Book pitch

This is a pitch for a new text that I’m thinking of writing. I want to post it here to solicit opinions from the general community before investing a lot of time into the actual writing.

Summary

There are a lot of students who ask me a question isomorphic to:

How do I learn to write proofs?

I’ve got this on my Q&A. For the contest kiddos out there, it basically amounts to saying “read the official solutions to any competition”.

But I think I can do better.

Requirements

Calling into question the obvious, by insisting that it be “rigorously proved”, is to say to a student, “Your feelings and ideas are suspect. You need to think and speak our way.”

Now there is a place for formal proof in mathematics, no question. But that place is not a student’s first introduction to mathematical argument. At least let people get familiar with some mathematical objects, and learn what to expect from them, before you start formalizing everything.

— Paul Lockhart

There was a while I tried to look around to find an introduction-to-proofs textbook that I liked. I specifically wanted to have the following requirements:

  • Pragmaticism: the textbook should not start with foundational issues like logical quantifiers or set theory. I have held a long belief that these are emphatically not the right way to start proofs, because in practice when one really does proofs, one is usually not thinking too much about the axioms of set theory.
  • Substantial: the results one proves as practice should feel interesting. They should have meat. For example, the statement that a tree always has one fewer edge than vertex is not obvious at first, so when one sees the proof it gives an idea. I believe this is important because I want to develop a student’s intuition, rather than try to teach them to work against it.
  • Intuitive: I reject the approach of some other instructors in which students start by proving basic results from first principles like the well-ordering principle, “all right angles are congruent”, etc. I think this is an experience that is worth having, but it should not be the first experience one has. (This is the same reason people’s first programming language is Python and not assembly.)
  • Combinatorial: for competition reasons. My currently recommended combinatorics textbook by Pascal96 is a bit on the difficult side. It would be nice to cover some ground here.

The closest I got was Jospeh Rotman’s Journey Into Mathematics textbook, which satisfies the first three conditions but not the fourth (the book draws from algebra, geometry, and number theory). I adore Rotman’s book and the copy I read at age 12 is tattered from extended use. I’d like to get the combinatorics in, too.

Picking a fight

I should state now this is against common wisdom. Terence Tao for example describes mathematical education in three parts: pre-rigorous, rigorous, post-rigorous. Relevant quotes:

[In the rigorous stage], one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. … The transition from the first stage to the second is well known to be rather traumatic.

My thesis is that for high school students with an enriched math background, the rigorous and post-rigorous stages should be merged or even inverted. Attending a math circle, going to math camps, or participating in competitions gives you a much better intuition than a typical starting undergraduate would otherwise have access to. I propose that we take advantage of this intuition, rather than ignore or suppress it.

Content

I’m eyeing graph theory as a topic to start off on, if not use wholesale. I think it is an amazing topic for teaching proofs with. Definitions that make sense, proofs that are intuitive but not obvious, lots of pictures that don’t lose rigor, and so on. I imagine I would start there and see where it takes me.

If I go through with it, I think it would take about a year for me to get some initial drafts available to the public.

Pay-what-you-want model

I want to try this out. I think it would look something like:

  1. You can download the nicely typeset PDF for 20 dollars;
  2. The entire source code is publicly readable on GitHub, so if you can’t pay or don’t want to pay just download the source and compile it. It might not have some formatting polishes or whatever but all the content is going to be there.
  3. If you don’t have a computer to compile things on, email me nicely and I’ll send you a copy.
  4. Pull requests welcome, and if you fix some sufficient number of typos or some major errors I’ll add your name to acknowledgments.

But I’m not sure yet.

Questions for the audience

  1. Is this something people would want to see?
  2. Is there any existing text that already satisfies my requirements?
  3. Is the payment model fair?
  4. Other comments or suggestions?

USA Special Team Selection Test Series for IMO 2021

A lot of people have been asking me how team selection is going to work for the USA this year. This information was sent out to the contestants a while ago, but I understand that there’s a lot of people outside of MOP 2020 who are interested in seeing the TST problems :) so this is a quick overview of how things are going down this year.

This year there are six tests leading to the IMO 2021 team:

  • USA TSTST Day 1: November 12, 2020 (3 problems, 4.5 hours)
  • USA TSTST Day 2: December 10, 2020 (3 problems, 4.5 hours)
  • USA TSTST Day 3: January 21, 2021 (3 problems, 4.5 hours)
  • RMM Day 1: February 2021 (3 problems, 4.5 hours)
  • APMO: March 2021 (5 problems, 4 hours)
  • USAMO: April 2021 (2 days, each with 3 problems and 4.5 hours)

Everyone who was at the virtual MOP in June 2020 is invited to all three days of TSTST, and then the top scores get to take the latter three exams as team selection tests for the IMO. Meanwhile, the RMM teams and EGMO teams are based on just the three days of TSTST.

Similar to past years, discussion of TSTST is allowed on noon Eastern time Monday after each day. That means you can look forward to the first set of three new problems coming out on Monday, November 16, and similarly for the other two days of TSTST.

To add to the hype, I’ll be doing a short one-hour-or-less Twitch stream at 8:00pm ET on Tuesday November 17 where I present the solutions to the TSTST problems of day 1.  If there’s demand, I’ll probably run a review session for the other two days of TSTST, as well.

EDIT: Changed stream time to Tuesday so more people have time to try the problems.

USEMO sign-ups are open

I’m happy to announce that sign-ups for my new olympiad style contest, the United States Ersatz Math Olympiad (USEMO), are open now! The webpage for the USEMO is https://web.evanchen.cc/usemo.html (where sign-ups are posted).

https://web.evanchen.cc/static/usemo/usemo-logo.png

The US Ersatz Math Olympiad is a proof-based competition open to all US middle and high school students. Like many competitions, its goals are to develop interest and ability in mathematics (rather than measure it). However, it is one of few proof-based contests open to all US middle and high school students. You can see more about the goals of this contest in the mission statement.

The contest will run over Memorial day weekend:

  • Day 1 is Saturday May 23 2020, from 12:30pm ET — 5:00pm ET.
  • Day 2 is Sunday May 24 2020, from 12:30pm ET — 5:00pm ET.

In the future, assuming continued interest, I hope to make the USEMO into an annual tradition run in the fall.

Napkin v1.5 (and more)

Careful readers of my blog might have heard about plans to have a second edition of Napkin out by the end of February. As it turns out I was overly ambitious, and (seeing that I am spending the next week in Romania) I am not going to make my self-imposed goal. Nonetheless, since I did finish a decent chunk of what I hoped to do, I decided the perfect is the enemy of the good and that I should at least put up what I have so far.

So since this is someplace between version 1 and the (hopefully eventually) version 2, it seems appropriate to call it version 1.5. The biggest changes include a complete rewrite of the algebraic geometry chapters, new parts on real analysis and measure theory, and a reorganization of many of the earlier chapters like group theory and topology, with more examples and problems. There’s also a new chapter 0 entitled “sales pitches” which gives an advertisement for each of the parts later. The obvious gaps: the chapters on probability are yet to be written, as is some more algebraic geometry. The updated flowchart from the beginning of the book is pictured below.

recent-flowchart

You can download the latest version from the usual page, or directly from https://usamo.files.wordpress.com/2019/02/napkin-v15-20190220.pdf. The number of errors has doubtless increased, and corrections are comments are more than welcome.

Incidentally, this seems as good a time as any to mention two more things:

That’s all.  Hope you all like it! Best wishes from the Zurich airport.

Some things Evan is working on for 2019

With Christmas Day, here are some announcements about my work that will possibly interest readers of this blog.

OTIS V Applications

Applications for OTIS V are open now, so if you are an olympiad contestant interested in working with me during the 2019-2020 school year, here is your chance. I’m hoping to find 20-40 students for the next school year. Note that the application has math problems in it, unlike previous years, so you have to start early.

OTIS Lecture Series

At the same time, I realize that I will never be able to take everyone for OTIS. So I am planning to post a substantial fraction of OTIS materials for public consumption, hopefully by late January, but no promises.

Napkin 2nd edition

The Napkin is getting a second edition which, if all goes well, should come out by the end of February (but that is a big “if”). Most chapters will be mostly unchanged modulo typos, but a few big changes:

  • I am hoping to add a new part on measure theory with an eye towards probability applications (e.g. law of large numbers, central limit theorem, stopped martingales).
  • There will be a bit of real analysis / calculus now. (Not much.)
  • Maybe two-ish bonus chapters on other topics being added.
  • The earliest chapters (on algebra and topology) are being re-organized significantly, though most of the content should remain the same.
  • The algebraic geometry chapters on schemes are getting a major facelift, because the old ones were terrible. They will still cover roughly the same content, but in a way that makes more sense, has more examples, and has more pictures.

This means that for the first time the numbering of the chapters is going to break with the new update. This also means there will be plenty of new typos and mistakes for readers to find. I’m looking forward to it!

SPARC 2019 applications

For high school students, SPARC applications will open soon. The deadline will probably be the end of February. This year SPARC will be held in the Bay Area from July 24 to August 2.

117(d): Please don’t tax PhD tuition waivers

This is a rare politics post; I’ll try to keep this short and emotion-free. If parts of this are wrong, please correct me. More verbose explanations here, here, here, here, longer discussion here.

Suppose you are a math PhD student at MIT. Officially, this “costs” $50K a year in tuition. Fortunately this number is meaningless, because math PhD students serve time as teaching assistants in exchange for having the nominal sticker price waived. MIT then provides a stipend of about $25K a year for these PhD student’s living expenses. This stipend is taxable, but it’s small and you’d pay only $1K-$2K in federal taxes (about 6%).

The new GOP tax proposal strikes 26 U.S. Code 117(d) which would cause the $50K tuition waiver to also become taxable income: the PhD student would pay taxes on an “income” of $75K, at tax brackets of 12% and 25%. If I haven’t messed up the calculation, for our single PhD student this means paying $10K in federal taxes out of the same $25K stipend (about 40%).

I think a 40% tax rate for a PhD student is a bit unreasonable; the remaining $15K a year is not too far from the poverty line.

(The relevant sentence is page 96, line 20 of the GOP tax bill.)

New algebra handouts on my website

For olympiad students: I have now published some new algebra handouts. They are:

  • Introduction to Functional Equations, which cover the basic techniques and theory for FE’s typically appearing on olympiads like USA(J)MO.
  • Monsters, an advanced handout which covers functional equations that have pathological solutions. It covers in detail the solutions to Cauchy functional equation.
  • Summation, which is a compilation of various types of olympiad-style sums like generating functions and multiplicative number theory.

I have also uploaded:

  • English, notes on proof-writing that I used at the 2016 MOP (Mathematical Olympiad Summer Program).

You can download all these (and other handouts) from my MIT website. Enjoy!

First drafts of Napkin up!

EDIT: Here’s a July 19 draft that fixes some of the glaring issues that were pointed out.

This morning I finally uploaded the first drafts of my Napkin project, which I’ve been working on since December 2014. See the Napkin tab above for a listing of all drafts.

Napkin is my personal exposition project, which unifies together a lot of my blog posts and even more that I haven’t written on yet into a single coherent narrative. It’s written for students who don’t know much higher math, but are curious and already are comfortable with proofs. It’s especially suited for e.g. students who did contests like USAMO and IMO.

There are still a lot of rough edges in the draft, but I haven’t been able to find much time to work on it this whole calendar year, and so I’ve finally decided the perfect is the enemy of the good and it’s about time I brought this project out of the garage.

I’d much appreciate any comments, corrections, or suggestions, however minor. Please let me know! I do plan to keep updating this draft as I get comments, though I can’t promise that I’ll be very fast in doing so.

Here’s a table of contents, in brief:

I. Basic Algebra and Topology
II. Linear Algebra and Multivariable Calculus
III. Groups, Rings, and More
IV. Complex Analysis
V. Quantum Algorithms
VI. Algebraic Topology I: Homotopy
VII. Category Theory
VIII. Differential Geometry
IX. Algebraic Topology II: Homology
X. Algebraic NT I: Rings of Integers
XI. Algebraic NT II: Galois and Ramification Theory
XII. Representation Theory
XIII. Algebraic Geometry I: Varieties
XIV. Algebraic Geometry II: Schemes
XV. Set Theory I: ZFC, Ordinals, and Cardinals
XVI. Set Theory II: Model Theory and Forcing

(I’ve also posted this on Reddit to try and grab a larger audience. We’ll see how that goes.)