ag.algebraic geometry - When is the canonical divisor of an algebraic surface smooth?

Any smooth projective surface with nonempty $K_X$ is obtained by blowing up finitely many points on its unique minimal model. From the formula $K_X=f^*K_Y+E$ for the blowup, you see that the exceptional divisors of the blowup are always in the base locus of $|K_X|$. Thus, the problem is reduced to the minimal model.

Then you have to go through the classification of the minimal models of surfaces, that's been known for a hundred years now. (Using a book such as van de Ven "Complex surfaces", or Shafarevich et al, or Beauville...)

For a minimal surface $Y$ Kodaira dimension 0 for example, $12K_Y=0$. So either $K_Yne 0$ and then $|K_X|=emptyset$, or $K_Y=0$ and then any divisor in $|K_X|$ is $sum a_i E_i$, where $E_i$ are the exceptional divisors of the blowups.

For a minimal surface $Y$ of Kodaira dimension 2, the question is still somewhat tricky. If looking at higher multiples $|mK_Y|$ suffices, then by a well known theorem (Bombierri? certainly I. Reider gave a very nice proof), $|5K_Y|$ is free, so a general element is smooth (in characteristic 0). For $|K_Y|$ I don't think the answer is known but why not search mathscinet.

Finally, for Kodaira dimension 1, an elliptic surface $pi:Yto C$, there is a well-known Kodaira's formula for the canonical class $K_Y=pi^*K_C + R$ with explicit rational coefficients in $R$. I'd play with that. Again, for higher multiples I think $|12K_Y|$ works.

Of course, to your example of a hypersurface in $mathbb P^3$ you can add the case of complete intersections, and other surfaces for which $K_X$ is either zero or $pm K_X$ is very ample.

co.combinatorics - Looking for an explicit formula for a limit of a binomial-like expression

Here's a probabilistic solution to your problem. Suppose $X_N$ is a binomial random variable with parameters $N$ and $s/N$ (ie. $X_N$ counts the number of heads in $N$ independent coin flips, each with probability $s/N$ of heads). Also let $P$ be a Poisson random variable with mean $s$, so that for all non-negative integers $k$:

$$Prob[P=k] = e^{-s}frac{s^k}{k!}.$$
A well-known result sometimes called the law of rare events implies that the distribution of $X_N$ converges to that of $P$ as $Nto +infty$. In particular, for any bounded real-valued function $f$ defined on the non-negative integers:
$$E(f(X_N)) = sum_{k=0}^N binom{N}{k}left(1-frac{s}{N}right)^{N-k}left(frac{s}{N}right)^{k}f(k)to E(f(P))=sum_{k=0}^{+infty}e^{-s}frac{s^k}{k!}f(k).$$

Apply this to $f$ satisfying $f(x)=1/x$ if $x>0$, $f(0)=0$, and the LHS becomes your expression. The RHS becomes:

$$sum_{k=1}^{+infty}e^{-s}frac{s^k}{k!times k} = e^{-s}int_{0}^{s}frac{e^{u}-1}{u}du,$$
which I guess is the same as the previous answer.

Added note: a quantitative version of the law of rare events gives the error bound:

$$forall f:Nto [0,1], |E(f(X_N))-E(f(P))|leq sleft(1-e^{-s/N}right);$$
this allows for simultaneous limits in N and $s$, and goes to $0$ iff $s^2/Nto 0$.

Unit in a number field with same absolute value at a real and a complex place

I was asked whether it was possible to produce a monic polynomial with integer coefficients, constant coefficient equal to $1$, having a real root $r > 1$ and a pair of complex roots with absolute value $r$, which are not $r$ times a root of unity. Bonus if the polynomial did not have roots of absolute value one. An answer (without the bonus) is:

$x^{12} - 4x^{11} + 76x^{10} + 156x^9 - 429x^8 - 2344x^7 + 856x^6 - 2344x^5 - 429x^4 + 156x^3 + 76x^2 - 4x + 1$.

I'd like an answer to the bonus question in the following strengthened form: Is there a unit $r$ in a number field such that $r$ has the same absolute value (bigger than one) at a real and a complex place (of $mathbb{Q}(r)$ to avoid trivial answers) but no archimedian place where
$r$ has absolute value $1$?

mathematical writing - Using TikZ in papers

It is not exactly answering either of your two questions, but here is another work-around. I had problems putting papers on the arxiv which used pgf/tikz because the version of pgf/tikz they used at the arxiv was not as up to date as my version. The admin at the arxiv told me to do the following. LaTeX your file with the option -recorder. This will create a .fls file containing a list of all of the FiLeS used by LaTeX when typesetting your document. Choose all of the files in the list containing "pgf" or "tikz" and move them into the directory containing your document. You can then send that directory to your collaborator/the arxiv/the journal without worrying about how up to date their set up is.

[Unfortunately, I then had the problem that the tikz graphics I produced required an enormous amount of working memory which was greater than the allocation on the arxiv server, so I resorted to using 'grab' on my mac to take a high resolution snap-shot of the graphic, which I then incorporated into the LaTeX file :-(. However, I have used this including-all-the-files technique for subsequent uploads to the arxiv.]

computational complexity - An Alternative to the Cook-Levin Theorem

In general to prove that a given problem is NP-complete we show that a known NP-complete problem is reducible to it. This process is possible since Cook and Levin used the logical structure of NP to prove that SAT, and as a corollary 3-SAT, are NP-complete. This makes SAT the "first" NP-complete problem and we reduce other canonical NP-complete problems (e.g. CLIQUE, HAM-PATH) from it.

My question is whether there is a way to prove directly from the definition/logical structure of NP that a different problem (i.e. not SAT) is NP-complete. A friend suggested that it would be possible to tailor the proof of the Cook-Levin Theorem to show that, for example, CLIQUE is NP-complete by introducing the reduction from SAT during the proof itself, but this is still pretty much the same thing.

gr.group theory - Criteria for Aut(G) to be simple

Here is an approximation of an answer to "For what finite groups is Aut(G) simple?"

As Daniel Miller mentioned, Inn(G) is a normal subgroup of Aut(G), so for Aut(G) to be simple either Inn(G) = 1, in which case G is abelian, or Inn(G) = Aut(G) is simple. The former case should be somewhat easy to handle assuming G is finite. In the latter case, we have that G/Z(G) is simple. If G is also perfect, then G is called quasi-simple. Of course, G need not be perfect as G ≅ A₅ × 2 shows. However, I believe this is the only obstruction, so ignoring a possible cyclic direct factor of order 2, G/Z(G) is simple, and G is quasi-simple. The finite quasi-simple groups and their automorphism groups are classified, but the classification is a bit long. For a fixed simple group, X = G/Z(G), there are only finitely many isomorphism classes of quasi-simple groups D such that D/Z(D) = X. In fact there is a unique largest one called the Schur cover, that I'll call D. If Z(D) is cyclic, then in fact Aut(G) = Aut(X) = Aut(D) does not pay any attention to the center. So all we need to do is find all X with Aut(X) = X [and each one works], and all X with Z(D) non-cyclic [and check which ones work].

Having done most, but not all, of that, I thought it might help to record the basic result:

If G = H×T where T=1 if H is abelian and T is cyclic of order dividing 2 otherwise, and where H is on the following list, then Aut(G) is simple:

• cyclic of order 3, 4, or 6


• elementary abelian of order 2ⁿ for n ≥ 3


• M11, 2.Sz(8), J1, 2.Sp(6,2), M23, M24, Ru, 2.Ru, Co3, Co2, Ly, Th, Fi23, Co1, 2.Co1, J4, B, 2.B, E7(2), M


• Ω(2n+1,2) for n ≥ 3


• Sp(2n,2) for n ≥ 3


• E8(p) for any prime p


• F4(p) for any prime p


• G2(p) for any prime p ≥ 5

Additionally if Aut(G) is simple, then G = H×T as above, except possibly H/Z(H) is on the following list:

• L3(4), U4(3), U6(2), 2E6(2)


• Ω⁺(4n,q) for certain q

These are groups with non-cyclic multiplier other than Sz(8) [definitely an example] and Ω⁺(8,2) [not an example]. The Ω⁺(4n,q) case should be mostly easy, as there are too many automorphisms to kill. The others would be easy in an ideal world, but as far as I know our computational knowledge of these groups is limited and/or flawed. Of course, I also need to check the abelian case carefully, but I think 3,4,6 and 2^n are the only abelian examples.

It would make another good answer: For what torsion abelian groups G is Aut(G) simple? This would handle the abelian groups here, as well as some of the original poster's interest, without delving into the nastier aspects of abelian groups.

ag.algebraic geometry - Canonical topology on the category of schemes?

following the answers given by Pantev, I will give you more example of canonical topology on some category(universally strict epimorphism)

if C is an abelian category or a topos, then canonical topology consists of all epimorphism
if C is a quasi abelian category,then canonical topology consists of all strict epimorphism(Note that strict epimorphism is subcanonical in general)

if C is a category of associative unital k-algebras(opposite category of affine schemes,not necessarily commutative). Canonical topology consists of all strict epimorphism which are precisely surjective morphism of algebras. In this case, univerally strict epimorphism coincides with strict epimorphism.

Rosenberg has a very detailed treatment for a category and 2-category(taken as category of spaces)with canonical topology(he called right exact structure).It is in MPIM preprint series," Homological algebra of noncommutative 'space' I"

What pantev mentioned is related to my answers in another question:
Does sheafification preserve sheaves for a different topology?

The effective descent topology is finer that fpqc topology, fppf topology,smooth topology(in Kontsevich-Rosenberg sense)

The descent topology on category of affine schemes(not necessarily commutative) coincides with subcanonical topology

The reference is Orlov's paper and Kontsevich-Rosenberg MPIM preprint series. Noncommutative stack and Noncommutative grassmannian and related construction

ct.category theory - Definition of forgetful functor

This is very difficult to do. The Stuff, Structure, Property approach coined by Jim Dolan, John Baez and Toby Bartels is the best formalization I've seen. Here is the synopsis from the nLab page:

Category theory frequently allows to give precise and useful formalized meanings to “everyday” terms, at least terms used everyday by practicing mathematicians.

It was indeed introduced originally in order to formalize the use of the notion “natural” in mathematics. Another frequently recurring pair of terms in math are “extra structure” and “extra properties”, to which we add the more general concept of “extra stuff”. In discussion among Jim Dolan, John Baez and Toby Bartels, the following useful formalization of these concepts in category theoretic terms was established.

---

Here is a counterpoint to some of the other responses. It's true that the abstract approach of Stuff, Structure, Property may seem sketchy (especially the passage to the core groupoid) but it is in fact surprisingly correct for some very broad classes of concrete categories with very rich notions of forgetfulness.

One such class (one that I am more familiar with) are the categories Mod(T) of models of a first-order theory T. There are three very distinct ways of forgetting things in Mod(T):

• Forgetting axioms of T (Forgetting Properties)


• Forgetting parts of the language (Forgetting Structure)


• Restricting to definable substructures (Forgetting Stuff)

In the absence of evil and under other ideal conditions, the functorial characterizations of Properties, Structure, and Stuff translate to important results in model theory (various definability, interpolation, and consistency theorems). The translations are sometimes a little on the weak side, but I think that with adjustments to account for type information not captured by the theory alone, the translation can be made broader and even more precise.

To me, this is strong evidence that Stuff, Structure, Property is indeed the correct way to translate these notions of forgetfulness from the concrete to the abstract. While it's true that talking about forgetfulness without knowing what you're forgetting is nonsensical in when looking at particular instances, this approach provides a way of abstracting and even reasoning about forgetfulness in a completely general setting.

PS: Note that I am not a category theorist, I'm just a very impressed outsider. I suspect that category theorists have even stronger intuitions for Stuff, Structure, Property approach.

gr.group theory - solving equations in the braid group

This group is a central extension of $PSL_2mathbb{Z}$, which is a virtually free group. There is an algorithm to solve equations in such groups, and parameterize the solutions. Since your equation is degree zero in $a,b,x$, if the lift of the solution in $PSL_2mathbb{Z}$ to $B_3$ solves the equation for one lift, it should work for any other lift. I'm not quite sure though how to determine this uniformly over all lifts of the solution. The solutions are given by Makanin-Razborov diagrams, and they are parameterized by various automorphisms. So I think you just need to check one solution in each equivalence class coming from each orbit.

fa.functional analysis - The role of completeness in Hilbert Spaces

The answers already posted are quite satisfying, I'd just like to add one more point of view (at the risk of making the thing more confused for the OP :). When Sobolev started solving PDEs, he did not have reasonable function spaces available: working in $C^2$ is a nightmare as soon as you want to do calculus of variations, and it is immediately clear that 'something is missing'. You naturally construct solutions by approximating them (with minimizing sequences, with smooth approximations etc. etc.). The original approach of Sobolev was: well, all I have is this approximating sequence, so THIS SEQUENCE is my solution, whatever that means. This was his original definition of 'weak solution'.

As you see, he was dispensing completely with completeness, and working only with functions in a dense subspace. This is perfectly fine, and I'm tempted to answer to the original question with the paradox: completeness is not really necessary, even from a theoretical standpoint, since of course you can embed every normed space in a complete one. But this is very awkward; it is vastly more economical to 'define' the limit of your approximating sequence. Indeed, this procedure is precisely what is called completion. Working in a complete space makes it possible to take the limit of your approximation and define a solution as a concrete object. 100 yeasr later, we find this approach totally natural. I think this was one of the driving forces behind the universal adoption of complete spaces in analysis.

motives - Kunneth formula for motivic cohomology

I now remember a nice argument, why there's no Kunneth formula for Chow groups of $X times X$ unless $X$ has a Tate motive. Let $X$ be smooth projective of dimension $d$.
We start with a decomposition of a diagonal:

$$[Delta] = sum_{i,j} alpha^i_j beta^{d-i}_j in oplus_i CH^i(X) otimes CH^{d-i}(X)$$
We can assume $alpha^i_j$ are linearly independent.
In this case we can show that $alpha^i_j$ form a basis of Chow groups and
$beta^{d-i}_j$ is the dual basis.

Indeed, as a correspondence $[Delta]$ acts as identity on Chow groups,
so for any class c,

$$c = [Delta]c = sum_{i,j} alpha^i_j deg(beta^{d-i}_j cup c),$$
and the claim follows if we substitute $c = alpha^i_j$.

Now $CH_i(X) = Hom(mathbb Z(i)[2i], M(X))$ and we can consider the set of $alpha^i_j$ as a morphism of motives

$$oplus_{i,j}mathbb Z(i)[2i] to M(X).$$
A simple computation shows that it is an isomorphism with the inverse given by $beta^i_j$.

And of course, on the other hand, if $X$ has a Tate motive, then Kunneth formula for Chow groups follows (it doesn't answer the question, since I only consider smooth projective varieties).

singularity theory - Is resolution of singularities effective?

Yes, in the sense that resolution of singularities is implemented in the computer algebra package Singular. See the manual of Singular for references. (There might be other/better references.) However, if I remember correctly the centers are not unique.

nt.number theory - Products of linear forms in 3 variables

This should be a comment to Robin's answer.

Take any irreducible polynomial $f in mathbb{Q}[x]$ of degree 3 with real roots, say $alpha, beta, gamma$. Set $f_1 = x + alpha y + alpha^2 z$, $f_2 = x + beta y + beta^2 z$, $f_3 = x + gamma y + gamma^2 z$.

You can find plenty of polynomials here.

co.combinatorics - Graphs of Tangent Spheres

The number of edges in such a graph is linear in the number of vertices, and they can be split into two equal-sized subgraphs by the removal of $O(n^{frac{d-1}{d}})$ vertices. See e.g.

A deterministic linear time algorithm for geometric separators and its applications.
D. Eppstein, G.L. Miller, and S.-H. Teng.
Fundamenta Informaticae 22:309-330, 1995.

Unlike in the 2d case (where we know that the maximum number of edges such a graph can have is 3n-6) the precise maximum edge density is not known even in 3d. There's a lower bound of roughly $frac{3828n}{607}approx 6.3064n$ in another of my papers,

Fat 4-polytopes and fatter 3-spheres.
D. Eppstein, G. Kuperberg, and G. Ziegler.
arXiv:math.CO/0204007.
Discrete Geometry: In honor of W. Kuperberg's 60th birthday, Pure and Appl. Math. 253, Marcel Dekker, pp. 239-265, 2003.

and an upper bound of $(4+2sqrt3)n approx 6.8284n$ in

Greg Kuperberg and Oded Schramm, Average kissing numbers
for non-congruent sphere packings, Math. Res. Lett. 1 (1994),
no. 3, 339–344, arXiv:math.MG/9405218.

co.combinatorics - The factorial of -1, -2, -3, ...

My question was intended somewhat along the line: Assume the Gamma function is not yet invented and Goldbach asks you the question: "What is (-n)! ?" You know that Goldbach expects a combinatorial answer in the domain of integer or rational numbers. What would you answer? I will give my answer in this sens.

Looking at GKP's ConMath, Table 253, the combined Stirling triangles in their dual form, we see: If we sum the columns in this triangle for k < 0 we get the factorial numbers, if we sum the rows for k > 0 we get the Bell numbers.

What about saying the Bell numbers are the factorial numbers at negative integers? Is the answer encoded in one of the most important triangles in combinatorics?

See what Knuth says about the origin of this duality (table on page 11).

{120}  
  . {24}  
  1,  .  {6} 
 10,  1,  .  {2} 
 35,  6,  1,  .   {1}  
 50, 11,  3,  1,  .  {1}
 24,  6,  2,  1,  1,  .
  0,  0,  0,  0,  0,  1,  
  0,  0,  0,  0,  0,  0,  1,.....{1}  
  0,  0,  0,  0,  0,  0,  1,  1,....{2}  
  0,  0,  0,  0,  0,  0,  1,  3,  1,....{5}  
  0,  0,  0,  0,  0,  0,  1,  7,  6,  1,....{15}  
  0,  0,  0,  0,  0,  0,  1, 15, 25, 10, 1,....{52}  

lo.logic - categorification of logic

has there be an effort to categorify first order logic? More particularly, structures in the sense of logic.

If so, then every structure of a first order theory is a category. so in particular, the universe of categories must be a (meta)-category itself. So I have another question: is there a development of a model theory of categorified logic?

The idea is like this. In modern set-theoretic based model theory, must of the interesting stuff comes by looking at different cardinalities. First order theory like Lowenheim-Skolem Theorem makes it easy to move up and down cardinalities, and after all the category SET is equivalent to CARDINALS. Very much this equivalence dictates the model theory.

So the universe of categories CAT, and whatever is a skeletal equivalent of it, will dictate the model theory of categorified logic.

Anyone aware of categorified logic?

fa.functional analysis - Application of bounded spectral theory.

I think Helge's answer cuts to the historical heart of the matter: solution operators for various differential equations tend to be bounded, non-compact operators (obtained in many cases from an unbounded differential operator via the functional calculus), and it is often quite useful from that point of view to know something about their spectra. This is one reason why the theory of elliptic operators over non-compact spaces is more complicated than the corresponding theory for compact spaces: one has to deal with the fact that the eigenvalues of the solution operators can accumulate at 0.

What I find more persuasive, however, are the ways in which spectral theory mediates the relationship between functional analysis and geometry. In many cases you will miss this relationship unless you ignore compact operators entirely. The celebrated Atiyah-Singer Index Theorem provides a particularly dramatic example of this phenomenon, but I'll focus on more digestible examples (A-S is more about spectral theory for unbounded operators anyway).

First, consider the classical Toeplitz operators. Given a complex valued function $g in C(S^1)$, the Toeplitz operator $T_g$ with symbol $g$ is defined as follows. Form the Hardy space $H^2(S^1)$ by considering the $L^2$-closure of the space of polynomial functions on $S^1$, regarding $S^1$ as a subspace of $mathbb{C}$, and let $P$ denote the orthogonal projection from $L^2(S^1)$ to $H^2(S^1)$. Then define $T_g: H^2(S^1) to H^2(S^1)$ to be $T_g(f) = T(fg)$. This is a bounded operator, and the classical Toeplitz index theorem asserts that its Fredholm index (the dimension of its kernel minus the dimension of its cokernel) is precisely the winding number of $g$. Thus an analytic invariant of the Toeplitz operator with symbol $g$ calculates a topological invariant of $g$.

That result by itself isn't heavy on the spectral theory. The connection with spectral theory is revealed by a more refined statement. Recall that the essential spectrum of a bounded operator $T$ on a Hilbert space $H$ is the spectrum of the image of $T$ in the Calkin algebra $Q(H)$ (which is the space of bounded operators on $H$ modulo the space of compact operators). A consequence of the Toeplitz index theorem (and its proof) is the fact that the essential spectrum of the Toeplitz operator $T_g$ is precisely the range of $g$. While this statement alone is scant evidence, this suggests a deep relationship between the essential spectrum of a bounded operator and geometry. This line of thinking culminates in the Brown-Douglas-Fillmore theorem, which makes the following startling assertion. Let $X$ be a nonempty subset of the complex plane, and define $Ext(X)$ to be the space of essential unitary equivalence classes of essentially normal operators with essential spectrum $X$ (here "essential" always means "modulo compact operators"). Direct sum of operators gives $Ext(X)$ the structure of a commutative semigroup, and the BDF theorem asserts that $Ext(X)$ is naturally isomorphic to the space $Hom(pi^1(X), mathbb{Z})$ of group homomorphisms between the first cohomotopy group of $X$ and $mathbb{Z}$. (Note: it is not even obvious that $Ext(X)$ has a zero element!) Thus spectral theory helps to classify certain kinds of bounded operators mod compacts in a particularly beautiful way (via algebraic topology).

As mentioned above, there are also fruitful interactions between functional analysis and geometry - mediated by spectral theory - which flow from analysis to geometry. Aside from Atiyah-Singer, there are fruitful generalizations of the Toeplitz index theorem along these lines. But let me give a different sort of example in the theory of hyperbolic diffeomorphisms.

Informally, a diffeomorphism $f: M to M$ on a smooth manifold $M$ is said to be Anosov (or uniformly hyperbolic) if $M$ admits transverse stable and unstable foliations for the action of $f$. Prototypical examples of Anosov diffeomorphisms on the 2-torus can be obtained by considering $2 times 2$ matrices with integer entries and irrational eigenvalues. It turns out that spectral theory has a great deal to say about smooth dynamical systems in general and Anosov diffeomorphisms in particular.

Given any diffeomorphism $f: M to M$, consider the bounded operator $f_*$ on the Banach space $Gamma^0(TM)$ of continuous vector fields on $M$ defined by

$(f_*v)(x) = df(v)(f^{-1}(x))$

If the non-periodic orbits of $f$ are dense in $M$, then a theorem of Mather asserts that the spectrum of $f_*$ is a disjoint union of finitely many annuli centered at the origin. If $H_i$ is the invariant subspace for $f_*$ corresponding to the $i$th annulus then the subspaces $E_i(x)$ of $T_x M$ consisting of the vectors $v(x)$ for $v in H_i$ form a $df$ invariant continuous distribution on $M$, and the direct sum of the $E_i(x)$'s gives the whole tangent space $T_x M$. So the Mather spectral theory of $f$ is very closely related to its dynamics. Indeed, one can characterize the Anosov diffeomorphisms as precisely those $f$ for which $1$ is not in the spectrum of $f_*$. Pesin used this idea to prove that Anosov diffeomorphisms are structurally stable, meaning they form an open subset of the full diffeomorphism group of $M$ (so that a small perturbation of an Anosov diffeomorphism is still Anosov). The same strategy also works for partially hyperbolic dynamical systems, which have a slightly different spectral characterization.

hyperbolic geometry - Poincaré disk model: is this locus a known curve?

In the Klein model, one may see that this is also a circle. Consider a line segment with one point on the center of the disk. One side of the triangle goes through the center. Then orthogonal lines to a line through the center are also orthogonal in the hyperbolic metric, e.g. since they are preserved by reflection. So one sees that a circle is traced out which goes through the origin. If you'd rather center the curve at the origin, then it will be an ellipse, since hyperbolic isometries of the Klein model are projective transformations.

To convert to the Poincare model, take a hemisphere sitting over the disk, and project vertically. The projection of the circle is given by the intersection of a cylinder over the circle with the upper hemisphere. This upper hemisphere is conformally equivalent to the Poincare model, e.g. by inversion through a sphere centered at the south pole of the lower hemisphere. I haven't computed the curve this traces out though.

at.algebraic topology - A list of machineries for computing cohomology

This is not a question, but I just hope to hear more from everyone here on it.

A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I have never seen this being made explicit.

What I have in mind at the moment:

"Basic" methods:

*) The definition: for example Simplicial cohomology makes the problem into one of pure linear algebra which can then be done by hand or by many computer program packages at the moment. For singular cohomology this is not really reasonable though.

*) The Axioms: Things Such as the Mayer–Vietoris sequence or the LES of a Pair. These two methods allow you to compute the cohomology of most cell complexes that you are likely to encounter early in your education. More detailed study of the maps in the sequences can get you even farther.

"Advanced" methods:

*) Spectral sequences. Leray-Serre seems to be the most commonly used, since many interesting spaces can be written in terms of fibrations.

*) Morse theory. Surprisingly effective for many difficult problems, especially if one can construct a good energy function, such that the critical sets and flows are simpler.

*) Weil conjecture. After Deligne's proof, one can go in the opposite direction and find Betti numbers by point-counting. Unfortunately it can not give the torsions as far as I know.

For the last two methods, I find Atiyah-Bott's celebrated paper on the moduli space of bundles an excellent demonstration.

Now I am looking forward to your inputs. How many important methods are missing here?

linear algebra - eigenvalues of edge regular graphs

In graph theory, an edge regular graph is defined as follows.
Let G = (V,E) be a regular graph with v vertices and degree k.
G is said to be edge regular if there is also integer λ such that:

Every two adjacent vertices have λ common neighbors.

A graph of this kind is sometimes said to be an er(v,k,λ).

I want know about eigenvalues of edge regular graph, how can we
find eigenvalue of this graph?

soft question - Quantitatively speaking, which subject area in mathematics is currently the most research active?

Sorry to add to the noise, but here it goes. With a little script-fu (and emacs, of course!) I retrieved the data from MSC corresponding to the last ten years in each of the Primary Classifications. Annoyingly the AMS changed their subject classification scheme recently, so that the numbers I queried were interpreted as MSC2010, whereas the papers are published from the year 2000.

43465     35 Partial differential equations
38151     62 Statistics
35994     81 Quantum theory
35633     68 Computer science
34474     65 Numerical analysis
28593     05 Combinatorics
28296     90 Operations research, mathematical programming
26406     34 Ordinary differential equations
26192     60 Probability theory and stochastic processes
23879     93 Systems theory; control
22361     11 Number theory
21689     76 Fluid mechanics
20787     91 Game theory, economics, social and behavioral sciences
19440     37 Dynamical systems and ergodic theory
18425     83 Relativity and gravitational theory
17323     94 Information and communication, circuits
17247     53 Differential geometry
16465     47 Operator theory
16134     03 Mathematical logic and foundations
15408     20 Group theory and generalizations
14225     92 Biology and other natural sciences
14051     82 Statistical mechanics, structure of matter
13663     46 Functional analysis
12894     74 Mechanics of deformable solids
11241     14 Algebraic geometry
10237     49 Calculus of variations and optimal control; optimization
10215     30 Functions of a complex variable
10154     16 Associative rings and algebras
 9801     01 History and biography
 9781     54 General Topology
 8014     42 Fourier analysis
 7103     58 Global analysis, analysis on manifolds
 6780     15 Linear and multilinear algebra; matrix theory
 6410     70 Mechanics of particles and systems
 6359     32 Several complex variables and analytic spaces
 6348     57 Manifolds and cell complexes
 6185     41 Approximations and expansions
 5935     39 Difference and functional equations
 5684     26 Real functions
 5349     17 Nonassociative rings and algebras
 5226     13 Commutative rings and algebras
 4840     78 Optics, electromagnetic theory
 4439     52 Convex and discrete geometry
 4418     33 Special functions
 4350     00 General
 3818     06 Order, lattices, ordered algebraic structures
 3511     28 Measure and integration
 3295     51 Geometry
 2948     22 Topological groups, Lie groups
 2944     55 Algebraic topology
 2538     86 Geophysics
 2089     45 Integral equations
 2052     18 Category theory; homological algebra
 1679     80 Classical thermodynamics, heat transfer
 1523     31 Potential theory
 1444     43 Abstract harmonic analysis
 1343     12 Field theory and polynomials
 1161     40 Sequences, series, summability
 1108     08 General algebraic systems
  898     44 Integral transforms, operational calculus
  775     19 K-theory
  534     85 Astronomy and astrophysics

Usual disclaimers apply. In particular, before concluding that nobody works in astrophysics, go and check the submission statistics for astro-ph: more than 11,000 submissions in 2009 alone! Clearly the AMS does not index very widely in this area.

Let me reiterate that I do not believe for a second that this data allows one to conclude anything of value about mathematics, just perhaps about mathematicians :)

Added (incorporating Gerald Edgar's summary in the comment below)

This is the summary of "pure maths" defined as classifications 00-60, with a total of 411902 articles reviewed in the decade that has just finished. That, in case you are wondering is 55.38% of all papers reviewed.

00--08      Logic and Combinatorics         63804    15.49%
11--20      Algebra and Number Theory       80689    19.59%
22--49,60   Analysis and Probability       216252    52.50%
51--58      Geometry and Topology           51157    12.42%

co.combinatorics - What does the generating function $x/(1 - e^{-x})$ count?

In the comments, Tom Copeland asked if, six years later, I had any further insights into the diagrammatics I proposed near the end of my question. So I figured I'd mention what I know, which is yet another (marginally) diagrammatic description of BCH multiplication and in particular the map $mathcal U mathfrak g to mathcal S mathfrak g$. What I'll describe is some part of http://dx.doi.org/10.1090/pspum/088 (also available at http://arxiv.org/abs/1307.5812). Some pictures are available in the last chapter of my thesis, but note that that chapter has a subtle error (the details of which I haven't yet sussed out) which I have corrected in the linked paper; the error does not affect the case of (duals of) Lie algebras, but does affect Poisson structures whose Taylor expansions include quadratic or higher terms.

I will describe some homological algebra, and then unpack into diagrams. Let $mathfrak G = mathfrak g otimes Omega_{mathrm{cpt}}(mathbb R)[1]$ denote the cochain complex of $mathfrak g$-valued compactly supported de Rham forms on $mathbb R$, shifted so that its cohomology is concentrated in degree $0$. Then $mathrm H^0(mathfrak G) = mathfrak g$, and the projection to homology is given by integrating de Rham forms — there is a cohomological-degree-$0$ map $int : mathfrak G to mathfrak g$. Note also that $Omega_{mathrm{cpt}}(mathbb R)$ has a non-unital graded-commutative multiplication ($wedge$), through which $mathfrak G$ picks up a graded-symmetric (!) cohomological-degree-$(+1)$ map $delta : mathfrak G otimes mathfrak G to mathfrak G$ by $delta(xotimes alpha,yotimes beta) = [x,y] otimes (alpha wedge beta)$ (up to a sign that depends on a choice of conventions about how to handle elements of shifted complexes). Consider the (graded-commutative) symmetric algebra $mathcal S mathfrak G$. I will denote its usual differential (which is the extension of de Rham-form differentiation as a derivation) by $partial_{mathrm{dR}}$.
Integration of de Rham forms extends to an algebra homomorphism $int : (mathcal Smathfrak G, partial_{mathrm{dR}}) to mathcal S mathfrak g$.

The map $delta$ has a canonical extension to a second-order differential operator on $mathcal S mathfrak G$ which I will also call $delta$. (A second order differential operator on a symmetric algebra is unique determined by its values on constant, linear, and quadratic terms. We declare that $delta$ vanishes on constants and linears, and set it to be the original $delta$ on quadratics.) Because $wedge$ plays well with $partial_{mathrm{dR}}$, $delta$ and $partial_{mathrm{dR}}$ graded-commute. The Jacobi identity implies $delta^2 = 0$. So we have a new differential $partial_q = partial_{mathrm{dR}} + delta$ on $mathcal S mathfrak G$.

Pick any $alpha in Omega_{mathrm{cpt}}^1(mathbb R)$ with $intalpha = 1$. (Henceforth I will call such one-forms bumps.) The map $x mapsto x otimes alpha$ from $mathfrak g to mathfrak G$ extends to an algebra homomorphism $mathcal S mathfrak g to mathcal S mathfrak G$ splitting $int$. There is a unique contracting homotopy $eta_alpha : Omega_{mathrm{cpt}}(mathbb R) to Omega_{mathrm{cpt}}(mathbb R)$ (of cohomological degree $-1$) such that the graded commutator $[partial_{mathrm{dR}},eta_{alpha}] = mathrm{id} - (alpha otimes) circ int$; it vanishes on $Omega_{mathrm{cpt}}^0$ and on $Omega_{mathrm{cpt}}^1$ satisfies $eta_alpha(beta) = partial_{mathrm{dR}}^{-1}(beta - (int beta)alpha)$; note that $int(beta - (int beta)alpha) = 0$, so the one-form $beta - (int beta)alpha$ has a unique antiderivative among compactly-supported functions.

We can extend $eta_alpha$ to $mathcal Smathfrak G$ in many ways, and the choice can be proven not to matter. To make a choice, declare that on the constants $mathcal S^0mathfrak G$ we have $eta_alpha = 0$, and on $mathcal S^nmathfrak G$ we have

$$eta_alpha(beta_1 odot dots odot beta_n) = frac1n sum_i beta_1 odot dots odot eta_alpha(beta_i) odot dots odot beta_n,$$
where $odot$ denotes the symmetric multiplication in $mathcal S$. Note that this is not the extension of $eta_alpha$ as a derivation.

Since $eta_alpha$ always attaches 1-forms and $delta$ involves wedge multiplication, $eta_alpha : mathcal S mathfrak g to (mathcal S mathfrak G,partial_q)$ is a chain map. The integration map $int : mathcal S mathfrak G to mathcal S mathfrak g$ splitting this map is not a chain map from $(mathcal S mathfrak G,partial_q)$. But with the above choices we can choose a different splitting, namely $int circ (mathrm{id} - delta eta_alpha)^{-1}$. (I have a 50% chance of getting that minus sign wrong.) I will leave checking that this is a chain map splitting $eta_alpha$ to you. Note that $(mathrm{id} - delta eta_alpha)^{-1} = sum_N (delta eta_alpha)^N$ converges on $mathcal S mathfrak G$, since $delta eta_alpha$ drops polynomial degree by $1$.

Pick bumps $alpha_1, dots,alpha_n$ such that the support of $alpha_i$ is in $[i-1,i]$, and pick one final bump $alpha$ arbitrarily. One can prove that the map $mathcal U mathfrak g to mathcal S mathfrak g$ is given on monomials $x_1 dots x_n$ (with multiplication in $mathcal U$) by

$$mathcal U mathfrak g ni x_1 dots x_n mapsto int circ (mathrm{id} - delta eta_alpha)^{-1} bigl( (alpha_1 otimes x_1) odot dots odot (alpha_n otimes x_n) bigr) in mathcal S mathfrak g$$
(or I might be off by a sign somewhere). In general, similar formulas describe the entire product on $mathcal Smathfrak g$ given by transporting the one from $mathcal U mathfrak g$ along the symmetrization isomorphism.

Let me now unpack this formula, or rather give the answer after some unpacking. (Proving that this is a valid unpacking is straightforward: you need to track the numerical factors coming from $eta_alpha$, understand how to apply a second-order differential operator to a monomial, and also include a brief "degree reasons" argument to get $eta_alpha$ and $delta$ to apply always to the same things at the same time.)

Define an $n$-leaf binary heighted forest, abbreviated forest, to be set of binary rooted trees whose leaves are put in bijection with the set ${1,dots,n}$ and whose nodes are totally ordered (I mean: totally order the set of all nodes) such that in a given tree, and path from root to leaf is increasing for the total ordering. Arbitrarily choose for each node which of its two branches is left and which right (the choice will cancel out).

Given a forest and the list $x_1,dots,x_n$ of elements in $mathfrak g$, there is an obvious element of $mathcal S mathfrak g$ given by putting the $x_i$s at the leaves and reading the forest as instructions of who to bracket with whom (then multiply the "root" outputs).

Now I will describe, for each forest, how to compute a number. Consider the map $Omega_{mathrm{cpt}}^1(mathbb R) otimes Omega_{mathrm{cpt}}^1(mathbb R) to Omega_{mathrm{cpt}}^1(mathbb R)$ given by $beta_1 otimes beta_2 mapsto beta_1 wedge eta_alpha (beta_2) - beta_2 wedge eta_alpha(beta_1)$. Place this map at each vertex, and $alpha_i$ at the $i$th leaf, and let the forest tell you how to apply this map to end up with a bunch of $1$-forms at the roots. Then integrate all these $1$-forms to get numbers, and multiply those numbers together.
Finally, suppose there are $k$ roots (and hence $(n-k)$ nodes). Then multiply by $frac1 n frac1{n-1} dots frac1{n-k}$.

Note that neither the number nor the element of $mathcal S mathfrak g$ determined by a forest depends on the height ordering. But I now want you to sum over all forests with total node-ordering of the product of these two numbers. That sum computes the map $mathcal U mathfrak g to mathcal S mathfrak g$ above. (If I had a good way to count the number of total orderings of the nodes for a given un-heighted forest, I would have used it.)

This is similar to, but not the same as, Kontsevich's star product. In particular, note that my forests have no wheels, whereas Kontsevich does not describe the symmetrization map $mathcal S mathfrak g cong mathcal U mathfrak g$, but rather this map twisted by some traces in the adjoint representation.

elliptic curves - Q-isogeny and Q-torsion subgroup

I assume you have an elliptic curve $E$ defined over $mathbb{Q}$ -- in simplest terms, this means a Weierstrass equation $y^2 = x^3 + Ax + B$ with $A,B in mathbb{Q}$.

Then a $mathbb{Q}$-isogeny is a finite morphism $varphi: (E,O) rightarrow (E',O')$ which is defined over $mathbb{Q}$: in other words, given locally by rational functions with $mathbb{Q}$-coefficients. An equivalent perspective is that an isogeny is essentially determined -- i.e., up to an automorphism on the target -- by its kernel $E[phi]$, a finite
subgroup of $E(overline{mathbb{Q}})$. Then the definedness over $mathbb{Q}$ is equivalent to invariance under the group $operatorname{Aut}(overline{mathbb{Q}}/mathbb{Q})$: for every Galois automorphism $sigma$, we want $sigma E[phi] = E[phi]$.

Similarly, the $mathbb{Q}$-torsion subgroup is the subgroup of the full torsion subgroup which is defined over $mathbb{Q}$. This means $E[operatorname{tors}] cap E(mathbb{Q})$: it is just the subgroup of $E(mathbb{Q})$ consisting of points of finite order. It can also (equivalently) be defined as the Galois invariants of $E[operatorname{tors}](overline{mathbb{Q}})$ (= $E[operatorname{tors}](mathbb{C})$).

Fundamental group of Lie groups

The Eckmann-Hilton argument is the correct answer, but it might be amusing to note that there is a very explicit homotopy as well. Suppose $alpha_1$, $alpha_2 in pi_1 G$, and define

$alpha : I^2 rightarrow G$

by $alpha(t_1,t_2) = alpha_1(t_1) cdot alpha_2(t_2)$, where $cdot$ is the product in $G$. Then along the diagonal, we have $alpha_1 cdot alpha_2$, the product using the group operation, while along the bottom edge followed by the right edge we have the composition $alpha_1 * alpha_2$, the product of loops in the fundamental group. Deforming the path shows they're homotopic. Similarly, along the left edge, followed by the top edge we get $alpha_2 * alpha_1$, so this product is commutative.

ct.category theory - Standard name of "atomic morphisms"?

As mentioned in the comments, I would probably call such a morphism "irreducible" or "prime." A "less evil," and perhaps more useful, version would be to ask that if $f = g circ h$, then either $g$ or $h$ is an isomorphism. In this form, if you regard the multiplicative monoid of a ring as a category with one object, the (noninvertible) irreducible morphisms are precisely the irreducible elements of the ring.

I agree that in "concrete categories" such morphisms are unlikely to be very common or useful, but one other situation in which they arise is free categories on directed graphs. In such a category, the nonidentity irreducible morphisms are precisely the generators (the images of the edges of the directed graph you started from).

algebraic k theory - Is Higher K-functor the derived functor of K0?

I don't think it's stupid, but I guess it depends what you mean by "derived functor." This is true in the weak sense that K-theory is naturally a space- or spectrum-valued functor, and the $K_i$ is the i-th homotopy of this functor. But it seems not to be the case that $K$-theory is a derived functor in the sense of Cartan-Eilenberg.

Let me discuss the question of the universality of $K$-theory:

I'll abuse terminology and refer to "categories" when I mean categories of a suitable kind, with appropriate added structure --- e.g., exact categories if you want to do Quillen K-theory, Waldhausen categories if you want to do Waldhausen K-theory, Waldhausen $infty$-categories if you want to do K-theory with them, etc. ...

Now if one translates the sense in which $K_0$ is universal as an abelian-group-valued functor on "categories" into the language of stable homotopy theory, one arrives at the universal property satisfied by K-theory as a spectrum-valued functor on "categories."

More precisely, we have additive $K_0$, denoted $K_0^{oplus}$, which is simply the functor that assigns to any "category" $mathcal{C}$ the group completion of the abelian monoid whose elements are isomorphism (or equivalence) classes of objects of $mathcal{C}$, where the sum is $oplus$. This functor is "inadequate" in the sense that there might be some exact (or fiber) sequences of $mathcal{C}$ that $K_0^{oplus}$ cannot see.

To address this, for any "category" $mathcal{C}$, we can build a new "category" $mathcal{E}(mathcal{C})$ whose objects are exact sequences. This "category" admits two functors to $mathcal{C}$ that send an exact sequence $[0to A'to Ato A''to 0]$ to either $A'$ or $A''$. For any functor $F$ from categories to abelian groups, we get an induced homomorphism $Fmathcal{E}(mathcal{C})to Fmathcal{C}oplus Fmathcal{C}$. Let's say that $F$ splits the exact sequences of $mathcal{C}$ if this morphism is an isomorphism, and let's say that $F$ is additive if $F$ splits the exact sequences of every "category."

Now $K_0$ has the following pleasant universal property. It is the initial object in the category of additive functors receiving a natural transformation from $K_0^{oplus}$.

Now to translate all this into stable homotopy. We have additive K-theory, denoted $K^{oplus}$, which is simply the functor that assigns to any "category" $mathcal{C}$ the spectrum corresponding to the group completion of the $E_{infty}$ space given by the (nerve of the) subcategory of $mathcal{C}$ comprised of the isomorphisms (or weak equivalences), where the sum is $oplus$. This functor is again "inadequate" in the sense that there might be some exact (or fiber) sequences of $mathcal{C}$ that $K^{oplus}$ cannot see.

Now for any functor $F$ from categories to spectra, we get an induced homomorphism $Fmathcal{E}(mathcal{C})to Fmathcal{C}vee Fmathcal{C}$. Let's say that $F$ splits the exact sequences of $mathcal{C}$ if this morphism is an equivalence, and let's say that $F$ is additive if $F$ splits the exact sequences of every "category."

Now $K$ has the following homotopy-universal property. It is the homotopy-initial object in the category of additive functors receiving a natural transformation from $K^{oplus}$.

So the universality of K-theory arises not from thinking of the disembodied K-groups, but rather from interpreting K-theory as a spectrum, and rewriting the universal property of $K_0$ in suitably homotopical language.

(References: Gonçalo Tabuada has a paper in which he characterizes K-theory by a similar universal property, and John Rognes and I have begun a similar paper in the context of Waldhausen $infty$-categories, an incomplete draft of which is on my webpage.)

ho.history overview - What proof of quadratic reciprocity is Hilbert referring to in this quote?

It could be Kronecker's determination of the sign of the Gauss sum by means of Cauchy's theorem. Already Gauss noted that the determination of the sign implies the law of quadratic reciprocity.

In response to the request for references:

Leopold Kronecker: Summirung der Gauss'schen Reihen ... J. Reine Angew. Math. 105 (1889), 267-268.

Also in volume 4 of his Werke, 297-300. (This was where I xeroxed it, so I can vouch for the page numbers, I have the pages in front of me right now).

Also in Landau's Elementare Zahlentheorie (together with two others, by Mertens and Schur),
near the end of the book.

Also supposed to be in Ayoub: Introduction to the Analytic Theory of Numbers, but I am not familiar with his book, so I cannot vouch for this.

There is a later determination of the sign of the Gauss sum by contour integration, due to Mordell, which is quite accessible; it is in Chandrasekharan's Introduction to Analytic Number Theory, page 35--39. Chandrasekharan does a more general case.

Now, I have not claimed that Kronecker's proof was the one that Hilbert was thinking of. I cannot read the mind of a dead man (nor that of a living one).

proof assistants - Is there a known way to formalise notion that certain theorems are essential ones?

Although your question is vague in certain ways, one robust answer to it is provided by the subject known as Reverse Mathematics. The nature of this answer is different from what you had suggested or solicited, in that it is not based on any observed data of mathematical practice, but rather is based on the provable logical relations among the classical theorems of mathematics. Thus, it is a mathematical answer, rather than an engineering answer.

The project of Reverse Mathematics is to reverse the usual process of mathematics, by proving the axioms from the theorems, rather than the theorems from the axioms. Thus, one comes to know exactly which axioms are required for which theorems. These reversals have now been carried out for an enormous number of the classical theorems of mathematics, and a rich subject is developing. (Harvey Friedman and Steve Simpson among others are prominent researchers in this area.)

The main, perhaps surprising conclusion of the project of Reverse Mathematics is that it turns out that almost every theorem of classical mathematics is provably equivalent, over a very weak base theory, to one of five possibilities. That is, most of the theorems of classical mathematics turn out to be equivalent to each other in five large equivalence classes.

For example,

• Provable in and equivalent to the theory RCA₀ (and each other) are: basic properties of the natural/rational numbers, the Baire Cateogory theorem, the Intermediate Value theorem, the Banach-Steinhaus theorem, the existence of the algebraic closure of a countable field, etc. etc. etc.




• Equivalent to WKL₀ (and each other) are the Heine Borel theorem, the Brouer fixed-point theorem, the Hahn-Banach theorem, the Jordan curve theorem, the uniqueness of algebraic closures, etc. , etc. etc.




• Equivalent to ACA₀ (and each other) are the Bolzano-Weierstraus theorem, Ascoli's theorem, sequential completeness of the reals, existence of transcendental basis for countable fields, Konig's lemma, etc., etc.




• Equivalent to ATR₀ (and each other) are the comparability of countable well orderings, Ulm's theorem, Lusin's separation theorem, Determinacy for open sets, etc.




• Equivalent to Π¹₁ comprehension (and each other) are the Cantor-Bendixion theorem and the theorem that every Abelian group is the direct sum of a divisible group and a reduced group, etc.

The naturality and canonical nature of these five axiom systems is proved by the fact that they are equivalent to so many different classical theorems of mathematics. At the same time, these results prove that those theorems themselves are natural and essential in the sense of the title of your question.

The overall lesson of Reverse mathematics is the fact that there are not actually so many different theorems, in a strictly logical sense, since these theorems all turn out to be logically equivalent to each other in those five categories. In this sense, there are essentially only five theorems, and these are all essential. But their essential nature is mutable, in the sense that any of them could be replaced by any other within the same class.

I take this as a robust answer to the question that you asked (and perhaps it fulfills your remark that you thought ideally the answer would come from proof theory). The essential nature of those five classes of theorems is not proved by looking at their citation statistics in the google page-rank style, however, but by considering their logical structure and the fact that they are logically equivalent to each other over a very weak base theory.

Finally, let me say that of course, the Reverse Mathematicians have by now discovered various exceptions to the five classes, and it is now no longer fully true to say that ALL of the known reversals fit so neatly into those categories. The exceptional theorems are often very interesting cases which do not fit into the otherwise canonical categories.

modular forms - Fourier coefficients for elliptic curves on average

As others have mentioned, if $p$ is fixed then you're really looking at elliptic curves over a fixed finite field.

From some points of view an interesting variant would be to look at elliptic curves say $E_{a,b}:y^2 = x^3 + ax + b$ where $a$ and $b$ vary over integers in a box, say $|a| leq A$ and $|b| leq B$ and relatively small compared to $p$. The one might try to find asymptotic results that hold as $p$, $A$, $B$ get large together. If $A$ and $B$ aren't too big then this is giving more information about individual curves. For example, in bounding the average analytic rank of elliptic curves it is important to get a good bound on

$$frac{1}{AB} sum_{p < P} sum_{|a| leq A} sum_{|b| leq B} a_P(E_{a,b})$$ with $A$ and $B$ as small as possible. For example, see A. Brumer, The average rank of elliptic curves. I, Invent. Math. 109(3), 445–472 (1992).

In a different but related direction, there is a paper of David and Pappalardi, Average Frobenius distributions of elliptic curves (it's the fourth from the bottom) on this subject. They get a kind of Lang-Trotter on average, so they are varying both $p$ and the coefficients defining the elliptic curves. Stephan Baier later made some improvements on this problem here.

ct.category theory - Equalizer objects in Set.

Given two parallel morphisms $f,g:Xto Y$ in some category $mathcal{C}$, let us consider the category $mathcal{E}_{f,g}$ :

Objects : all pairs $(E,e)$, where $E$ is an object of $mathcal{C}$ and $e:Eto X$ is a morphism in $mathcal{C}$ such that $fcirc e=gcirc e$,

Morphisms : from $(E',e')$ to $(E,e)$ are just morphisms $varphi:E'to E$ in $mathcal{C}$ such that $e'=ecircvarphi$.

Now an equaliser of $f,g$ is just a final object in the category $mathcal{E}_{f,g}$. Final objects in any category are unique (provided they exist), up to a unique morphism; we may then talk of "the" equaliser of $f,g$.

When $mathcal{C}$ is the category of sets, the equaliser always exists (and is therefore uniquely unique); as you say, it is the largest subset of the common source where the two maps coincide.

It is fine to think of it as a "maximal" object in $mathcal{E}=mathcal{E}_{f,g}$, but one must realise that it is also a "minimal" object in the opposite category $mathcal{E}^circ$ in the sense of being an initial object therein.

soft question - Favorite popular math book

I know this is a little late for Christmas, but nevertheless, I have a few (some of which have already been mentioned) books I've read that I've quite enjoyed. For the sake of brevity, I'll let you search the titles on Amazon for reviews and better descriptions.

Title: Everything & More: A Compact History of Infinity
Author: David Foster Wallace

Title: The Mathematical Experience
Author(s): Philip J Davis & Reuben Hersh

Title: One, Two, Three...Infinity
Author: George Gamow

Title: Pi in the Sky
Author: John D. Barrow

Title: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
Author: John Derbyshire

Title: Strength in Numbers
Author: Sherman Stein

Title: e: The Story of a Number
Author: Eli Maor

Title: A History of Pi
Author: Petr Beckmann

Title: Nature's Numbers
Author: Ian Stewart

Title: Mathematics: The Science of Patterns
Author: Keith Devlin

Title: Zero: The Biography of a Dangerous Idea
Author: Charles Seife

Title: How to Enjoy Calculus
Author: Eli S. Pine
(Not really a "popular" book, per se', but still pretty good)

Title: How to Think About Weird Things
Author(s): Theodore Schick & Lewis Vaughn
(Not really about mathematics, but not so far out of the way that you wouldn't enjoy it if you also enjoy mathematics)

fa.functional analysis - Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?

The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $mathbb{R}$ or $mathbb{C}$ for this question). The continuous dual is the subspace of all continuous linear functionals. Under certain conditions, it's quite easy to find an element of the algebraic dual of a space that is not in the continuous dual:

1. We can start with a particular vector space, say $ell^0$, and put on it two inequivalent norms, say $|-|_1$ and $|-|_2$. As they are inequivalent, there will be some functional that is continuous with respect to the one but not the other, in this case the functional $(x_n) to sum x_n$ will do.




2. We can start with a normed vector space which has a topological basis which we then extend to a Hamel basis. Assuming that this extension is non-trivial, we can define a functional which is zero on the original topological basis but non-zero on some element of the extension. Unfortunately, the existence of a Hamel basis depends on some version of choice (not sure if it's equivalent to AC or not, but that's not important).




3. We can modify the previous construction to avoid Hamel bases by using the Hahn-Banach theorem.

What I would like is an example of a non-continuous functional on a Banach space that can be explicitly written down. So I don't want to use choice or HBT. If there were an example of a normed vector space with two inequivalent norms (may as well assume that $|-|_a$ is strictly weaker than $|-|_b$) so that the space is a Banach space with respect to the weaker norm then method (1) would work, but that would contradict the open mapping theorem (unless there's some subtlety that I can't see right now).

This builds on What’s an example of a space that needs the Hahn-Banach Theorem?. I'm not interested in throwing choice or HBT out of the window, but it's useful to know (from a pedagogical angle if nothing else) exactly where they are needed and how far one can go without them. One thing that surprised me a little in the answers to that question was that you need choice/HBT to see that $ell^1$ is not the continuous dual to $ell^infty$. That made me wonder about the algebraic dual and whether the same is true for that, or not.

homological algebra - Is ΩΣ in {simplicial commutative monoids} group completion?

I think an answer is given by the arguments that Segal gives in Section 4 of his paper on "Categories and Cohomology Theories" (aka, the $Gamma$-space paper), in Topology, v.13. I'll try to sketch the main idea, translated into the context of simplicial commutative monoids. I'll show that if $M$ is a discrete simplicial commutative monoid, then it's group completion is homotopically discrete; according to the comments, this should answer the question.

Given a commutative monoid $M$, we can define a simplicial commuative monoid $M'$ as the nerve of the category whose objects are $(m_1,m_2)in Mtimes M$, and where morphisms $(m_1,m_2)to (m_1',m_2')$ are $min M$ such that $m_im=m_i'$. We can prolong this to a functor on simplicial commutative monoids.

Let $H=H_*|M|=H_*(|M|,F)$ (the homology of the geometric realization of $M$, with coefficients in some field $F$), viewed as a commutative ring under the pontryagin product. Then Segal shows that $H_*|M'|approx H[pi^{-1}]$, where $pi$ denotes the image of $pi_0|M|$ in $H_0|M]$. His proof amounts to computing the homology spectral sequence for a simplicial space whose realization is $M'$, and whose $E_2$-term is $mathrm{Tor}_i^H(Hotimes H,F)$, and observing that the higher tor-groups vanish.

This means that if $M$ is discrete, then $H_*|M'|$ is concentrated in degree $0$. Since $|M'|$ is a grouplike commutative monoid, the Hurewicz theorem should tell us that $|M'|$ is weakly equivalent to a discrete space, namely the group completion of the monoid $M$.

Segal goes on to show that $BMto BM'$ is a weak equivalence, using the above homology calculation and another spectral sequence. Since $M'$ is weakly equivalent to a group, $Omega BMapprox Omega BM'approx M'$.

ct.category theory - "Category" of Nonempty Metric Spaces and Contractive Maps?

The usual way of getting a category of metric spaces is to take metric spaces as objects, and the nonexpansive maps (ie, functions $f : A to B$ such that $d_B(f(a), f(a')) leq d_A(a, a')$) as morphisms.

However, for my purposes I'd like to use the Banach fixed point theorem to get a category with a trace structure or Conway operators on it, which means I want to consider the contraction mappings on nonempty metric spaces -- that is, there should be $q < 1$ for each mapping $f$ such that $d_B(f(a), f(a')) leq q cdot d_A(a, a')$.

But nonempty metric spaces and contraction mappings don't form a category, since the identity function is not a contraction map! Is there some way of defining this kind of setup as a category? I'm happy to play games with the metrics (e.g., use ultrametrics, but bounds on them, that sort of thing), if it helps.