[Directiva] documentos para mañana

[Claudia Alfonzo]

Les adjunto las últimas resoluciones y el orden del día para mañana.
También los documentos relacionados con la renovación de efectividad de 
Richard para que puedan adelantar la lectura, porque son unos cuantos.

Saludos!

-- 
Secretaría
Centro de Matemática
Facultad de Ciencias

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Referee Report: Singular Instantons and Painleve VI, by Richard Muniz
Manasliski

Recommendation: Accept (marginal), subject to revision and repeat
refereeing

This manuscript is a continuation of the author's works [16,17] of which I
was not aware. These works contain an obscure but (I think) remarkable
result, describing an equivalence between certain solutions of the
self-duality equations for SU(2) gauge fields and certain solutions of the
Painleve VI equation. There have been a number of papers exploring links
between the self-duality equations and Painleve VI, starting from
"Self-duality and the Painleve transcendents" by L J Mason and N M J
Woodhouse in Nonlinearity, Volume 6 (1993). (Cited 50 times on Google
Scholar - definitely should also be cited in the current paper.) The author
is fortunate that I did not get to referee [17]. If I had, I might well
have said that it is not enough to write "Then, with the above expressions
it is possible to find y(x) in terms of the instanton", it is necessary to
complete the calculation. In my opinion the most significant part of the
current paper is that this is done, arriving at Proposition 3.1. giving a
completely explicit map from a solution of equations (2.1) in the paper to
a solution of Painleve VI. Once proposition 3.1 is known, the fact that the
map works also for non-integer values of theta (corresponding, by the paper
[19] of Sadun, to singular instantons) is pretty much immediate and I would
not have said it is worth a paper of its own. (Also the fact that the
corresponding solutions of Painleve VI are not algebraic is of limited
interest - and I think follows from other general results on values of the
parameters in Painleve VI for which there exist algebraic solutions. The
author should investigate this.) But given that Proposition 3.1 did not
appear in [16,17], the current paper should probably be accepted, subject
to revision.

Unfortunately the paper is not well written and there are some errors. The
most significant of these is that there is an error in (3.5) - the
parentheses in the denominator are not balanced. I wish to check
Proposition 3.1 by direct calculation and cannot do this!

I list examples of other points that should be dealt with (there are also
language issues):

* Painleve VI is arguably the most important second order ordinary
differential equation with the Painleve property, but to claim it is the
most important equation is overdoing it.

* In Painleve VI on page 2, several extra + signs!

* Suddenly on p.3 "Sadun's instantons" are mentioned, without any
explanation of what they are. Bad writing.

* Equations (2.1). We are told to see [3,16]. I am not sure why the author
sometimes only refers to his work [16] and not [17]. But rather more
importantly I cannot see equations (2.1) in [3], but I do so very similar -
but different! - equations in [4]. The relationship must be explained.

* A critical fact about system (2.1) is that there is a conserved quantity
of the form s1(t) a1^2 + s2(t) a2^2 + s3(t) a3^2 where s1(t),s2(t),s3(t)
are simple rational functions (s1(t) = (t^2-1)/(t^2-9), s2(t) =
-(t+1)/(t(t-3)), s3(t) = (t-1)/(t(t+3)) I think, maybe I have some signs
wrong). This is what allows this third order system to be reduced to a
second order equation with a parameter.

* Equation (3.2): ct on the right hand side. The author means "constant", t
is a variable!

* Theorem 5.2: what is meant by "resonant values" of the parameters? This
term is used without definition. It seems to me the proof is only that the
solutions are not algebraic if the parameters do not take special values,
the current wording implies that for the special values the solutions are
algebraic.

* In contrast to Theorem 5.2, the wording of the abstract implies that the
solutions are never algebraic!!

* The abstract should be changed to say that this is a continuation of
[16,17] (why only mention [16]?), giving an explicit version of the map
between instantons and solutions of Painleve VI (this is the most important
result), extending this to the case of the singular instantons described by
Sadun, and looking at the possibility of the resulting solutions of
Painleve VI being algebraic.

* As already mentioned, the author should look at, and cite, the wider
literature relating solutions of the self duality equations and solutions
of Painleve VI, and works on the values of parameters of Painleve VI for
which there exist algebraic solutions (I do not know this literature well -
but a search on Google scholar immediately gave several references, for
example arXiv/0809.4873).

In general the writing could be made clearer by such standard things as
outlining the structure of the paper at the end of the introductory
section, starting each section with a clear statement of what is done in
that section and so on.
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