### Experiments in Learning Math

#### by ayushkhaitan3437

I’ve been away for some time. I went to India for a month and a half to attend my sister’s wedding. I could only access internet regularly from my mobile phone, and hence couldn’t blog about what I’ve been up to. Now that I’m back in the United States, I intend to make up for that deficit.

During the holidays, at least in the first half, I got some time to reflect on the Math that I know and understand. I suddenly realized that although I do have a passing familiarity with some areas of Math, there are very few things in it that I deeply understand. Most of my mathematical knowledge comprises of words that I may have seen somewhere but do not understand, and theorems whose statements I may have crammed but don’t know how to prove. I would stick my neck out and say that this is true for many math students, at least in the initial stages.

I spent a lot of time studying Hartshorne’s Chapter 2 on schemes, and other books including Bredon’s book on Geometry. After a couple of weeks of reading those, I can safely say that I retain only a skeleton of the statements of the theorems. This statement is broadly true for most math that I’ve studied.

I then adopted a slightly different approach. I would take a topic that I had a passing familiarity with, and then try to develop it on my own. I spent a considerable amount of time stating and proving Louiville’s theorem, the fact that holomorphic functions defined on an open set have a zero integral around a triangle, etc. I felt that for the first time in my life, I really understood these fairly elementary theorems. I have been trying this method for about a month. I can safely say that I understand Math much better than I used to. I try to make minimal use of books to learn facts. I first try and develop concepts, definitions, theorems and proofs, and then refer to books to try and see where I went wrong.

This approach has especially come in handy in understanding more abstract forms of Math like Algebraic Geometry and Category Theory. These disciplines are infamous for seemingly arbitrary definitions. In developing these concepts on my own, I could sometimes see the need for said definitions by comparing them with other possible definitions. Dry and unmotivated Math often pushes students away from certain fields, which is a tragedy as those techniques could have been useful in many fields of research. Trying to develop those concepts independently somewhat attenuates this problem, and gives the prospective mathematician his/her own perspective on things. Which is perhaps the most useful thing he/she could have.

More important than understanding Math better, is the fact that now I enjoy Math and other disciplines more. I feel an urge to take an exercise book, develop whatever trivial/obvious concepts I can, and see how far I can go. I don’t yet know whether this is a good way to learn Math….only time will tell. However, this certainly is a much much more enjoyable way.

You probably already know this, but an old edition of Lang’s Algebra included the exercise “Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book.” This was partly because he was too lazy or too busy to write the chapter on homological algebra, but it is also because the advice is not that bad.

Proving all the theorems in a book or completely recreating a field for yourself is incredibly time-intensive, however, and so I think it ought to be used sparingly. Trying to prove theorems while reading a book before looking at the proof is a good idea. Striving to understand material to the point that definitions, proof steps, and even results seem intuitive is an excellent idea as well. I frequently find myself at a blackboard for hours playing with some idea to achieve this end. This has been my experience with lectures, textbooks, course notes, research monographs, and papers.

This sort of thing is also good practice for reading research works, which often contain no exercises, can skip many steps, and sometimes, sadly, fail to motivate or to explain adequately their ideas. Making a new idea seem obvious for yourself, sketching your own examples, and developing your own conjectures are all incredibly powerful aids to understanding.