ag.algebraic geometry - Question about a family of semistable curves

I don't think this is quite right. Here is the right statement: let $E_1, ..., E_g$ be the tails, with maps $q_i: E_i longrightarrow B$. Then

$pi_* omega_{C/B} = bigoplus (q_i)_* omega_{E_i/B}$.

So, if your tails don't vary with B, this bundle is trivial.

Explanation: $omega_{C/B}$ can be described explicitly: a section of $omega_{C/B}$ is a one-form on each component of $C$, with simple poles at the nodes of $C$, so that at every node the residues of the form on the two components match.

Now, on a curve of genus $1$, a one-form with only a single simple pole, must in fact have no poles. So the sections of $omega_{C/B}$, restricted to the $E_i$, are sections of $omega_{E_i/B}$. Moreover, the sections of $omega_{C/B}$ restricted to the rational components are one-forms with no poles, and are hence $0$. So to give a section of $omega_{C/B}$ is simply to give a section of $omega_{E_i/B}$ on each $E_i$. QED.

symmetric functions - When the splitting fields of shifted generic polynomials are linearly disjoint?

Let me start by rigorously pose my question.

Let $K$ be an algebraically closed field of characteristic $2$, let $n$ be an even integer number, let $f(X) = X^n + T_1 X^{n-1} + cdots + T_n$, be the generic polynomial, that is, $T = (T_1, ldots, T_n)$ is a tuple of algebraically independent variables over $K$.

Let $Omega = $ { $omega_1, ldots, omega_m$} be a finite subset of $K$, let $f_i(X) = f(X) - omega_i$, and let $F_i$ be the splitting field of $f_i$ over $K(T)$ ($i=1,ldots, m$).

Question: For which $Omega$ the splitting fields $F_1, ldots, F_m$ are linearly disjoint over $K(T)$?

Remarks:

1. If the characteristic of $K$ is NOT $2$, or if $n$ is odd, then the splitting fields are linearly disjoint for arbitrary $Omega$. Thus, I pose the question the specific case of $p=2$ and $n$ even.




2. The answer cannot be ALWAYS, as in the previous remark. Indeed, one can show that if $p=n=2$, $m=4$, and $omega_1 + omega_2 + omega_3 + omega_4 = 0$, then the splitting fields are not linearly disjoint. In fact, if $p=n=2$, the answer is that the splitting fields are linearly disjoint if and only if the sum of any even number of elements of $Omega$ does not vanish.




3. How one proves 1 + 2: The linear disjointness of the splitting fields can be reduced to the linear independent of the discriminant as elements in $H^1(K,mathbb{Z}/2mathbb{Z})$. If $pnmid n$, then one can use ramification theory to achieve this, if $pneq 2$ but divides $n$, one can calculate this by hand using the formula given by the determinant of the Sylvester matrix. If $p=2$, I know of no formula for the discriminant in terms of the coefficients. However when $p=n=2$ situation is simple enough to do calculations and hence get 2.

Motivation: The linear disjointness of the splitting fields allows one to calculate a Galois group of a composite of polynomials, which in turn yields arithmetic features of the ring of polynomials over large finite fields. Let me not elaborate on that here

at.algebraic topology - understanding Steenrod squares

The Steenrod square is an example of a cohomology operation. Cohomology operations are natural transformations from the cohomology functor to itself. There are a few different types, but the most general is an unstable cohomology operation. This is simply a natural transformation from $E^k(-)$ to $E^l(-)$ for some fixed $k$ and $l$. Here, one regards the graded cohomology functors as a family of set-valued functors so the functions induced by these unstable operations do not necessarily respect any of the structure of $E^k(X)$.

Some do, however. In particular, there are additive cohomology operations. These are unstable operations which are homomorphisms of abelian groups.

In particular, for any multiplicative cohomology theory (in particular, ordinary cohomology or ordinary cohomology with $mathbb{Z}/2mathbb{Z}$ coefficients) there are the power operations: $x to x^k$. These are additive if the coefficient ring has the right characteristic. In particular, squaring is additive in $mathbb{Z}/2mathbb{Z}$ cohomology.

Given an unstable cohomology operation $r: E^k(-) to E^l(-)$ there is a way to manufacture a new operation $Omega r: tilde{E}^{k-1} to tilde{E}^{l-1}(-)$ using the suspension isomorphism (where the tilde denotes that these are reduced groups):

$E^{k-1}(X) cong E^k(Sigma X) to E^l(Sigma X) cong E^{l-1}(X)$

This is quite straightforward and is a cheap way of producing more operations. When applied to the power operations it produces almost nothing since the ring structure on the cohomology of a suspension is trivial: apart from the inclusion of the coefficient ring all products are zero.

What is an interesting question is whether or not this looping can be reversed. Namely, if $r$ is an unstable operation, when is there another operation $s$ such that $Omega s = r$? And how many such are there? Most interesting is the question of when there is an infinite chain of operations, $(r_k)$ such that $Omega r_k = r_{k-1}$. When this happens, we say that $r$ comes from a stable operation (there is a slight ambiguity here as to when the sequence $(r_k)$ is a stable operation or merely comes from a stable operation).

One necessary condition is that $r$ be additive. This is not, in general, sufficient. For example, the Adams operations in $K$-theory are additive but all but two are not stable.

However, for ordinary cohomology with coefficients in a field, additive is sufficient for an operation to come from a stable operation. Moreover, there is a unique sequence for each additive operation. This means that the squaring operation in $mathbb{Z}/2mathbb{Z}$ cohomology has a sequence of "higher" operations which loop down to squaring. These are the Steenrod squares.

The sequence stops with the actual squaring (rather, becomes zero after that point) because, as remarked above, the power operations loop to zero.

One important feature of these operations is that they give necessary conditions for a spectrum to be a suspension spectrum of a space. If a spectrum is such a suspension spectrum then it's $mathbb{Z}/2mathbb{Z}$-cohomology must be a ring. That's not enough, however, it must also have the property that, in the right dimensions, the Steenrod operations act by squaring. (Of course, this is necessary but not sufficient.)

fa.functional analysis - Are smooth functions on an uncountable sum continuous?

Not all smooth functions are continuous. It is a fact of the Frölicher−Kriegl−Michor theory that bounded multilinear maps are smooth. For example the canonical bilinear evaluation $Etimes E'tomathbb R$ given by $(x,u)mapsto u(x)$ is bounded, hence smooth, but discontinuous when $E=sum_{mathbb R}mathbb R$. I too quickly thought that this would give the required smooth discontinuous map as a composite $Eto Etimes Eto Etimes E'tomathbb R$.

Using Jarchow's notation, and istead considering the space $F=mathbb R^{ mathbb N}timesmathbb R^{ (mathbb N)}=prod_{mathbb N}mathbb Rtimessum_{mathbb N}mathbb R$ , then one has the Frölicher−Kriegl smooth discontinuous map $Ftomathbb R$ given by $(x,y)mapstosum_{iinmathbb N}(x_icdot y_i)$ .

It should be noted that this discontinuity is with respect to the locally convex topology. Frölicher−Kriegl smooth maps are always continuous with respect to the Mackey−closure topology whose open sets are precisely the $Usubseteq F$ such that for every $xin U$ and every bounded set $B$ in $F$ there is $varepsilon>0$ with $varepsilon Bsubseteq U-x$ .

The Frölicher−Kriegl theory is essentially a bornological theory. One may observe that in Frölicher's and Kriegl's book one uses a canonical topology corresponding to the bornology, namely the strongest, bornological one, whereas in Kriegl's and Michor's book one allows any locally convex topology with the same bounded sets. In this sense, the KM−approach to smoothness is a bit floppy since the spaces are topological but bornology is the only one that matters.

nt.number theory - What is inter-universal geometry?

In a research statement, he says:

"The essence of arithmetic geometry lies not in the various specific schemes that occur in a specific arithmetic-geometric setting, but rather in the abstract combinatorial patterns, along with the combinatorial algorithms that describe these patterns, that govern the dynamics of such specific schemes."

Regarding this, he then talks about how his main motivations are monoids, Galois categories, and dual graphs of degenerate stable curves, which leads him to talking about his geometry of categories stuff, and then to "absolute anabelian geometry." He then links to a bunch of papers that I would assume elaborate a bit on it. He then goes on to talk about extending Teichmuller Theory.

Generally, his research statement is fairly readable (and consider that I'm very much a nonspecialist in arithmetic anything) and seems to link to things with more details.

ag.algebraic geometry - Behaviour of Zeta-function under Finite Morphism

You should be looking not at just zeta functions, but at the L-functions.

Then yes, for a finite etale Galois morphism the identity should be

        Z(Y) = Z(X) * L(X, pi_1) * L(X, pi_2) * ...

(where the product is over summands of the regular representation of Galois group of the morphism, Z being the L-function of the trivial representation.) This is in no way restricted to finite fields — in fact the idea as well as the notation comes from theorem about Dirichlet L-functions.

The proof is that by a definition of what is L-function it can be written either for a trivial mixed sheaf on Y (LHS) or for its pushforward on X (RHS).

ac.commutative algebra - How much theory works out for "almost commutative" rings?

Don't get too excited about the theory of algebraic geometry for almost commutative algebras. A ring can be almost commutative and still have some very weird behavior. The Weyl algebras (the differential operators on affine n-space) are a great example, since they are almost commutative, and yet:

• They are simple rings. Therefore, either they have no 'closed subschemes', or the notion of closed subscheme must correspond to something different than a quotient.


• The have global dimension n, even though their associated graded algebra has global dimension 2n. So, global dimension can jump up, even along flat deformations.


• There exist non-free projective modules of the nth Weyl algebra (in fact, stably-free modules!). Thus, intuitively, Spec(D) should have non-trivial line bundles, even though it is 'almost' affine 2n-space.

Just having a ring of quotients isn't actually that strong a condition on a ring. For instance, Goldie's theorem says that any right Noetherian domain has a ring of quotients, and that is a pretty broad class of rings.

Also, what sheaf are you thinking of D as giving you? You have all these Ore localizations, and so you can try to build something like a scheme out of this. However, you start to run into some problems, because closed subspaces will no longer correspond to quotient rings. In commutative algebraic geometry, we take advantage of the miracle that the kernel of a quotient map is the intersection of a finite number of primary ideals, each of which correspond to a prime ideal and hence a localization. In noncommutative rings, there is no such connection between two-sided ideals and Ore sets.

Here's something that might work better (or maybe this is what you are talking about in the first place). If you have a positively filtered algebra A whose associated graded algebra is commutative, then A_0 is commutative, and so you can try to think of A as a sheaf of algebras on Spec(A_0). The almost commutativity requirement here assures us that any multiplicative set in A_0 is Ore in A, and so we do get a genuine sheaf of algebras on Spec(A_0). For D_X, this gives the sheaf of differential operators on X. Other algebras that work very similarly are the enveloping algebras of Lie algebroids, and also rings of twisted differential operators.

co.combinatorics - Reachability in digraphs

I have a problem that is reducible to (efficiently) determining the reachability of a node in a fully dynamic planar digraph.

I'm aware of "A fully dynamic data structure for reachability in planar digraphs" which provides O(n^(2/3) log n) query with a O(n)-space data structure.

Can this be / has this been bettered?

If all my queries have the same source node, is there a more efficient (in time/space/both) way?

Are there any other related literature that deals with more efficient queries albeit with more restrictions imposed on the digraph?

Thanks!

ag.algebraic geometry - Different definitions of the dimension of an algebra

In a non-commutative ring, you need to be careful with what you even mean by a prime ideal, and usually there are very few two-sided ideals you might call prime. Oh, and even in the cases when there is a nice ring of fractions, it won't be a field, and so transcedence degree is still bad.

My personal favorite notion of dimension is 'global dimension', the maximum projective dimension of any module of the ring. This concept exists for any ring, and in fact for any abelian category (though, if there aren't enough projectives, you need to play with the definition). The only problem is that it can often be infinity, even for relatively mild rings, like C[x]/x^2. It still makes for a pretty good theory of 'smooth dimension', however.

From a conceptual perspective, Krull dimension seems best suited for geometric perspectives, since it is measuring chains of irreducible closed subsets. The easiest times to work with Krull dimension is when you are in a Cohen-Macaulay ring, and then Krull dimension is equivalent to depth, which is easier to prove things about, since you only need to produce a maximal regular sequence.

shimura varieties - What is the image of complex conjugation under Siegel Galois representations?

Let $G$ be the reductive group $operatorname{GSp}_{4}$. Let $pi$ be a smooth admissible cuspidal representation of $operatorname{GSp}_{4}(mathbb{A}^{(infty)})$ of dominant weight. Assume, for caution, that $pi$ satisfies a multiplicity one hypothesis.

Fix $p$ an odd prime. To $pi$ is attached a $p$-adic representation $rho$ of the absolute Galois group of $mathbb{Q}$ unramified outside a finite set of finite places and such that the characteristic polynomial of the Frobenius morphisms $Frell$ for $ell$ outside this set coincides with the Euler factor at $ell$ of the degree 4 $L$-function of $pi$. This Galois representation occurs in the degree 3 cohomology of the étale cohomology of a Siegel-Shimura variety.

The image of complex conjugation under $rho$ is semi-simple so can be chosen to be diagonal with eigenvalues 1 and -1. How many $-1$ are there?

ac.commutative algebra - Local complete intersections which are not complete intersections

Varieties in an affine space.

• in characteristic zero any lci $Xsubseteqmathbb{A}^n$ is a set-theoretic complete intersection.


• In characteristic $p$ any lci curve $Csubseteqmathbb{A}^n$ is a set-theoretic complete intersection.

Projective varieties

Let $Xsubseteqmathbb{P}^n$ be a smooth non-degenerate degree $p$ (a prime number) variety of codimension $c$. Then $X$ is not a scheme-theoretic complete intersection. Indeed, if $X = H_1cap H_2cap...cap H_c$, then $deg(H_2)=...=deg(H_c) = 1$ and $deg(H_1) = p$ by Bezout's theorem because $p$ is prime. Therefore $X$ would be degenerate. An example is again the twisted cubic $Csubsetmathbb{P}^3$. However $C$ is a set-theoretic complete intersection. There exist a quadric surface $Q$ and a cubic surface $S$ such that $Qcap S = 2C$ (i.e. $Q$ and $S$ are tangent along $C$).

Hartshorne Conjecture: If $Xsubseteqmathbb{P}^N$ is a smooth variety of dimension $n$, codimesnion $c$ and $cgeq 2n+1$ then $X$ is a scheme-theoretic complete intersection.

Hartshorne Conjecture has been proven for Fano varieties of codimension two and quadratic varieties (i.e. varieties that can be defined just by quadratic polynomials).

Thanks to Barth’s result: Barth, W.: ”Transplanting cohomology classes in complex-projective
space”, Amer. J. Math., 92, 951-967 (1970), and since no indecomposable rank
two vector bundle on $mathbb{P}^N$, $Ngeq 5$, is known, it is generally believed that any
smooth, codimension two subvariety of $mathbb{P}^N$, $N geq 6$, is a complete intersection. The main results for codimension two subvarieties can be summarized as follows: let $omega_Xcong mathcal{O}_X(e)$, $d$ the degree of $X$ and $s$ the minimal
degree of an hypersurface containing $X$. if $e leq N + 1$ or if $d < (N − 1)(N + 5)$ or if $s leq N − 2$, then $X$ is a complet intersection. For $N = 5,6$ we can something more: let $X subset mathbb{P}^6$ be a smooth, codimension two subvariety, if $sleq 5$ or if $d leq 73$, then $X$ is a complete intersection. Let $X subset mathbb{P}^5$ be a smooth, subcanonical threefold. If $s leq 4$, then $X$ is a complete intersection. This is Theorem 1.1 of http://arxiv.org/abs/math/9909137.

ag.algebraic geometry - SGA1 Chapter XIII (tamely ramified sheaves)

If you look in the text, $y$ is not a closed point but a maximal point, meaning a maximally generic point. Then $O_{X_{bar{s}y}}$ is a discrete valuation ring, and so it makes sense to talk about tameness of extensions. Also, surely she's working in the etale topology, otherwise it would be kind of silly to try and represent functors by etale covers, but I didn't actually see where she says that. It's got to be written somewhere, though!

I suppose you're right about a sheaf of sets being a stack with no non-identity maps, but taking this as a definition is exactly as sensible as defining a set to be a category with no non-identity maps.

"torseur sous F" = "torsor under F" = "F-torsor"

pr.probability - Looking for a version of Itô's Lemma

Hi Everyone

I have some difficulties deriving the Stochastic Differential Equation for the following problem, any help or reference would be appreciated.

We are given a Brownian Motion $B_t$ and we note $M_t=sup_{sle t}B_s$.
Moreover we have a smooth real valued function $F(t,x,y)$ (for example a $C^{1,2,1}$) over $mathbb{R}^+times mathbb{R} times mathbb{R}^+$ and we are looking for the SDE followed by $F(t,M_t-B_t,M_t)$

I have a hard time trying to express $dF$.

Best regards

examples - What is the first interesting theorem in (insert subject here)?

In most students' introduction to rigorous proof-based mathematics, many of the initial exercises and theorems are just a test of a student's understanding of how to work with the axioms and unpack various definitions, i.e. they don't really say anything interesting about mathematics "in the wild." In various subjects, what would you consider to be the first theorem (say, in the usual presentation in a standard undergraduate textbook) with actual content?

Some possible examples are below. Feel free to either add them or disagree, but as usual, keep your answers to one suggestion per post.

• Number theory: the existence of primitive roots.


• Set theory: the Cantor-Bernstein-Schroeder theorem.


• Group theory: the Sylow theorems.


• Real analysis: the Heine-Borel theorem.


• Topology: Urysohn's lemma.

Edit: I seem to have accidentally created the tag "soft-questions." Can we delete tags?

Edit #2: In a comment, ilya asked "You want the first result after all the basic tools have been introduced?" That's more or less my question. I guess part of what I'm looking for is the first result that justifies the introduction of all the basic tools in the first place.

lo.logic - Reasons for success in automated theorem proving

You may be interested in the wonderful little book ``The Efficiency of Theorem Proving Strategies: A Comparative and Asymptotic Analysis'' by David A. Plaisted and Yunshan Zhu. I have the 2nd edition which is paperback and was quite cheap. I'll paste the (accurate) blurb:

``This book is unique in that it gives asymptotic bounds on the sizes of the search spaces generated by many common theorem-proving strategies. Thus it permits one to gain a theoretical understanding of the efficiencies of many different theorem-proving methods. This is a fundamental new tool in the comparative study of theorem proving strategies.''

Now, from a critical perspective: There is no doubt that sophisticated asymptotic analyses such as these are very important (and to me, the ideas underlying them are beautiful and profound). But, from the perspective of the practitioner actually using automated theorem provers, these analyses are often too coarse to be of practical use. A related phenomenon occurs with decision procedures for real closed fields. Since Davenport-Heinz, it's been known that general quantifier elimination over real closed fields is inherently doubly-exponential w.r.t. the number of variables in an input Tarski formula. One full RCF quantifier-elimination method having this doubly-exponential complexity is CAD of Collins. But, many (Renegar, Grigor'ev/Vorobjov, Canny, ...) have given singly exponential procedures for the purely existential fragment. Hoon Hong has performed an interesting analysis of this situation. The asymptotic complexities of three decision procedures considered by Hong in ``Comparison of Several Decision Algorithms for the Existential Theory of the Reals'' are as follows:

(Let $n$ be the number of variables, $m$ the number of polynomials, $d$ their total degree, and $L$ the bit-width of the coefficients)

CAD: $L^3(md)^{2^{O(n)}}$

Grigor'ev/Vorobjov: $L(md)^{n^2}$

Renegar: $L(log L)(log log L)(md)^{O(n)}$

Thus, for purely existential formulae, one would expect the G/V and R algorithms to vastly out-perform CAD. But, in practice, this is not so. In the paper cited, Hong presents reasons why, with the main point being that the asymptotic analyses ignore huge lurking constant factors which make the singly-exponential algorithms non-applicable in practice. In the examples he gives ($n=m=d=L=2$), CAD would decide an input sentence in a fraction of a second, whereas the singly-exponential procedures would take more than a million years. The moral seems to be a reminder of the fact that a complexity-theoretic speed-up w.r.t. sufficiently large input problems should not be confused with a speed-up w.r.t. practical input problems.

In any case, I think the situation with asymptotic analyses in automated theorem proving is similar. Such analyses are important theoretical advances, but often are too coarse to influence the day-to-day practitioner who is using automated theorem proving tools in practice.

(* One should mention Galen Huntington's beautiful 2008 PhD thesis at Berkeley under Branden Fitelson in which he shows that Canny's singly-exponential procedure can be made to work on the small examples considered by Hong in the above paper. This is significant progress. It still does not compare in practice to the doubly-exponential CAD, though.)

Learning Algebra & Group Theory on my own

I've enjoyed Peter Cameron's exposition of permutation groups; most have a very CS feel and are almost exclusively devoted to solving some sort of combinatorial problem (often of real use).

He has recently written some lecture notes on a problem in synchronization^{(course page)} that make non-trivial use of permutation groups to understand finite automata. One important tool in hardware design is a "reset word" that takes the automaton from any state and brings it back to a fixed initial state. The key point being that you send the same message no matter what state the automaton is currently in, and no matter what it ends up in the fixed initial state, ready for new commands.

The goal of the course is to make progress towards solving a conjecture that if an n-state automaton has a reset word at all, then it has a short one, that is, one of length at most (n−1)².

He also has a nice encyclopedia about "design theory". Block designs are highly symmetric arrangements that allow for efficient and accurate statistical experiments (I believe originally in agriculture, but now widely used in many areas, especially medicine) as well as dense codes in coding theory. I first learned about them in the local cryptography seminar where they were used to give better understanding of some algebraic stream ciphers.

If you do not currently have problems you want to solve using group theory, but want to learn to solve some beautiful problems in and using group theory, then I found Butler's Fundamental Algorithms of Permutation Groups to be quite good. It uses spanning trees to solve a fundamental problem in permutation groups, and shows several very good examples of how permutation groups let you very naturally prune an exponential search tree down to something that will work like a charm in practice. The open source GAP has many of these algorithms implemented in fairly easy to read procedural language (like C, but with garbage collection). It of course also has many of the modern solutions to the same problems, so you can also see how things have improved.

matrices - Area of a surface in terms of the densitized triad

Hi,
I need to know if this relation is correct for a metric:

$g_{a[b}g_{c]d}=frac{1}{2}epsilon_{ace}epsilon_{bdf}gg^{ef}$

I know that :

$frac{1}{2}epsilon_{ace}epsilon_{bdf}g^{ef}=g_{b[a}g_{c]d}$

but I don't see how the determinant $g$ of the metric could appear.

Edit:

Ok so the previous relation emerged when computing the area of a surface $S$ in terms of the "densitized" triad $E_{i}^{a}=ee_{i}^{a}$ where $a,b,c,...$ are the spatial coordinates and $i,j,k,...$ are $SU(2)$ coordinates, e the determinant of the triad matrix defined by $g_{ab}=e_{a}^{i}e_{b}^{j}delta_{ij}$ where $g_{ab}$ is the spatiale metric. So, since the computation of the area uses the determinant of the the metric $h_{alphabeta}$ induced by $g_{ab}$ on $S$: ($alpha,beta,... =1,2;and; a,b,..=1,2,3$)

$h_{alphabeta}=g_{ab}frac{partial x^{a}}{partialsigma^{alpha}}frac{partial x^{b}}{partialsigma^{beta}}$

So in computing the determinant $h$ explecitely on finds the term

$g_{a[b}g_{c]d}$ which needs to equal to $frac{1}{2}epsilon_{ace}epsilon_{bdf}gg^{ef}$
in order to obtain the final result:

$h=E_{i}^{a}E_{j}^{b}delta^{ij}n_{a}n_{b}$ where $n$ are normal vectors $n_{a}=epsilon_{abc}frac{partial x^{b}}{partialsigma^{1}}frac{partial x^{c}}{partialsigma^{2}}$

EDIT2:

After the notification of Willie Wong, I decided to put my original problem as a question, i.e: deriving the expression of the determinant of the induced metric on $S$ in terms of the densitized triad.

ho.history overview - How do we know that Fermat wrote his famous note in 1637?

Not only do we not know the date, we don't even know whether he wrote the remark at all.
For all we know it might have been invented by his son Samuel, who published his father's comments.

In his letters, Fermat never mentioned the general case at all, but quite often posed the problem of solving the cases $n=3$ and $n=4$. I am almost certain that Fermat discovered infinite descent around 1640, which means that in 1637 he did not have any chance of proving FLT for exponent 4 (let alone in general).

In 1637, Fermat also stated the polygonal number theorem and claimed to have a proof; this is just about as unlikely as in the case of FLT -- I guess Fermat wasn't really careful in these early days.

Let me also mention that Fermat posed FLT for $n=3$ always as a problem or as a question, and did not claim unambiguously to have a proof; my interpretation is that he did not have a proof for $n = 3$, and that he knew he did not have one.

Edit Let me briefly quote two letters from Fermat:

I. Oeuvres II, 202--205, letter to Roberval Aug. 1640 Fermat claims that if $p = 4n-1$ be prime, then $p$ does not divide a sum of two squares $x^2 + y^2$ with $gcd(x,y) = 1$. Then he writes

I have to admit frankly that I have found nothing in number theory
that has pleased me as much as the demonstration of this proposition,
and I would be very pleased if you made the effort of finding it, if
only for learning whether I estimate my invention more highly than it
deserves.

This looks as if Fermat had just discovered "his method" of descent.
Starting from $x^2 + y^2 = pr$ one has to show that there is a prime
$q equiv 3 bmod 4$ dividing $r$ which is strictly less than $p$.

II. In his letter to Carcavi from Aug. 1659 (Oeuvres II, 431--436), Fermat writes:

I then considered certain questions which, although negative, do
not remain to receive a very great difficulty, for it will be easily
seen that the method of applying descent is completely different from
the preceding [questions]. Such cases include the following:

1. There is no cube that can be divided into two cubes.


2. There is only one square number which, augmented by $2$,
   makes a cube, namely $25$.


3. There are only two square numbers which, augmented by $4$,
   make a cube, namely $4$ and $121$.


4. All squared powers of $2$ augmented by $1$ are prime numbers.

My interpretation of this is that Fermat lists four results which he
believes can be proved using his method of descent. In my opinion
this implies that Fermat did not have a proof of FLT for exponent $3$
in 1659.

Edit 2
In light of the discission at wiki.fr let
me add a couple of additional remarks along with a promise that a nonelectronic
publication of my views on Fermat will appear within the next two years (if I can
find a publisher, that is).

A search in google books for "hanc marginis" and Fermat for the
years up to 1900 reveals several hits, none of which claims that
the remark was written around 1637; in particular there are no dates
given in Fermat's Oeuvres or in Heath's Diophantus. Starting with
Dickson's history, this changes dramatically, and nowadays the date
1637 seems to be firmly attached to this entry.

The dating of the entry seems to come from a letter written by
Fermat to J. de Sainte-Croix via Mersenne mentioned in Nurdin's
answer; this letter is not dated, but since Descartes, in a letter
to Mersenne from 1638, refers to a result he credits to Sainte-Croix,
but which Fermat claims he has discovered, it is believed that Fermat's
letter to Mersenne was written well before that date. The reasons for
dating it to September 1636 are not explained in Fermat's Oeuvres.

In this letter, Fermat poses the problem of finding two fourth
powers whose sum is a fourth power, and of finding two cubes whose
sum is a cube. The reasoning seems to be that in 1636, Fermat
had not yet found (or believed to have found) a proof of the general
theorem, so the entry must have been written at a later date.
Since he did not refer to the general theorem in any of his
existant letters, it is also believed that he soon found his
mistake, so the entry cannot have been written at a time when
Fermat was mature enough to find sufficiently difficult proofs.

Let me also add that the following dates can be deduced from
Fermat's letters:

• 1638 Numbers 4n-1 are not sums of two rational squares


• 1640 Fermat's Little Theorem


• 1640 Discovery of infinite descent; used for showing that
  (1) primes 4n-1 do not divide sums of to squares.


• 1640 Statement of the Two-Squares Theorem


• 1641 - 1645 Proof of (2) FLT for exponent 4


• later: Proof of (3) the Two-Squares Theorem

It is impossible to attach any dates between 1644 and 1654 to
Fermat's discoveries since he either wrote hardly any letter
in this period, or all of them are lost.

Fermat claimed to have discovered infinite descent in connection
with results such as (1), and that he at first could apply it
only to negative statements such as (2), whereas it took him a
long time until he could use his method for proving positive
statements such as (3). Thus the proofs of (1) - (2) - (3) were
found in this order.

This means in particular that if Fermat's entry in his Diophantus
was written around 1637, then the marvellous proof must have been
a proof that does not use infinite descent.

I would also like to remark that the Fermat equation for exponents
3 and 4 had already been studied by Arab mathematicians, such as
Al-Khujandi and Al-Khazin, who both attempted proving that there
are no solutions. The cubic equation also shows up in problems
posed by Frenicle and van Schooten in response to Fermat's
challenge to the English mathematicians.

at.algebraic topology - classification of smooth involutions of torus

Let $mathbb{Z}_2={1,g},T^2={(e^{itheta_1},e^{itheta_2})}$ and place $T^2$ in $mathbb{R}^3$ as the locus of the rotation of $2pi$ rads of the circle${(y,z)|(y-2)^2+z^2=1}$ around $z$ axis.

It is known that there are 5 nonequivalent smooth involutions on torus,and they are:

1.$g(e^{itheta_1},e^{itheta_2})=(e^{i(theta_1+pi)},e^{itheta_2})$ (rotation$pi$ rads around $z$ axis) with null fixed point set and orbit space $T^2$

2.$g(e^{itheta_1},e^{itheta_2})=(e^{-itheta_1},e^{itheta_2})$(reflection along $x=0$) with fixed point set $S^1times S^0$ and orbit space an annulus

3.$g(e^{itheta_1},e^{itheta_2})=(e^{itheta_2},e^{itheta_1})$(switch the two coordinates) with fixed point set the diagonal circle and orbit space Mobius band

4.$g(e^{itheta_1},e^{itheta_2})=(e^{i(theta_1 +pi)},e^{-itheta_2})$(restriction of the involution $(x,y,z,mapsto (-x,-y,-z)$ of $mathbb{R}^3$ to torus)with null fixed point set and orbit space klein bottle

5.$g(e^{itheta_1},e^{itheta_2})=(e^{-itheta_1},e^{-itheta_2})$(reflection along $x=0$ plus reflection along $z=0$)
with fixed point set 4 points and orbit space $S^2$

i want to know how to derive the result above.for the free case it seems easy.since the action is free,the orbit space must be a manifold also,and has euler char 0,hence must be torus or klein bottle.
for the nonfree case,the orbit is not manifold,but "orbifold".
and we have Riemann-Hurwitz Formula:

$chi(O)=chi(X_O)-sum_{i=1}^n (1-frac{1}{q_i})-frac{1}{2}sum_{j=1}^m (1-frac{1}{r_j})$

here$chi(O)$ is the orbifold euler char and $chi(X_o)$ is the euler char of the underlying space associated to the orbifold $O$,and $q_i$and $r_j$ denote the angles for sigular points(cone points and reflector corners
can we determine the remaining 3 involutions by using this formula?Thank you!

at.algebraic topology - Are there two non-homotopy equivalent spaces with equal homotopy groups?

The simplest examples I know are the $3$-dimensional lens spaces $L(p,q)$. They display many oddities.

Consider the lens spaces $L(p,q_0)$ and $L(p,q_1)$, $gcd(p,q_0)=gcd(p,q_1)=1$. Their fundamental groups are isomorphic to the cyclic group $newcommand{bZ}{mathbb{Z}}$ $bZ/pbZ$. Since both these lens spaces have the same universal cover $S^3$, their higher homotopy groups are also isomorphic.

A theorem of Franz-Rueff-Whitehead (see Theorem 2.60 of these notes) shows that $L(p,q_0)$ and $L(p,q_1)$ are homotopy equivalent if and only

$$q_1equiv pm ell^2 q_0bmod p, $$

for some $ellinbZ$. This reduces the problem to a number theoretic one. For example, $L(5,1)$ is not homotopy equivalent to $L(5,2)$ since $pm 2$ is not a quadratic residue mod $5$.

On the other hand, the lens spaces $L(7,1)$ and $L(7,2)$ are homotopy equivalent, but they are not homeomorphic.

real analysis - Convergence of Fourier Series of $L^1$ Functions

I recently learned of the result by Carleson and Hunt (1968) which states that if $f in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me that if $f in L^p$ for $1 < p < infty$, then the Fourier series of $f$ converges to $f$ in $L^p$. Either of these results implies that if $f in L^p$ for $1 < p < infty$, then the Fourier series of $f$ converges to $f$ in measure.

My first question is about the $p = 1$ case. That is:

If $f in L^1$, will the Fourier series of $f$ converge to $f$ in measure?

---

I also recently learned that there exist functions $f in L^1$ whose Fourier series diverge (pointwise) everywhere. Moreover, such a Fourier series may converge (Galstyan 1985) or diverge (Kolmogorov?) in the $L^1$ metric.

My second question is similar:

Do there exist functions $f in L^1$ whose Fourier series converge pointwise a.e., yet diverge in the $L^1$ metric?

---

(Notes: Here, I mean the Fourier series with respect to the standard trigonometric system. I am also referring only to the Lebesgue measure on [0,1]. Of course, if anyone knows any more general results, that would be great, too.)

graph theory - Conditions for subgraph relationship in circulant Cayley digraphs

I have two circulant Cayley digraphs: that is, Cayley digraphs X = Cay(ℤ/m, S) and Y = Cay(ℤ/n, T), for odd integers m < n, and sets with sizes |S| = (m − 1)/2, and |T| = (n − 1)/2.

These digraphs are antisymmetric, in that S is disjoint from −S, and T is disjoint from −T. (It follows that for each distinct pair of vertices a,b in either graph, there is either an arc from a to b, or vice versa.)

Question. What conditions on m, n, S, and T must hold for X to be an induced directed subgraph of Y?

Verifying a technical lemma regarding homotopy pushouts in the theory of simplicial model categories

Important Edit: I e-mailed Jacob Lurie, and he said that the statement of condition (*) is incorrect as printed.

Here is the correct statement of (*):

For any cofibration $f:Ato B$ and any trivial fibration $g:Xto Y$ in $C$, the induced morphism:

$$operatorname{Map}(B,X)to operatorname{Map}(B,Y)times_{operatorname{Map}(A,Y)} operatorname{Map}(A,X)$$

is a trivial Kan fibration. (Where $operatorname{Map}$ is the (sSet)-enriched $operatorname{Hom}$).

Let $C$ be a simplicially-enriched category with a model structure, not necessarily compatible with the simplicial enrichment.

Suppose that all objects of $C$ are cofibrant, that $C$ is tensored and cotensored over $SSet$, and that the class of weak equivalences of $C$ is closed under filtered colimits.

(*)Suppose further that for any cofibration $f:Ato B$ and any fibration $g:Xto Y$ in $C$, the induced morphism:

$$operatorname{Map}(B,X)to operatorname{Map}(B,Y)times_{operatorname{Map}(A,Y)} operatorname{Map}(A,X)$$

is a Kan fibration. (Where $operatorname{Map}$ is the (sSet)-enriched $operatorname{Hom}$).

Lastly, assume that $Aotimes Delta^ntilde{to}Aotimes Delta^0=A$ is a weak equivalence for any object $A$ in $C$ and any $nin mathbf{N}$. (Here, the tensor $Aotimes K$, where $A$ is in $C$ and $K$ is in $sSet$, is the object of $C$ representing the functor $Map(A,-)^K$).

Let $Lsubseteq K$ be an inclusion of simplicial sets. Suppose $sigma:Delta^nhookrightarrow K$ is a nondegenerate simplex of $K$ with all of its faces living in $L$. That is, we can factor the map $partialsigma:=sigma|_{partialDelta^n}$ through the inclusion $Lsubseteq K$ (in fact, we will assume that the target of this map actually is $L$).

Then for any object $D$ in $C$, the pushout $$Dotimes Delta^ncoprod_{DotimespartialDelta^n} Dotimes Lcong Dotimes (Delta^ncoprod_{partialDelta^n}L)$$ is a homotopy pushout. Now, the question here is, why is this the case?

The proof I'm reading says that it follows from the line marked (*) above (and the fact that $C$ is left-proper (which follows from the fact that all objects of $C$ are cofibrant)), but it's not clear to me how to apply that hypothesis.

That is, how does the line marked (*) imply anything relevant?

If you'd like to look up the original source, it is Higher Topos Theory Proposition A.3.1.7 (in the appendix).

Edit: It's probable that a few of the hypotheses are unrelated to the actual question. I included everything because I'm not sure what's important.