My Math Beginnings

I have been doing mathematics since I am very young because parents were both teachers and researchers in science. My mother herself studied a bit of an undergrad in mathematics before changing paths. I tried to read several math books that were in the shelf of my grandfather and there are some that I remember quite distinctively: The Theory of Matrices of Gantmacher and Principles of Real Analysis from Rudin.

The challenge to understand these two books was what marked my math childhood: for me math books had to be like this and became an aspiration. This transformed into a rejection of “introductions”, books should go to the point demanding everything from the reader. It took me several years to overcome this idea. If a book said Introduction to…I rejected it immediately.

However, the result I remember the most from those years was presented in some of the math books we had from MIR publishers. They had very interesting titles and I spent a very large portion of my teenhood collecting their books. Back then, when I was around nine years old, I was reading Equivalent and Equidecomposable Figures by V.G. Boltianski. The result being that two plane figures with the same area can be cut into a finite number of pieces and reordered one into the other. This is called Bolyai-Gerwien Theorem. I found it quite impressive. This led me to read about Hilbert’s Third Problem on the similar questions on polyhedrons and the very famous Dehn Invariant. I found back then all these sorts of topics quite amusing.

However, myself as a student of mathematics, I wasn’t particularly good. I remember I struggled to death with division with decimal when I was in sixth grade of primary school.

The Olympiads

During my teenhood years I started to compete in Mathematics Olympiads. When I was in junior highschool I used to study a lot by myself: I usually would be reading books on physics, chemistry and algebra or geometry. I would also read a lot of history.

As I’ve been saying I found the topics I used to read interesting, but I cannot say I really understood the depth of it, let alone use it myself. It was like superficially understanding a novel but then forgetting the plot. In second year (what we called eight grade) we started studying analytic geometry and I just did not understand what the point of it was. I could plot points in the (x, y)-plane but who knows for what reason that was relevant…

However, one day I was returning from school and I mentioned this to my father, and he told me: well, this is just the equation of a line. That didn’t make sense to me, so I pushed back. After a back and forth I understood what he meant and then he asked me: so, if you solve the system of equations, what is the solution? I had no idea. While he ate his salad he told me: well, it’s the point where the two lines intersect. I think that has been one of the most brutal mind-blowns I have had, suddenly all algebra and geometry made sense to me.

This led me to study more deeply analytic geometry and conics and higher degree equations, as well as deeper classical and non-classical geometry, and eventually I reached calculus. I devoured the books of James Stewart of Calculus (which people for some really dislike many times, but I like them…). I used to compete with my father to see who could integrate or evaluate a series faster: I lost every single time. Sometimes I was so annoyed at it that I had meltdowns (I was fifteen!).

In any case, in school this change of demeanor toward the subject was noticed. They knew I was good at math but not that I cared about it. Once they saw I did care, they sent me to participate in some contests. These were very weird questions with very weird formatting, and I cannot say I understood what they were trying to achieve. I do have two vivid memories of this, however. On one occasion I went, completely unprepared, to the state math Olympiad (I did not know that is what I was going to) and we were asked “which of the following triples cannot be the side lengths of a triangle?”. I had no idea how to answer that question, so I approached a professor (which I can only described as very Einstein looking) about it. He explained: well, it doesn’t close, it’s the triangle inequality, right? *Sips coffee and leaves*

The other event I remember vividly was when I participated in a math contest organized by a university. The winners would get a full scholarship. While I was trying to solve this exam one question required an algebra trick, that is very easy if you have done these things before, but I hadn’t. I discovered the trick during the test and almost pass out of excitement. I started to explain my thought process, in what I cannot describe as something else but an essay on madness. Later in the month I was declared the first-place winner and professors at the university told my father that they were very impressed by my explanation methods. I didn’t use the scholarship though.

When participating in one of these contests we saw that the International Mathematical Olympiad (IMO) was going to take place in Mexico. We have never heard of this before. I thought it was an event you just register to and go, so I wanted to try and see what one does there. I discovered very soon that is not how you get into the IMO and that there is a whole process to follow. Thus, it became my teenhood math goal to participate in this Olympiad.

I succeeded in this, and I participated in the International Mathematical Olympiad 2018 in Madrid, Spain. I got an honorable mention. I was close to the bronze medal, but the second day I was so unbelievably intoxicated from some bad food the previous day that I thought I would die on the Olympiad table. They actually told me to go to the doctor instead of taking the test, but I refused. I would die on the table if needed to. I continue to believe I would have won silver medal (or at least be very close) if I had not almost died from tartar meat poisoning.

Number Theory, Geometry and the quest for Fermat’s Last Theorem

During the Olympiad trainings I was told I should read two books. One of them was Fermat’s Last Theorem by Simon Singh and the other was Whom the God’s Love by Leopold Infeld. These were very exciting books for me to read, specially the second one, which is unbelievably thrilling. I wanted to know more about the topics and stories they shared.

Of course, within mathematics, the story of Fermat’s Last Theorem and the eventual dramatic proof by Andrew Wiles in 1994 is very romantic. I was quite interested in learning about the proof and how it worked. While still in the Olympiad I devised a plan (which now I can say was very naive…but I was seventeen) of how we would study slowly the proof of this result. I would organize a seminar every three or four months on some different topic, every time moving toward the proof and its requirements.

In this way I taught so many seminars, which were very intense: we would meet at 9 am and work until 2 pm, with only a 20-minute break at noon. And this could be every day, including weekends sometimes, for up to two weeks. I organized seminars like this on: linear algebra, complex analysis, group theory, set theory, differential geometry, differential equations, algebraic number theory, combinatorics, Riemannian geometry. It became a common thing for my friends to receive an email from me with the title: I want to organize a seminar. I was so intense that sometimes I would even put them tests and design homeworks for them. Of course, we never really approached the real proof of Fermat’s Last Theorem but we did learn a lot about mathematics and hard work.

In this process I myself became very interested as well in geometry. In my undergrad there was no number theory, but geometry was big. Thus, I started to engage with differential geometry and soon with spin geometry. I ended working in spin geometry for my bachelor thesis, following the suggestion of my advisor to extend the formula of Schroedinger-Lichnerowicz to other cases he had been studying. That thesis won me the best undergraduate thesis award in Mexico and also led to my first paper (all my thesis was one paragraph withing a larger paper) with my advisor. To be fair it was his paper with a paragraph on my thesis 🙂

I thought for a long time I would be a geometer.

Knot Theory

In the Olympiad we were taught many different ideas of mathematics. One of them was the search of invariants to prove that two objects are really different (i.e. invariants as a way to distinguish objects). One of the classical examples of this comes from Knot theory and it consists of proving that the trefoil is indeed not the trivial knot. It uses the fact that tricolorability is a knot invariant.

For us, back then, it was just one example within an enormous set of examples on this idea. However, I found the idea of knots quite fascinating. I started to study knots by myself from sources I found online. I think the theory of knots is incredible. Furthermore, it was an area that is very active in the Research Center where I studied undergrad. While I was there, I studied knot theory and participated in some of the events around the area.

I still find them captivating: both for their mathematical versality, but also because of their potential to be drawn. The beautiful pictures, and sculptures, one can create out of knots is very attractive to me. Through the years I continue to study these objects and to engage newcomers to the math world with knot theory, which I find is an area that is welcoming without being dismissive of the talents of the public (i.e. it is not childish).

Fractals and Dynamics

In the Olympiads it is common to talk about the hot problems there are in math. One of the classical ones to discuss is Collatz Conjecture: pick a positive integer n, if it is even divide it by 2 and if it is odd multiply it by 3 and add 1. Prove that no matter the value of n, if you repeat this you reach 1 eventually. This was my first exposure to a discrete dynamics problem, and I spent a lot of time doing cycles and trying to find patterns on this and achieving absolutely nothing out of it!

However, when studying this I reached the famous Collatz Fractal, which is the fractal you obtain when you expand this system to the complex numbers. It is a very impressive fractal and, as far as I remember, the first one I saw that was a drawing in the complex plane. By then, of course, I had knowledge of the Sierpinski Carpet, the Koch Curve, and the Cantor Sets because they are usually put as exercises to compute areas and perimeters by induction (i.e. exercises of induction and also to impress young students with objects of limited area but infinite perimeter!).

Fractals have a beauty that I find hard to describe. I really like them, but it is the facts that one can say about them what is incredibly remarkable. Again, in my undergrad, Dynamics and Fractals are areas that are studied by the faculty of the research center. In truth, I was supposed to take complex variables with the undergrads, but I asked permission (and got it granted) to take it with the Ph.D. students because the teacher was a researcher on Dynamics and the Mandelbrot Set.

In the second part of that course, we followed the book One Dimensional Dynamics by Welington De Melo. It was here where I learned about the Wandering Domain questions and the Non-wandering Domain theorem of Sullivan. I also learned about the theory of the Fatou Components and their classification. The struggle to find a Herman ring was fascinating to me! The idea of the wandering domain has motivated through the years in doing cartoons but also to write poetry. When you remove the mathematics from the wandering domains, what remains is a deeply human concept of very strong nostalgic dimension.

I still read about fractals and try to find them whenever I can.

Toronto and The Trace Formula

When I arrived to the University of Toronto I did not know what I was going to end up doing. I knew I could follow Knot theory here as well as Dynamics and Fractals. I thought however that I would do geometry, which is also a big area here. However, in the second term of my master’s degree I took a course related to the Langlands Program. I originally took this course because its name was Representations of semisimple Lie groups, and I though it related very well with geometry.

A friend of mine told me that it actually was a course on the Langlands Program. I have heard of this program many times before, because of Fermat’s Last Theorem and Number Theory, but I did not really know what it was. I had also heard the story of the Fundamental Lemma superficially, but I did not understand what it meant. I took the course and then continued to do my masters project with the same material. The topic was on The Langlands Classification.

This turn out to be the first of a series of courses in The Trace Formula and Beyond Endoscopy (I had no idea what any of this was). However, when I looked for what The Trace Formula was, I found out that it was crucially used (in the way of the Jacquet Langlands Correspondence) for the proof of Fermat’s Last Theorem. I didn’t need more convincing.

I started to work on this almost immediately after my master’s degree concluded and the PhD started. At the end of my first term of the PhD I had a mission: understand and generalize, whichever ways are possible, the work of Altug. Since then, I have been studying the Langlands program and Beyond Endoscopy in many different ways.

Mathematics, Art and Storytelling

Great part of what informs my mathematics is the dimension it has to be colorful (in the literal sense of the word). Also, I feel that mathematics has a great potential for arts in a way that shows the endeavor of the mathematician and its mythologies, as opposed to only the “beauty” of what you can do with them.

This has motivated several of the cartoons and short stories I have created. It has also helped me to develop characters in what I like to think is the mythology of mathematics. To the very least it has helped me to give language, even if it is only to myself, to what I see within mathematics that are not the mathematics themselves. Rather: the mathematicians, their environment, their ways, fears and joys.

In particular, one of the conclusions that I have had through my experience is that mathematics is worth doing despite the fact that they are not always beautiful, and they are not always fun. The mathematics path is extremely painful at times, in truth, and yet there is something about them that is luring. I think this dimension is better explored with art and stories than with the mathematics itself.