… as she teaches me the special care necessary when playing variations (don’t study them linearly! focus on structural similarities not visible in the melody! play in a sequence of different ways (eyes closed, fingers lingering not pressing the keys, air playing, repeating note names, mute playing, etc.) each passage…) I start to see the potential dreariness of variations not well played out, the possible drift into vapidness … and by symmetry, the extreme richness and brutally meditative mind state that may be attained when really playing variations linking the various possibilities opposing richness and structural similarity…

the final movement of Hob. XVI 24 (cf. Richter)

Enigma Variations (not the Elgar orchestral piece, but the Aciman novel) is a long-winded, extremely well-crafted extended novella. Aciman takes up the main subject of his now very famous Call Me By Your Name and literally unfolds it through variations in later life, variations of an early, burgeoning sensual/sexual experience of ¿love? that leaves a boy, a man, marked throughout his entire life, and whose many additional loves are lived as variations of some sort of the first (unaware) one. Paolo falls in love (without really knowing it, without even being able to detect it, let alone phrase it, without as much as a language for his feeling of infatuation) with a cabinet-maker, a falegname in an island off the coast of Italy where his family spends summers. Paolo, at twelve, slowly discovers his own love for twenty-something year-old Gianni, for his hands and nails, for his trim frame and green eyes, for his face he doesn’t dare look directly – and in uncovering his own outsidification and othernessifaction ends up building from rough pieces a language for what his eyes, his racing heartbeat, his breath, his arms, his skin hair raising, his balls tickling, his ¿unwanted? erection have already given him the knowledge he cannot yet phrase… This first theme, so reminiscent of Elio’s story in Call Me By Your Name, has later some variations. Alternating love for women and for men, in a kind of odd nod to Virginia Wolff’s Orlando, the rest of Paolo (later Paul in New York)’s loves continue playing a note of untold arousal, mental courting, projection of images, smells, textures that Paul knows are often best left unexpressed. A triumph of the non-explicit (made explicit in Aciman’s prose, of course). An endless set of variations of his early theme.

Beethoven’s Diabelli Variations (mentioned often in Paolo’s conversation with his father in the island in Italy, hummed by both to the exasperation of the mother, as a secret key-code between father and son) – and then Paolo’s understanding of his own father’s infatuation with the same young man that he as a young adolescent lived through – Paolo’s un-judgmental and again implicit camaraderie with the memory of his own father. And the Diabelli underscoring those memories.

Photographic variations (on Finnish glass geometries):

Mathematical variations are always tricky. In some unacknowledged sense, whole swaths of math are really sophisticated variations on themes. But we do not really, we do not truly call them that, we do not truly think in those terms. Usually.

(I feared when first seeing this overhanging Möbius strip that it would be too contrived, too cliché. The Möbius strip is an almost immediate image coming to mind when evoking the main theme of the Salón Nacional de Artistas this year, “the reverse/back of the threading/of the weaving” (el revés de la trama) and the special exhibition Aracne’s Fable under the curatorship of Alejandro Martín. Yet on second view I found this variation on a classical theme, by Adrián Gaitán, very powerful. The heavy physicality provided by the used mattresses, apparently taken from some whorehouse in Cali (at least according to our guide at the exhibition). And that seems to be the case. The mattresses, made of cheap polyester-like material, woven and rewoven and repaired after many uses, bear stains and traces of bodily exertion, of many possible sexual acrobatics but also of sweat and blood, sperm and urine, vaginal and anal secretions, saliva and tears; all those human fluids and traces of people (and suffering and moaning and exploitation and delight, money transactions and childbirths and hopes for the lives of those children) also woven into the fabric, also immanent and impossible to efface. All that heaviness turned by Adrián Gaitán’s variation into a floating symbol of a primal kind of reversion, into a pristine and ideal and immaterial shape.)

Threads and fragments of an inspiring visit.

Boris Zilber visited us for about a week. His was an inspiring, illuminating visit. Conversations took place at the two universities, but also (and most wonderfully) on the mountains surrounding Bogotá: on Saturday up La Cruz from Chía, on Tuesday early morning up to the páramo via La Vieja, on the eastern side of Bogotá. Conversations went from math to life events to math to physics to non-commutative geometry to model theory to math to physics to poetry to math to books to math to nature to math to historical events to geometry to continuity to physics to discontinuity to geometry to life events to history of Russia to Oxford to math to model theory to logic to infinitary logics to abstract elementary classes to number theory to modular functions to math to physics to life events to poetry to Heisenberg to probability to math to model theory to unbounded operators to operator algebras to {\mathbb F}_1 to model theory to categories to model categories to homotopy to physics to math to Oxford to Wiles to Clay to continuous model theory to approximation structures to sheaves to sieves to flavors to Zariski. And so on.

Here are a few fragments and visual threads:

Un seminario tri-direccional

El seminario de Geometría y Teoría de Modelos este semestre está en modo jónico: en contraste con el modo dórico del semestre pasado, cuando todo estaba super-anunciado de antemano y muy organizado, este semestre son puras jam-sessions, improvisaciones según lo que vaya saliendo. Esa alternación entre modos dóricos y jónicos (o mayores y menores, o visibles y bajo perfil, si se quiere) me gusta. Permite mostrarse un poco en los semestres visibles, hacer algo de pasarela matemática, invitar gente, ser visible. Pero luego alterna uno con el trabajo yamsesionado, más libre, con unos pocos estudiantes y el visitante ocasional, y avanza uno en ideas matemáticas durante los semestres menos visibles. Somos unos cinco o seis, pero es realmente muy peculiar que es un seminario que jala en tres direcciones distintas.

Ayer mi charla se llamaba Haces de tipos y tipos de haces y daba los primeros pasos hacia lo que deberá venir después de nuestro artículo con Gabriel Padilla y nuestro artículo con Maicol Ochoa. Iniciamos ahora sí el camino hacia estabilidad (pero desde el principio tratando los espacios de tipos como lo hace la gente de Grupos Compactos: como espacios-moduli o foliaciones o cocientes de haces cuyas secciones son los tipos orbitales, tratando de usar al máximo la geometría desde el principio). Ayer inicié en lo más básico del mundo con un test de Tarski-Vaught para extensiones elementales de haces, luego seguí a tipos-haces y teorías-haces (surgieron varias preguntas técnicas naturales: exactitud y coherencia de los prehaces asociados, universalidad local y \kappa-saturación local de prehaces – y conexiones posibles con el trabajo de Will Boney en su teorema de Łoś generalizado a clases elementales abstractas bajo grandes cardinales).

Pero lo más interesante fue contar con las preguntas de los geómetras (Gabriel y su estudiante Walter Páez, que está realmente a mitad de camino, pues anda estudiando estabilidad en teoría de modelos también) y las preguntas de teoría de Ramsey de Goyo – Goyo logró ver similitud entre mis construcciones y trabajos de Natasha Dobrinen con Stevo Todorcevic – trabajos en que acumulan espacios de Ramsey, van dando mejores y mejores aproximaciones (obviamente, muy combinatorias), controlan ultrafiltros con el orden de Tukey-reducibilidad (aparentemente ortogonal a nuestras clasificaciones, si le creemos a Artem y a John)… y el punto de vista de sistemas dinámicos.

El 20 27 de mayo hablará mi estudiante Johan García (que vendrá de Medellín) sobre formalismo tannakiano – Johan viene de la teoría de la representación (trabajaba con Zavadskyy hasta la muerte súbita de ese profesor) y decidió aterrizar en la teoría de modelos, vía Hrushovski y Kamensky. Creo que será interesante contrastar.

Óscar Muñoz – Autorretrato en agua sobre piedra

It is impossible not to think about geometry (really, some very twisted version, with very variable metric) when riding a taxi in Taxco.

The only cars that fit those streets are “vochos”, i.e. little old Volkswagen beetles that used to pepper Mexico City and now have been banned from there. They gather enormous speed and momentum considering their size. The slopes of Taxco seem to be of at least 45º of inclination, and with so many bends and twists and possibilities of meeting right in front another such taxi, going as well at high speed, particle collision comes to mind. Pedestrians have about 30 cm (really, 20 cm or even less) to avoid the vochos going up so quickly and passing by them without decelerating, at LHC-like speeds. It does take some geometry to master that, I am sure, but also to be a passenger in such little cars and keep sang froid.