A mathematical Christmas

Je songe par example qu’à travers toute mon activité enseignante de mathématicien, j’ai eu à cœur de “faire passer l’étincelle” de la création mathématique, en faisant d’emblée crédit de créativité aux élèves qui eux-mêmes me faisaient confiance en venant faire leur apprentissages avec moi, et en m’efforçant de leur transmettre quelque chose de plus grand prix qu’un savoir-faire et un métier. Force m’est de constater que cet enseignement a été un échec sur toute la ligne, alors même que certains de mes élèves sont devenus des mathématiciens célèbres. Et je me rends compte que ma défaillance, comme celle de chacun sans exception de ceux qui furent mes élèves, ne se place nullement au niveau intellectuel, mais bien au niveau spirituel. C’est la situation que je ne cesse de découvrir et de sonder sous toutes ses faces tout au long de Récoltes et Semailles. Quant à cette “étincelle” que je n’ai su transmettre a aucun, je sais bien qu’elle n’est nullement de nature intellectuelle, qu’elle ne réside ni dans une vivacité ni dans une puissance, ni dans des dons extraordinaires ni dans une méthode irrésistible, main qu’elle es, elle aussi, d’essence spirituelle. - Alexander Grothendieck in La Clef des Songes, footnote 170, Section 46.

A literally allegorically philosophical appeal

It has been a long time since we first met here at the end of November, but understand that mathematical results, as fruits 🍇 🍉 🍋 🍊 🍈 🍎 🍍 🍌 🍑 🍐 🍏 🥝 🍓 🍒 🥭  slowly growing in the branches of a tree 🌳 planted over a rich soil, take their own time to flourish (and even more to ripen). During this period ❄️ , I have been trying several approaches facing the Generalized Lax Conjecture (GLC, around which my project gravitates). Some of them still look promising nowadays in my mind (although not well developed) while others lost part of their attractiveness during time even though we have to remark that every approach followed deserves to be remembered.

And this is true for several reasons. First of all, we are in debt with the next generations of mathematicians that will come after us with the shinning light of the future as much as we are with those that came before us behind the shadows of the past. The explanation for this is that, although having to be done by us linearly as a consequence of our own human limitations, we cannot ignore the fact that the deep, far-reaching interconnectedness of mathematics as well as the jumping nature that its behaviour presents sometimes in contrast with the linear one that we expect to find when we approach to it at a first sight shape a drawing of the panorama of the mathematical landscape that breaks our primary intuitions. Could we seriously deny that Poincaré is in debt with Perelman as much as Perelman is with Poincaré (even though they never met and the French never knew about the Russian)? This question is easier to answer if we interpolate: could we seriously deny that Thurston is in debt with Perelman as much as Perelman is with Thurston? Thus, we owe to those who will read our work (if any) a detailed explanation of our achievements pursuing our goals but also of our failures in the way towards our objectives so they can understand the best and amplest possible our perspectives and choices, and these (all together and at once without hiding anything) have to be the results that we bequeath to the (mathematical) world with which we will always be in debt for all the beauty that showed and brought to us in the first instance. And yes, the beauty is also in the partial, incomplete, erroneous or diverging paths and not only in the clean, ordered, neat, tidy ones that reach to the result immediately and thoroughly. Moreover, nobody knows where an incomplete path leads to and this place in addition to be inherently interesting could be interested for other different purposes.

These people who will follow us in the history of the development of mathematics will have to understand and read what we did as much as we struggle today with the thoughts of those who precede us in this endeavour for the understanding that pushes always our way far from the borders that we inherited, beyond what we take for granted in every generation. And those who will follow us deserve to know that we usually ran in the darkness of the unknown trying to find a switch that could turn on the bulb 💡 of our time enlightening a path that originally was supposed to lead us to a different place or in a distinct direction. However, as the time passes inside this darkness and the failures come, we discover the beauty of the path itself and the revealing fact that mathematics is more a labyrinth of paths without any end than a maze of ends reached through paths. In the path we find the beauty and the truth: the truth of what can and cannot be done. And, when mathematics is understood as a bunch of paths, it is easy to give a sense to what are the theorems, the lemmas and the propositions: they are the flora surrounding the way.

The theorems are the big trees in the middle of the view that can be perceived from far points in the distance or even sometimes from the remoteness of your space (understandable universe) of understanding (we are all able to understand the Fermat's Last Theorem even though its proof is far away from the interval of mathematics we move around usually; by the way, could we seriously deny that Fermat is in debt with Wiles as much as Wiles is with Fermat (even though a wider margin could have abruptly stopped their necessity of intergenerational collaboration and the French never knew about the English)?), and when you are closer to one of these theorems, over the earth deeply dug by its roots, looking directly in front of its big, tall, strong, wide and thick trunk you feel the presence of the robust knowledge of generations of mathematicians condensed in its sap; the lemmas are these small bushes which grow around the big trees giving the stability necessary to the soil so it can support all the weight of the big trees (theorems); and, finally, the propositions are the small fruit trees that provide easily reachable food to the animals that dare to venture into the depth of the forest where there is yet no path.

In all the previous analogy is easy to set a place for the definitions as the same soil of the landscape under which the trees grow roots to keep connected with the earth exactly as theorems fix in the field of what is known through proofs while the nutrients of the soil are the axioms which, from the death of reason and the inert of logic that they represent as the first statements assumed without proof, give birth to the vigorous life of mathematical theories. Every tree has branches that serve as miniaturized samples of its power hiding in their cells the genetic material of the big tree, these are the examples, of course. Then we have some warning posters and remarks that recommend some paths over others whose view is less pleasant or whose walk is more torturous for the traveler.

However, as usually happens when we are follow a path in a forest, we tend to forget about how arduous was the process to make that path and how painful were at the beginning the pricks provoked by the thorns on the legs of the first hikers that crossed through that way arriving sometimes to the depth of the forest with no more than some red stains in the clothes hiding bleeding scars in their broken skin after crossing the new path. And we tend to forget the blood 🩸 and hide the pain in the history as if it never happened: but this is never the real story. The errors, the mistakes are also part of the endeavour and they deserve to be remembered.

Additionally, I affirm that it is healthy for the discipline to point them out and expose them. Not only because this will avoid other people in the future committing the same errors but also as a matter of pursuing the truth, which is our main objective: we cannot hide how mathematics is done in tidy books composed by neat pages of spruced reasonings trimmed away from the bad and less regular (i)logic with which our human brain 🧠 fools and lie us through its pervasive power of persuasion equipped with erroneous but natural-looking intuitions (evil aqueous deception machine!). And, sometimes, more often than we think, some of these erroneous paths hide a beauty that cannot be appreciate by the first people walking through it because the time of the first wayfarer does not usually coincide with the necessary period of maturity of mathematics as a whole to appreciate what lies in front of the eyes of the observer, exactly as happens when children reject the intense bitterness of pure chocolate 🍫 or pure coffee ☕️  in favour of sweeter options (which are basically sugar) because their gustation is still undeveloped or undereducated to appreciate the high quality of a broader family of tastes. I know what you are thinking and, yes, this leads to a point where we can affirm that some mathematics are pure chocolate made under the effect of pure coffee, and that makes me hungry... so I will try to finish this post.

What did I mean with this all this?

Well, the answer should be evident, but let me summarize. During the next posts I will briefly expose my attempts and ideas to attack the GLC. I confess that I have no taste of the trueness of this statement (maybe I can dedicate another post to discuss and explain this position). Hence, I had ideas both to prove and to disprove the GLC. Usually, ideas to prove the GLC are based on the fact that the Lax Conjecture (LC) is true and the hope that then maybe it is possible to build up from there to a proof of the GLC while ideas to disprove the GLC are based mainly in the fact that the collection of rigidly convex sets looks apparently richer than the collection of spectrahedra so it should contain candidates showing a broader variety of behaviours than the ones that are reachable through the use of linear matrix inequalities, i.e., those having spectrahedral representations. However, at the same time, these attempts fail for two different reasons. If there is a way to extend the proof of the LC to the GLC it has to be subtle because the relaxation of the original conjecture that allow this problem to be still open is also subtle and the LC deals with determinantal representation of polynomials while the GLC deals with spectrahedral representation of sets, concepts which are not exactly the same. In the opposite side, the attempts to find rigidly convex sets that are not spectrahedral is connected to the difficulty imposed by the degree and number of variables where it would be possible to find such a counterexample (if it exists) as this polynomial have to be, at least, of degree three in three variables; this complicates even the fact of finding real zero polynomials and determine that in fact they are so. The problem is even bigger when we have to keep control over some specificities of some specific points of a specific rigidly convex that eliminates the possibility for this set to be spectrahedral. Another, tactics that I follow are natural: staying in the border, as it looks more manageable, and going towards more abstract, exotic and/or esoteric structures and theories that could give new information and insights into the problem because those closer to the natural ways to look for an answer seem to be already deeply explored, visited and studied by people with more experience and knowledge than me and who were not able to find a solution within these places (theories). So, all in all, I continue looking for a switch in the darkness, do you want to join me in my endeavour?

A photo before going

The next picture is intended as a clue about the post to come in a few days. I will try to avoid making it very evident… But I think I will not manage to.

«Godi, Fiorenza, poi che se' sì grande
che per mare e per terra batti l'ali,
e per lo 'nferno tuo nome si spande!»