"We are not that much smarter than each other." - Richard Feynman

- Introduction
- Papers and presentations
- Expii (education startup)
- Student-run math contests
- Resources/notes
- Classes and other math-related activities, including SCUM information.

Hi! You've reached my old webpage; as of Fall 2017, I'm a first-year graduate student at Princeton.

If I were a Springer-Verlag Graduate Text in Mathematics, I would be William S. Massey's I am intended to serve as a textbook for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unecessary definitions, terminology, and technical machinery. Wherever possible, the geometric motivation behind the various concepts is emphasized. Which Springer GTM would |

See also arXiv and Google Scholar Citations.

1-color-avoiding paths, special tournaments, and incidence geometry (with Jonathan Tidor and Ben Yang); written during MIT SPUR 2016. We thought about a recent question of Loh (2015): must a 3-colored transitive tournament on N vertices have a 1-color-avoiding path of vertex-length at least N^(2/3)? This question generalizes the classical Erdos--Szekeres theorem on monotone subsequences (1935). To me this problem seems natural and surprisingly rich. Note: Gowers and Long just uploaded a very nice preprint (September 2016) on this problem and its natural generalizations.

On Hilbert 2-class fields and 2-towers of imaginary quadratic number fields, J. Number Theory 160 (2016), 492--515; written at Duluth REU 2015. I thought about a question of Martinet (1978): must every imaginary quadratic number field K/Q have an infinite Hilbert 2-class field tower when the discriminant of K has 5 prime factors? Naturally one tries to use class-field-theoretic constructions and inequalities, but difficulties arise from the combinatorics of certain 5 x 5 (or smaller) binary matrices of quadratic symbols. I proved some new cases and found some precise reasons and examples explaining some of the difficulties.

Simultaneous core partitions: parameterizations and sums, Electronic J. Combin. 23(1) (2016), #P1.4, 34 pages; written at Duluth REU 2015. This showed me a beautiful side of partitions and hook lengths that I had not seen before, and gave me a nice opportunity to interact with the vibrant core partitions community. Specifically, I re-interpreted some number-theoretic stabilizer sizes of Fayers in a friendlier way, which led to proofs of some enumerative-combinatorial conjectures of Fayers, as well as a simpler proof of a foundational structural result of Fayers.

Local-global principles in number theory (Hasse--Minkowski for quadratics), MIT Student Colloquium for Undergraduates in Math (11/2016).

Riemann--Roch with Hodge theory, MIT Curves Learning Seminar (9/2016).

1-color-avoiding paths, special tournaments, and incidence geometry (presented with Jonathan Tidor), MIT SPUR Conference (8/2016); slides available here.

Martinet's question on Hilbert 2-class field towers, AMS Contributed Papers Session in Number Theory, Joint Mathematics Meetings, Seattle, WA (1/2016); slides available here.

Roots of unity filter (finite Fourier analysis), MIT Student Colloquium for Undergraduates in Math (10/2015).

18.100p (p-adic analysis), MIT Student Colloquium for Undergraduates in Math (11/2014).

I have been closely involved with most of the math concept maps (see here for concrete topics and here for meta heuristics and strategies) and discussions about curriculum, problems, tags, and related user design, as well as moderation and some concrete content such as Expii Solve (one of whose goals is to illustrate why people should care about math in the first place). I have enjoyed working with several wonderful current and former teammates. If you have feedback/suggestions or otherwise want to talk about Expii, please email me.

Problem-writing is a great opportunity for positive math propaganda. I believe contest organizers should do everything in their power to benefit students as much as possible (mathematically or otherwise). You can read more about my problem-writing philosophy here.

I was a problem czar (along with Calvin Deng and Bobby Shen) for the HMMT February 2015 tournament, with problems, solutions, and commentary here (including the proof-based HMIC). (Some of my favorite problems from the contest are highlighted on Facebook.) During my freshman year, I was a problem czar for the HMMT November 2013 tournament, with problems, solutions, and commentary here. You might also be interested in the HMMT forum on AoPS.

In high school, I was (along with Evan Chen, Ray Li, David Yang, Calvin Deng, Alex Zhu, and others) heavily involved with the ELMO (Shortlist and semi-official page a la IMO-official) and Online Math Open, two student-run contests with slightly different audiences. If you're a middle or high school student (or someone else with free time) interested in problem solving, I recommend you check the problems out; we put a lot of love into them!

I was a grader at MOP 2014; right before it I also got to teach a couple black/IMO classes (handout, tex), where I tried to pull and organize problems from some less common themes (e.g. interlacing real roots). Here's a random handout on Dipohantine equations I helped write a while ago.

If you'd like to discuss any of the problems here, have feedback, etc. you can find my email address above.

*Problems from the Book* (PFTB), by Gabriel Dospinescu and Titu Andreescu. Easily the most inspiring book I read in all of high school. (The sequel, *Straight from the Book*, is just as good, with not just solutions to the first half of PFTB, but also beautiful appendices full of exposition on related topics. The authors are really gracious too---I got a free copy just for posting some solutions on AoPS!) I certainly do have to credit the Art of Problem Solving (AoPS) series and website for getting me started in middle school (and so much more), though.

For lots of advanced undergraduate analysis (including Fourier analysis, complex analysis, and measure theory/integration), I think it is hard to beat the *Princeton Lecture Series* by Stein--Shakarchi.

For (modern) number theory, Ireland--Rosen and Neukirch are both nice.

For exposure to lots of different areas and ideas in math, see the *Princeton Companion to Mathematics*. I might recommend some specific articles in the future.

In **Spring 2017**, I took 6.207/14.15, 18.117, and 18.906.

In **Fall 2016**, I took 18.116, 18.905, 18.965, and 21A.500/STS.075, and graded 18.A34.

In **Summer 2016**, I participated in SPUR, working on a 1-color-avoiding Ramsey project (slides here) with Jonathan Tidor and Ben Yang (mentor); I may include further thoughts on the problem at some point.

In **Spring 2016**, I took 18.726 (Algebraic Geometry 2, Davesh Maulik), 18.099 (Seminar in Discrete Analysis, Peter Csikvari), 8.04 (Quantum Physics I, Barton Zwiebach), and 21M.250 (Beethoven to Mahler, Teresa Neff).

18.726 helped me appreciate 18.725 much better, but I still have a lot to learn. At some point I might post my expository paper on the formal functions theorem and the connectedness version of Zariski's main theorem; apparently this historically provided one of the first examples of the clean power of scheme theory and cohomology when compared to Zariski's original proof for varieties. 18.099 was pretty fun; I'm glad Peter was willing to run it after Mark Sellke and I asked him. Mark and I wrote an expository paper on Gowers' Fourier-analytic proof of Szemeredi for 4-term progressions (which Gowers also exposes in much greater depth in a section of his recent arXiv article: Generalizations of Fourier analysis, and how to apply them). Re: 8.04 (and physics in general) I'm still lacking a lot of physical intuition, but maybe I'll find time to read one of those "physics for mathematicians" books someday. 21M.250 introduced me to some great music and interesting musical anecdotes, but for some reason the writing assignments (concert reviews) took me a long time to finish.

In **IAP 2016**, I went to the Joint Math Meetings in Seattle, WA (slides here), and did the Directed Reading Program (DRP) with Sammy Luo and Augustus Lonergan (mentor).

In **Fall 2015**, I took 18.725 (Algebraic Geometry 1, Roman Bezrukavnikov), 18.994 (Seminar in Geometry, William Minicozzi), 18.901 (Intro Topology, Marc Hoyois), and 21M.303 (Writing in Tonal Forms 1, Charles Shadle). And maybe at some point I want to self-study/group-study some of the following topics (let me know if you're also interested in anything): analytic number theory, class field theory, discrete analysis (esp. Erdos discrepancy problem or Szemeredi's theorem).

I might eventually add comments and some writeups; it was a really tough semester for me due (mostly) to 18.725 but I don't think it would be productive to comment until I understand more AG. For 18.994 I might post some notes on the Gauss-Bonnet theorem. I really enjoyed Shadle's teaching and encouragement in 21M.303; I might post my two projects at some point if I decide it isn't too embarrassing.

In **Summer 2015**, I went to the Duluth REU. I worked on two projects: simultaneous core partitions and Hilbert 2-class field towers (slides here).

In **Spring 2015**, I took 18.786 (Number Theory 2; Bjorn Poonen; official Tate's thesis notes on the Stellar link; full notes by Eva Belmont (again) here; my spotty notes here), 18.318 (Extremal Combinatorics; Choongbum Lee; webpage has excellent lecture notes; my (mostly pset) writeups here), 18.821 (Project Lab in Math, with Gary (Ka Yu) Tam and Soohyun Park; project drafts on Overleaf: Project 1 (points on conics: Hasse--Minkowski theorem for quadratics), mentored by Yifeng Liu; and Project 2 (efficiency of generating matrices: lengths of matrix algebras), mentored by Clark Barwick), and 21M.302 (Harmony and Counterpoint 2; Keeril Makan). I'm listening in on Harvard Math 271 (Arithmetic Statistics; Arul Shankar). At some point I should take some non-math classes, but there are too many interesting math classes this year... I should also remember that classes are only a small part of the story, and not necessarily the best way to learn/grow mathematically either.

It was a busy semester (esp. due to Expii and HMMT stuff) but overall I enjoyed it very much! My main regret (these past two years actually) is not talking more to professors and other students, even about math---I guess I'll just have to try to gradually open up to people. But my classes and professors exposed me to lots of cool new material I might not have thought of seeking on my own. As my sister has also noticed by now, it's both frustrating and inspiring how much is out there...

Also, this semester I co-organized (along with Soohyun Park and Peter Haine) the Student Colloquium for Undergraduates in Mathematics (SCUM), co-sponsored by HMMT, UMA, and USWIM. (For now, here's SCUM Spring 2015 information, and you can find older links in the "Outreach" section on Carl Lian's page.) Let us know if you (or someone you know) might like to give a talk, or want to discuss SCUM- or math community- related things in general (e.g. suggestions for SCUM or ideas for other possible synergistic math activities).

In **IAP 2015**, I did the Directed Reading Program (DRP) with Jane Wang (mentor) on the ergodic approach to additive combinatorics, with (informal) notes/writeups/comments hopefully here. I may also try to do/find/ask/start/apply for UROP/SPUR/REUs.

In **Fall 2014**, I took 18.103 (Fourier Analysis; Larry Guth), 18.705 (Commutative Algebra; Yifeng Liu), 18.785 (Number Theory 1; Bjorn Poonen; notes by Eva Belmont here), and 21M.301 (Harmony and Counterpoint 1; Travis Alford). I also audited 18.112 (Complex Analysis; Jonathan Kelner), but self-studied Ch. 1-3,5,6,8 of the text (Stein and Shakarchi---a great book IMO) over the summer.

Overall I enjoyed my classes a lot; 18.103 and 18.785 in particular complemented and expanded my previous concrete/problem-based exposure to the topics in the aforementioned PFTB/SFTB (and also 18.702). The synergy of 18.705 and 18.785 was also nice (see e.g. blog post on classification of finitely generated modules over Dedekind domains). I might add more comments later, especially if anyone asks, but probably my informal personal notes/writeups/comments from the semester will do for most purposes: commutative algebra, number theory, Fourier analysis.

I also gave a SCUM talk this semester titled "18.100p" (p-adic analysis). I tried to cover too much...

In **Spring 2014**, I took 8.022 (Physics E&M; Nuh Gedik, TA Min Chen), 18.03 (Differential Equations; Bjorn Poonen), 18.100B (Real Analysis; Toby Colding), 18.702 (Algebra 2; Michael Artin), and 24.900 (Intro Linguistics; Sabine Iatridou, TA Ruth Brillman). But I was fat and didn't take any gym classes.

8.022 was slightly painful and a little hand-wavy for me, but overall made way more sense than AP physics (and as recommended by David Yang, I may take more physics classes in the future to gain computational intuition). 18.03 was pretty good with Poonen (my undergraduate advisor!), but it may or may not have been a good idea to take the 18.152 (PDE) route instead if I'd heard about the possibility earlier (I actually heard about it from my CPW pre-frosh, Mark Sellke---it's on one of the MIT math pages somewhere, but IIRC notably missing from the main page). Oh well. 18.100B had fairly nice psets, and I found it instructive to prove all the theorems in Rudin (well, up to Chapter 7-8, where the course stopped). Although he did provide some additional motivation, I felt Colding mostly read from the book and went relatively slow (although I didn't go to too many classes... 9:30 is a bit early for me), but for a more basic class like this one it might not matter too much. 18.702 was a lot more fun than 18.701 (again, Artin's awesome), but I don't think I really learned any representation theory (a lot of the stuff on characters, etc. seemed really unmotivated, so I failed to prove most of the theorems in the chapter; EDIT: after talking to David, I think the issue is that representation theory of *algebras* is more natural than representation theory of *groups*, and in fact the former more or less subsumes the latter). After that (more commutative stuff) I found everything quite instructive (rings/factoring, quadratic number fields, modules, fields/Galois theory), although I've heard from others that the Galois chapter is not too good (personally I enjoyed reconciling the group-theoretic/automorphism-based perspective from the book/course with the more field-theoretic/minimal-polynomial-based perspective I had previously from PFTB (mentioned above)). 24.900 was not as fun as I'd expected (to be fair it is a CI-H, so I'm now done with those, though in retrospect maybe I should've waited to take 11.124/11.125 in junior/senior year or something like that), but I did learn several interesting ideas from the first two-thirds or so of the course (semantics/pragmatics/syntax, especially syntax). (Unfortunately, I found the remaining third on phonetics/phonology/morphology less exciting, and slacked off more than I maybe should have. Prioritization is hard.) (Random interesting idea from near the end of the course: X-bar theory, which pretty reasonably seems to apply to sentences (we parse sentences according to a tree structure), also seems to apply to music. So it would be interesting if people instead learned music the way they do spoken languages...)

In **Fall 2013**, I took 5.111 (Intro Chemistry; Catherine Drennan, Alexander Klibanov), 7.012 (Intro Biology; Eric Lander, Robert Weinberg, Michelle Mischke), 18.A34 (Math Problem Solving/Putnam Seminar; Abhinav Kumar, Henry Cohn), 24.00 (Intro Philosophy; Caspar Hare, TA Matt Mandelkern), and 18.701 (Algebra 1; Michael Artin). Oh, also badminton classes both quarters, for PE.

5.111 was a little painful for me, but I'd guess that it's slightly easier in the fall due to the curve. 7.012 was much better than expected; Lander's quite good, so I recommend taking it in the fall. 18.A34 was pretty fun; the problems, students, and professors are great. 24.00 was not very painful for a CI-H, and there were a few interesting ideas, though I can't say I learned too much (even though analytic philosophy, the focus of the class, should in principle have more to say than continental philosophy, the latter has its own charm). 18.701 was pretty easy by itself, but Artin's a great instructor and the material (basic linear algebra and group theory) is pretty important/fundamental (so I think trying to motivate everything and prove all the theorems yourself can be a good way to learn the material; on the other hand, depending on your style, it may be more inspiring to see the material in action in more advanced subjects closer to your heart).