# Friday February 25th

## Crawling through the trenches and Validation

## This week was exhausting

- I was supposed to take this evening off, but somehow that goal didn’t end up happening, but I did make it halfway through a 57-page paper.
- Today we had this snowstorm that pretty much made all of my classes online, which was great because I got to take a power nap just before a meeting. I had a thought that if I ran out of food, I could always eat ice, I guess. I caught up with one of my mentors and one of the things I walked away thinking about was validation. Another mentor who happens to work at the same company said to me this week that they prefer to do more writing versus experiments, and that they choose people who complement the skills they have when working on research. That startled me because in grad school, you constantly feel like you should do it all.
- You are judged in interviews and in giving talks on whether you have technical depth, whether your communication is clear, how many experiments you have done, and on your proofs. I remember most recently silently kicking myself a few weeks ago because I had named the wrong author when someone had asked a question about a paper I had referenced. And this was because the night before, I had been reading quite a few papers and the other author’s name had stuck, from a different paper. Argh.
- You can say that I’m
*relieved*that the more senior you are, the more you learn to embrace your strengths. I definitely have yet to get there. But then again, I’m still working through my specialization, which is hard because I’ve always been one of those persons who draws from multiple things. But already, I’ve felt a lot more focused. One of the things I was also encouraged to do is figure out what “rigour” means to me. One of my fears, as I have told quite a few mentors, is being released out into the world and finding out that I am not where I should be in terms of training. A friend of mine who teaches Theatre Design at a school in NYC told me that she once told her Master’s student that yes, she will go out into the world and make mistakes, but their hope is that they have prepared them enough to bounce back and recover from those mistakes, and to thrive. So my mentor asked me to figure out what is the thing that I really would like to obtain in that sense? I’m still figuring that out, but so far, it is specifically mathematical proofs (nope; not PL or any of that other stuff). In the past, it was tied to “magical thinking”, but certainly the attitude in this department (thankfully) is that anyone can learn it, and it’s a worthwhile skill to learn. I’m almost a bit frustrated that I bought in to this idea that it was “special” to learn and understand proofs in things like PL and Haskell, because in may ways, struggling to learn those meant I was distracted from seeing and learning the beauty of proofs and all the things they*could*be used for (which is sorry to say it, not just for things like Programming Languages and Haskell). And I felt deficient because failing at understanding something that was “obviously a Profunctor” (although no one could really explain it) meant I missed out on the opportunity and space to be wrong and fully understand things. But I have that space here now, with these people. And I feel great, even though it’s a lot of work and yes, it’s exhausting. I feel like, even though I’m still learning, just doing Pure mathematical proofs daily has translated to rigour in other areas of my research and work in general, given me a lot more focus, and I feel legitimately more satisfied in my progress and in feeling challenged. I’ve really enjoyed it because it’s also a pretty common language; there isn’t a need to know things about Haskell or compilers or whatever. And in many ways, a part of me asked “why wasn’t I introduced to it this way before?”. In the past, it has been introduced with noise like having to learn about compilers, types, programming, etc. But it is really a language, as one of my advisors said. You struggle with it and you learn to speak it. It feels pretty universal, even though there is a learning curve, and it translates in a usable way to other aspects of research in a way I haven’t necessarily seen in other skills I have had to pick up. It’s essentially like the yoga of academic training or something. It improves your stamina for problem-solving, too! - So yeah, even if you don’t care about Haskell or functional programming or compilers or any of that stuff, learning to write proofs in general is just a great skill, and you shouldn’t let anyone make you feel badly about learning it. If you want to learn it, just do it!
- One of the things that I have seen is a sliding scale of dependence and that expectation of dependence within schools and departments. In some schools, there is less hand-holding. Some schools have the knowledge and experience to contextualize what setting a student up for success looks like. Others don’t. And I think that this can result in a tipping of the scales in the direction of schools that know what works. And if you are not in one of those institutions, or those departments, or with those advisors, it’s that much harder to catch your stride. But it can be done. I have been realizing the longer I am in grad school that I can’t beat myself up for the hand I was given; I just have to make the best out of it to achieve my goals. In the beginning, it felt like the odds were slim to none; a well-known advisor, a fancy school and coming in with publications already put you at the top of the heap for awards, notoriety, basically a guarantee of future success in grad school. As time progresses, a lot of that stuff matters less. Sticking it out, persisting and finding your people (and “making your own luck”) are worth more.

## Short week

- So this week was short, but we covered:
- Algebra: Chinese Remainder Theorem (R-Algebras, Ideals), Cones, CoCones, Diagrams, Limits, Colimits, equalizers, terminal and initial objects, representable functors, adjoint pairs, tensors (doing tensor product functor next time),germs.
- Number Theory: we covered units, invertibility, zero divisors, equivalence relations.
- Random Probabilistic Graphs: Branching process (generally and in graphs), Galton-Watson. I was a few days ahead on the homework.
- Elliptic Curves: maps of isogenies, degrees of char of function fields, well-behavedness of isogenies (proving), and separable vs inseparable degree, invertibility, injectivity relating to extension over field, Galois, equivalence of categories, Velu’s Formula and began Hurwitz.
- Research with advisor
- I was accepted to this Winter school and placed in a virtual study group on Algebraicity and automorphic forms on unitary groups. I just began reading through the lecture notes, even though the actual seminar doesn’t begin until March. So far things that came up were Automorphic forms as analytic functions on Hermitian symmetric spaces, PEL (Polarization, Endomorphism, Level structure) type data / moduli, Rosati condition (I found a post on the Rosati involution, but still reading through the first pass of the paper), hyperspecial subgroups, tuples that satisy Kottwitz’s determinant condition, (Toroidal) Compactifications, Conditions of the automorphic form that require our function to be holomorphic, the quadratic imaginary field as it relates to the Picard modular group and Picard modular forms and notes on the adelic ring.

## Other things

- I saw an awesome talk on Quantum ML at LXAI at AAAI (which made me think I had an hour more than I actually did, which made me late).
- I saw a really interesting talk on Knot Theory and AI.
- I was invited to review for a conference and connected with two mentors at Msft, who I’ll meet next week!
- I am on a website!
- It might be someone’s birthday very soon :)

## I should go to bed, so that’s it.

Written on February 25, 2022