One of the coolest (pure math) facts in Conway’s book ONAG is the explicit construction of the algebraic closure $\overline{\mathbb{F}_2}$ of the field with two elements as the set of all ordinal numbers smaller than $(\omega^{\omega})^{\omega}$ equipped with nimber addition and multiplication.

Some time ago we did run a couple of posts on this. In transfinite number hacking we recalled Cantor’s ordinal arithmetic and in Conway’s nim arithmetics we showed that Conway’s simplicity rules for addition and multiplication turns the set of all ordinal numbers into a field of characteristic zero : $\mathbb{On}_2$ (pronounced ‘Onto’).

In the post extending Lenstra’s list we gave Hendrik Lenstra’s effective construction of the mystery elements $\alpha_p$ (for prime numbers $p$) needed to do actual calculations in $\mathbb{On}_2$. We used SAGE to check the values for $p \leq 41$ and solved the conjecture left in Lenstra’s paper Nim multiplication that $(\omega^{\omega^{13}})^{43} = \omega^{\omega^7} + 1$ and determined $\alpha_p$ for $p \leq 67$.

Aaron Siegel has now dramatically extended this and calculated the $\alpha_p$ for all primes $p \leq 181$. He mails :

“thinking about the problem I figured it shouldn’t be too hard to write a dedicated program for it. So I threw together some Java code and… pushed the table up to p = 181! You can see the results below. Q(f(p)), excess, and alpha_p are all as defined by Lenstra. The “t(sec)” column is the number of seconds the calculation took, on my 3.4GHz iMac. The most difficult case, by far, was p = 167, which took about five days.

I’m including results for all p < 300, except for p = 191, 229, 263, and 283. p = 263 and 283 are omitted because they involve computations in truly enormous finite fields (exponent 102180 for p = 263, and 237820 for p = 283). I'm confident that if I let my computer grind away at them for long enough, we'd get an answer... but it would take several months of CPU time at least.

p = 191 and 229 are more troubling cases. Consider p = 191: it's the first prime p such that p-1 has a factor with excess > 1. (190 = 2 x 5 x 19, and alpha_19 has excess 4.) This seems to have a significant effect on the excess of alpha_191. I’ve tried it for every excess up to m = 274, and for all powers of 2 up to m = 2^32. No luck.”

Aaron is writing a book on combinatorial game theory (to be published in the AMS GSM series, hopefully later this year) and will include details of these computations. For the impatient, here’s his list

Reineke’s observation that any projective variety can be realized as a quiver Grassmannian is bad news: we will have to look at special representations and/or dimension vectors if we want the Grassmannian to have desirable properties. Some people still see a silver lining: it can be used to define a larger class of geometric objects over the elusive field with one element $\mathbb{F}_1$.

In a comment to the previous post Markus Reineke recalls motivating discussions with Javier Lopez Pena and Oliver Lorscheid (the guys responsable for the map of $\mathbb{F}_1$-land above) and asks about potential connections with $\mathbb{F}_1$-geometry. In this post I will ellaborate on Javier’s response.

The Kapranov-Smirnov $\mathbb{F}_1$-floklore tells us that an $n$-dimensional vectorspace over $\mathbb{F}_1$ is a pointed set $V^{\bullet}$ consisting of $n+1$ points, the distinguished point playing the role of the zero-vector. Linear maps $V^{\bullet} \rightarrow W^{\bullet}$ between $\mathbb{F}_1$-spaces are then just maps of pointed sets (sending the distinguished element of $V^{\bullet}$ to that of $W^{\bullet}$). As an example, the base-change group $GL_n(\mathbb{F}_1)$ of an $n$-dimensional $\mathbb{F}_1$-space $V^{\bullet}$ is isomorphic to the symmetric group $S_n$.

This allows us to make sense of quiver-representations over $\mathbb{F}_1$. To each vertex we associate a pointed set and to each arrow a map of pointed sets between the vertex-pointed sets. The dimension-vector $\alpha$ of quiver-representation is defined as before and two representations with the same dimension-vector are isomorphic is they lie in the same orbit under the action of the product of the symmetric groups determined by the components of $\alpha$. All this (and a bit more) has been worked out by Matt Szczesny in the paper Representations of quivers over $\mathbb{F}_1$.

Oliver Lorscheid developed his own approach to $\mathbb{F}_1$ based on the notion of blueprints (see also part 2 and a paper with Javier).

Roughly speaking a blueprint $B = A // \mathcal{R}$ is a commutative monoid $A$ together with an equivalence relation $\mathcal{R}$ on the monoid semiring $\mathbb{N}[A]$ compatible with addition and multiplication. Any commutative ring $R$ is a blueprint by taking $A$ the multiplicative monoid of $R$ and $\mathcal{R}(\sum_i a_i,\sum_j b_j)$ if and only if the elements $\sum_i a_i$ and $\sum_j b_j$ in $R$ are equal.

One can extend the usual notions of prime ideals, Zariski topology and structure sheaf from commutative rings to blueprints and hence define a notion of “blue schemes” which are then taken to be the schemes over $\mathbb{F}_1$.

What’s the connection with Reineke’s result? Well, for quiver-representations $V$ defined over $\mathbb{F}_1$ they can show that the corresponding quiver Grassmannians $Gr(V,\alpha)$ are blue projective varieties and hence are geometric objects defined over $\mathbb{F}_1$.

For us, old-fashioned representation theorists, a complex quiver-representation $V$ is defined over $\mathbb{F}_1$ if and only if there is an isomorphic representation $V’$ with the property that all its arrow-matrices have at most one $1$ in every column, and zeroes elsewhere.

Remember from last time that Reineke’s representation consisted of two parts : the Veronese-part encoding the $d$-uple embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^M$ and a linear part describing the subvariety $X \hookrightarrow \mathbb{P}^n$ as the intersection of the image of $\mathbb{P}^n$ in $\mathbb{P}^M$ with a finite number of hyper-planes in $\mathbb{P}^M$.

We have seen that the Veronese-part is always defined over $\mathbb{F}_1$, compatible with the fact that all approaches to $\mathbb{F}_1$-geometry allow for projective spaces and $d$-uple embeddings. The linear part does not have to be defined over $\mathbb{F}_1$ in general, but we can look at the varieties we get when we force the linear-part matrices to be of the correct form.

For example, by modifying the map $h$ of last time to $h=x_0+x_7+x_9$ we get that the quiver-representation

is defined over $\mathbb{F}_1$ and hence that Reineke’s associated quiver Grassmannian, which is the smooth plane elliptic curve $\mathbb{V}(x^3+y^2z+z^3)$, is a blue variety. This in sharp contrast with other approaches to $\mathbb{F}_1$-geometry which do not allow elliptic curves!

Oliver will give a talk at the 6th European Congress of Mathematics in the mini-symposium Absolute Arithmetic and $\mathbb{F}_1$-Geometry. Judging from his abstract,he will also mention quiver Grassmannians.

A standard Grassmannian $Gr(m,V)$ is the manifold having as its points all possible $m$-dimensional subspaces of a given vectorspace $V$. As an example, $Gr(1,V)$ is the set of lines through the origin in $V$ and therefore is the projective space $\mathbb{P}(V)$. Grassmannians are among the nicest projective varieties, they are smooth and allow a cell decomposition.

A quiver $Q$ is just an oriented graph. Here’s an example

A representation $V$ of a quiver assigns a vector-space to each vertex and a linear map between these vertex-spaces to every arrow. As an example, a representation $V$ of the quiver $Q$ consists of a triple of vector-spaces $(V_1,V_2,V_3)$ together with linear maps $f_a~:~V_2 \rightarrow V_1$ and $f_b,f_c~:~V_2 \rightarrow V_3$.

A sub-representation $W \subset V$ consists of subspaces of the vertex-spaces of $V$ and linear maps between them compatible with the maps of $V$. The dimension-vector of $W$ is the vector with components the dimensions of the vertex-spaces of $W$.

This means in the example that we require $f_a(W_2) \subset W_1$ and $f_b(W_2)$ and $f_c(W_2)$ to be subspaces of $W_3$. If the dimension of $W_i$ is $m_i$ then $m=(m_1,m_2,m_3)$ is the dimension vector of $W$.

The quiver-analogon of the Grassmannian $Gr(m,V)$ is the Quiver Grassmannian $QGr(m,V)$ where $V$ is a quiver-representation and $QGr(m,V)$ is the collection of all possible sub-representations $W \subset V$ with fixed dimension-vector $m$. One might expect these quiver Grassmannians to be rather nice projective varieties.

However, last week Markus Reineke posted a 2-page note on the arXiv proving that every projective variety is a quiver Grassmannian.

Let’s illustrate the argument by finding a quiver Grassmannian $QGr(m,V)$ isomorphic to the elliptic curve in $\mathbb{P}^2$ with homogeneous equation $Y^2Z=X^3+Z^3$.

Consider the Veronese embedding $\mathbb{P}^2 \hookrightarrow \mathbb{P}^9$ obtained by sending a point $(x:y:z)$ to the point

\[ (x^3:x^2y:x^2z:xy^2:xyz:xz^2:y^3:y^2z:yz^2:z^3) \]

The upshot being that the elliptic curve is now realized as the intersection of the image of $\mathbb{P}^2$ with the hyper-plane $\mathbb{V}(X_0-X_7+X_9)$ in the standard projective coordinates $(x_0:x_1:\cdots:x_9)$ for $\mathbb{P}^9$.

To describe the equations of the image of $\mathbb{P}^2$ in $\mathbb{P}^9$ consider the $6 \times 3$ matrix with the rows corresponding to $(x^2,xy,xz,y^2,yz,z^2)$ and the columns to $(x,y,z)$ and the entries being the multiplications, that is

$$\begin{bmatrix} x^3 & x^2y & x^2z \\ x^2y & xy^2 & xyz \\ x^2z & xyz & xz^2 \\ xy^2 & y^3 & y^2z \\ xyz & y^2z & yz^2 \\ xz^2 & yz^2 & z^3 \end{bmatrix} = \begin{bmatrix} x_0 & x_1 & x_2 \\ x_1 & x_3 & x_4 \\ x_2 & x_4 & x_5 \\ x_3 & x_6 & x_7 \\ x_4 & x_7 & x_8 \\ x_5 & x_8 & x_9 \end{bmatrix}$$

But then, a point $(x_0:x_1: \cdots : x_9)$ belongs to the image of $\mathbb{P}^2$ if (and only if) the matrix on the right-hand side has rank $1$ (that is, all its $2 \times 2$ minors vanish). Next, consider the quiver

and consider the representation $V=(V_1,V_2,V_3)$ with vertex-spaces $V_1=\mathbb{C}$, $V_2 = \mathbb{C}^{10}$ and $V_2 = \mathbb{C}^6$. The linear maps $x,y$ and $z$ correspond to the columns of the matrix above, that is

$$(x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9) \begin{cases} \rightarrow^x~(x_0,x_1,x_2,x_3,x_4,x_5) \\ \rightarrow^y~(x_1,x_3,x_4,x_6,x_7,x_8) \\ \rightarrow^z~(x_2,x_4,x_5,x_7,x_8,x_9) \end{cases}$$

The linear map $h~:~\mathbb{C}^{10} \rightarrow \mathbb{C}$ encodes the equation of the hyper-plane, that is $h=x_0-x_7+x_9$.

Now consider the quiver Grassmannian $QGr(m,V)$ for the dimension vector $m=(0,1,1)$. A base-vector $p=(x_0,\cdots,x_9)$ of $W_2 = \mathbb{C}p$ of a subrepresentation $W=(0,W_2,W_3) \subset V$ must be such that $h(x)=0$, that is, $p$ determines a point of the hyper-plane.

Likewise the vectors $x(p),y(p)$ and $z(p)$ must all lie in the one-dimensional space $W_3 = \mathbb{C}$, that is, the right-hand side matrix above must have rank one and hence $p$ is a point in the image of $\mathbb{P}^2$ under the Veronese.

That is, $Gr(m,V)$ is isomorphic to the intersection of this image with the hyper-plane and hence is isomorphic to the elliptic curve.

The general case is similar as one can view any projective subvariety $X \hookrightarrow \mathbb{P}^n$ as isomorphic to the intersection of the image of a specific $d$-uple Veronese embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^N$ with a number of hyper-planes in $\mathbb{P}^N$.

One of the nicer tools around is bookworm arXiv which ‘is a collaboration between the Harvard Cultural Observatory, arxiv.org, and the Open Science Data Cloud. It enables you to explore lexical trends in over 700,000 e-prints, spanning mathematics, physics, computer science, and statistics’ posted on the arXiv.

One possible use is to explore the popularity of certain topics. Below is the graph of the number of papers submitted monthly to the arXiv in noncommutative geometry, quantum groups, cluster algebras and symplectic reflection (algebras).

The default gives the graphs in the percentage of all papers submitted, but it is better to change this to the number of papers (I think). Sadly, at present one can only search for one- and two-word phrases.

Extremely useful is that it gives you the full list of papers (with direct links to the papers) containing the search terms when you click on that months point in the graph. For example, there are 4 sheets of papers in noncommutative geometry for october 2011

Clearly, there are plenty of other fun uses for this bookworm. For example, you can graph the number of papers in a topic in function of the nationality of the submitter. Here are the papers in noncommutative geometry, submitted by people from the US, France, the UK and Italy.

Or, you can use it for vanity reasons, giving you the list of all papers containing a reference to your work, which may not always be a good idea, blood-pressure wise…

The dinosaurs among you may remember that before this blog we had the ‘na&g-forum’ to accompany our master-class in noncommutative algebra & geometry.

That forum ran on an early flat-panel iMac G4 which was, for lack of a better name, baptized ‘the matrix’.

The original matrix did survive the unification of the three Antwerp universities and a move to a different campus but then died around bloomsday 2007 and was replaced by an intel iMac.

This second matrix did host a number of blogs and projects started (and usually ended rather quickly) such as ‘MoonshineMath’, a muMath-site called noncommutative.org, the ‘F-un Mathematics’ blog dedicated to the field with one element and, of course, this blog.

About a month ago matrix-II was replaced by a state-of-the-art iMac running 10.7. The transition went smooth apart from the fact that 10.7 doesn’t like ‘localhost’ but prefers ’127.0.0.1′ in setting up wordpress blogs.

Besides neverendingbooks, matrix-III runs angs@t – angs+ which is the blog of the antwerp noncommutative geometry seminar. It will be revamped over the summer and will probably be the website for our renewed master-class, starting next year.

The ‘F-un Mathematics’ blog was dropped in the transition but still survives at Ghent University where it is managed by Koen Thas.

As far as NeverendingBooks is concerned i hope to make a fresh start with blogging and will try to get more structure in this site by changing to a responsive wordpress theme (‘These responsive, fluid, or adaptive WordPress themes, automatically adjust according to the screen size, resolution and device on which they are being viewed’).

As a result this page will look weird from time to time over the next week or so. My apologies.

This week i was at the conference Noncommutative Algebraic Geometry and its Applications to Physics at the Lorentz center in Leiden.

It was refreshing to go to a conference where i knew only a handful of people beforehand and where everything was organized to Oberwolfach perfection. Perhaps i’ll post someday on some of the (to me) more interesting talks.

Also interesting were some discussions about the Elsevier-boycot-fallout and proposals to go beyong that boycot and i will certainly post about that later. At the moment there is still an embargo on some information, but anticipate a statement from the editorial board of the journal of number theory soon…

I was asked to talk about “algebraic D-branes”, probably because it sounded like an appropriate topic for a conference on noncommutative algebraic geometry claiming to have connections with physics. I saw it as an excuse to promote the type of noncommutative geometry i like based on representation schemes.

If you like to see the slides of my talk you can find the handout-version here. They should be pretty self-exploratory, but if you like to read an unedited version of what i intended to tell with every slide you can find that text here.

Almost three decades ago, Yuri Manin submitted the paper “New dimensions in geometry” to the 25th Arbeitstagung, Bonn 1984. It is published in its proceedings, Springer Lecture Notes in Mathematics 1111, 59-101 and there’s a review of the paper available online in the Bulletin of the AMS written by Daniel Burns.

In the introduction Manin makes some highly speculative but inspiring conjectures. He considers the ring

$$\mathbb{Z}[x_1,\ldots,x_m;\xi_1,\ldots,\xi_n]$$

where $\mathbb{Z}$ are the integers, the $\xi_i$ are the “odd” variables anti-commuting among themselves and commuting with the “even” variables $x_j$. To this ring, Manin wants to associate a geometric object of dimension $1+m+n$ where $1$ refers to the “arithmetic dimension”, $m$ to the ordinary geometric dimensions $(x_1,\ldots,x_m)$ and $n$ to the new “odd dimensions” represented by the coordinates $(\xi_1,\ldots,\xi_n)$. Manin writes :

“Before the advent of ringed spaces in the fifties it would have been difficult to say precisely what me mean when we speak about this geometric object. Nowadays we simply define it as an “affine superscheme”, an object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}_2$-graded supercommutative rings.”

Here’s my own image (based on Mumford’s depiction of $\mathsf{Spec}(\mathbb{Z}[x])$) of what Manin calls the three-space-2000, whose plain $x$-axis is supplemented by the set of primes and by the “black arrow”, corresponding to the odd dimension.

Manin speculates : “The message of the picture is intended to be the following metaphysics underlying certain recent developments in geometry: all three types of geometric dimensions are on an equal footing”.

Probably, by the addition “2000″ Manin meant that by the year 2000 we would as easily switch between these three types of dimensions as we were able to draw arithmetic schemes in the mid-80ties. Quod non.

Twelve years into the new millenium we are only able to decode fragments of this. We know that symmetric algebras and exterior algebras (that is the “even” versus the “odd” dimensions) are related by Koszul duality, and that the precise relationship between the arithmetic axis and the geometric axis is the holy grail of geometry over the field with one element.

For aficionados of $\mathbb{F}_1$ there’s this gem by Manin to contemplate :

“Does there exist a group, mixing the arithmetic dimension with the (even) geometric ones?”

Way back in 1984 Manin conjectured : “There is no such group naively, but a ‘category of representations of this group’ may well exist. There may exist also certain correspondence rings (or their representations) between $\mathsf{Spec}(\mathbb{Z})$ and $x$.”

I’ve LaTeXed $48=2 \times 24$ posts into a 114 page booklet Monsters and Moonshine for you to download.

The $24$ ‘Monsters’ posts are (mostly) about finite simple (sporadic) groups : we start with the Scottish solids (hoax?), move on to the 14-15 game groupoid and a new Conway $M_{13}$-sliding game which uses the sporadic Mathieu group $M_{12}$. This Mathieu group appears in musical compositions of Olivier Messiaen and it can be used also to get a winning strategy of ‘mathematical blackjack’. We discuss Galois’ last letter and the simple groups $L_2(5),L_2(7)$ and $L_2(11)$ as well as other Arnold ‘trinities’. We relate these groups to the Klein quartic and the newly discovered ‘buckyball’-curve. Next we investigate the history of the Leech lattice and link to online games based on the Mathieu-groups and Conway’s dotto group. Finally, preparing for moonshine, we discover what the largest sporadic simple group, the Monster-group, sees of the modular group.

The $24$ ‘Moonshine’ posts begin with the history of the Dedekind (or Klein?) tessellation of the upper half plane, useful to determine fundamental domains of subgroups of the modular group $PSL_2(\mathbb{Z})$. We investigate Grothendieck’s theory of ‘dessins d’enfants’ and learn how modular quilts classify the finite index subgroups of the modular group. We find generators of such groups using Farey codes and use those to give a series of simple groups including as special members $L_2(5)$ and the Mathieu-sporadics $M_{12}$ and $M_{24}$ : the ‘iguanodon’-groups. Then we move to McKay-Thompson series and an Easter-day joke pulled by John McKay. Apart from the ‘usual’ monstrous moonshine conjectures (proved by Borcherds) John McKay also observed a strange appearance of $E(8)$ in connection with multiplications of involutions in the Monster-group. We explain Conway’s ‘big picture’ which makes it easy to work with the moonshine groups and use it to describe John Duncan’s solution of the $E(8)$-observation.

I’ll try to improve the internal referencing over the coming weeks/months, include an index and add extra material as we will be studying moonshine for the Mathieu groups as well as a construction of the Monster-group in next semester’s master-seminar. All comments, corrections and suggestions for extra posts are welcome!

If you are interested you can also download two other booklets : The Bourbaki Code (38 pages) containing all Bourbaki-related posts and absolute geometry (63 pages) containing the posts related to the “field with one element” and its connections to (noncommutative) geometry and number theory.

I’ll try to add to the ‘absolute geometry’-booklet the posts from last semester’s master-seminar (which were originally posted at angs@t/angs+) and write some new posts covering the material that so far only exists as prep-notes. The links above will always link to the latest versions of these booklets.

Last fall, Matilde Marcolli gave a course at CalTech entitled Oral Presentation: The (Martial) Art of Giving Talks. The purpose of this course was to teach students “how to effectively communicate their work in seminars and conferences and how to defend it from criticism from the audience”.

The lecture notes contain basic information on the different types of talks and how to prepare them. But they really shine when it comes to spotting the badasses in the public and how to respond to their interference. She identifies 5 badsass-types : the empreror and the hierophant (see below), the chariot (the one with a literal mind, asking continuously for details), the fool (the one who happens to sit in the talk but doesn’t belong there) and the magician (the quick smartass).

I’ll just quote here the description of, and most effective strategy against, the first two badass-types. Please have a look at the whole paper, it is a good read!

“The Emperor is the typical figure of power and authority in a given field. It refers to those people who have a tendency to think that the whole field is their own private property, and in particular that only what they do in the field is important, that the work of all others is derivative and that in any case they are not being quoted enough. These are typically pathological narcissists, so one needs to take this into account in interacting with them.
The trouble of having The Emperor in your audience is that he (it is rarely she) can very easily disrupt your presentation completely, by continuous interruptions, by running his own commentary while you are trying to stay focused on delivering your talk and by distracting the rest of the audience.
The Emperor is by far one of the most dangerous encounters you can make in the wilderness of the conference rooms.”

Counter-measure : “Keep in mind that the Emperor is a pathological narcissist: part of the reason why he keeps interrupting your talk is because he cannot stand the fact that, during those fifty minutes, the attention of the audience is focused on you and not on him. His continuous interruptions and complaints are a way to try to divert the attention of the audience back to him and away from you. That your talk gets disrupted in the process, he could not care the less.
A good way to try to avoid the worst case scenario is to make sure (if you know in advance you may be having the Emperor in the audience) that you arrange in your talk to make frequent references to him and his work. In this way, he will hopefully feel that his need to be at the center of attention is sufficiently satisfied that he can let you continue with your talk. Effectiveness: high.”

“The Hierophant represents a priestly figure. What this refers to here is the type of character who feels entitled to represent (and defend) a certain “orthodoxy”, a certain school of thought, or a certain group of people within the field.
Typically the hierophants are the minions and lackeys of the Emperor, his entourage and fan club, those who think that the Emperor represents the only and true orthodoxy in the field and that anything that is done in a different way should be opposed and suppressed.
These characters are generally less disruptive than the Emperor himself, as they are really only fighting you on someone else’s behalf. Nonetheless, they can sometime manage to seriously disrupt your presentation.”

Counter-measure : “This is essentially the same advise as in the case of the Emperor. To an objection that substantially is of the form: “This is not the right way to do things because this is not what what we do (= what the Emperor does)”, which is what you expect to hear from the Hierophant, you can reply along lines such as: “There is also another approach to this problem, developed by the Emperor and his school, which is a very interesting approach that gave nice and important results. However, this is not what I am talking about today: I am talking here about a different approach, and I will be focusing only on the specific features of this other approach…”
Something along these lines would recognize “their” work without having to make any concession on their approach being the only game in town.
Effectiveness: high (unless the Emperor is also present and is delegating to his hierophants the task of attacking you: in that case they won’t give up so easily and the effectiveness of this line of defense becomes medium/low).”

I’ve never apologized for prolonged periods of blogsilence and have no intention to start now.

But, sometimes you need to expose the things holding you back before you can turn the page and (hopefully) start afresh.

Long time readers of this blog know I’ve often warned against group-think, personality cults and the making of exaggerate claims as possible threats to the survival of noncommutative geometry (for example in the group think post).

However, I was totally unprepared for this comment left on the noncommutative geometry blog, begin October:

Noncommutative Geometry is a field whose history is unpredictable.
When should I expect the pickaxe? Yours, Leon Trotsky

After sharing this on Google+ someone emailed suggesting I’d better have a look at some ‘semi-secret’ blogs. I did spend the better part of that friday going through more than 3 years worth of blogposts and cried my eyes out.

It is sad to read a message in a bottle and notice that after more than two years the matter is still far from resolved.

I wish you all a healing and liberating 2012!