and bigger bound on the number of times the loop is executed.

Note that for this algorithm, it is important to use n mod b instead of nb/M, because the check for uniformity (n < M - (M mod b)) is simple.

ag.algebraic geometry - GW invariants for varieties with negative first Chern class

EDITED.

The first version of this answer contained correct information, but was answering not the question that was asked:))

So here is the new version of the answer.

1) If a variety with ample canonical bundle does not have rational curves then, indeed, its RATIONAL GW invariants vanish. Because GW invariants can be counted via algebraic geometry and the corresponding moduli spaces will be empty.

But this will not imply that all GW invariants vanish. Indeed, you can take
minimal complex surfaces of general type with $b_+>1$. Then the GW invariant is non-zero, the canonical curve gives a non-zero contribution. But of course it has a non-zero genus.

---

The following is a discussion of vanishing and non-vanishing of GW invariants of varieties of general type. This was my previous answer. This is not immediately related to the question, but I decided to leave it here because it may be helpfull (for somebody).

Hypersurfaces of degree $2n-1$ and less in $CP^n$ always contain lines. And for $2n-1>n+1$ this will be an example of a manifold with ample canonical bundle.
I think it should be possible to show that for these hypersurfaces their rational GW invariants are non-zero, though I am not aware if such a calculation has been done for all these examples.

There was a related discussion here:

Why is a variety of general type hyperbolic?

Nethertheless "morally" it is ture that for a large part of varieties of general type of dimesnion at least 4 with ample canonical bundle GW invarinats vanish. For example, it is conjectured that a "generic" hypersurfaces of degree $dge 2n+1$ in $CP^n$ does not contain rational curves (you need generic here becasue for every d there will be a hypersurface that contains a line). This is a hard conjecture. But it implies that rational GW invariant of such varieties vanishes.

A different thing that can be said about varieties of general type is that they are not projectively unirulled and this can be seen as vanishing of certain rational GW invariants. This follows from a result of Kollar and Ruan. You can see the discussion on page 4 of the following paper:

SYMPLECTIC BIRATIONAL GEOMETRY

http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.3265v1.pdf

model categories - Equivariant map preserves stabilizer

Let $G$ be a group and $X$ a set equipped with a transitive right $G$-action. Further, let $c: Xto X$ be a $G$-equivariant map. Is it true that $text{Stab}(x) = text{Stab}(c(x))$ for all $xin X$?

This doesn't seem to be an interesting mathoverflow question on its own, but the reason I ask is the following: In Hovey's book on Model Categories Hovey proves some sort of five lemma for pointed model categories (Thm.6.5.3). In the end of the proof, he seems to conclude from $alphatext{Stab}(x)alpha^{-1}subsettext{Stab}(x)$ that equality holds; I don't understand how this can be done. The above statement comes from a try to fix this gap.

evolution - How many times did life emerge from the ocean?

I presume you mean how many times did life emerge from the ocean? ("how often" implies you want to know the regularity). Anyway, great question. I really enjoyed reading and thinking about it.

I doubt we know the precise number, or even anywhere near it. But there are several well-supported theorised colonisations which might interest you and help to build up a picture of just how common it was for life to transition to land. We can also use known facts about when different evolutionary lineages diverged, along with knowledge about the earlier colonisations of land, to work some events out for ourselves. I've done it here for broad taxonomic clades at different scales - if interested you could do the same thing again for lower sub-clades.

As you rightly point out, there must have been at least one colonisation event for each lineage present on land which diverged from other land-present lineages before the colonisation of land. Using the evidence and reasoning I give below, at the very least, the following 9 independent colonisations occurred:

• bacteria


• cyanobacteria


• archaea


• protists


• fungi


• algae


• plants


• nematodes


• arthropods


• vertebrates

Bacterial and archaean colonisation
The first evidence of life on land seems to originate from 2.6 (Watanabe et al., 2000) to 3.1 (Battistuzzi et al., 2004) billion years ago. Since molecular evidence points to bacteria and archaea diverging between 3.2-3.8 billion years ago (Feng et al.,1997 - a classic paper), and since both bacteria and archaea are found on land (e.g. Taketani & Tsai, 2010), they must have colonised land independently. I would suggest there would have been many different bacterial colonisations, too. One at least is certain - cyanobacteria must have colonised independently from some other forms, since they evolved after the first bacterial colonisation (Tomitani et al., 2006), and are now found on land, e.g. in lichens.

Protistan, fungal, algal, plant and animal colonisation
Protists are a polyphyletic group of simple eukaryotes, and since fungal divergence from them (Wang et al., 1999 - another classic) predates fungal emergence from the ocean (Taylor & Osborn, 1996), they must have emerged separately. Then, since plants and fungi diverged whilst fungi were still in the ocean (Wang et al., 1999), plants must have colonised separately. Actually, it has been explicitly discovered in various ways (e.g. molecular clock methods, Heckman et al., 2001) that plants must have left the ocean separately to fungi, but probably relied upon them to be able to do it (Brundrett, 2002 - see note at bottom about this paper). Next, simple animals... Arthropods colonised the land independently (Pisani et al, 2004), and since nematodes diverged before arthropods (Wang et al., 1999), they too must have independently found land. Then, lumbering along at the end, came the tetrapods (Long & Gordon, 2004).

Note about the Brundrett paper: it has OVER 300 REFERENCES! That guy must have been hoping for some sort of prize.

References

• Battistuzzi FU, Feijao A, Hedges SB. 2004. A genomic timescale of prokaryote evolution: insights into the origin of methanogenesis, phototrophy, and the colonization of land. BMC Evol Biol 4: 44.


• Brundrett MC. 2002. Coevolution of roots and mycorrhizas of land plants. New Phytologist 154: 275–304.


• Feng D-F, Cho G, Doolittle RF. 1997. Determining divergence times with a protein clock: Update and reevaluation. Proceedings of the National Academy of Sciences 94: 13028 –13033.


• Heckman DS, Geiser DM, Eidell BR, Stauffer RL, Kardos NL, Hedges SB. 2001. Molecular Evidence for the Early Colonization of Land by Fungi and Plants. Science 293: 1129 –1133.


• Long JA, Gordon MS. 2004. The Greatest Step in Vertebrate History: A Paleobiological Review of the Fish‐Tetrapod Transition. Physiological and Biochemical Zoology 77: 700–719.


• Pisani D, Poling LL, Lyons-Weiler M, Hedges SB. 2004. The colonization of land by animals: molecular phylogeny and divergence times among arthropods. BMC Biol 2: 1.


• Taketani RG, Tsai SM. 2010. The influence of different land uses on the structure of archaeal communities in Amazonian anthrosols based on 16S rRNA and amoA genes. Microb Ecol 59: 734–743.


• Taylor TN, Osborn JM. 1996. The importance of fungi in shaping the paleoecosystem. Review of Palaeobotany and Palynology 90: 249–262.


• Wang DY, Kumar S, Hedges SB. 1999. Divergence time estimates for the early history of animal phyla and the origin of plants, animals and fungi. Proc Biol Sci 266: 163–171.


• Watanabe Y, Martini JEJ, Ohmoto H. 2000. Geochemical evidence for terrestrial ecosystems 2.6 billion years ago. Nature 408: 574–578.

gr.group theory - Characteristic subgroups and direct powers

Solved question: Suppose H is a characteristic subgroup of a group G. Is it then necessary that, for every natural number n, in the group $G^n$ (the external direct product of $G$ with itself $n$ times), the subgroup $H^n$ (embedded the obvious way) is characteristic?

Answer: No. A counterexample can be constructed where $G = mathbb{Z}_8 times mathbb{Z}_2$ (here $mathbb{Z}_n$ is the group of integers modulo $n$) with $H$ the subgroup

$${ (0,0), (2,1), (4,0), (6,1) }$$

This subgroup sits in a weird diagonal sort of way and just happens to be characteristic (a quirk of the prime $2$ because there isn't enough space). We find that $H times H$ is not characteristic in $G times G$.

(ASIDE: The answer is yes, though, for many important characteristic subgroups, including fully invariant subgroups, members of the upper central series, and others that occur through typical definitions. Since for abelian groups of odd order, all characteristic subgroups are fully invariant, the answer is yes for abelian groups of odd order, so the example of order $2^4$ has no analogue in odd order abelian groups.)

My question is this:

1. Strongest: Is it true that if H is characteristic in G and $H times H$ in $G times G$, then each $H^n$ is characteristic in $G^n$ [NOTE: As Marty Isaacs points out in a comment to this question, $H times H$ being characteristic in $G times G$ implies H characteristic in G, so part of the condition is redundant -- as explained in (2)]?


2. Intermediate: Is there some finite $n_0$ such that it suffices to check $H^n$ characteristic in $G^n$ for $n = n_0$? Note that if $H^n$ is not characteristic in $G^n$ for any particular n, then characteristicity fails for all bigger $n$ as well. I'd allow $n_0$ to depend on the underlying prime of G if we are examining $p$-groups.


3. Weakest: Is there a test that would always terminate in finite time, that could tell, for a given H and G, whether $H^n$ is characteristic in $G^n$ for all n? The "try all n" terminates in finite time if the answer is no, but goes on forever if the answer is yes. In other words, is there a finite characterization of the property that each direct power of the subgroup is characteristic in the corresponding direct power of the group?

ADDED: My intuition, for what it's worth, is that those subgroups H of G that can be characterized through "positive" statements, i.e., those that do not make use of negations or $ne$ symbols, would have the property that $H^n$ is characteristic in $G^n$. On the other hand, those whose characterization requires statements of exclusion (not a ...) would fail because negative statements are difficult to preserve on taking direct powers. But I don't know how to make this rigorous.

human biology - What is the "lifecycle" of an average eschar and what types of cells are involved in each stage?

Response to cutaneous tissue damage occurs is several distinct but overlapping phases. First an scab is formed as the blood is 'allowed' to clot. A matrix is formed as the platelets adhere to one another, which contracts and 'dries' (forces out the serum) to form a scab. This process clearly happens prior to tissue regeneration/formation to reduce further blood loss.

Fibrinogen

In response to damage to blood vessels a signaling cascade is initated that results in the massive production of fibrin, which is deposited at the site of wounding and subsequently forms blood clots by activating platelets and stabilising the matrix by cross-linking (see PDB's protein of the month 2006).

So this is why clots do not appear to be attached to the surrounding tissue; the fibrin causes the blood to form a matrix that is very tightly bound to itself. Only after clot formation does the tissue remodeling occur, and it is during this stage that the scab will 'bind' more stongly to the surrounding tissue.

The Cell Types and Processes of Wound Healing

This figure (from (Li, 2007)) shows the various stages that occur and cells that are involved in acute wound healing (wounds such as burns and cuts that heal "in a timely fashion");

algorithms - Reconstructing a graph given access to its cut function

Given an unweighted graph $G = (V, E)$, let the cut function on this graph be defined to be:
$C:2^V rightarrow mathbb{Z}$ such that:
$$C(S) = |{(u,v) in E : u in S wedge v notin S}|$$
Suppose you have the ability to query $C$, but otherwise have no knowledge of the edge set $E$. Is it possible to reconstruct $G$ by making only polynomial (in $|V|$) many queries to the cut function?

ca.analysis and odes - An integral arising in statistics

The integral I need:
$$t(x)=int_{-K}^{K}frac{exp(ixy)}{1+y^{2q}}dy$$

$K<infty$, q natural number

For q=1 this integral is
$$pi/2-int_{Arc}frac{exp(ixy)}{1+y^{2}}dy $$
Where Arc has radius $K$

Upper bound is $$Kpi/(K^2-1)^2$$

Can I obtain a better expression for the integral?

One more question about this integral. For K<1 this integral is just
$$-int_{Arc}frac{exp(ixy)}{1+y^{2}}dy?$$

ag.algebraic geometry - Interpretation of elements of H^1 in sheaf cohomology.

$H^1(V;mathcal{F})$ is the space of bundles of affine spaces modeled on $mathcal{F}$. An affine bundle $F$ modeled on $mathcal{F}$ is a sheaf of sets that $mathcal{F}$ acts freely on as a sheaf of abelian groups (i.e., there is a map of sheaves $Ftimes mathcal{F}to F$ which satisfies the usual associativity), and on a small enough neighborhood of any point, this action is regular (i.e., the action map on some point gives a bijection). You should think of this as a sheaf where you can take differences of sections and get a section of $mathcal{F}$.

This matches up with what Anweshi said as follows: given such a thing, you can try to construct an isomorphism to $mathcal{F}$. This means picking an open cover, and picking a section over each open subset and declaring that to be 0. The Cech 1-cochain you get is the difference between these two sections on any overlap, and if an isomorphism exists, the difference between the actual zero section and the candidate ones you picked is the Cech 0-chain whose boundary your 1-cochain is.

Another way of saying this is that a Cech 1-cycle is exactly the same sort of data as transition functions valued in your sheaf, so if you have anything that your sheaf acts on (again, as an abelian group), then you can use these transition functions to build a new sheaf; a homology between to 1-cycles (i.e. a 0-cycle whose boundary is their difference) is exactly the same thing as an isomorphism between two of these.

I'll note that there's nothing special about line bundles; this works for any sheaf of groups (even nonabelian ones). For example, if you take the sheaf of locally constant functions in a group, you will classify local systems for that group. If you take continuous functions into a group, you will get principal bundles for that group. If you take the sheaf $mathrm{Aut}(mathcal{O}_V^{oplus n})$, you'll get rank $n$ locally free sheaves. A particularly famous instance of this is that line bundles are classified by $H^1(V;mathcal{O}_V^*)$.

gr.group theory - Automorphism group objects

Consider a monoidal category C with operation $otimes$, unit object $1$, and diagonal map $delta:A to A otimes A$ for all $A in C$ (with naturality conditions on the diagonal map).

We can define a notion of "group object" in the category, where a group object is an object G of C along with a map from $1$ to G (playing the role of the identity element), a map from G to G (playing the role of the inverse map) and a map $G otimes G to G$ (playing the role of the multiplication). Then, we put compatibility conditions on these operations that are commutative diagrams corresponding to associativity, identity elements and inverses. Note that we need to use the diagonal morphism to formulate the condition on inverses.

Wikipedia defines group objects only in the case where the monoidal operation is the categorical product, so if necessary, we can restrict attention to just those cases.

We can define group object morphisms, etc. Some examples of group objects are: topological groups (category of topological spaces with continuous maps and Cartesian product), Lie groups (category of differential manifolds with smooth maps and manifold product), abelian groups (category of groups with group homomorphisms and direct product, by the Eckmann-Hilton principle), and groups (category of sets).

We can then proceed to define a "group object action" of a group object G on an object A by the analogues of the two conditions for a group action. My questions:

1. Under what situations does there exist a group object that plays the role played by the symmetric group on a set? i.e., A group object $operatorname{Sym}(X)$ for each X in C with an action on X such that any group object action of G on X corresponds to a group object homomorphism $G to operatorname{Sym}(X)$ with the desired compatibility conditions?


2. How does this group object $operatorname{Sym}(X)$, if it exists, relate to the automorphism group $operatorname{Aut}_C(X)$ (which is an actual group, not a group object)?


3. We can also define a "group object action by automorphisms" as the action of one group object on another satisfying the additional condition of being group automorphisms. (We need to use the diagonal morphism to formulate this compatibility condition). Is there some object called the "automorphism group object of a group object" such that any group object action corresponds to a group object morphism to the automorphism group object?


4. How does this automorphism group object relate to the automorphism group in the group object category?

at.algebraic topology - Properties of the class of topological spaces possessing a CW-structure

Let ${mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).

Is it true that for a covering map $Estackrel{f}{to} B$ with $Ein{mathcal C}$ we have $Bin{mathcal C}$, too?

It is true that the total space of a covering lies in ${mathcal C}$ if the base space does, but the reverse implication is not clear to me.

Edit

As Algori pointed out, the quotient space is not even Hausdorff in general. What about finite regular coverings, i.e. those which come from a free action of a finite group on the total space? Is it true then that the quotient space carries a CW-structure, too?

I'm interested in that because this would imply that given a free group action of a finite group on a "nice" space like a CW-complex, one can always choose a CW-structure with respect to which $G$ just permutes cells. Then the corresponding cellular complex would be a (possibly nice) complex of ${mathbb Z}G$-modules (for example, if the space was a sphere, then this procedures can be used to construct a periodic ${mathbb Z}G$-resolution of the trivial module ${mathbb Z}$, showing that the group has to have periodic invariants like homology and cohomology; in this particular case, however, things behave well as the quotient space ${mathbb S}^n/G$ is still a compact manifold).

Thank you.

co.combinatorics - Generalizations of Planar Graphs

A well ordering, $leq$, on a set $S$ is a WELL-QUASI-ORDERING if and only if every sequence $x_iin S$ there exists some $i$ and $j$ natural numbers with $i < j$ with $x_ileq x_j$. (See wikipedia article at bottom)

Robertson-Seymour Theorem: The set $S=Graphs/isomorphism$ are well-quasi-ordered under contraction.

The corollary of this theorem is that any property $P$ of graphs which is closed under the relation of contraction (meaning if $P(G_2)$ and $G_1leq G_2$ then $P(G_1)$) is characterized by a finite set of excluded minors (Which is explained below). An example of such a $P$ is planarity, or linkless embeddability of a graph into R^3. i.e. Every contraction of a planar graph is planar.

Suppose $P$ is a property closed under $leq$.

***If $Bleq G$ and $B$ is not $P(B)$, Then not $P(G)$.

The idea is to characterize $P$ by a collection of bad $B$'s. The finiteness of the set of excluded minors comes from the well-quasi ordering and doesn't use the idea of a graph:
Assuming we have well-quasi-ordered set $(S,leq)$. One can prove that every property $P$ which is closed under the relation is characterized by a Finite set of excluded minors.That is, there exists some $X=lbrace x_1,ldots,x_nrbrace subset S $ such that for all $sin S$, $$ mbox{ not } P(s) iff exists i, x_i leq s $$.

The existence of a finite set $X$ is implicitly in 12.5 of Diestel (link at bottom, see the corollary of Graph Minor Theorem in Diestel). First convince yourself that there exists a set of $B's$ (not necessarily finite) as in * that characterize property $P$. Then consider the smallest such set of $B$'s and using the property of well-quasi ordering show that is is finite. Note that as in the second wikipedia article, we can say any set of elements $Asubset S$ such that for all $a,bin A$ we have $a nleq b$ must be a finite set (provided $leq$ is a well-quasi-ordering).

Actual work done showing stuff in a topological direction has be done by Eran Nevo http://www.math.cornell.edu/~eranevo/

I suspect that Matroids have a well-quasi-ordering and that there is work being done toward proving an analogous theorem for them.

I have a limit on links:

en.wikipedia.org/wiki/Well-quasi-ordering

en.wikipedia.org/wiki/Robertson-Seymour_theorem

diestel-graph-theory.com/GrTh.html

rt.representation theory - Finite dimensional spherical representation of $SO(n,1)(mathbb{R})$

Bonsoir Ludo! I am puzzled by the fact that your title asks something more restrictive than the OP, since the latter does not contain the word "spherical". Let me answer the latter first. Any finite-dimensional representation of $SO(n,1)(mathbb{R})$ extends to a representation of the complexification, which is $SO_{n+1}(mathbb{C})$. By Weyl's unitary trick, those are in 1-1 correspondence with unitary finite-dimensional representation of the maximal compact subgroup of the complexification, here $SO_{n+1}$. The finite-dimensional, unitary, irreducible representations of such a group are parametrized by their highest weight, and can be described via Verma modules, see Chapter IV in Knapp's "Representation theory of semi-simple groups" (Princeton UP, 1986).

Now, if you need only spherical irrep's, this amounts to consider irreducible $SO_{n+1}$-representations having non-zero $SO_n$-invariant vectors; or, equivalently (by an easy case of Frobenius reciprocity), irreducible $SO_{n+1}$-sub-representations of $L^2(S^n)$ (where $S^n=SO_{n+1}/SO_n$ is the $n$-sphere). These correspond to homogeneous harmonic polynomials in $n+1$ variables.

mg.metric geometry - How far can the analogy between a Cayley graph and a symmetric space be pushed?

The Cheeger constants for graphs and Riemannian locally symmetric spaces are closely related. Via inequalities of Buser and Cheeger, these are also related to eigenvalues of the laplacians for each. This analogy led to the first construction of expander graphs, by Margulis, via Property (T). More recently, this analogy has been exploited by several people, notably Marc Lackenby, to study finite-sheeted coverings using Cayley graphs of finite quotients as a finite simplicial approximation.

The point, roughly, is the following. Let $Gamma$ be a group with generating set $S$, and suppose $Gamma = pi_1(M)$ for some Riemannian manifold $M$. Then any finite quotient $F$ under a homomorphism $phi$ has a generating set $phi(S)$, so we can form the corresponding Cayley graph $mathcal{G}(F, phi(S))$. Properties of $mathcal{G}(F, phi(S))$ like girth, spectrum, expansion constants, Cheeger constant, and so forth are closely related to the analogous concept for the finite-sheeted covering $M_phi$ of $M$ corresponding to the subgroup $mathrm{kernel}(phi)$ of $Gamma$. This analogy is most potent when you consider a family {$mathcal{G}(F_j, phi_j(S))$} of Cayley graphs corresponding to a family $F_j$ of finite quotients of $Gamma$.

References for all these concepts are the books On Property ($tau$) by Lubotzky and Zuk (unpublished, but on Lubotzky's website), Discrete Groups, Expanding Graphs and Invariant Measures by Lubotzky, Elementary Number Theory, Group Theory and Ramanujan graphs by Davidoff, Sarnak, and Valette, and Marc Lackenby's paper Expanders, ranks and graphs of groups, Israel J. Math. 146 (2005) 357-370.

Fourier transform for dummies

One of the main uses of Fourier transforms is to diagonalize convolutions. In fact, many of the most useful properties of the Fourier transform can be summarized in the sentence "the Fourier transform is a unitary change of basis for functions (or distributions) that diagonalizes all convolution operators." I've been ambiguous about the domain of the functions and the inner product. The domain is an abelian group, and the inner product is the L² inner product with respect to Haar measure. (There are more general definitions of the Fourier transform, but I won't attempt to deal with those.)

I think a good way to motivate the definition of convolution (and thus eventually of the Fourier transform) starts with probability theory. Let's say we have an abelian group (G, +, -, 0) and two independent random variables X and Y that take values in G, and we are interested in the value of X + Y. For simplicity, let's assume G = {x₁, ..., x_n} is finite. For example, X and Y could be (possibly biased) six-sided dice, which we can roll to get two independent elements of Z/6Z. The sum of the die rolls mod 6 gives another element of the group.

For x ∈ G, let f(x) be the probability P(X = x), and let g(x) = P(Y = x). What we care about is h(x) := P(X + Y = x). We can compute this as a sum of joint probabilities:

h(x) = P(X + Y = x) = Σ_y+z=xP(X = y & Y = z)

However, since X and Y are independent, P(X = y & Y = z) = P(X = y)P(Y = z) = f(y)g(z), so the sum is actually

h(x) = Σ_y+z=xf(y)g(z) = Σ_y∈Gf(y)g(x-y).

This is called the convolution of f and g and denoted by f*g. In words, the convolution of two probability distributions is the probability distribution of the sum of two independent random variables having those respective distributions. From that, one can deduce easily that convolution satisfies nice properties: commutativity, associativity, and the existence of an identity. Moreover, convolution has the same relationship to addition and scalar multiplication as pointwise multiplication does (namely, bilinearity). In the finite setting, there's also an obvious L² inner product on distributions, with respect to which, for each f, the transformation g -> f * g is normal. Since such transformations also commute, recalling a big theorem from finite-dimensional linear algebra, we know there's an orthonormal basis with respect to which all of them are diagonal. It's not difficult to deduce then that in such a basis, convolution must be represented by coordinatewise multiplication. That basis is the Fourier basis, and the process of obtaining the coordinates in the Fourier basis from coordinates in the standard basis (the values f(x) for x ∈ G) is the Fourier transform. Since both bases are orthonormal, that transformation is unitary.

If G is infinite, then much of the above has to be modified, but a lot of it still works. (Most importantly, for now, the intuition works.) For example, if G = Rⁿ, then the sum Σ_y∈Gf(y)g(x-y) must be replaced by the integral ∫_y∈Gf(y)g(x-y)dy to define convolution, or even more generally, by Haar integration over G. The Fourier "basis" still has the important property of representing convolution by "coordinatewise" (or pointwise) multiplication and therefore of diagonalizing all convolution operators.

The fact that the Fourier transform diagonalizes convolutions has more implications than may appear at first. Sometimes, as above, the operation of convolution is itself of interest, but sometimes one of the arguments (say f) is fixed, and we want to study the transformation T(g) := f*g as a linear transformation of g. A lot of common operators fall into this category. For example:

• Translation: T(g)(x) = g(x-a) for some fixed a. This is convolution with a "unit mass" at a.


• Differentiation: T(g)(x) = g'(x). This is convolution with the derivative of a negative unit mass at 0.


• Indefinite integration (say on R): T(g)(x) = ∫^x_-infinityg(t)dt. This is convolution with the Heaviside step function.

In the Fourier basis, all of those are therefore represented by pointwise multiplication by an appropriate function (namely the Fourier transform of the respective convolution kernel). That makes Fourier analysis very useful, for example, in studying differential operators.

Tate Cohomology via Stable Categories

Situation

Let $G$ be a finite group and provide $Gtext{-mod} := {mathbb Z}Gtext{-mod}$ with the Frobenius structure of ${mathbb Z}$-split short exact sequences. Denote by $underline{Gtext{-mod}}$ the associated stable category with loop functor $Omega$.

For any Frobenius category $({mathcal A},{mathcal E})$ and a complete projective-injective resolution $P_{bullet}$ of some $Xin{mathcal A}$, we have for any $Yin{mathcal A}$ a canonical isomorphism of abelian groups

$H^n(text{Hom}_{mathcal A}(P_{bullet},Y))cong [Omega^n X,Y]$,

where $[-,-] := text{Hom}_{underline{{mathcal A}}}(-,-)$.

Applying this to $Gtext{-mod}$ yields an isomorphism

$widehat{H}^k(G;M)cong [Omega^k{mathbb Z},M]$,

where $widehat{H}^k(G;M)$ denotes the Tate-Cohomology of $G$ with values in $M$.

If I didn't mix things up, in this language Tate-Duality should mean that the canonical map

$[{mathbb Z},Omega^k{mathbb Z}]otimes_{mathbb Z}[Omega^k{mathbb Z},{mathbb Z}]to[{mathbb Z},{mathbb Z}]cong{mathbb Z}/|G|{mathbb Z}$

is a duality.

Question

I'd like to know sources which introduce and treat Tate cohomology in the way described above, i.e. using the language of Frobenius categories and its associated stable categories. In particular, I would be interested in a proof of Tate Duality using this more abstract language instead of resolutions.

Does anybody know such sources?

Remark

It seems to be more difficult to work over the integers instead of some field, for in this case, the exact sequences in the Frobenius structure $Gtext{-mod}$ are required to be ${mathbb Z}$-split, which is not automatic. As a consequence, there may be projective/injective objects in $(Gtext{-mod},{mathcal E}^{G}_{{e}})$ which are not projective/injective as ${mathbb Z}G$-modules. Further, the long exact cohomology sequence exists only for ${mathbb Z}$-split exact sequences of $G$-modules (not good, because Brown uses the exact sequence $0to {mathbb Z}to{mathbb Q}to{mathbb Q}/{mathbb Z}to 0$ in his proof of Tate duality); of course, one can choose particular complete resolutions of ${mathbb Z}$ consisting of ${mathbb Z}G$-projective modules, and such a resolution yields a long exact cohomology sequence for any short exact sequence of coefficient modules, but this seems somewhat unnatural and doesn't fit into the picture right now.

Partial Results

(1) For any subgroup $Hleq G$ there are restriction and corestriction morphisms

$[Omega^k {mathbb Z},-]^{underline{G}}=widehat{H}^*(G;-)leftrightarrowswidehat{H}^*(H;-)=[Omega^k{mathbb Z},-]^{underline{H}}$

defined as follows: for any $G$-module $M$, the abelian group $[{mathbb Z},M]^{underline{G}}$ is in canonical bijection with $M^G / |G| M^G$, and there are restriction and transfer maps

$text{res}: M^G / |G| M^Glongrightarrow M^H / |H| M^H,quad [m]mapsto [m]$,

$text{tr}: M^H / |H| M^Hlongrightarrow M^G / |G| M^Gquad [m]mapstoleft[sumlimits_{gin G/H} g.mright]$,

respectively. Now

$[Omega^k{mathbb Z},M]^{underline{G}}cong [{mathbb Z},Omega^{-k}M]^{underline{G}}stackrel{text{res}}{longrightarrow} [{mathbb Z},Omega^{-k}M]^{underline{H}}cong[Omega^k{mathbb Z},M]^{underline{H}}$

$[Omega^k{mathbb Z},M]^{underline{H}}cong [{mathbb Z},Omega^{-k}M]^{underline{H}}stackrel{text{tr}}{longrightarrow} [{mathbb Z},Omega^{-k}M]^{underline{G}}cong[Omega^k{mathbb Z},M]^{underline{G}}$

seems to be the natural thing to define restriction and transfer. (This is very similar to the usual method of giving a morphism of $delta$-functors only in degree $0$ and extend it by dimension shifting, though a bit more elegant in my opinion)

Note that it was implicitly used that $Omega^k$ commutes with the forgetful functor $Gtext{-mod}to Htext{-mod}$

(2) For any subgroup $Hleq H$, $gin G$ and a $G$-module $M$ there is a map

$g_*: widehat{H}^*(H;-)towidehat{H}^*(gHg^{-1};M)$

extending the canonical map

$M^H/|H|M^Hlongrightarrow M^{gHg^{-1}}/|H|M^{gHg^{-1}},quad [m]mapsto [g.m]$.

(1) and (2) fit together in the usual way; there is a transfer formula and a lifting criterion for elements of Sylow-subgroups.

(3) The cup product on $widehat{H}^*(G;{mathbb Z})$ is given simply by composition of maps:

$[Omega^p{mathbb Z},{mathbb Z}]otimes_{mathbb Z}[Omega^q{mathbb Z},{mathbb Z}]stackrel{Omega^qotimestext{id}}{longrightarrow}[Omega^{p+q}{mathbb Z},Omega^q{mathbb Z}]otimes_{mathbb Z}[Omega^q{mathbb Z},{mathbb Z}]longrightarrow [Omega^{p+q}{mathbb Z},{mathbb Z}]$

Does anybody see why this product is graded-commutative?

molecular biology - Is there a Reverse Transcription optimization for long, 9kb, transcripts?

Has anyone optimized RT for long transcripts (9kb)? The downstream application will be PCR amplification and Illumina library prep. It will be trivial to make internal primers sets for the PCR that are specific as long as there are no chimeric sequences. If there are, they will probably get primed also. If anyone knows of an optimization and/or other potential pitfalls, I would love to hear them.

ct.category theory - Are bicategories of lax functors also bicategories of of pseudofunctors?

Yes, there is. A relevant general framework is the following: for any
2-monad T, we can define notions of pseudo and lax morphism between
T-algebras, and there is a forgetful functor from the 2-category of
T-algebras and pseudo morphisms to the 2-category of T-algebras and
lax morphisms. If T is well-behaved, this forgetful functor has a
left adjoint; see for instance this paper.

There is a 2-monad on the 2-category of Cat-graphs whose algebras are
bicategories, whose pseudo morphisms are pseudofunctors, and whose lax
morphisms are lax functors. Therefore, the above applies to
bicategories. If you trace through the construction, you'll see that
it is given essentially by the recipe you proposed. (This case of the construction can probably be found elsewhere in the literature as well, in more explicit form, but this is the way I prefer to think about it.)

The caveat is that the 2-cells in the 2-categories
defined above are not any of the the usual sort of transformations
between bicategories, only the
icons. (This is what allows you to
have a 2-category containing lax functors.) However, the usual sorts of
transformations are "corepresentable," that is, for any bicategory D
there is a bicategory Cyl(D) such that pseudo or lax functors into
Cyl(D) are the same as pairs of pseudo or lax functors into D and a
pseudo (or lax, with a different definition of Cyl) natural
transformation between them, and likewise we have 2Cyl(D) for
modifications. I believe one can use this to show that in this case,
the construction coming from 2-monad theory does have the property you
want.

Of course, by Chris' question, it seems that this version of L cannot
itself be described as a left adjoint, since there is no 2- or 3-category
containing lax functors and arbitrary pseudo/lax transformations.

ca.analysis and odes - Measure 0 sets on the line with Hausdorff dimension 1

I use $dim_H(E)$ to denote the Hausdorff dimension of a set $E subseteq mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $dim_H(E) < 1$, then $|E| = 0$. The converse is not true, and there are many cases where $dim_H(E) = 1$ yet $|E| = 0$. So the question:

What was the first (or most elementary) example of this phenomenon?

After some looking around, I was able to prove that a central Cantor set $C$ with ratio of dissection $r_k = 1/(2+frac{1}{k})$ satisfies the condition I want. It is easy to see that $|C| = 0$ since at step $n$ of the process to construct this Cantor set, it has measure $2^n(r_1 cdots r_n)$ which in this case limits to 0, but for the Hausdorff dimension I required a non-trivial result from the paper Sums of Cantor sets (Cabrelli, Hare, Molter) that gave the formula

$dim_H(C) = liminf_n frac{n ln 2}{ln r_1 cdots r_n}$.

This result is fairly recent and sophisticated, and I feel that there should be older and simpler examples.

fa.functional analysis - Need help with references on the status of a "Littlewood Problem"

The "Littlewood Problem" in the title asks for a characterization of finite sequences

n₁< ...< n_k of integers such that z^n₁+z^n₂+...+z^n_k≠0
for any complex number z of unit modulus.

Does anybody know about the current status of this problem?

1)I came to know this Littlewood Problem through the paper of Casazza & Kalton, http://www.jstor.org/pss/2699467.

2)For k=2,3,4, by some simple geometric argument, a complete characterization can be easily obtained. I wonder if such a result has already appeared in the literature.

3)Furthermore, I wonder if at least for the case of k=5, (or indeed, similarly for any k),the following is true? And if it is, whether it is in the literature somewhere.

Suppose that for some complex number z of unit modulus and some integers n₁< ...< n₅,

z^n₁+z^n₂+...+z^n₅=0

then either z^n₁,z^n₂,..,z^n₅
are evenly distributed on the unite circle (i.e., they look like the 5th roots of unit
after a certain rotation is applied to each)
or three pounts among z^n₁,z^n₂,..,z^n₅
are evenly distributed on the unite circle.

ag.algebraic geometry - Easiest way to determine the singular locus of projective variety & resolution of singularities

Concerning your first question:

For many questions, the easiest way to see the nuts and bolts of a projective variety $V subseteq P^n$ is to look at its cone $CV subseteq A^{n+1}$. After all, the graded ring whose Proj is $V$ is the same as the ungraded ring whose Spec is $CV$. Obviously, there is almost always a singularity at the origin; but if you ignore that point, the other singular points all correspond between $V$ and $CV$. You can also think of the grading as geometrically represented by multiplication by $k^*$, if you are working over an algebraically closed field $k$. (Because the homogeneous polynomials are then eigenvectors of that group action.) You can think of $V$ as obtained from $CV$ by and then dividing by scalar multiplication.

The atlas-of-charts analysis of a projective variety is certainly important, but to some extent it is meant as an introduction to intrinsic algebraic geometry rather than as the best computational tool.

---

Your second question is reviewed in Wikipedia. As Wikipedia explains, Hironaka's big theorem was that it is possible to resolve all singularities of a variety by iterated blowups along subvarieties. I do not know a lot about this theory, but if so many capable mathematicians went to so much trouble to find a method, then surely there is no simple method.

On the other hand for curves, there is a stunning method that I learned about (or maybe relearned) just recently. Again according to Wikipedia, taking the integral closure of the coordinate ring of an affine curve, or the graded coordinate ring of a projective curve, solves everything. The claim is that it always removes the singularities of codimension 1, which are the only kind that a curve has.

gn.general topology - Galois Groups vs. Fundamental Groups

I saw this question a while ago and felt something in the way of a (probably misguided) missionary zeal to make at least a few elementary remarks. But upon reflection,
it became clear that even that would end up rather long, so it was difficult to find the time until now.

The point to be made is a correction: fundamental groups in arithmetic geometry are not the same as Galois groups, per se. Of course there is a long tradition of
parallels between Galois theory and the theory of covering spaces, as when Takagi writes of being misled by
Hilbert in the formulation of class field theory essentially on account of
the inspiration from Riemann surface theory. And then, Weil was fully aware that
homology and class groups are somehow the same, while speculating that a sort of
non-abelian number theory informed by the full theory of the 'Poincare
group' would become an ingredient of many serious arithmetic investigations.

A key innovation of Grothendieck, however, was the formalism for refocusing attention on the
base-point. In this framework, which I will review briefly below, when one says
$$pi_1(Spec(F), b)simeq Gal(bar{F}/F),$$
the base-point in the notation is the choice of separable closure
$$b:Spec(bar{F})rightarrow Spec(F).$$
That is,

Galois groups are fundamental groups with generic base-points.

The meaning of this is clearer in the Galois-theoretic interpretation of the fundamental group of
a smooth variety $X$. There as well, the choice of a separable closure
$k(X)hookrightarrow K$ of the function field $k(X)$ of $X$ can be viewed as a base-point
$$b:Spec(K)rightarrow X$$
of
$X$, and then
$$pi_1(X,b)simeq Gal(k(X)^{ur}/k(X)),$$
the Galois group of the maximal sub-extension $k(X)^{ur}$ of $K$ unramified over $X$.
However, it would be quite limiting to take this last object as the definition of the fundamental group.

We might recall that even in the case of a path-connected pointed topological space $(M,b)$ with universal covering space $$M'rightarrow M,$$
the isomorphism $$Aut(M'/M)simeq pi_1(M,b)$$ is not canonical. It comes rather
from the choice of a base-point lift $b'in M'_b$. Both $pi_1(M,b)$ and $Aut(M'/M)$
act on the fiber $M'_b$, determining bijections
$$pi_1(M,b)simeq M'_bsimeq Aut(M'/M)$$
via evaluation at $b'$. It is amusing to check that the isomorphism of groups obtained thereby is independent of
$b'$ if and only if the fundamental group is abelian. The situation here is an instance of the choice involved in the isomorphism
$$pi_1(M,b_1)simeq pi_1(M,b_2)$$
for different base-points $b_1 $ and $b_2$.
The practical consequence is that when fundamental groups are equipped with natural
extra structures coming from geometry, say Hodge structures or Galois actions, different base-points give rise to enriched groups that are
are often genuinely non-isomorphic.

A more abstract third group is rather important in the general discussion of base-points. This is
$$Aut(F_b),$$
the automorphism group of the functor
$$F_b:Cov(M)rightarrow Sets$$
that takes a covering $$Nrightarrow M$$ to its fiber $N_b$. So elements of $Aut(F_b)$ are
compatible collections $$(f_N)_N$$ indexed by coverings $N$ with each $f_N$ an automorphism of the set $N_b$.
Obviously, newcomers might wonder where to get such compatible collections, but
lifting loops to paths defines a natural map
$$pi_1(M,b)rightarrow Aut(F_b)$$
that turns out to be an isomorphism. To see this, one uses again the fiber
$M'_b$ of the universal covering space, on which both groups act compatibly.
The key point is that while $M'$ is not
actually universal in the category-theoretical sense, $(M',b')$ is universal
among pointed covers. This is enough to show that an element of $Aut(F_b)$ is completely determined by its action
on $b'in M'_b$, leading to another bijection $$Aut(F_b)simeq M'_b.$$
Note that the map $pi_1(M,b)rightarrow Aut(F_b)$ is entirely canonical,
even though we have used the fiber $M'_b$ again to prove bijectivity, whereas the identification with $Aut(M'/M)$
requires the use of $(M'_b,b')$ just for the definition.

Among these several isomorphic groups, it is $Aut(F_b)$ that ends up most relevant for the
definition of the etale fundamental group.

So for any base-point $b:Spec(K)rightarrow X$ of a connected scheme
$X$ (where $K$ is a separably closed field, a 'point' in the etale theory), Grothendieck defines
the 'homotopy classes of etale loops' as
$$pi^{et}_1(X,b):=Aut(F_b),$$
where $$F_b:Cov(X) rightarrow mbox{Finite Sets}$$ is the functor that sends a finite etale covering
$$Yrightarrow X$$ to the fiber $Y_b$. Compared to a construction like
$Gal(k(X)^{ur}/k(X))$, there are three significant advantages to this definition.

(1) One can easily consider small base-points, such as might come from
a rational point on a variety over $mathbb{Q}$.

(2) It becomes natural to study the variation of $pi^{et}_1(X,b)$ with $b$.

(3) There is an obvious extension to path spaces $$pi^{et}_1(X;b,c):=Isom(F_b,F_c),$$ making up a two-variable
variation.

This last, in particular, has no analogue at all in the Galois group approach to
fundamental groups. When $X$ is a variety over $mathbb{Q}$, it becomes possible, for example, to study $pi^{et}_1(X,b)$ and
$pi^{et}_1(X;b,c)$ as sheaves on $Spec(mathbb{Q})$, which encode rich information about
rational points. This is a long story, which would be rather tiresome to expound upon here
(cf. lecture at the INI ).
However, even a brief contemplation of it might help you to appreciate the arithmetic perspective that general $pi^{et}_1$'s
are substantially more powerful than Galois groups. Having read thus far, it shouldn't surprise you that
I don't quite agree with
the idea explained, for example, in this post
that a Galois group is only
a 'group up to conjugacy'. To repeat yet again, the usual Galois groups are just fundamental groups with specific large base-points.
The dependence on these base-points as well as a generalization to small base-points
is of critical interest.

Even though the base-point is very prominent in Grothendieck's definition, a curious fact is that it took quite a long time for even the experts to fully metabolize its significance.
One saw people focusing mostly on base-point independent constructions
such as traces or characteristic polynomials associated to representations. My impression is that the initiative for allowing the base-points a truly active role
came from Hodge-theorists like Hain, which then was taken up by arithmeticians like Ihara and Deligne.
Nowadays, it's possible to give entire lectures just about base-points, as Deligne has actually done on several occasions.

Here is a puzzle that I gave to my students a while ago: It has been pointed out that
$Gal(bar{F}/F)$ already refers to a base-point in the Grothendieck definition. That is,
the choice of $Fhookrightarrow bar{F}$ gives at once a universal covering space and a base-point.
Now, when we turn to the manifold situation $M'rightarrow M$, a careful reader may have noticed a hint above that there is
a base-point implicit in $Aut(M'/M)$ as well.
That is, we would like to write $$Aut(M'/M)simeq pi_1(M,B)$$ canonically for some base-point $B$. What is $B$?

Added:

-In addition to the contribution of Hodge-theorists, I should say that Grothendieck himself urges attention to many base-points in his writings from the 80's, like 'Esquisse d'un programme.'

-I also wanted to remark that I don't really disagree with the point of view in JSE's answer either.

Added again:

This question reminds me to add another very basic reason to avoid the Galois group as a definition of $pi_1$. It's rather tricky to work out the functoriality that way, again because the base-point is de-emphasized. In the $Aut(F_b)$ approach, functoriality is essentially trivial.

Added, 27 May:

I realized I should fix one possible source of confusion. If you work it out, you find that the bijection $$pi_1(M,b)simeq M'_bsimeq Aut(M'/M)$$ described above is actually an anti-isomorphism. That is, the order of composition is reversed. Consequently, in the puzzle at the end, the canonical bijection $$Aut(M'/M)simeq pi_1(M,B)$$ is an anti-isomorphism as well. However, another simple but amusing exercise is to note that the various bijections with Galois groups, like $$pi_1(Spec(F), b)simeq Gal(bar{F}/F),$$ are actually isomorphisms.

Added, 5 October:

I was asked by a student to give away the answer to the puzzle. The crux of the matter is that
any continuous map $$B:Srightarrow M$$ from a simply connected set $S$ can be used as a base-point for the
fundamental group. One way to do this to use $B$ to get a fiber functor
$F_B$ that associates to a covering $$Nrightarrow M$$the set of splittings of the covering $$N_B:=Stimes_M Nrightarrow S$$ of $S$.
If we choose
a point $b'in S$, any splitting is determined by its value at $b'$, giving
a bijection of functors
$F_B=F_{b'}=F_b$ where $b=B(b')in M$. Now, when $$B:M'rightarrow M$$ is the universal
covering space, I will really leave it as a (tautological) exercise
to exhibit a canonical anti-isomorphism
$$Aut(F_B)simeq Aut(M'/M).$$ The 'point' is that $$F_B(M')$$ has a canonical
base-point that can be used for this bijection.

moduli spaces - Is the Torelli map an immersion?

Respectfully, I disagree with Tony's answer. The infinitesimal Torelli problem fails for $g>2$ at the points of $M_g$ corresponding to the hyperelliptic curves. And in general the situation is trickier than one would expect.

The tangent space to the deformation space of a curve $C$ is $H^1(T_C)$, and the tangent space to the deformation space of its Jacobian is $Sym^2(H^1(mathcal O_C))$. The infinitesimal Torelli map is an immersion iff the map of these tangent spaces

$$ H^1(T_C) to Sym^2( H^1(mathcal O_C) )$$

is an injection. Dually, the following map should be a surjection:

$$ Sym^2 ( H^0(K_C) ) to H^0( 2K_C ), $$

where $K_C$ denotes the canonical class of the curve $C$. This is a surjection iff $g=1,2$ or $g=3$ and $C$ is not hyperelliptic; by a result of Max Noether.

Therefore, for $gge 3$ the Torelli map OF STACKS $tau:M_gto A_g$ is not an immersion. It is an immersion outside of the hyperelliptic locus $H_g$. Also, the restriction $tau_{H_g}:H_gto A_g$ is an immersion.

On the other hand, the Torelli map between the coarse moduli spaces IS an immersion in char 0. This is a result of Oort and Steenbrink "The local Torelli problem for algebraic curves" (1979).

F. Catanese gave a nice overview of the various flavors of Torelli maps (infinitesimal, local, global, generic) in "Infinitesimal Torelli problems and counterexamples to Torelli problems" (chapter 8 in "Topics in transcendental algebraic geometry" edited by Griffiths).

P.S. "Stacks" can be replaced everywhere by the "moduli spaces with level structure of level $lge3$ (which are fine moduli spaces).

P.P.S. The space of the first-order deformations of an abelian variety $A$ is $H^1(T_A)$. Since $T_A$ is a trivial vector bundle of rank $g$, and the cotangent space at the origin is $H^0(Omega^1_A)$, this space equals $H^1(mathcal O_A) otimes H^0(Omega^1_A)^{vee}$ and has dimension $g^2$.

A polarization is a homomorphism $lambda:Ato A^t$ from $A$ to the dual abelian variety $A^t$. It induces an isomorphism (in char 0, or for a principal polarization) from the tangent space at the origin $T_{A,0}=H^0(Omega_A^1)^{vee}$ to the tangent space at the origin $T_{A^t,0}=H^1(mathcal O_A)$. This gives an isomorphism
$$ H^1(mathcal O_A) otimes H^0(Omega^1_A)^{vee} to
H^1(mathcal O_A) otimes H^1(mathcal O_A). $$

The subspace of first-order deformations which preserve the polarization $lambda$ can be identified with the tensors mapping to zero in $wedge^2 H^1(mathcal O_A)$, and so is isomorphic to $Sym^2 H^1(mathcal O_A)$, outside of characteristic 2.