gt.geometric topology - Teichmuller theory and moduli of Riemann surfaces

I'll discuss things which are more applications of the mapping class group to moduli space rather than Teichmuller theory per se, but of course this is all tightly connected.

One of the big applications of this point of view is to the cohomology of moduli space. The moduli space of curves is not quite a classifying space for the mapping class group because the action of the mapping class group on Teichmuller space is not free, but the problem all comes from finite order elements. One can think of moduli space as a "rational classifying space" or an "orbifold classifying space" for the mapping class group. The upshot is that the group cohomology of the mapping class group is identical to the cohomology of moduli space with $mathbb{Q}$ coefficients.

I will try to give a brief survey of this field, but it is huge and I will omit a lot of important work.

There is now a lot known about the group cohomology of the mapping class group. The most spectacular is the resolution by Madsen-Weiss of the Mumford conjecture giving the rational cohomology ring in a stable range. This is certainly not known via algebro-geometric methods.

This was proceeded by many older results. The most germane come from a series of papers by Harer in the '80's which (among other things) do the following:

1) Show that the cohomology stabilizes as the genus increases.

2) Calculate the Euler characteristic. This really is not a theorem about the mapping class group, as the proof uses a certain triangulation of moduli space rather than group theory. However, this triangulation definitely comes from Teichmuller theory rather than algebraic geometry, and it is still part of this same circle of ideas.

3) Make a number of low-dimensional calculations (up to degree 3 in published work and 4 in unpublished work).

The calculation of $H_2$ by Harer in particular is the key to calculating the Picard group of moduli space.

These low-dimensional cohomology calculations can now be (basically) done via algebraic geometry. See the paper "Calculating cohomology groups of moduli spaces of curves via algebraic geometry" by Arbarello and Cornalba. Thus the Picard group of moduli space can now be calculated via algebraic geometry.

A more recent application of this point of view comes from work of myself which calculates the Picard groups of the moduli spaces of curves with level structures (see my paper of the same title). I think it would be very interesting to try to make this same calculation using algebro-geometric methods, but I have no idea how to do so.

ho.history overview - Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?

Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:

The theory of commutative normed rings [i.e., (complex) Banach algebras], created by Gelfand in the late 1930s, has become today one of the most active areas of functional analysis. The key idea in Gelfand's theory -- that maximal ideals are the underlying "points" of a commutative normed ring -- not only revolutionized harmonic analyis but had an enormous impact in algebraic geometry. (One need only look at the development of the concept of the spectrum of a commutative ring and the concept of scheme in the algebraic geometry of the 1960s and 1970s to see how far beyond the borders of functional analysis Gelfand's ideas penetrated.)

I was skeptical when reading this, which led to the following:

Basic Question: Did Gelfand's theory of commutative Banach algebras have an enormous impact, or any direct influence whatsoever, in algebraic geometry?

I elaborate on the question at the end, after some background and context for my skepticism.

In the late 1930s, Gelfand proved the special case of the Mazur-Gelfand Theorem that says that a Banach division algebra is $mathbb{C}$. In the commutative case this applies to quotients by maximal ideals, and Gelfand used this fact to consider elements of a (complex, unital) commutative Banach algebra as functions on the maximal ideal space. He gave the maximal ideal space the coarsest topology that makes these functions continuous, which turns out to be a compact Hausdorff topology. The resulting continuous homomorphism from a commutative Banach algebra $A$ with maximal ideal space $mathfrak{M}$ to the Banach algebra $C(mathfrak{M})$ of continuous complex-valued functions on $mathfrak{M}$ with sup norm is now often called the Gelfand transform (sometimes denoted $Gamma$, short for Гельфанд). It is very useful.

However, it is my understanding that Gelfand wasn't the first to consider elements of a ring as functions on a space of ideals. Hilbert proved that an affine variety can be considered as the set of maximal ideals of its coordinate ring, and thus gave a way to view abstract finitely generated commutative complex algebras without nilpotents as algebras of functions. On the Wikipedia page for scheme I find that Noether and Krull pushed these ideas to some extent in the 1920s and 1930s, respectively, but I don't know a source for this. Another related result is Stone's representation theorem from 1936, and a good summary of this circle of ideas can be found in Varadarajan's Euler book.

Unfortunately, knowing who did what first won't answer my question. I have not been able to find any good source indicating whether algebraic geometers were influenced by Gelfand's theory, or conversely.

Elaborated Question: Were algebraic geometers (say from roughly the 1940s to the 1970s) influenced by Gelfand's theory of commutative Banach algebras as indicated by Guillemin and Sternberg, and if so can anyone provide documentation? Conversely, was Gelfand's theory influenced by algebraic geometry (from before roughly 1938), and if so can anyone provide documentation?

rt.representation theory - Springer corresponding for nullcones other than the standard nilpotent cone

A couple of things: Springer theory does not give a bijection between orbits of the nilpotent cone and irreducible representations of the Weyl group outside of type A: in general, there is an injective map from the irreducible representations of the Weyl group to the set of equivariant irreducible local systems on nilpotent orbits (but it is not necessarily surjective).

For the action on other representations, there is a paper by Misha Grinberg called "A generalization of Springer theory using nearby cycles" which generalizes Springer theory to a class of polar representations V, which behave sufficiently like case of the adjoint representation. From this point of view, you do not need the resolutions.

In fact in a sense his results show that sometimes there cannot be a resolution: he proves that the Fourier transform of the nearby cycles sheaf is an intersection cohomology sheaf, and this is the analogue of the sheaf you get from the Grothendieck resolution in the ordinary Springer theory. However, in the case of symmetric spaces, he also computes the monodromy action on the local system determining the intersection cohomology sheaf, and shows that it does not have to be semisimple. A consequence of this is that the local system cannot arise from a finite cover, and thus does not come from some sort of resolution. A similar phenomenon was notice by Grojnowski in his thesis on character sheaves on symmetric spaces.

ag.algebraic geometry - complex singularity exponent, lct

Hi everybody,

I have a question about log canonical thresholds / complex singularity exponents.

If I understood well, this invariant sees more things than the multiplicity, for example, the cusp in dimension 1 ($x^3+y^2=0$) has $c_0 = 5/6$ and the ordinary quadratic singularity has c_0=1.

I have several questions :

1) Is it true that for the singularity $x^a+y^b+z^c=0$, the complex singulary exponent is $min(1;1/a+1/b+1/c)$ ?

2) If it's true, how to distinguish $x^3+y^2+z^2=0$ and the ordinary quadratic surface singularity ? Is there another invariant (besides the Milnor number) ?

3) If I have a variety with isolated singularities, does it make sense to try to measure its 'singularity' by adding the exponents at all the singular points ? Or taking the inf ?

4) compared to the Milnor number, what is the advantage of working with this invariant ? I thought it would be a way to say that a curve with a cusp is 'more singular' than a curve with two ordinary quadratic points (with the Milnor number they are 'equaly singular'), but I don't know if that makes sense.

Thanks in advance,

J-B B

Super-linear time complexity lower bounds for any natural problem in NP?

Sorry I am so late to the discussion, but I just registered...

There are non-linear time lower bounds on multitape Turing machines for NP-complete problems. These lower bounds follow from the fact that the class of problems solvable in nondeterministic linear time is not equal to the class of those solvable in (deterministic) linear time, in the multitape Turing machine setting. This is proved in:

Wolfgang J. Paul, Nicholas Pippenger, Endre Szemerédi, William T. Trotter: On Determinism versus Non-Determinism and Related Problems (Preliminary Version) FOCS 1983: 429-438

In fact, unraveling the proof shows that there must be some problem solvable in nondeterministic linear time that is not solvable in $o(n cdot (log^* n)^{1/4})$ time (again, on a multitape Turing machine). Note the * in the logarithm; this is just "barely" above linear. One known application of this result is that a natural NP-complete problem in automata theory cannot be solved in $o(n cdot (log^* n)^{1/4})$ time:

Etienne Grandjean: A Nontrivial Lower Bound for an NP Problem on Automata. SIAM J. Comput. 19(3): 438-451 (1990)

Unfortunately the lower bound of Paul et al. relies crucially on the geometry that arises from accessing one-dimensional tapes. We don't know how to prove a non-linear lower bound even if you allow the Turing machine to have a constant number of two-dimensional tapes. We can prove time lower bounds for NP problems on general computational models if you severely restrict the extra workspace used by the machine. (This is getting into my own work so I won't say more unless you're truly interested.)

As for the comment above me: the sorting lower bound holds only in a comparison-based model, which is extremely restricted. The claim that sorting requires Omega(n log n) time on general computational models is false. There are faster algorithms for sorting integers. See for example:

Yijie Han: Deterministic sorting in O(n log logn) time and linear space. J. Algorithms 50(1): 96-105 (2004)

nt.number theory - Reference for the expected number of prime factors of n larger than n^alpha is -log alpha

Do you mean that $log(1/alpha)$ is the expected number of prime factors in $(x^alpha,x]$ when $alphato0$? For fixed $alphain(0,1)$ what you are claiming is not true. For example,

$|{nle x:exists p|n;{rm with};sqrt{x}lt ple x}|sim xlog2 $

and

$|{nle x:plesqrt{x};{rm for all};p|n}|sim(1-log2)x$.

When $alphato0$ it is not hard to prove what you need, but I am not sure where you can find a precise reference. For example, setting $omega(n;y,z)=|{p|n:ylt ple z}|$ and following the proof of Theorem 6 in page 311 in Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" gives that

$|{nle x:|omega(n;x^alpha,x)-log(1/alpha)|ge(1+delta)log(1/alpha)}|ll xalpha^{Q(1+delta)},$

where $Q(1+delta)=int_1^{1+delta}log tdt$.

nt.number theory - Expressability of an electrical circuit with probabilistic switches

Here is a purely number theoretical question that I got to know from our electrical engineering department.

Call a number $qin mathbb{N}$, good if one can do the following:

Given a set of "probabilistic" switches, each of which is open with probability $frac{a}{q}$, $a=1,2,dots,q-1$ (you have infinitely many of each type), and two nodes $U,V$. Then for every $n,bin mathbb{N}$ such that $ble q^n-1$ one can build a simple series parallel circuit (where one can use each type of switch more than once) connecting $U$ to $V$ where the probability of $Uto V$ being open is exactly $frac{b}{q^n}$.

The question is which numbers are good? I think the conjecture is that only numbers which are multiples of $2$ or $3$ are good. $5$ for example is not good as one can not construct a circuit which is open with probability exactly $frac{7}{25}$.

P.S. A "simple series parallel" circuit is one that can be build recursively by the operation of placing a switch in series with our circuit or placing a switch in parallel with our circuit. For example the wheatstone bridge is not simple series parallel. Also if one for example, connects between $U,V$ two switches with probabilities $p_1,p_2$ (of being open) in series one gets a probability of $p_1p_2$ of the section $UV$ being open, while if we connect them in parallel we get a probability $1-(1-p_1)(1-p_2)$ of it being open.

EDIT: I will rephrase the question in simple mathematical terms, as the original question is poorly phrased.

Let $qin N$. A set $S_qsubset mathbb{Q}$ contains all numbers of the form $frac{a}{q}$, with $a=1,2,dots q-1$. It also satisfies the property $xin S_qimplies frac{ax}{q}in S_q$ and $x+frac{a-ax}{q}in S_q$ for any $a=1,2,dots q-1$.

For which $q$ does $S_q$ contain every number of the form $frac{b}{q^n}$ (where $b < q^n$)?

matrices - Linear Algebra Problems?

Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I should be able to solve, but don't see any obvious approach to get started.

Here's an example of the type of problem I am thinking of: Let $A, B$ be $ntimes n$ matrices, set $C = AB-BA$, prove that if $AC=CA$ then $C$ is nilpotent. (I saw this one posed on the KGS Go Server)

Ideally, such a reference would also contain challenging problems (and techniques to solve them) about orthogonal matrices, unitary matrices, positive definiteness... hopefully, all harder than the one I wrote above.

nt.number theory - Two questions about units in Number Fields

Dirichlet's unit theorem is a relatively straightforward extension of the well known proof that the Pell equation has a nontrivial solution by a clever use of Dirichlet's box principle. The "machinery" is only needed to control the combinatorial explosion, or,
as far as geometry of numbers is concerned, as a natural generalization of the box principle.

As for your second question I first thought that the answer is no. Take for example a biquadratic number field $K = {mathbb Q}(sqrt{m},sqrt{n},)$, with $m,n > 0$ chosen in such a way that the unit index (units of K : units from the subfields) is 1. This implies that the only units in K are multiples of those coming from the three subfields. But you still get generators of the field by taking the product of two units coming from different subfields, so your question is still open.

ag.algebraic geometry - A versal deformation of a simple node

I have a passing familiarity with moduli theory, which gets me in trouble when I want to understand specific examples.

The basic question I would like to understand is how to prove something is a versal deformation of the simple node. To be specific, let
$$ X_0 := Spec(mathbb{C}[x,y]/xy) $$
and let
$$ X_{ver} := Spec(mathbb{C}[x,y,t]/xy-t) $$
Note that there is a natural map $X_{ver}rightarrow mathbb{A}^1$ which sends a point to its $t$-coordinate, whose fiber over zero is $X_0$.

I want to claim that $X_{ver}$ is a versal deformation of $X_0$. What I mean by this is as follows. Consider a faithfully flat morphism $pi:X rightarrow B$, together with a distinguished point $pin B$ such that $pi^{-1}(p)$ is isomorphic to $X_0$. Then there is

• an open neighborhood $B'subset B$ of $p$,


• a (non-unique) map $f_0:B'rightarrow mathbb{A}^1$ which takes $p$ to $0$,


• and a (non-unique) map $f_1:pi^{-1}(B')rightarrow X_{ver}$

such that the natural diagram
$$pi^{-1}(B') rightarrow X_{ver} $$
$$downarrow ;;;;;;;;;;;;;;;;;;;;;;;;; downarrow $$
$$ B' ;;;;; rightarrow ;;;;; B $$
is commutative, and is a pullback diagram (ie, a fibered product). Conceptually, this says that any (sufficiently small) faithfully flat family which contains a copy of $X_0$ must be gotten from pullback from the family $X_{ver}rightarrow mathbb{A}^1$. (Note that when I refer to preimages and fibers, I mean the scheme-theoretic ones)

However, I am having trouble showing this. I have been approaching this from the algebraic perspective, by considering formal deformations of the algebra $mathbb{C}[x,y]/xy$. These aren't hard to understand, and it is straight-forward to construct maps from $mathbb{C}[x,y,t]/xy-t$ to a given formal deformation. However, I can't figure out how to make these maps surjective; ie, to keep the corresponding scheme maps from being multi-sheeted covering maps. In any event, I suspect this algebraic approach is wrong anyway, since it will, at best, only prove its a formal versal deformation.

I should also mention I am not sure if `faithfully flat' is the idea I want here. I am bad at knowing what modifiers to a flat family prevent pathological fibers, so if this is the wrong notion, please correct me.

Reference for Unitary Group attached to $E/k$

For discussions on forms of classical groups you can look at:

-- André Weil, Algebras with involutions and the classical groups, J. Indian Math. Soc. 24 (1961), 589-623 (also in Oeuvres Complètes).

-- Platonov-Rapinchuk, Algebraic groups and number theory, Ac. Press, 1994.

-- The book of involutions, AMS Coll. Publ., vol. 44, 1998.

-- Kneser, Lecture on Galois cohomology of classical groups, Tata Inst. of Fund. Research, Bombay, 1969.

I've written a summary of Weil's theorems and proofs (following Platonov-Rapinchuk). It is in French ... and only deals with local base fields.

lo.logic - Generalized Cox Theorems, valuations on boolean sets, bayesian probabilities and posets

Bayesian probabilities are usually justified by the Cox theorems, that can be written this way:

Under some technical assumptions (continuity, etc, etc...), given a set $P$ of objects $A, B, C, ldots$, with a boolean algebra defined over it with operations $A wedge B$ (and) and $A | B$ (or) such that:

1) $A wedge B = B wedge A$

2) $A wedge (B wedge C) = (A wedge B) wedge C$

3) $A | (B wedge C) = (A|B) wedge (A|C)$

and a "valuation":

$f : P rightarrow mathcal{R}$

there is a strictly monotonic "regraduation function" $R : mathcal{R} rightarrow mathcal{R}$ such that, for:

$R(f(Awedge B)) = R(f(A)) + R(f(B))$ (sum rule)

and

$R(f(A|B)) = R(f(A) ) R(f(B))$ (product rule)

This theorem allows one to show that any system designed to "evaluate" boolean expressions consistently with a single real number redunds in the laws of classical probability (this can be seen shortly here: arxiv:physics/0403089 and more thoroughly here: arxiv:abs/0808.0012)

Recently this has been extended for valuations of the type $f : P rightarrow mathcal{R}^2$ in http://arxiv.org/abs/0907.0909 and they proved that there are just 5 canonical valuations compatible with the underlying Boolean algebra (one of them giving a complex field structure to the "valuation" field).

My question/proposal is: is it possible/interesting/feasible to classify at least a class of valuations of the type:

$f : P rightarrow W$

where W is a continuous manifold? If we retrict our attention to $mathcal{R}^n$ for example, is there, for each n, a set of canonical valuations to which all others can be reduced after a regraduation?

If this can be done, are those nice rules for inference in some sense? Are they useful as inference tools in specific situations?

dg.differential geometry - Important results that use infinite-dimensional manifolds?

As a counterpoint to some of the other answers, actually Banach (and Hilbert) manifolds aren't quite as useful as might be hoped. Two indicators of this are:

1. All Hilbert manifolds are diffeomorphic to an open subset of the model space. (I think this also holds for Banach manifolds but am not sufficiently sure.) In particular, any Hilbert manifold is parallelisable so the tangent space contains no information.




2. If a Banach Lie group acts faithfully on a finite dimensional manifold then the group is finite dimensional. So there is no Banach model of diffeomorphisms (as a Lie group).

So Banach and Hilbert manifolds tend to be used as places to put other things. They are big enough to contain just about everything but simple enough that they don't add any extra complications. This leads to their use as extreme examples of "very big spaces" and means that for any particular situation one can often chop the infinite down to the finite (but very big). So, for example, with regard to representing K-theory, for any particular finite dimensional manifold there's a finite dimensional Grassmannian that will do but if you want to represent K-theory for all finite dimensional manifolds then you need an infinite dimensional Grassmannian.

However, the situation changes once you allow other model spaces (the largest category of such is the category of convenient vector spaces, the introduction of which is a very interesting read on calculus in infinite dimensions). There you can and do get much more interesting behaviour and you discover that you can study infinite dimensional manifolds as objects in their own right.

One example of this is the notion of semi-infinite structure. This is, almost by definition, only available in infinite dimensions (there are shadows in finite) and has proved an important source of ideas, if not actual techniques.

Of course, many of the techniques involve bringing things back down to finite dimensions in the final analysis but that's because we want to actually compute something and so end up with a number; and the easiest way to get a number is to count a finite number of things. But that's no different to any other computation, so shouldn't be seen as a disadvantage.

So back to the original question. Well, I don't really have a good answer to that because I work with infinite dimensional manifolds so I don't spend any time worrying about what others want to apply this work to, I just get on with it. But nonetheless, one theme that I see a lot is that of the infinite dimensional "picture" being the right one and the one that gives the intuition for how the finite dimensional approximations fit together.

Thus we see that Floer theory is really Morse theory applied to loop spaces, but not many will compute it as such. We see that the elliptic genus is (was originally!) really index theory applied to loop spaces, but again that's not useful for computations!

In general, anywhere where you've got functions that can vary, you've got an infinite dimensional manifold and it behoves you to remember it because it can give you important insights on how to proceed.

gn.general topology - Morse theory and Euler characteristics

You could have a smooth function $f : Bbb R to Bbb R$ whose critical point set is a Cantor set (minima) and the centres of the complementary intervals (local maxima) -- let $f$ be some suitable smoothing of the distance function from the Cantor set (or you could use the smooth Urysohn lemma to construct the function), say. The two sets (local max / local min) don't have the same cardinalities so you've got no hope of a formula.

I suppose the most direct case where it should fail for the critical point set a manifold, is when the Hessian is non-degenerate in the normal directions but having non-constant signature over an individual critical component. Off the top of my head I don't have an example but the proof (Bott's proof) fundamentally breaks down in this situation so this is where I would look first.

reference request - Space of derivations of holomorphic (analytic) functions

Let $M$ be a (real) smooth manifold, and $p in M$.

The space of (linear) derivations $D:C^{infty}(p) to mathbb{R}$ (i.e., maps satisfying $D(f+g) = D(f)+ D(g)$ and $D(fg)=D(f)g(p) + f(p)D(g)$) on the algebra $C^{infty}(p)$ of differentiable functions defined on some neighbourhood of $p$) is then a $n$-dimensional vector space (this is one way to define the tangent space $T_p M$ after all).

It is easy to see that if we consider instead derivations $D:C(p) to mathbb{R}$ on the space $C(p)$ of continuous functions, then the space of derivations is trivial.

My question is: when $M$ is a complex (or analytic) manifold, what is the dimension of the space of derivations on holomorphic (or analytic) functions defined near p?

I've once heard that this space is infinite dimensional. Is this true? (and if it is, is there a simple proof or some reference material?)

pr.probability - Methods for choosing a result from a multiple output node Neural Network

Regarding your intuition, it may or may not be true. In many cases people do the opposite, i.e. downweight classes with low membership. This happens any time someone uses a prior p(c) on the class membership.

If you still think you need to boost the probability of small classes, then using different weights for different classes may help. Define the weight of class $i$ to be $W_i sim 1/p(c_i)$ ($p(c_i)$ is the prior probability of class $i$). Then pick $i$ that minimizes something like $W_i cdot p(c_i|MLP)$. Designing the weights has to be done empirically, I think.

For Tom:
MLP is multi-layer perceptron, a type of neural network.

The problem can be reformulated as follows. Suppose you have some data (like an image of a glyph). It can belong to one of several classes (say, an English character, 'a' to 'z', so 26 classes in this case). You magically get a set of probabilities $p(c_i | data)$ (the probability that your data is in class $c_i$). In our case, this is given by the MLP. You need to decide on a single class. The most obvious rule is to pick $c_i$ for which $p(c_i | data)$ is maximal. But OP feels that if one class has very few members, it should be given advantage. Another option is to have weights and pick the class that maximizes $W_i p(c_i | data)$. The disadvantage of this is that we don't have a principled method of choosing the weights $W_i$.

One possibility is to use $W_i sim 1/p(c_i)$. In this case, $p(c_i | data) / p(c_i) simeq p(data | c_i)$, which seems somewhat principled (not completely arbitrary). But maybe there are better ways of doing that.

real analysis - Dependence of error on mesh for Riemann sums

If $f(x) = x$ on $[0,1]$, then a mesh less than $delta$ means the Riemann sum differs from the integral by about $delta/2$, so $delta(epsilon)$ is about $2epsilon$. Thus, I presume that you mean $epsilon$ instead of $epsilon^2$.

A function with bounded variation satisfies a linear upper bound for the error in terms of the mesh size and total variation, hence a linear lower bound for the required $delta(epsilon)$.

A function without bounded variation may require a smaller mesh. For example, consider the continuous extension of
$f(x) = x cos(pi/x^n)$ to $[0,1]$. The difference in $f$ between sampling $f$ at $x=(2k+1)^{-1/n}$ and at
$x=(2k)^{-1/n}$ is about $2x$ at locations differing by about $1/n~x^{n+1}$. That means with mesh $delta$, you can sample $f$ to be either $1$ or $-1$ up to about $x=c delta^{1/{(n+1)}}$, for an error of $cdelta^{2/(n+1)}$.

Therefore, $delta(epsilon)$ can't be larger than $c epsilon^{(n+1)/2}$.

lo.logic - Non-constructive proofs of decidability?

When I teach computability, I usually use the following example to illustrate the point.

Let $f(n)=1$, if there are $n$ consecutive $1$s somewhere in the decimal expansion of $pi$, and $f(n)=0$ otherwise. Is this a computable function?

Some students might try naively to compute it like this: on input $n$, start to enumerate the digits of $pi$, and look for $n$ consecutive $1$s. If found, then output $1$. But then they realize: what if on a particular input, you have searched for 10 years, and still not found the instance? You don't seem justified in outputting $0$ quite yet, since perhaps you might find the consecutive $1$s by searching a bit more.

Nevertheless, we can prove that the function is computable as follows. Either there are arbitrarily long strings of $1$ in $pi$ or there is a longest string of $1$s of some length $N$. In the former case, the function $f$ is the constant $1$ function, which is definitely computable. In the latter case, $f$ is the function with value $1$ for all input $nlt N$ and value $0$ for $ngeq N$, which for any fixed $N$ is also a computable function.

So we have proved that $f$ is computable in effect by providing an infinite list of programs and proving that one of them computes $f$, but we don't know which one exactly. Indeed, I believe it is an open question of number theory which case is the right one. In this sense, this example has a resemblence to Gerhard's examples.

Making sense of the Fourier transform of the product of two functions

The Fourier transform of the product of two functions f(x) and g(x) is
given as:

$mathcal{F}[ f(x)g(x)] = int_{-infty}^{+infty} F(omega^prime) G(omega - omega^prime) domega^prime ; = ; mbox{convolution of} ; ; F(omega^prime )G(omega^prime)$

where $F(omega^prime)$ and $G(omega^prime)$ are the Fourier transforms of $f(x)$ and $g(x)$ respectively.

Although I understand the derivation of this formula, I've got difficulty making sense of two frequency terms $omega$ and $omega^prime$. I'm fine with $omega^prime$ but I don't know what to make of $omega$. Should I treat it as a constant, or should I set it to zero?

I'm really interested in the Fourier transform of the square of the second derivative of a function e.g. $mathcal{F}[ f^{primeprime}(x)^2 ]$. Because this problem does not involve a shift, I don't know what to make of the shift term $omega$.

ag.algebraic geometry - On the Clifford index of a curve

when c= 0 Clifford's them includes the fact that any divisor with Clifford index 0 is a multiple of the hyperelliptic fiber, ie: a sum of fibers of the hyperelliptic map. If c=1 then the curve is either trigonal or a plane quintic- I believe that it is an exercise in A-C-G-H. Kind of a folk lore result. I have not heard of anyone explicating all the possible cases when c=2.

algorithms - On Clenshaw's summation formula

When one has a finite sum of the form

$S=sum_{k=0}^{n}{c_k F_k(x)}$

where $F_k(x)$ satisfies a two-term recurrence relation

$F_{k+1}(x)=alpha_k F_k(x)+beta_k F_{k-1}(x)$

the standard algorithm used for computing $S$ is Clenshaw summation:

$y_{n+2}=y_{n+1}=0$

$y_k=alpha_k y_{k+1}+beta_{k+1}y_{k+2}+c_k; ; ;k=n;mathrm{to};1;mathrm{step};-1$

$S=y_1F_1(x)+(beta_1 y_2+c_0)F_0(x)$

I am also very familiar with the concept of "minimal" and "dominant" solutions of a two-term recurrence; to review, the dominant solution is the solution most stably propagated when the two-term recurrence is run in the forward direction, while the minimal solution is the solution that is stably computed when the recurrence is run backwards. As an example, for the recurrence relation of the Bessel functions, $J_n(x)$, the function of the first kind, is the minimal solution while $Y_n(x)$, the other solution to the Bessel ODE, is the dominant solution to the two-term recurrence.

Mystifying to me, however, is this remark given in the book "Numerical Recipes" without justification of any sort:

"...Clenshaw's recurrence is always stable, independent of whether the recurrence for the functions... is stable in the upward or downward direction."

My question now, is why would the stability properties of $F_k$ (i.e. its being minimal or dominant) not spoil the recurrence, if this is true? If not, I would be interested in seeing a counterexample.

What would a graduate course on systolic geometry typically cover?

This is interesting. I imagine that any course would vary quite a bit depending on who taught it.

Any course should probably contain Gromov's proof of the systolic inequality for essential manifolds. Other than that, I am not sure. The course could dive into systoles on surfaces and some of the arithmetic constructions in Teichmuller theory, or it could develop harmonic maps and scalar curvature rigidity theorems, or it could take a dynamical systems approach and discuss the relationship between volume entropy and closed geodesics.

I have no idea. You should come up with a curriculum and post it here.

nonasymptotic complexity results

The general fact surrounding some of the other answers is
the following:

Every computably axiomatizable consistent theory $T$
containing trivial arithmetic admits very short theorems
requiring extremely long proofs.

The basic fact is that there can be no total computable
bound on the length of the proof required. To see this,
suppose that there is a computable total function $f$ such
that whenever statement $psi$ is a statement of size at
most $n$ provable from $T$, then there is a proof of size
at most $f(n)$. In this case, the question of whether $T$
proves $psi$ will be decidable, since we can simply
inspect all proofs of length $f(n)$, where $n=|psi|$ and
check if any of them are proofs of $psi$. But it is
impossible that $T$ proves $psi$ is decidable, since we
could then produce a consistent completion of $T$, by the
usual process of completing a theory, which is effective
for decidable theories. This would produce a computable
complete consistent theory of arithmetic, in contradiction
to the incompleteness theorem.

One can make the conclusion fairly concrete via the halting
problem. If $T$ is true and contains some trivial
arithmetic, then $T$ will prove all true instances of the
assertion program $p$ halts on input $m$, and prove no
false instances. If there were a total computable function
$f(p,m)$ such that whenever $p$ halted on input $m$, then
there was a proof of this of length at most $f(p,m)$, then
we could decide the halting problem: on input $(p,m)$,
compute $f(p,m)$ and then look at all proofs of that length
and determine if there is a proof of halting or not.

In conclusion:

Theorem. For any computably axiomatizable true theory $T$ and for any total computable
function $f$, there is a program $p$ and input $m$ such
that $T$ proves that $p$ halts on input $m$, but there is
no proof of this in fewer than $f(p,m)$ steps.

Since as we all know, the computable functions can grow
quite outrageously, this means that for any theory there
will be short theorems that require extremely long proofs.

nt.number theory - What does Faltings' theorem look like over function fields?

The "function field analogues" of Faltings' theorem were proved by Manin, Grauert and Samuel: see

29/PMIHES_1966_29_55_0/PMIHES_1966_29_55_0.pdf">http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1966_29/PMIHES_1966_29_55_0/PMIHES_1966_29_55_0.pdf

especially Theorem 4. (The quotation marks above are because all of this function field work came first: the above link is to Samuel's 1966 paper, whereas Faltings' theorem was proved circa 1982.)

The statement is the same as the Mordell Conjecture, except that there is an extra hypothesis on "nonisotriviality", i.e., one does not want the curve have constant moduli. For some discussion on why this hypothesis is necessary, see e.g. p. 7 of

http://math.uga.edu/~pete/hassebjornv2.pdf

An effective height bound in the function field case is given in Corollaire 2, Section 8 of

---

Szpiro, L.(F-PARIS6-G)
Discriminant et conducteur des courbes elliptiques. (French) [Discriminant and conductor of elliptic curves]
Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988).
Astérisque No. 183 (1990), 7--18.

---

Note that effectivity on the height is much better than effectivity on the number of rational points (Faltings' proof does give the latter). This is not to be confused with uniform bounds on the number of rational points, for which I believe there are only conditional results known in any case.

ag.algebraic geometry - Is the fixed locus of a group action always a scheme?

The question gives the "wrong" definition of Fix(T), hence the resulting confusion.

A more natural definition of the subfunctor X^G of "G-fixed points in X" is
(X^G)(T) = {x in X(T) | G_T-action on X_T fixes x}
               = {x in X(T) | G(T')-action on X(T') fixes x for all T-schemes T'}.
(Of course, can just as well restriction to affine T and T' for "practical" purposes.)

By way of analogy with more classical situations, if the base is a field k then a moment's reflection with the case of finite k shows that
{x in X(k) | G(k) fixes x}
is the "wrong" notion of (X^G)(k), whereas
{x in X(k) | G-action on X fixes x}
is a "better" notion, and is what the above definition of (X^G)(k) says.

From this point of view, if (for simplicity of notation) the base scheme is an affine Spec(k) for a commutative ring k then the "scheme of G-fixed points" exists whenever G is affine and X is separated provided that k[G] is k-free (or becomes so after faithfully flat extension on k). So this works when k is a field, or any k if G is a k-torus (or "of multiplicative type"). See Proposition A.8.10(1) in the book Pseudo-reductive groups.

knot theory - Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus with 24 triangles, respectively. The 4x5 chessboard complex is only a pseudomanifold — each vertex has the 3x4 torus as its link, rather than a spherical link that a proper manifold would have — but if you delete its 20 vertices you get a bona-fide cusped hyperbolic 3-manifold, triangulated by 120 regular ideal tetrahedra. It sort of looks like the kind of manifold you might get as the complement of a 20-component link in Euclidean space. Is it a link complement? And if so, which link is it the complement of?

Edit: here are a couple of general references on chessboard complexes.

Ziegler, G. M. (1994). Shellability of chessboard complexes. Israel J. Math. 87: 97–110.

Björner, A.; Lovász, L.; Vrecica, S. T.; Zivaljevic, R. T. (1994). Chessboard complexes and matching complexes. J. London Math. Soc. 49: 25–39.

They also have important applications to the proof of colored Tverberg theorems in discrete geometry: see, e.g.

Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler (2009). Optimal bounds for the colored Tverberg problem. arXiv:0910.4987.

nt.number theory - Distinguishing congruence subgroups of the modular group

This question is something of a follow-up to
Transformation formulae for classical theta functions .

How does one recognise whether a subgroup of the modular group
$Gamma=mathrm{SL}_2(mathbb{Z})$ is a congruence subgroup?

Now that's too broad a question for me to expect a simple answer
so here's a more specific question. The subgroup $Gamma_1(4)$
of the modular group is free of rank $2$ and freely generated by
$A=left(
begin{array}{cc}
1&1\
0&1
end{array}right)$
and
$B=left(
begin{array}{cc}
1&0\
4&1
end{array}right)$. If $zeta$ and $eta$ are roots of unity there is a
homomorphism $phi$ from $Gamma_1(4)$ to the unit circle group
sending $A$ and $B$ to $zeta$ and $eta$ resepectively. Then the kernel $K$
of $phi$ has finite index in $Gamma_1(4)$. How do we determine whether $K$
is a congruence subgroup, and if so what its level is?

In this example, the answer is yes when $zeta^4=eta^4=1$. There are
also examples involving cube roots of unity, and involving eighth
roots of unity where the answer is yes. I am interested in this example
since one can construct a "modular function" $f$, homolomorphic on
the upper half-plane and meromorphic at cusps such that $f(Az)=phi(A)f(z)$
for all $AinGamma_1(4)$. One can take $f=theta_2^atheta_3^btheta_4^c$
for appropriate rationals $a$, $b$ and $c$.

Finally, a vaguer general question. Given a subgroup $H$ of
$Gamma$ specified as the kernel of a homomorphism from $Gamma$ or
$Gamma_1(4)$ (or something similar) to a reasonably tractable target group,
how does one determine whether $H$ is a congruence subgroup?

complex geometry - Holomorphic vector fields acting on Dolbeault cohomology

The question.

Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) decomposition and also that it commutes with $bar{partial}$. From this it follows that $u$ acts infinitesimally on the Dolbeault cohomology groups $H^{p,q}(X)$ of $X$. My question is, does anyone know of an example in which this action is non-trivial?

Some context.

To give some context, first note that the analgous action for de Rham cohomology is always trivial: If $M$ is any smooth manifold and $v$ any vector field, then the formula $L_v = d circ i_v + i_v circ d$ shows that the infinitesimal action of $v$ on de Rham cohmology is trivial. (This is an instance of the more general fact that homotopic maps induce the same homomorphisms on singular cohomology. The field $v$ generates diffeomorphisms which are by construction isotopic to the identity map.)

Returning to Dolbeault cohomology, suppose we know that each Dolbeault class is represented by a $d$-closed form. (For example, this is true if $X$ is a compact Kähler manifold, by Hodge theory.) Then the action is necessarily trivial. The proof is as follows. Let $alpha$ be a $bar{partial}$-closed (p,q)-form which is also $d$-closed. Then we know that $L_u alpha = d(i_u alpha)$ is also of type (p,q). So,
$$
L_ualpha = bar{partial}left((i_ualpha)^{p, q-1}right) + partialleft((i_u alpha)^{p-1, q}right)
$$
and the other contributions $bar{partial}((i_ualpha)^{p-1,q}$) and $partial((i_ualpha)^{p,q-1})$ vanish. Now the fact that $barpartial((i_ualpha)^{p-1,q}) = 0$ and our hypothesis imply that there is a (p-1, q-1)-form $beta$ such that
$$
(i_ualpha)^{p-1,q}+ barpartial beta
$$
is closed. Hence
$$
partial left((i_ualpha)^{p-1,q}right) = barpartial partial beta
$$
and so
$$
L_ualpha = bar partial left( (i_u alpha)^{p,q-1} + partial betaright)
$$
which proves the action of $u$ on $H^{p,q}(X)$ is trivial.

nt.number theory - lower bound for torsion of abelian varieties

Now let me address your last question. For the sake of simplicity, let us assume that $K$ is a global field of characteristic $p>2$ and the ring $End(A)$ of all endomorphisms of $A$ (over an algebraic closure of $K$) is the ring $Z$ of integers. Then my old results (Math. Notes: 21 (1978), 415--419 and 22 (1978), 493--498) imply that for all but finitely many primes $ell$ the Galois module $A[ell]$ is absolutely simple and the Galois group $Gal(K(A[ell])/K)$ is noncommutative.

I claim that for all but finitely many primes $ell$ the order of
$Gal(K(A[ell])/K)$ is divisible by $ell$ and therefore $[K(A[ell]):K] > ell$.

Indeed, suppose that for a given $ell$ the order of
$Gal(K(A[ell])/K)$ is not divisible by $ell$. Let us call such $ell$ exceptional. Then the natural representation of $Gal(K(A[ell])/K)$ in $A[ell]$ can be lifted to characteristic zero, i.e.,
$Gal(K(A[ell])/K)$ is isomorphic to a (finite) subgroup of $GL(2dim(A),C)$ where $C$ is the field of complex numbers. Now, by a theorem of Jordan, there exists a positive integer $N$ that depends only on $dim(A)$ anf such that $Gal(K(A[ell])/K)$ contains a normal commutative subgroup, whose index does not exceed $N$. This means that $K(A[ell])/K$ contains a Galois subextension $K_{0,ell}/K$ such that $[K_{0,ell}:K] le N$ and the field extension $K(A[ell])/K_{0,ell}$ is abelian.

Let $S$ be the (finite) set of places of bad reduction for $A$. The field extension $K(A[ell])/K$ is unramified outside $S$; the same is true for the subextension $K_{0,ell}/K$. (In the number field case one should add the divisors of $ell$ but we live in characteristic $p$!). Recall that the Galois extensions of global $K$ with degree $le N$ and ramification only at $S$ constitute a finite set. Let $L/K$ be the compositum of all such extensions. Clearly, $L$ is also a global field while $L/K$ is a finite Galois extension that contains $K_{0,ell}$. In addition, the Galois group $Gal(L(A[ell])/L)$ is commutative. Now considering $A$ as an abelian variety over $L$ and applying previously mentioned results, we obtain that the set of exceptional primes $ell$ is finite.

qa.quantum algebra - Is there a good differential calculus for quantum SU(3)?

For quantum $SU(2)$, Woronowicz gave a well differential calculus. If we denote the generators of quantum $SU(2)$ by $a,b,c,d$, then the ideal of ker($epsilon)$ corresponding to this calculus is
$$
< a+ q^2d - (1+q^2),b^2,c^2,bc,(a-1)b,(d-1)c>.
$$
This calculus can be shown to generalise the classical calculus on $SU(2)$ when $q=1$. Does anyone know of a (good) calculus (and its ideal) for quantum $SU(3)$?

dg.differential geometry - Prescribing a Riemannian metric along a given geodesic

Let $gamma : [0,1] to mathbb R^2$ be a finite $C^2$-curve in the plane which does not intersect itself. Let $p(z)$ be a second-degree polynomial in $z in mathbb R^2$. Can we construct a Riemannian metric $g$ along $gamma$ such that

• $gamma$ is a geodesic of $g$,



• $gamma$ has length 1, and



• $p(z)$ is the 2-jet of $g$ at $gamma(0)$? (i.e. this prescribes $g$ and its first and second derivatives at $gamma(0)$) Edit: As per Sergei's comment, assume that $p$ is chosen so that $gamma$ does in fact solve the geodesic equation.

I think so, and my sketch of an argument follows the proof of existence of Fermi coordinates in reverse. I haven't worked through this in detail yet, though, because I'm more concerned about the next question:

Let $gamma$ and $eta$ both be finite curves $[0,1] to mathbb R^2$ which do not intersect themselves, and such that $gamma(0) = eta(0)$ and $gamma(1) = eta(1)$ with no other intersections (i.e. $gamma cup eta$ is a piecewise, simple, closed $C^2$-curve in the plane). Can we construct a metric $g$ as above? Note that if so, $gamma(0)$ and $gamma(1)$ will be conjugate points along $gamma$.

ag.algebraic geometry - For a line bundle L on a smooth projective variety X, what is meant by Pic^L(X)

Let me address the last part of your question.

Let $X$ be a smooth, projective variety over an arbitrary ground field $k$.

I want to write $Pic^{[L]}(X)$ instead of $Pic^L(X)$ -- i.e., to make explicit that the variety depends only on the Neron-Severi class of $L$ -- for reasons which will become clear shortly.

Suppose first that $L$ is algebraically equivalent to $0$. Then $Pic^{[L]}(X) = Pic^0(X)$, so certainly it is an abelian variety.

Next suppose that $L$ is a $k$-rational line bundle on $X$. Then $Pic^{[L]}(X)$ is not literally an abelian variety, because it is a nonidentity coset of a group rather than a group itself. However, it is canonically isomorphic to the abelian variety $Pic^0(X)$ just by mapping a line bundle $M$ to $M - L$. So it might as well be an abelian variety, really.

Finally, supose that $L$ is not itself $k$-rational but that its Neron-Severi class $L$ is rational -- i.e., $L$ is given by a line bundle over the algebraic closure which is algebraically equivalent to each of its Galois conjugates. Then $Pic^{[L]}(X)$ is a well-defined principal homogenous space of the Picard variety $Pic^0(X)$ but need not have any $k$-rational points. For instance, suppose that $X$ is a curve. Then the Galois action on the Neron-Severi group is trivial, so taking $L/overline{k}$ to be any degree $n$ line bundle, we get $Pic^{[L]}(X) = Pic^n(X) = Alb^n(X)$, a torsor whose $k$-rational points parameterize $k$-rational divisor classes of degree $n$. (Note that here when I write $Pic^0(X)$ I am talking about the Picard variety rather than the degree $0$ part of the Picard group. More careful notation would be $underline{operatorname{Pic}}^0(X)$.)

In particular, if $X$ is a genus one curve, then there is a canonical isomorphism $X cong Pic^1(X)$, so $Pic^1(X)$ can be endowed with the structure of an abelian variety iff $X$ has a $k$-rational point.

Some further material along these lines can be found in Section 4 of

http://math.uga.edu/~pete/wc2.pdf

big list - Counterexamples in Algebra?

Harry Hutchins "Examples of commutative rings" may be of interest.
It is based on his 1978 Chicago Ph.D. thesis under
Kaplansky, and not surprisingly it serves as a useful complement to
Kaplansky's excellent textbook Commutative Rings (most references
to proofs refer to Kaplansky). There is also a 3 page list of
errata, updates,... dated July 1983, which is distributed with the book.

Hutchins, Harry C. 83a:13001 13-02
Examples of commutative rings. (English)
Polygonal Publ. House, Washington, N. J., 1981. vii+167 pp. $13.75. ISBN 0-936428-05-8

The book is divided into two parts: a brief sketch of commutative ring theory
which includes pertinent definitions along with main results without proof
(but with ample references), and Part II, the 180 examples. The examples do
cover a very large range of topics. Although most of them appear elsewhere,
they are enhanced by a fairly complete listing of their properties. Example
67, for instance, is M. Hochster's counterexample to the polynomial
cancellation problem, and it lists a number of properties of the two rings
that were not given in the original paper Proc. Amer. Math. Soc. 34 (1972),
no. 1, 81 - 82; MR 45 #3394. Some of the examples appear more than once,
since many rings exhibit more than one interesting property. (R=Kx, y, z is
used in Examples 6 and 22.) The examples are grouped into areas, but a
drawback is that these have not been labeled and separated off. In addition,
the Index is for Part I and definitions only, and this means that searching
for a specific example with certain properties can be time consuming. The book
can be used as a supplement to one of the standard texts in commutative ring
theory, and it does appear to complement the monograph by I. Kaplansky
Commutative rings, Allyn and Bacon, Boston, Mass., 1970; MR 40 #7234;
second edition, Univ. Chicago Press, Chicago, Ill., 1974; MR 49 #10674.
--Reviewed by Jon L. Johnson

gt.geometric topology - Triangulating surfaces

[Three years later …]

All the published proofs of triangulability of surfaces that I am aware of use the Schoenflies theorem, which is not exactly an easy thing to prove. There is however another line of proof which avoids the Schoenflies theorem and instead uses the Kirby torus trick that underlies Kirby-Siebenmann theory in higher dimensions. There is a 1974 paper by A.J.S.Hamilton that gives much simpler proofs of Moise's theorems on triangulability of 3-manifolds using the torus trick, and the same ideas can be applied even more simply for surfaces. Instead of the Schoenflies theorem one needs a few results about surfaces strictly in the PL (or smooth if one prefers) category. Namely, one needs to know that PL structures are unique up to PL homeomorphism in the following four cases: $S^1times S^1$, $S^1times{mathbb R}$, $[0,1]times{mathbb R}$, and $D^2$. These can be regarded as special cases of the usual classification theorem for compact PL surfaces, extended to include a few noncompact cases.

I haven't seen this proof in the literature, so I've written it up and posted it on the arXiv here, working in the smooth category rather than the PL category.

It's not clear how suitable this proof would be for an undergraduate course. Besides the ingredients mentioned above, a little basic covering space theory is also needed. If one were in the fortunate position of already having covered these things, then this proof might be accessible to undergraduates. On the other hand, it could be of some interest to go through a proof of the often-quoted-but-seldom-proved Schoenflies theorem. (In this connection I might mention a paper by Larry Siebenmann on the Schoenflies theorem in the Russian Math Surveys in 2005, giving history as well as a proof.)

gn.general topology - covers of $Z^infty$

The answer is NO even if we replace $4$ by $3$.
Let me sketch a proof. This is based upon the following lemma.

Lemma. Fix $S>0$ and for an integer $k$ conisder in $mathbb Z^k$ sets $X$ of diameter at most $S$. Denote by $Vol(X)$ the number of points in $X$ and denote by $X1$ the set of points of distance at most $1$ from $X$. Now let $delta(S,k)$ be the supremum over all $X$ of diameter at most $S$ of the ratio:

$$r(S,k)=sup_{Xsubset mathbb Z^k}frac{Vol(X)}{Vol(X1)}.$$

I claim that for a fixed $S$, $lim_{kto infty}r(S,k)=0$.

Let us skip the proof of the lemma and instead deduce the claim. Suppose by contradiction that the answer is positive. Then for every $k$ we will get a solution to the problem in $mathbb Z^k$ with the fixed number of sets ($U^0,...,U^n$) such that each $U^i_j$ is of the diameter at most $S$. Now, chose such $k$ that $r(S,k)<frac{1}{2n}$ and let us deduce the contradiction.

From Lemma it follows that the supremum of asimptotic density of each set $U^i$ in $mathbb Z^k$ is less than $frac{1}{n+1}$. Indeed, since the distance between different components of $U^i$ is $4$, every point of $U^i1$ that does not belong to $U^i$ is on distance one from at most one component of $U^i$. And lemma gives us the inequality (that should be understood as assymptotic in $mathbb Z^k$)
$$Vol(U^i)<frac{1}{2n} Vol(U^i1)le frac{1}{2n}Vol(mathbb Z^k)$$
Hence $mathbb Z^k$ can not be covered by $U^0,...,U^n$.

It is clear where this proof breakes if we conisder $2$-disjoint sets. In this case one point of $U^i1$ can be on distance $1$ to many components of $U^i$ and the above inequality will not hold. But for $3$-disjoint sets this works.

As for the proof of the lemma, I think, it should be rather standard.

class field theory - Topological Langlands?

In a workshop about the geometry of $mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands Correspondence!

I had never heard about this program; I have found this page via Google:

http://www.math.jhu.edu/~asalch/toplang/

I only know a bit about the Number-Theoretic Langlands Program, and I still have a hard time trying to understand what is happening in the Geometrical one, so I cannot even start to draw a global picture out of the information dispersed in that site.

So, the questions are: What do you know (or what can you infere from the web) about the Topological Langlands Correspondence? Which is the global picture? What are its analogies with the (original) Langlands Program? Is it doable, or just a "little game" for now? What has been proved until now? What implications would it have?

(Note: It is somewhat difficult to tag this one, feel free to retag it if you have a better understanding of the subject than I have!)

ac.commutative algebra - What is the transcendence degree of Q_p and C over Q?

In both cases the transcendence degree is the cardinality of the continuum. CH is not needed.

This is a corollary of the following result: let $K$ be any infinite field, and let $L/K$ be any extension. Then

$# L = operatorname{max} (# K, operatorname{trdeg}_K L)$.

To prove this, in turn it suffices to establish the following two results (each of which is straightforward):

1) If $K$ is infinite and $L/K$ is algebraic, then $# L = # K$.

2) If $K$ is any infinite field, $T = {t_i}_{i in I}$ is an arbitrary set of indeterminates and $K(T)$ is a purely transcendental function field in the indeterminates $T$, then $ # K(T) leq # T + # K$.

ho.history overview - Problem of Mouton

The problem studied by the seventeenth century astronomer Gabriel Mouton was to construct logarithmic and trigonometric tables using only addition and subtraction, avoiding multiplication and division (so that the calculations could be performed by unskilled workers). Mouton derived an interpolation formula using successive numerical differences, that would later be used by Gaspard de Prony and others.

See: "The history of mathematical tables", by Martin Campbell-Kelly (Oxford UP, 2003).
or see http://youryuboku2.web.fc2.com/eBooks/Hairdressers.pdf

dg.differential geometry - Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex?

I don't think that's true. If the dimension of $M$ is bigger than that of $P$, then a necessary condition is that every tangent hyperplane $Psubset T_pM$ develops locally into a totally geodesic submanifold of $M$. This is not true for arbitrary manifolds. Some examples of manifolds in which this is true include those of constant sectional curvature. Symmetric spaces being a special subclass which verifies this conditions. Someone more awake can probably come up with a sufficient criterion.

I can't think of a very good example right now, but I am pretty sure that if take the 3-dimensional Riemannian Schwarzschild solution, start from a $r$-orthogonal plane outside of the apparent horizon, you'd get a counterexample.

If the dimension of $P$ is the same as dimension of $M$, however, then you should be okay as long as you make $U$ small enough.

Edit: Ah, for $P$ 2 dimensional and $M$ 3, by applying the Codazzi equations one sees that a necessary condition for locally developing the hyperplanes to totally geodesic submanifolds is that $Ric(X,Y) = 0$ whenever $g(X,Y) = 0$. This is obviously a very strong condition that is not satisfied by most manifolds.

ct.category theory - "adjoint" =?= "inverse of composite endofunctor is uniform bi-composition"

Understanding adjoints has always been (and continues to be) a bit of a struggle for me.

Today I stumbled upon a property of adjoint functors which seemed extremely intuitive to me. I was wondering why this property isn't mentioned more often in introductory category theory literature, and whether or not it completely characterizes adjunctions.

If two functors $F:Cto D$ and $U:Dto C$ are adjoint $Fdashv U$, then for every $f:F(Y)to X$ in $D$ there exists an $hat f:Yto U(X)$ in $C$ such that

$$ U(f)circ eta_Y = hat f$$
$$ epsilon_Xcirc F(hat f)=f$$

If we substitute the top equation into the bottom, we get

$$ epsilon_Xcirc F(U(f)circ eta_Y)=f$$

and by functoriality we get

$$ epsilon_Xcirc F(U(f))circ F(eta_Y)=f$$
$$ epsilon_Xcirc (Fcirc U)(f)circ F(eta_Y)=f$$

What the last equation says is that we can recover any morphism $f$ from the action of the "round trip endofunctor" $Fcirc U$ on it by pre-composing with $epsilon_X$ and post-composing with $F(eta_Y)$. These two morphisms are determined only by the domain and codomain of $f$ -- we only needed to know $X$ and $F(Y)$ in order to pick the two morphisms. We would have picked the same two morphisms for some $gneq f$ as long as $g:F(Y)to X$.

So, I believe it is correct to say that "if the domain of a morphism is within the range of a functor which has a right adjoint, then it can be recovered from the action of the composite endofunctor on it by pre-composition with some morphism and post-composition with some other morphism, where the choice of these two morphisms is completely determined by the domain and codomain of the original morphism". There is, of course, an equivalent statement for morphisms with a codomain in the range of a functor with a left adjoint.

So, my three questions are: (1) is this correct, (2) if so, why isn't it used to explain adjunctions to beginners (I certainly would have caught on quicker!) and (3) does the condition completely characterize adjoint functors?

Thanks,

ho.history overview - Great mathematicians born 1850-1920 (ET Bell's book ≲ x ≲ Fields Medalists)

Motivated by an e-mail from grshutt, I looked myself at the Princeton Companion to Mathematics. I discovered that the editors of the PCM answered the same question as the one that I posed here. Part VI is a collection of biographies of 95 mathematicians, plus one for Bourbaki, and the fact that the Bernoullis share a slot. They chose 37 mathematicians born between 1850 and 1920, exactly in the middle of the range that I suggested. See the table contents to see who they picked. They also wrote nice biographies, although they are much shorter and more sedate than E.T. Bell's single-author work.

It is a little embarrassing that we had this whole discussion without noticing that the PCM, with Timothy Gowers as the chief editor, had already done the same work. But it is also useful. It seems that when informed people carefully make lists of influential mathematicians of comparable length, they'll arrive at roughly the same list. Typically the intersection is about 1/2 if the lists are exactly the same length. PCM's list is longer than Bell (in the corresponding period) and mine, and shorter than Pete's, so we can look at who I (or Harrison) included but they did not, or who they included but Pete did not.

PCM minus Pete: Fredholm, Vallée-Poussin, Russell, Sierpinski*, Littlewood*, Skolem, Ramanujan*, Courant, Kolmogorov*, Church, Hodge* (* = honorable mention in Pete's list)

me minus PCM: Picard, Hurwitz, Lefschetz, Siegel, Chevalley, Gelfand

Harrison minus PCM: Henri Cartan, MacLane, Erdos, Feynman

Bell minus PCM: Zeno, Eudoxus, Poncelet, Monge

PCM $cap$ Pete $cap$ Harrison $cap$ me: Hilbert, Minkowski, Hausdorff, Cartan, Noether, Weyl, Banach, von Neumann, Godel, Weil

Some of the differences can be explained as a somewhat different emphasis. In listing Russell and Church, the PCM gave more credit to the philosophical end of logic. (This is also grshutt's point in in his answer posted here and in his letter.) Whether Feynman should "count" is another question with two different reasonable answers. (As I said in a comment, the argument for Feynman is that he is an architect of quantum field theory, and quantum field theory is arguably a fundamental mathematical theory, even though parts of it haven't been made rigorous.)

And I suppose that some of the differences indicate the ill-defined side of the question, or they are just random fluctuations, and aren't really worth arguing over.

elliptic curves - The class number formula, the BSD conjecture, and the Kronecker limit formula

The following observation suggests that perhaps the analogy between
class groups and Tate-Shafarevich groups is not as close as one might
think, and that, at least in the quadratic case, the right object is
the group of ideal classes modulo squares.

Let $Q_0$ denote the principal binary quadratic form with
discriminant $d$, and let ${mathcal P}: Q_0(X,Y) = 1$ be the
associated Pell conic. For each prime power $q = p^r$, denote
the number of ${mathbb F}_q$-rational points on ${mathcal P}$
by $q - a_q$; it is easily checked that $a_q = chi(q)$, where
$chi = (frac{d}{cdot})$ is the quadratic character with
conductor $d$.

Define the local zeta function at $p$ as the formal power series
$$ Z_p(T) = expBig(sum_{r=1}^infty N_r frac{T^r}r Big), $$
where $N_r$ denotes the number of ${mathbb F}_q$-rational points
on ${mathcal P}$. A simple calculation shows that
$$ Z_p(T) = frac{1}{(1-pT)(1-chi(p)T)}. $$

Set $P_p(T) = frac1{1 - chi(p)T} $ and define the global $L$-series as
$$ L(s,chi) = prod_p P_p(p^{-s}). $$
This is the classical Dirichlet L-series, which played a major role in
Dirichlet's proof of primes in arithmetic progression, and was almost
immediately shown to be connected to the class number formula.

Class groups do not occur in the picture above; like their big brothers,
the Tate-Shafarevich group, they are related to the global object we
started with: the Pell conic. The integral points on the affine Pell
conic form a group, which acts (in a more or less obvious way - think
of integral points as units in some quadratic number fields) on the
rational points of curves of the form
$$ Q(x,y) = 1, $$
where $Q$ is a primitive binary quadratic form with discriminant $d$.
This action makes $Q$ into a principal homogeneous space ("over the
integers"), and the usual action of SL$_2({mathbb Z})$ on quadratic
forms respects this structure. The equivalence classes of such spaces
form a group with respect to taking the Baer sum, which coincides with
the classical Gauss composition of quadratic forms.

Principal homogeneous spaces with an integral point are trivial in the
sense that they are equivalent to the Pell conic ${mathcal P}$. The
spaces with a local point everywhere (i.e. with rational points) form
a subgroup Sha isomorphic to the group $Cl^+(d)^2$ of square classes.
Defining Tamagawa numbers for each prime $p$ as $c_p = 1$ or $=2$
according as $p$ is coprime to $d$ or not, we find that the usual class
number (in the strict sense) is the order of Sha times the product of
all Tamagawa numbers (the latter is twice the genus class number).

Now we can use Dirichlet's class number formula for proving the BSD
conjecture for conics:
$$ lim_{s to 0} s^{-r} L(s,chi) = frac{2hR}{w} =
frac{|Sha| cdot R^+ cdot prod c_p}
{| {mathcal P}({mathbb Z})_{tors}|}. $$
Observe that $R^+$ denotes the regulator of the Pell conic, i.e. the logarithm
of the smallest totally positive unit $> 1$.

The proof of Dirichlet's class number formula uses the class group, which
is a group containing Sha as a quotient, and a group related to the Tamagawa
numbers as a subgroup. It remains to be seen whether such a group exists in
the elliptic case.

It might be possible to make advance without having such a group:
in the case of Pell conics, the zeta functions of ideal classes are,
if I recall it correctly, closely related to the series defined by
summing over all $1/Q(x,y)^s$ for integers $x$, $y$. The question
remains how to imitate such a construction for the genus 1 curves
representing elements in Sha.

Remark: The Tamagawa numbers may be defined as certain $p$-adic integrals;
see an unpublished masters thesis (in Japanese) by A. Iwaomoto, Kyoto 2005.
For more info on the above, see
here.

For ideas pointing in a different direction, see

• D. Zagier, The Birch-Swinnerton-Dyer conjecture from a naive point of view,
  Prog. Math. 89, 377-389 (1991)

topological "milnor's conjecture" on torus knots.

Here's a question that has come up in a couple of talks that I have given recently.

The 'classical' way to show that there is a knot $K$ that is locally-flat slice in the 4-ball but not smoothly slice in the 4-ball is to do two things

1. Compute that the Alexander polynomial of $K$ is 1, and so by results of Freedman's you know that $K$ is locally-flat slice.




2. (due to Rudolph) Somehow obtain a special diagram of $K$ (or utilize a more subtle argument) to show that you can present $K$ as a separating curve on a minimal Seifert surface for a torus knot. Since we know (by various proofs, the first due to Kronheimer-Mrowka) that the genus of torus knot is equal to its smooth 4-ball genus (part of Milnor's conjecture), the smooth 4-ball genus of $K$ must be equal to the genus of the piece of the torus knot Seifert surface that it bounds, and this is $geq 1$.

Boiling the approach of 2. down to braid diagrams, you come up with the slice-Bennequin inequality.

Well, here's the thing. I have this smooth cobordism from the torus knot to $K$, and then I know that $K$ bounds a locally-flat disc. This means that the locally-flat 4-ball genus of the torus knot must be less than its smooth 4-ball genus. So if you were to conjecture that the locally flat 4-ball genus of a torus knot agrees with its smooth 4-ball genus, you would be wrong.

My question is - are there any conjectures out there on the torus knot locally-flat genus? Even asymptotically? Any results? Any way known to try and study this?

Thanks, Andrew.

gr.group theory - Database of finite presentations of used groups

Do You know any kind of database of presentations of groups?

It may be on-line or off-line in form of tables, ideally case would be integrated in some Computer Algebra System. I am interested the most in infinite group presentations, but feel free to put here also information about tables of presentations of finite groups.

Maybe this thread should be wiki-type because probably there is many good answers to this question, and it is hard to compare for example software system with some kind of book or publication about this matter? I add biglist tag in a hope that it would appear;-)

co.combinatorics - Flipping Hilbert series of semigroup rings

I'll first give intuition, and then give a precise statement.

For $|z|<1$, we have $sum_{i geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two functions

$phi(i) = begin{cases} 1 quad i geq 0 \ 0 quad i<0 end{cases}$ and $psi(i) = begin{cases} 0 quad i geq 0 \ -1 quad i<0 end{cases}$

have the "same" Fourier transform. This question is about generalizations of this phenomenon to higher dimensions.

---

Let $phi : mathbb{Z}^n to mathbb{R}$ be a function such that (1) $phi$ can be written as a rational combination of the characteristic functions of finitely many rational cones and (2) there is a linear function $lambda$ so that $phi$ vanishes on ${ e : lambda(e) leq 0 } setminus { 0 }$. We define
$$h(phi) = sum phi(e) z^e.$$
This sum converges somewhere, and gives a rational function of $z$.

Given $phi$ as above, and given a generic linear functional $zeta$, one can show that there is a unique function $phi^{zeta}$ such that (1) $phi$ can be written as a rational combination of the characteristic functions of finitely many cones (2) $phi$ vanishes on ${ e : zeta(e) leq 0 } setminus { 0 }$ and (3) $h(phi^{zeta}) = h(phi)$.

For example, the above computation shows that $phi^{(-1)} = psi$. If $phi$ is the characteristic function of a simplicial cone, this operation is easy to describe. In principle, this means we can always calculate $phi^{zeta}$ by writing $phi$ as a linear combination of simplicial cones.

If $phi$ is the characteristic function of a cone, what is known about $phi^{zeta}$? For example, is it true that (a) all the values of $phi^{zeta}$ have the same sign (NO, counter-example below), or that (b) $phi^{zeta}$ only takes the values $-1$, $0$ and $1$?

soft question - Your favorite surprising connections in Mathematics

Although it is not all that spectacular, since it does not really relate two different fields of mathematics, it has always been surprising to me that the gradient flow equation for the Chern-Simons functional on a (closed, oriented) 3-manifold $Y$ turns out to be the ASD(=Yang-Mills) equation on the cylinder $Ytimesmathbb{R}$.

---

The next thing is not really a connection, but definitely one of my favorite surprises in mathematics. By the work of Michael Freedman, the classification of closed, oriented, simply-connected topological 4-manifolds is basically equivalent to the classification of unimodular, symmetric bilinear forms (uSBFs) over the integers. As nice as this is, it comes with the grain of salt that the classification of uSBFs is not an easy task. Specifically, the classification of definite uSBFs is a hard problem and far from being solved.

And now comes the surprise: Simon Donaldson tells us that if we look at smooth 4-manifolds, then the only definite uSBFs that can occur are the trivial ones!

Riemann surfaces: explicit algebraic equations

You are trying to relate the periods of the curve (which are analytic invariants), to algebraic invariants, so the most you can get is some power series. Suppose you get from Gamma to the period matrix of the Jacobian: tau, then:

In genus 1 - which you are not interested in - you have the j-invariant, which is a function of tau.

In genus 2 you have the Igusa invariants.

In genus 3 you don't have a formula (too complicated), but you have an algorithm: there are 28 tangents to the theta divisor at the 28 odd 2-torsion points, these are the 28 bitangents of the canonical curve, and you can (effectively) reconstruct the curve from the bitangents.

In higher genera you can start in a similar way: map the 4-torsion points to some grassmanian (which is an embedding of A_g plus some level: Grushevsky and Salvati-Mani), but I'm not aware of a reconstruction algorithm.

measure theory - Discontinuous convolutions

I believe we get the stronger statement that the convolution of an infinitely differentiable integrable function $f$ with any integrable function $g$ will result in $f*g$ being infinitely differentiable. In fact, $(f*g)' = f'*g$ (This can be found in The Fourier Transform and Its Applications, Bracewell) Thus the convolution of two infinitely differentiable, integrable functions will necessarily be (infinitely) differentiable and so also continuous.

EDIT: Also, if my math is correct, we can check this formula via the Fourier transform:

$widehat{((f*g)')}(r) = irwidehat{f*g}(r) = irwidehat{f}(r)widehat{g}(r) = widehat{f'}(r)widehat{g}(r) = widehat{(f'*g)}(r)$

Apologies, this only applies in the case that $f$ is compactly supported. Back to the drawing board...

nt.number theory - Geometric meaning of fiber of modular parameterization over a point of an elliptic curve?

As Stankewicz explains, although elliptic curves appear in two guises in the modular parameterization $X_0(N) to E,$ first because $E$ is an elliptic curve, and secondly because $X_0(N)$ parameterizes elliptic curves, it is something of a red herring to think of these two appearances of elliptic curves as having anything to do with one another.

The reason that $X_0(N)$ appears in the problem of describing elliptic curves is because
elliptic curves have two dimensional $H^1$, and $X_0(N)$ is the Shimura variety over $mathbb Q$ associated to the group $GL_2$. Thus, as Stankewicz notes, Shimura curves (which are the Shimura varieties attached to twisted forms of $GL_2$) can equally well give parameterizations of elliptic curves.

Now the way we prove things about $X_0(N)$ (e.g. properties of its Heegner points, as in Pete Clark's answer) is using its moduli interpretation. But there are two things to bear in mind:

First, most (maybe all?) properties of the special points on $X_0(N)$, such as the Heegner
points, are special cases of general aspects of the theory of Shimura varieties (so although the proofs use the moduli interpretation, the statements can be formulated in a way that
doesn't refer to the moduli-theoretic interpretation, but instead refers to the interpretation
of $X_0(N)$ as a Shimura variety).

Second, the transfer of information is always from $X_0(N)$ to $E$. So while Heegner points give certain interesting points on $X_0(N)$ defined over class fields of quadratic imaginary fields, which can be mapped down to $E$ to give interesting points on $E$ defined over such fields, if one takes a random point on $E$ defined over a class field of an imaginary quadratic field and pulls it back to $X_0(N)$, it is not so easy to say what is going on with the preimages in general.

Finally, I think remark (3) in Pete Clark's answer is an interesting one. In the Mazur and Swinnerton-Dyer paper that David Hansen refer's to in his first comment, if I am remembering correctly, they also suggest that the images in $E$ of the critical points of the map from $X_0(N)$ to $E$ that lie on the geodesic arc joining $0$ to $infty$ in the upper half-plane may be worth studying. As with Birch's suggestion of Weierstrass points, I'm not sure how much has been done on this.