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.

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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 $Ytimesmathbb{R}$.

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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.