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.

at.algebraic topology - Serre spectral sequence with spectra

Since the ordinary Serre spectral sequence is about a fibration of spaces, I don't think you can just talk about a "fibration of spectra" and expect that to be a generalization, since the suspension spectrum functor doesn't preserve fibration sequences. However, there is a version of the Serre spectral sequence involving parametrized spectra, which one can think of as a fibration whose base is an ordinary space and whose fibers are spectra. It can be found in section 20.4 of May-Sigurdsson, Parametrized Homotopy Theory.