My UCLA life
Spring Quarter 2024
A Mathematician’s Delight
Today is one of the days I feel like I am a mathematician. I immersed myself from 9am to midnight in math without getting disturbed other than getting food, and when I realized it was late night, I actually felt slightly sad I was running out of time because I could go on for another 12 hours digging into papers and books and there were such an abundance of resources out there that I felt bad for not having enough time to go through all of them.
During the afternoon I spent my time sitting down and actually working on my complex analysis homework without rushing to look at solutions. It was about automorphisms of the unit disc and upper half plane, and the Schwarz lemma which I found to be really powerful. I felt really good in actually working the questions out on my own and I thought the time was well spent since I needed to strengthen myself in being able to come up with independent solutions.
For the rest of the day I read some exposition papers by J.P. May and discovered his wonderful archive full of useful notes in algebra and topology, which I will see myself relishing in the summer. I don’t exactly remember how I came across his notes (p.s. I remember it now, I was reading his REU student’s article on proofs of the freeness of subgroups of free groups, and I searched up May’s website), but I remembered curiousity directed me to search up his profile because I had had always heard of his name from Logan and others but I knew embarassingly not too much from him. Anyways, his notes are extremely terse and concise - each 10-page article could make up a quarter’s length of course on the subject. I will see myself learning a lot from these notes and returning to them in months or years as a more experienced mathematics student.
After cruising on May’s notes and various internet resources, I feel like my goals in math for the rest of the year is in their formation. Before stating them, I want to take a moment to reflect on what I learned this quarter.
The subject that I spent most intellectual power in the quarter is Algebraic Topology (the homological algebra I have learned are, strictly speaking, a subset of Algebraic Topology). In particular, I looked at the fundamentals of classical homotopy theory, homology theory, derived categories, and triangulated catgeories. Unashamedly, I fell into the slippery slope of wanting to learn more about (co)homological theories, higher algebra and homotopical theory, number theory, etc. The classes really opened my eyes to another level of abstraction and theory building, but at the same time I feel like 1) jeez, there’s so much more to learn, how did all the graduate students do that?!!, and 2) my foundation is still quite shaky. (for example, if you tell me to compute the induced map between homologies of a singular chain complex, I might likely be getting miserably lost.) I don’t know how much my math skills have improved, but at least I am now much more comfortable when people throw around terms like tensor product, derived catgories, (co)homologies, or other abstract nonsense.
My side quest in math for this quarter is learning classical complex analysis theory. I have at least learned them once before in Terry’s 246A, but this time actually taking a class on complex analysis and working on the homework made me more at ease in using these classical results (admittedly, I didn’t know how to do contour integration before taking 132H). Although complex analysis is not directly related to my interests right now, it is tied closely with algebraic geometry and functional analysis, whose complexity and beauty never fails to amaze me, so the time building up the foundation before getting into those topics was well spent. Afterall, I think I need to diversify my knowledge in mathematics before diving into abstract nonsense, in the sense that at least I am comfortable with measure theory, complex analysis, functional analysis, fourier analysis - at least the basics of them. For areas more related to algebraic topology, I will also need to read on algebraic number theory, p-adic theory, manifold theory, and combinatorics (very important!!). I am guessing that the rule of thumb of choosing classes, summarizing from Logan and my experience, is at least 1 class every quarter in something not-too-tangentially-related to my field of interest.
Before getting back to my math goals for the rest of the year, I want to mention that my biggest takeaway this quarter is not how much I learned in math, but the connections I formed with people in the mathematical community. Taking graduate classes opened the opportunities to me to hangout and discuss mathematics with graduate students. Every graduate student is obviously way advanced than me, but they are willing to accept me as a peer to learn with them as equals, which I am very grateful of. I was intimidated by them at first when I knew nothing, but when I start taking graduate classes I realized that oh, actually I am somewhat starting to qualify in forming my own opinions in math, and that gave me the confidence to go up and meet many of the graduate students. I also went to several really fun graduate student socials (thanks, Logan!), with the most unforgettable one being a day-long seminar in Algebraic Topology that included a dinner served with fine wine and a hike in the Malibu mountains with Ben and some other graduate students. It reminded me of the response I receive when in an occasion I asked a physics graduate student an advice they wish to know in undergrad, “be silly”. Math is a vast subject that takes multiple decades to even master a very specialized field, so it is very important to chill and have a good time while doing math. Not to mention that the benefit of connecting with fellow math-minded people is that you get to know what people in math are doing nowadays and keep up with the trend, and also learn about more opportunities.
Okay, that was a long excursion. After a quarter of exploring, the things I want to learn becomes clear: overarchingly, I want to study Algebraic Toplogy at the 227A level (Hatcher Ch.3,4 + Spectral Sequences). With Merkurjev’s class in the bag, I should be equipped with enough homological algebra machinery, and the biggest constraint for me is just time. I will also be doing basic exam and the AT part of the G&T qual past problems after Summer begins. On the side, I kind of want to keep a little side quest in analysis going, so I will try to work on Tao’s measure theory notes. The goal is not to rush through the theory fast, but rather to sit down and think and work out all the exercises. These three endeavours should occupy me most of the time outside the four classes I will be taking in summer, including an intro to computer science class which I am looking forward to (should be pretty interesting!), and I will continue these endeavours into fall. They will accumulate into the goals of 1) taking 227B in Winter 2025 and 2) passing the Geometry and topology qual and the basic next March.
Winter Quarter 2024
Math Wrap: Exploring the Majesty of Modern Mathematics
“The more I learned, the more conscious did I become of the fact that I was ridiculous. So that for me my years of hard work at the university seem in the end to have existed for the sole purpose of demonstrating and proving to me, the more deeply engrossed I became in my studies, that I was an utterly absurd person.” - Fyodor Dostoevsky, The Dream of a Ridiculous Man
In reflecting on my mathematical adventures this quarter, I have this paradoxical feeling: while I’m proud my knowledge did greatly advance both conceptually and mathematically, I feel truly like an imposter here because there were too much opportunities in UCLA that I was not able to make use of (alas, 24 hours a day is very limited), and the more I learned, the more I felt ignorant in front of the vast universe.
I took 3 math classes this quarter: 110BH Rings and Modules by Dr. Alexander Merkurjev, 32BH Integral Multivariable Calculus by Dr. Richard Wong, and 121 Topology by Dr. Ko Honda. In retrospect, these classes were quite solid and suited my level, but I did not feel satisfied during the quarter of my classes, because I wanted to take graduate class instead. I partially succeeded: I enrolled in Dr. Michael Hitrik’s 255A Functional Analysis, but it eventually became too overwhelming for me that I decided to drop the class. Functional Analysis is a very interesting subject though and it has a special place in my heart. After dropping Hitrik’s class though, I freed up so much time that I did a lot of learning and could have time to further my hobbies outside school.
My personal triumph in math this quarter came from my attempt to understand K-theory, under a directed reading program led by Ben Spitz, an excellent mentor. We met weekly to talk about Weibel’s K-book, which I think we went through around 30% of the first two chapters. K-theory is extremely hard but also extremely rewarding. It concerns the groups K0, K1, K2, …, which apparently are important invariants in many areas of algebra and number theory. The grand goal of algebra concerns classification of spaces and objects, and you want to tell if one space is “the same” as the other by looking at “invariants”, i.e. properties that do not change. You could tell, for example, that a donut is different from a sphere because the former has one hole but the latter does not. This is one of the invariants (Euler Characteristic) in topology. K-theory studies an important invariant that influences algebraic geometry, algebraic topology, and much more. I spent a lot of time understanding sheaves and projective modules, and the subject is really quite beautiful once the different perspectives align.
Besides this, I was so pleased to be able to attend seminars and colloquia from a world-class university. They supplied me with immense knowledge and joy that I could not have conceived of if I stayed in a local university in Hong Kong. I could list some of them here,
Symmetric Tensor Categories, by Pavel Etingof from MIT. It’s a new approach to study representations by considering the category of representations altogether instead of individual representations. I think this notion of considering the entire structure of an object reoccurs extremely frequently, for example in algebraic topology one considers the group of covering transformations of a covering space instead of a single covering space.
An Introduction to the Langlands Program, by Jukka Keranen from UCLA.
Homotopy Groups of Spheres, by Zhouli Xu from UCSD. He gave an excellent introduction to stable homotopy theory (namely, it arises from looking at the patterns of the higher homotopy groups of S^n) and motivic homotopy. Homotopy groups of spheres in general are extremely hard to compute (a ton of spectral sequences…), but they are important because of geometric considerations: the Generalized Poincare conjecture states that any closed smooth manifold of dimension n is diffeomorphic to S^n. The case n = 7 is false, demonstrated by Milnor’s exotic 7-sphere which has 28 different smooth structures. Thus the questions that homotopy groups of spheres could answer are, for which n is there a unique smooth structure on S^n, and how many smooth structures are there? I think this is a very fascinating area of mathematics which is extremely deep.
Algebraic Topology Participating Seminar Series, run by Undergraduate and Graduate students every Friday. The topic was Algebraic K-Theory, and I remember Zach talking about Milnor’s definition of higher K-groups and Logan talking about Topological Hochschild Homology and much more, all of them intensively my brain power.
Graduate Student Seminars on every Thursday.
I could elaborate on every talk I went to, but it would be too time-consuming for me and would defeat layman readers. Sometimes I would walk into an algebra seminar and understand literally 0% of what they were saying, but it is okay. It might sound masochistic to say I “enjoyed” those talks, but I certainly do think they bulked me up in my mathematical maturity and made me more antifragile when I come across an entirely new topic. I still think they’re great to go, suffer, and learn.
This quarter can, I think, be best summarized by my wrestling with Algebra. I struggled learning it, but also had an existential crisis in doing pure math that still affects me. In my mind, the uselessness of algebra outside of pure math has made me reflect countless times on whether I should continue in the path of understanding the great aesthetic platonic universe when the world is burning and people are getting killed by war and disaster. It has also given me many moments of happiness when I discovered marvelous connections, but it also kept tormenting me when I thought I could study something more “useful”.
Update (03/31/2024): I believe that “good” mathematics has to be of some value to the well-being to humanity. This feeling of the disattachment between “mind” and “body” dissipates, when a few days after writing this blog, I encounter John Baez’s wonderful webpage. He works in mathematical physics, but is also very knowledgeable in Algebraic Topology. He uses his research in category theory to build networks and help others, and also uses his knowledge to deal with environmental issues. I want to be the kind of person like him. I realized that actually the things that I am studying can be useful to humanity. This makes me motivated into going into pure mathematics again.
Fall Quarter 2023
Thrilling New World: Math at UCLA
I took 4 classes in this quarter - one class more than what a typical freshman would take. All 3 of my math classes (32AH, 110AH, 115AH) were honors, 2 of them were upper divisions and demanded maturity with proof writing. Even though I was concerned about completing them, I managed to stick to my schedule and clutch my finals to get a decent grade.
I do not advise anyone to follow my path without sufficient preparation, for the mastery of these materials requires dedication of time, but I think my act of taking these hard classes illustrated just my general attitude of studying in UCLA - craving for knowledge! UCLA is a wonderful academic institution to study math. We have objectively the best professors in the world, very helpful graduate students, and advanced and enthusiastic undergraduate students. There is simply no excuse not to absorb knowledge greedily like a sponge. These three months were more intellectually stimulating than all of my previous 17 years of life combined. I tried to make the most of learning in my busy schedule by auditing graduate classes. I audited Prof. Paul Balmer’s 210A Algebra for 4 weeks: we went through category theory and I still have a headache when thinking of the Yoneda Lemma and (co)limit/adjoints… the hell to who invented that abstract nonsense! That said, I do like Balmer’s lecture style, in front of the blackboard he is the artist and showman, displaying everything aesthetically. Of all professor’s boardwork I like Prof. Alexander Merkurjev the most. He did not write much on the board: my friend joked that he only needed to bring one paper to the class because he could copy down everything from the board in that paper. However, every word he writes must carry meaning; none is superfluous. He was clear and concise, which I found the most pleasant because everything was in perfect order, nothing more, nothing less. In fact, that is the attitude I think one should approach math, writing, or any other subject. Mozart once remarked that his favorite sound is “silence”, and in his music, he distills emotions into simple melody but profound clarity, and that is what I think a mathematician should similarly work their style on.
I audited Prof. Mario Bonk’s 245A Real Analysis for 3 weeks, but stopped going because I needed time to socialize with my fellow freshmen. He is a good lecturer though, I remember his class went up to constructing measures from premeasure, and it was highly interesting but, alas, one must make sacrifices when choosing what to study, for one’s life is short. Bonk’s other remarkable feature was his handwriting. I loved his way of writing “B” in mathcal or mathfrak, whatever it was, and his Greek letters were elegant.
The class I made the most out of was Prof. Tao’s 246A complex analysis - yes, that professor Terry Tao. His lectures were clear, although he stuttered in many of his sentences (maybe that was because he thinks faster than he speaks?). He was a great explainer of concepts (always sketching the big picture before going into the definition, which gave me great intuition), and always visualized the concepts by repeatedly drawing on the same picture. To borrow from William Butler Yeats quote on his blog page, “Think like a wise man, but communicate in the language of people”, I think Terry lived it fully. His lectures were always easy to understand, I would simply walk into the lecture unprepared and still understand all the content. And the nature of complex analysis - marvelous results following one after another - makes his lectures like an addicting novel or television series, always making me look forward to the next lecture. I always sat in the first row - which was usually empty because other students could not stand the invisible pressure coming from the GOAT mathematician in this age, but I somehow did not have that feeling - and I did have several interesting interactions with him. He did not like his lecture getting interrupted during class, so it was usually after class that questions were raised. One time I asked about whether there was an analogy of the generalization of the Great Picard Theorem from taking the entire complex plane as the domain to taking the domain from open sets, he answered, “I don’t know”. It struck me that Terry - a mathematician probably the most honorable and reputable in the field - would be so open to admit his ignorance to a student much junior to him. He is very humble, and I can feel that he is truly enjoying math, doing math for math’s sake. In my diary I noted down many times I was “inspired”, “elevated” after his lecture. Sometimes when I was low in motivation in math, his enthusiasm (very contagious) would keep me going. I once wrote “His class was more than just a class” because I learned so much from his way of approaching problems, his attitude, and his profound insights on many areas of mathematics. We went through many classical results like Cauchy’s Theorem, Cauchy Integral Formula, Residue Theorem, Riemann Mapping Theorem, just to mention a few, and we also defined objects like complex manifolds and Riemann surfaces in full generality and studied the lifting problem and the classification of Riemann surfaces (briefly), results that were not in the undergraduate complex analysis curriculum. The undergraduate complex analysis class probably covers 80% of 246A, so I think as long as one has the mathematical maturity to handle abstract concepts, 246A would be one of the most accessible introductory graduate courses and one can take it directly without taking the undergraduate version (132H).
I think one of the most important factors that would make one a successful mathematician is to always strive for opportunities and learn as much as possible. This is especially true for an American university: if one sticks to the official schedule of mathematics one would learn nothing with his bachelor’s degree of mathematics (c.f. British universities, where there is much less flexibility but an average student will get sufficient spoon-feeding to actually know how to do math). My experience in this quarter (and Summer 2023 quarter) helped me understand how to “cheat” the system by taking harder classes (you can replace some lower division coursework with upper division coursework, effectively learning more), which I hope to illuminate future fellow math majors here if they happen to come across my post. I have also started going to math seminars and colloquiums about twice a week, listening to students presenting their reading project on a math paper (e.g. one presentation was on a proof of James Maynard’s result on the small gaps of primes related to the twin prime conjecture) or listening to postgrads attempting to describe their PhD thesis. The former I can usually follow the train of thoughts because it is less demanding in technicality, and the latter I only understand about 1%, but I still keep going to the latter because I think math must be learned in a non-linear way: getting prior exposure to concepts and abstractions as early as possible even though one has a knowledge gap in between, so one can get more comfortable with the next encounter, and also sufficient motivation to study the tool which one might have no idea what applications it might have before. And last but not least, I think these talks are also a great way to connect with the mathematical circle and understand the general trend in mathematics.