Math Year One (of two)

At last, I have started a Master’s degree in Mathematics at university (Autónoma de Madrid.) After years of longing to do this, the stars aligned enough.

As happened with my Master’s degree in Computer Science, I’ve been admitted despite coming from a different undergraduate degree. The Master’s academic committee required that those of us coming from engineering or physics degrees should take a prequel year to fill mathematical gaps. In my technical blog I go over a summary of the classes I’ve taken this prequel year. In this little post I want to keep things less detailed.

The subjects this year are at a level of sophistication that proves this is no longer math as number crunching, or as service to physics/engineering. I share these classes with junior and senior undergraduates, and I can see the disappointment in a few of them, as professors define strange new objects and theories.
They probably did not know that mathematics becomes more verbal, and stranger, as it goes deeper.

There are some recurring themes in these classes that are hallmarks of higher math.

• Using sets: Many of us suffered sets during primary education. Venn diagrams, bijections, arrows. To what end?
  In higher math, set-based thinking shifts your point of view in useful ways.

• Inverse functions: as one of my professors noted: we tend to think of functions carrying points to points, or numbers to numbers. It’s natural to think of inverse functions as carrying sets to sets.
  Inverse functions are essential in the definition of continuity, and of integrability, for example.

• Equivalence relations and quotients: what kind of numbers and operations would we get if we made 6 equal to 0? Quotients are used all the time to build new mathematical objects from existing ones. Once you get used to them, they seem natural.

• New objects all the time: In high school and even the first couple of years of university, new mathematical objects appeared only after a lengthy motivation. The gloves are off now, and one routinely creates new mathematical objects.

My professors were good. They knew their stuff. Some of them were friendlier, some more serious, some drier. All of them prepared a coherent syllabus, provided sound bibliography and exercise sets, and made an effort to motivate their classes.
This puts them worlds ahead of the professors I suffered during my undergraduate degree at the Universidad Politécnica de Madrid (said experience was so bad that I decided to palate-cleanse by moving to the US to attend Columbia University.)

Math has a good thing going for it, academically: given math’s emphasis on deduction and logic, professors tend to prove most of the theorems they enunciate. This forces them to slow down and pay attention to their language.

I’m still a full-time employee at a startup, I still write code for a living, but I attended almost all available hours of class. Even after the COVID-19 pandemic forced us to move the lectures online, I attended almost all available classes.

There are many good things about classrooms. There’s listening to professors talking; especially answering questions, and calibrating from that what is assumed to be common knowledge, and what is thought to be a subtle point. There’s hearing classmates talking between classes, there’s seeing panicky expressions after an exam. Ways of talking that you absorb naturally.
And, perhaps, the feeling that you are not out of place.