I was hoping to write this over the Summer, but between studying for my PDE qualifying exam, struggling to make progress in my reading course, and being alive, I did not have much time to do much else, and as usual, I had lots of ideas of things I could do in my free time that did not come to fruition.

(This may be my messiest episode so far because of how much I’m going to attempt summarizing and talking about so please forgive me for that. Also, I’m trying to write these without rereading my previous entries, so if I contradict my past self, it is my personal hope that this means that I am learning and growing from some kind of mistake or misjudgement. Either way, it’ll be more interesting to look back upon one day. )

Thoughts after finishing the coursework portion of the program

I’m actually very glad that the core sequences are over and my qualifying exams are passed (update: I did pass PDEs). Looking back, there are definitely things I could have done to better learn the material and excel in the courses as one would try to do for undergraduate courses. However, between juggling life and TA responsibilities while trying to to get a headstart on reading courses and finding an advisor, I don’t think the time investment is worth it over learning how to better manage all these things together, i.e. understanding and being okay with not doing the best in courses, but beginning to mature into the program/department and finding your style of TAship and reading/communicating with professors. I think the first year is perfectly fine to buckle down in courses and leave other things for later, but the difficulty adjusting and huge ramp up in difficulty made that kind of impossible for me my first year.

I do still feel bad about Algebra because of just how unwilling I was to be vulnerable with my professors and classmates at the time. I’m now more accepting of how it played out, but given the popularity of stuff like algebraic geometry and how far reaching it is in topology (which in turn affects analysis), I do regret not understanding just a little more of it. One day, I’ll go thank Dr. Peter Samuelson for absolutely saving my life that very unfortunate Winter quarter where I learned absolutely nothing. I don’t even think I know what a module is.

I’ve come to understand that to really truly understand the fundamentals of a subject, a huge time investment is absolutely needed for someone like me. To build up deep understanding, one requires both an intuitive understanding of the major results, but also a strong technical understanding of common practices and techniques. These come hand in hand. I came into grad school looking for that post-rigorous and intuitive understanding, but I forgot that you can only reach that through technical mastery. To this, I mean that in order to feel comfortable with a major theorem or result in the sense that you know:

  • Under what situations does it feel correct to try to invoke it
  • How small adjustments to its hypothesis may affect it
  • Intuitively why the theorem is true
  • Under what context led to the development of the theorem

The effort must start with knowing how to prove it by heart. This doesn’t mean coming up with the proof on your own, but it very likely means memorizing the proof to the finest detail and understanding why such details are necessary. Such memorization allows one to then look past the technical detail and begin to build intuition in the sense above. In other words, it’s about no longer needing to constantly reference back to the proof, or a text, so that you can move ahead in your understanding. This also isn’t just about the theorem at hand. Proofs of major results often contain techniques and ideas that are commonplace in that subject.

These ideas came from conversations and comments heard from Dr. Estela Gavosto and Dr. Patricio Gallardo and to some extent my community college professor Raj Misra, who never failed to remind us of the value and importance of memorization. I’m sorry to say that I indeed forgot this. The very blunt way that I re-understood this was during the math education seminar where we were discussing aspects of learning mathematics from the perspective of undergrads. The aspect of “repetition” came up and I understood it in the sense that students do need to grind problem sets to build muscle memory in order to build intuition about patterns or properties of the objects they’re working with (like integrals or limits). I asked how this extended to proofs and Dr. Gavosto’s response was that there are many proofs that fit into certain categories which reuse similar techniques (template problems). I then asked about graduate level material such as understanding a big theorem, and her response was to be able to prove it without looking.

At another time Dr. Gallardo affirmed this stance by talking about this idea of grinding. Certainly, it’s rote and often unenjoyable, but his approach was that it is unfortunately necessary. Mathematics is prized and valued in society. For many students from where he came from (and in many places in general), mathematical ability can literally save them from poverty or a poor trajectory in life. Grinding for understanding, but also speed was a method of survival for some students, and this was another aspect that I could not see at the time. I will speak more about this in the next section.

That being said, I am not trying to say that an eager student must memorize an entire textbook to get anywhere. Major theorems and their proofs are already more than enough on top of regular definitions. The idea is that once memorization occurs, and the inuition arrives, the eventual departure of technical ability and memories won’t be as bad since the inuition generally sticks and so do major techniques and ideas. Thus, proving big results eventually sounds something like “We start with this kind of object, use this technique and this idea and the result is achieved”. The details are lost, but that is now okay.

As for the other aspects of courses, I am still convinced that typing up notes helps me understand them better, if only because I get a second pass at them. It also gives me something nicer to read than my own handwriting and I just like LaTeX. I am still convinced that this advice on homework is likely the optimal approach for me. The only change I’d make is that, eventually, you may have to call it quits and search online or “cheat”. Dr. Amir Moradifam said it well; some problems and exercises are literally results that were developed over decades (or centuries) of professional mathematicians struggling. To then expect a regular kid to come to the idea of the proof sounds ridiculous. Of course, we have the advantage of structured curriculum and scaffolding and it is vital to attempt things independently at first, for some amount of time while attempting the usual good practices (direct, contradiction, contrapositive, reducing the problem to a special case, computing examples, etc). This time is important because you’ll run into trouble and get stuck and frustrated. It is exactly here that you should write down notes about why you’re stuck. Some examples (from an analyst) of what I mean:

  • I can’t quite get past this bound
  • Maybe I don’t understand this definition as well as I think I do
  • I can’t control the size of this object
  • I don’t have enough information about this set

In a similar sense, you should also write down your hopes and dreams about the problem:

  • If only this inequality went the other way
  • I wish this object had this property I could take advantage of
  • I hope this set contains this type of element

By writing these notes, you give structure and shape to the point at which you can no longer make progress without a frustrating amount of time. I don’t mean to say that you’ll never spend a long period of time working on a problem; that’s exactly what research is. But to spend such long frustrating amounds of time with nothing to show and only negative feels building; that will only lead to burnout. Use this process as a form of catharsis and then go out and find answers to these. Look up solutions, talk to classmates and professors. Then go back and respond to your list.

  • I couldn’t get past this bound because I was comparing to an object that was too big
  • I misread/misinterpretted a definition or had the wrong technique in mind
  • There was a stupid trick, idea, or lemma that came out of the blue, but now I know it!
  • This hope I had turned out to be true with this justification. Perhaps I should keep this inutition in mind for similar situations later on.

The last two are especially important. If you have a taste for something, even if you don’t know how to prove it, once the proof in known, it only confirms that you had the correct “sense”, which will be more important for research later on. Also sometimes, there are things we’ll just never had a good inutuition about so we just trust those that came up with it and add it to our arsenal. Everyone’s brains work differently in math and that is part of the beauty of it (unless it’s not beautiful).

Thoughts on my reading course and moving slowly towards research

I’m still reading with Dr. Yat Tin Chow and I’m enjoying it a lot more than at first. I was kind of scared at first with Evan’s optimal control notes because it felt weirdly not rigorous as if it was intended for engineering students or something (sorry to any engineers). It turned out I was correct. The notes on viscosity solutions to the Hamilton Jacobi equation was more interesting, but I still have a distaste for straight up PDE theory (regularity and existence kind of stuff). We’ve since moved onto the topic of optimal transport, which is especially more difficult than the prior two set of notes. The book is well written and a lot of background is included, in part because it’s graduate level material, but also potentially because it’s an applied subject. The variety of tools we’re using feels very large and overwhelming at times which often leaves me “lost in the trees”, unable to see the forest. Realistically, it’s nothing too unusual like a lot of measure theory, a good amount of convex analysis and some basic notions from differential geometry (i.e. the topic of manifolds and nothing too crazy beyond that). The convex analysis was new to me, but it’s not quite graduate level stuff, so it’s been very approachable with the exception of understanding Rockafellar’s duality result.

As I’m reading, I’m trying to go through all the proofs written, adding details for myself where necessary, and prove everything that doesn’t have a proof. Dr. Chow has stated that this is the only way to build mastery in a field you intend to do research in as opposed to doing more skimming and grasping just the big ideas for fields relatively distant to what they want to specialize in (certainly you could try to specialize in everything). I’d actually be happy sticking with optimal transport since it does have a good amount of measure theory and I’m starting to appreciate convex analysis, but I’m interested in seeing where Dr. Chow will take me.

As of recently, he has suggested I try talking to an additional advisor, Dr. Alpar Meszaros from the University of Durham in the UK (as of me writing this). Apparently, they were postdocs at UCLA together under Wilfrid Gangbo. I’ve seen one research talk by Dr. Meszaros where I understood nothing and zoned out, but I did email him a long time ago asking for references on geometric measure theory (at a time when I thought I knew what that was and was interested in it), and he pointed me towards a kind of hybrid of that with calculus of variations (which is one of the things he specializes in). Ironically, the author of the optimal transport book is Dr. Meszaros’ advisor, Filippo Santambrogio. It’s all very exciting and anxiety-inducing, so I’m now more motived and terrified to hurry and finish understanding the first chapter of this book.

Some concerns I currently have:

  • Of course, I have great fear of sounding clueless in front of Dr. Meszaros, so I’ll declare myself clueless the first chance I get.
  • I’m well aware that I’m a slow student, but I still get frustrated at times with my pace. I wish I could be just a bit faster.
  • As I said above, I feel like I’m getting lost in the trees without seeing the forest of what I’m reading. I like to attribute a story to the mathematics I’m doing, combining intuition with history (which are often the same), but other than rereading the wikipedia page for optimal transport, I still don’t feel like I’m standing on a particularly dense foundation. It still feels like I was just handed this particular problem and started reading about the theoretical underpinnings of it without much motivation as to why this problem is important and why it takes the form it does. Most of this has to do with my reluctance to let go of structured curriculum. Everything built on itself throughout undergrad mathematics, or at least started from a trivial enough point where I didn’t need the “backstory”. Reading for research definitely gives off that feeling of being lost among trees, and I know it’s part of my job to fill in the forest and get a good wide view of it all, but I’ve never developed that skill.
  • In the previous section, I mentioned grinding for the sake of understanding. I’m not sure if there are things I should be doing or not in that context. Should I be rewriting important proofs for stuff like the Arzela-Ascoli theorem, or the duality result and really taking those to heart. Or are there computational things I should be doing to gain some kind of intuition about certain objects, like the support of a measure, weak, weak-* convergence of sequences of measures, or calculating the c-transform or convex conjugate of various functions. I feel like Dr. Chow would tell me a yes and no to this as it seems to be a big waste of time, but it is also important that I have a good understanding of these things.
  • In the same context, I still struggle, at times very embarrassingly, to come up with simple examples of things (like a measurable map whose preimage isn’t surjective onto the full source sigma algebra). I’ve been trying to compile a list of examples of various phenomena in analysis, but again, it’s slow-going.

To finish off, there was one thing I was particularly proud of accomplishing in my reading course, but I’ll link to that here.

Thanks.

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