some bookmarks

It is worth looking at this link


This article is very interesting - the conclusion is that tumour
growth is primarily a space competition occurring at the surface.

There was a boy
a very strange enchanted boy
they say he wandered very far, very far
over land and sea
a little shy and sad of eye
but very wise was he

And then one day
a magic day he passed my way
and while we spoke of many things
fools and kings
this he said to me
"the greatest thing you'll ever learn
is just to love and be loved in return"

by Eden Ahbez sung by Nat King Cole (among many others)


hmmm - now i know where that came from.
sigh!

Prettty pissed

friday

It being a Friday - and now evening - there is not a soul in the building.
No one to talk to - which is the single largest source of my frustration
in Seoul. Especially, academics.

Fundamental density

I just heard Mateos talk at Strings 2007. He clearly mentions that at finite density the ME branes do not exist. The argument being the F1-force balance.
Its a bit funny - because from our work, we know that at fixed density and quark mass, there are no black hole type embeddings for low enough temperature (this is because of the temperature dependence in the expression for the physical density in terms of the charge density q).

I am going to just try to compute the fluctuation spectrum - and then drop the whole thing.

A work diary

I am trying to study what happens when we take string corrections to the gauge theory confinement-deconfinement phase transition a la Herzog.

The method to be adopted is:
First: find the contribution of the D7-brane to the total free energy (coming from Born-Infeld action). This contribution makes sense only if N_c is finite.

Second: If N_c is finite, then the D7-brane contributes a tadpole to the dilaton and axion (the latter via the Bianchii identity for F^{(1)} _{RR}). Similarly, the metric also picks up some back reaction.

Therefore, there are corrections to the bulk supergravity action as well, at O (N_f/N_c).

Thus, add all of these contributions, for a truncated geometry, a la Herzog (or Pando-Zayas) for each of thermal AdS, and AdS-Black hole backgrounds.

Then estimate the correction to the phase transition temperature - in particular what does it depend on ?

In this, we must keep T and quark mass m_q fixed.

WIP-2

I need to understand clearly, the results of Atish - because he is able to get a whole series to agree with the sugra results.
This would be a useful thing to keep in mind too.

Today's agenda is to do the R^4 terms - using Gross and Sloan. Note that in type II this is the leading correction in 10D - there is no R^2 correction in 10D (the R^4 can give rise to a 4D term provided the compact space has non-zero Riemann...)

Work in progress

I am trying to look at higher derivative corrections to ashoke's entropy function. My idea is whether one can reverse the usual line of argument starting from the knowledge of the terms, to arrive at the entropy function.
I intend to first assume some coefficients for the higher derivative terms and fix them by requiring that the entropy come out right.

This is a little tricky partly because of field redefinitions. Firstly, AdS X S spaces are symmetric, which makes the coefficients not ALL determinable.

First fix the metric that appears in front of the field-strengths. And express all quantities in terms of this metric (this is because, we would like the charges to be fixed - these are computed in terms of the metric and the field-strengths). Now we have no more freedom to perform field redefinitons of the metric at least.

Then, we require that the entropies match - along with the requirement that there be no additional graviton poles. This fixes the coefficients nicely to quadratic order (for heterotic large black holes with four charges).
This means the following: If we had started with some other choice for the higher derivative curvature terms, then by a suitable field redefinition, one could bring it to the above form. In this case, the metric changes, and hence what one calls charges will change. Thus keeping the charges the same, will (partially) fix the field redefinition freedom.

I guess that it would be possible to do better, if I find some rotating black holes for which the metric is no longer that of a symmetric space examples are Horowitz+Sen. In this case, I expect that i will be able to relax the requirement of having no additional graviton poles.

The key point in this assumption is that AdS times S is expected to be an exact solution, so that order by order in alpha' also, it is a solution. It may happen that for rotating black holes, the corresponding "near horizon" geometry is not as simple - thus thwarting any attempt at determining coefficients in this manner beyond quadratic order.

Serre Reciprocity Conjecture

These are notes from C. S. Dalawat's lecture on the Serre Reciprocity Conjecture at H.R.I and from the draft of an article written by him.

The conjecture relates representations of Gal(Qbar/Q) and modular forms. The "level 1" case of this has been proved by Chandrashekhar Khare, "by carrying out a strategy he had worked out in collaboration with Jean-Pierre Wintenberger, and using results by Luis Dieulefait and others". This apparently breaks " a psychological barrier".

In the following we shall attempt to present, somewhat concisely, the statement of this conjecture. For a pedagogical article suitable for (advanced) high school students, click here (postscript format).

Let us get a few terms defined: Everyone knows what a field is;

• the field G of Gaussian numbers is the set {a+bi, a,b rational} and that Aut(G)=Z/2Z (observe that since polynomials are inv under Aut, i-> -i)

• the set of algebraic numbers A, i.e., those that are roots of polynomials also forms a field and what about Aut(A)=? - this is the question. For instance, all finite order elements in Aut(A) are order 2 !
  

A (sub)field is called Galoisian if the only numbers invariant under all of Aut are the rationals. (there is apparently a conjecture that every finite group G appears as the Aut of some galoisian subfield of A)

Finite Fields and Local Fields