Escoltant... [26/01/17]

• Grandes ciclos, Cuerda y órgano (03/01/17) -- RTVE Radio Clásica
• pastoral en fa mayor (lovely)
• benedictus en re-bemol mayor (choral)
• te deum en la menor
• (Reger, 12 piezas para organo)
• Greisengesang D778 (Schubert, orch. Reger, text by Rückert)
• "Und nur dem Duft der Träume gib Dach und Fach!"

Escoltant... [25/01/17]

• Solo jazz, Jimi Hall revivido (10/01/17) -- RTVE Radio Clasica
• Solo jazz, John Coltrane en el Village Vanguard (09/01/17) -- RTVE Radio Clasica

A few lines from Mill's Autobiography

There was one cardinal point in this training, of which I have already given some indication, and which, more than anything else, was the cause of whatever good it effected. Most boys or youths who have had much knowledge drilled into them, have their mental capacities not strengthened, but overlaid by it. They are crammed with mere facts, and with the opinions or phrases of other people, and these are accepted as a substitute for the power to form opinions of their own; and thus the sons of eminent fathers, who have spared no pains in their education, so often grow up mere parroters of what they have learnt, incapable of using their minds except in the furrows traced for them. Mine, however, was not an education of cram. My father never permitted anything which I learnt to degenerate into a mere exercise of memory. He strove to make the understanding not only go along with every step of the teaching, but, if possible, precede it. Anything which could be found out by thinking I never was told, until I had exhausted my efforts to find it out for myself. [...] In this he seems, and perhaps was, very unreasonable; but I think, only in being angry at my failure. A pupil from whom nothing is ever demanded which he cannot do, never does all he can.

John Stuart Mill, Autobiography (1873) 

I happen to be reading a few works which have sparked again my perpetual curiosity in education and learning at a young age. This time, it began (as always) during one of my many hours spent digging through Wikipedia; I found the article on the Romanes lecture, and the one titled Humanism in Education (given by Sir Richard Claverhouse Jebb in 1899) caught my eye. I read through it, and an urgent desire to gain a more in-depth understanding of Renaissance history overflowed past other lowly duties of the day.

The History of the Renaissance World (by Susan Wise Bauer)² was my choice, and I have been diligently reading it for the past few days. I have also just recently started to translate the aforementioned lecture to Catalan; it is amazing how homesickness reveals a longing for something I hadn't missed before; how I feel much more strongly linked to my homeland, its past and, particularly, its future. These recent explorations have once more inspired me to pursue a betterment of education, not only for my hypothetical child (oh, how dreams of Socratic discourses with an eight-year-old creature trump my desires for his non-existence!) but for the whole of my country. The longer I live in Chicago, the more I feel my roots call, in a truly historical echo of sorts. Not to say that I don't desire such a change in education for the whole world, but the later seems even less likely to occur, sadly.

It was interesting to realize that Susan is a proponent of classical education, a concept which I have not known by its proper name until now, but which happens to align with my views on education. Also, she wrote The Well-Educated Mind, a book I'll keep in mind for my next cultural pursuits in the form of books. Things seem to fall into place, by chance--or by our ignorance of the complex machinery of causality, as Borges once said ¹.

We'll see how it goes; how long I maintain this productivity streak (of which, I shall remember myself, is not measured by quantity nor speed) is an indicator of my true intentions.

Some musical discoveries

I have been doing a summer project at ICFO for the past few days. The two hours of commute allow me to focus on music and nothing else, so I have been listening to some classical music podcasts by RTVE Radio Clásica, mostly Sala de Cámara and Sinfonía de la mañana.

Here is a list of pieces that I have enjoyed, with some updates planned for the near future.

• Sonata violoncelo e piano, Luís de Freitas Branco
• String quartet no. 14 in A# major, Dvořak
• Trio in D minor: Élegie, Arensky
• Introduction and Allegro for Harp, Flute, Clarinet and String Quartet, Ravel
• Todd und Verklärung, Richard Strauss
• Langsamer Satz, Webern
• String quartet no. 6 in F minor, Mendelssohn
• String quartet no. 2: Notturno, Borodin

Where am I and where am I going

'Tis a struggle often fought in my minds.

I have finally fully digested and accepted (or so I like to believe) that I will study at the University of Chicago for the next four years of my life. On a global scale, four years are merely a breeze, a blink of an eye, but I am in years, merely so, a pale leaf whirled about by this great winds. I am excited to jump into adulthood with such a challenge, both intellectual and personal, but there are always buts.

Great hopes and expectations, and doubts and fears are equally distributed in my dreams for the future. I desire to grow into the person that I dream of becoming, and leaving home is by necessity part of the whole package. I have felt many things that I didn't think I would, prompting me to read more into my inner and unconscious thoughts. Homesickness, or the very thought of predicted homesickness is, sadly, very real.

I am reading literature and math, trying to figure out what Love is in a relationship (capitalized to avoid forgetting), and gathering forces to lose all fears.

I end this short update, hopefully with more to come (as always, I strive to write more, but never quite strike), with a small poem by John Keats, always lyrical and evocative:

‘O thou whose face hath felt the Winter’s wind;
Whose eye has seen the Snow clouds hung in Mist
And the black-elm tops ’mong the freezing Stars
To thee the Spring will be a harvest-time—
O thou whose only book has been the light
Of supreme darkness which thou feddest on
Night after night, when Phoebus was away
To thee the Spring shall be a tripple morn—
O fret not after Knowledge—I have none
And yet my song comes native with the warmth
O fret not after Knowledge—I have none
And yet the Evening listens—He who saddens
At thought of Idleness cannot be idle,
And he’s awake who thinks himself asleep.’

"The Dream of a Ridiculous Man"

And yet it's so simple: in one day, in one hour - it could all be set up at once! The main thing is -- love others as yourself, that's the main thing, and it's everything, there's no need for anything else at all: it will immediately be discovered how to set things up. And yet this is merely an old truth, repeated and read a billion times, but still it has never taken root! "The consciousness of life is higher than life, the knowledge of the laws of happiness is higher than happiness" -- that is what must be fought! And I will. If only everyone wants it, everything can be set up at once.
And I found that little girl . . . And I'll go! I'll go!

-- The Dream of a Ridiculous Man, F.M. Dostoyevski

I have been working on some book typesetting, and a few things are almost ready to be shared. This is not one of them though; at least not the English translation.

Understanding convex polytopes (III)

This entry marks the end of my short series of posts regarding convex polytopes and graph-associahedra. I've had lots of fun and learned a lot during this month of research. Here follows a summary of the conclusions of our work.

Understanding the polytope made from $\text{Cycle}_n$ in a constructive manner proved to be harder than anticipated, and we summed up our work in this area with a conjecture relating the $\mathcal{B}$-trees of the cycle on $n+1$ vertices with the lattice paths on the $n \times n$ grid. Recall that the $\mathcal{B}$-trees of the cycle are just binary trees with an extra root at $n+1$, with a certain $\pmod{n+1}$ shift applied to the labels. We can define an equivalence relation based on this operation, and say that two trees $T_1$ and $T_2$ are congruent if they are generated from the same binary tree (with an extra root) $T$. Let $[T]$ denote the equivalence class of $n+1$ trees generated by a canonical binary tree $T$ and the shifting operation. We proceed to define the other side of the relation, lattice paths.

There is a direct construction of lattice paths from $(0,0)$ to $(n,n)$ on the $n \times n$ grid, in particular Dyck paths (those which do not cross the diagonal), from binary trees by expressing a binary tree as a balanced parenthesis sequence. We need to define a parallel of descent for lattice paths, for that is the center of our conjecture. Let $\text{peaks}(P)$ be the number of left-peaks of a lattice path; from the parenthesized construction, a left-peak in the Dyck path is equivalent to a descent in the binary tree. Next, we need to find an operation which serves as parallel for our label shifting on trees. There is a cool operation on Dyck paths (defined by Chen in [1]) which generates a set of $n+1$ lattice paths, and so we define again an equivalence relation based on this operation, with $[P]$ denoting the equivalence class of lattice paths generated from a Dyck path $P$. For distinct $P$ and $P'$, we can show that $[P] \cap [P'] = \varnothing$, and equivalently for our canonical binary trees.

As stated, there is a map $\Phi : \text{binary trees} \rightarrow \text{Dyck paths}$, and we conjecture that the descent generating function of the equivalence class $[T]$ is equal to the descent generating function of the equivalence class $[\Phi(T)]$:

$$\sum_{T' \in [T]} x^{\text{des}(T')} = \sum_{P' \in [\Phi(T)]} x^{\text{peaks}(P')}$$

We verified the conjecture for $n$ up to 13.

On the other hand, we found a combinatorial interpretation for the $\gamma$-polynomial of the cyclohedron. Following the approach of Posntnikov et al. in [2], we defined yet another delicate operation on lattice paths, allowing us to arrive to the desired $\gamma(x) = \sum_{r=0}^{\left \lfloor{\frac{n}{2}}\right \rfloor}\binom{n}{r,r,n-2r} x^r$ in a nice combinatorial way, in contrast to the previous work based on hyper-geometric series manipulations on the $h$-polynomial.

In conclusion, I am extremely happy of the work we've done, considering my very limited exposure to advanced mathematics (and even less to research level mathematics!). I enjoyed the research experience, and the constant unknown factor was such a motivator to keep me going; what's this? What if I do that? Are we doing the right thing? Is there another point of view? I am certainly convinced to pursue mathematics for as long as I can.

References

[1] Chen, Y. M. (2008), The Chung–Feller theorem revisited. Discrete Mathematics, 308(7), 1328-1329.
[2] Postnikov, A., Reiner, V., & Williams, L. (2008). Faces of generalized permutohedra. Doc. Math, 13(207-273), 51.