Water Waves with Angled Crests: Preprint Available

At some point, I hope I’ll have time to write a longer exposition of my thesis research for the general public. For now, though, I’d like to announce that a preprint of the paper containing my research, A Priori Estimates for Two-Dimensional Water Waves with Angled Crests, jointly written with my advisor, Sijue Wu, is now available on the arxiv.1

Let me give two quick summaries, one for mathematicians and one for everyone else.

For mathematicians, here’s the abstract2:

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-right angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with the inward-facing normal derivative of the pressure non-negative, instead of the strong Taylor stability criterion, which requires that the inward-facing normal derivative of the pressure be bounded below by a strictly positive constant.

For everyone else:

Draw the cartoon version of water in the ocean. What does it look like? You probably have drawn something with waves containing sharp, pointed crests. (Something like this.) This is our intuitive notion of what water waves look like. And yet, despite the tremendous advances in water waves over the past two decades, most current research3 cannot handle such waves, because those angled crests make the problem harder. (For those of you who remember calculus, this is because the sharp angle means that things aren’t differentiable. Given that the problem we’re solving uses [partial] differential equations, this is bad.) Our paper is a step in the direction of understanding that problem; it does this by proving a technical result suggesting that, in some sense, that picture can be handled by our basic mathematical model of water waves in the ocean. In addition, this research helps move towards an understanding of the role of bottoms and coasts in how water moves in oceans, by considering a simplified model of the effect of a vertical wall on water waves.

Thanks to everyone who supported me during grad school!


Update: Also, you can see the video and slides of a talk my co-author Sijue Wu gave about this and related papers.


  1. This preprint contains a few minor refinements of the dissertation, so it’s probably more useful. But if you want to see a copy of the dissertation, email me and I’d be happy to send you a copy. 
  2. Slightly modified only to remove the dependence on mathematical symbols. 
  3. I should note that I’m referring specifically to the mathematical research in water wave equations from a theoretical perspective; I’m not familiar with numerical and experimental research. 

Links: Collapsing NYC Skyscraper, NCAA Bagmen, Yahoo, Health Insurance, Etc.

Now that my PhD is done—for the water wave aficionados out there, I will post a link to the preprint of my research sometime soon, I hope1—it’s time to start catching up on things. The writeup on my math, linguistics and writing freshman seminar is in the works. For now, here are assorted links.

  • This is really scary/fascinating: there was a design flaw in the CitiCorp Center tower in Manhattan that could have led to its collapse. For more, see the BBC documentary (part 2 and part 3) and the New Yorker article (paywall).

  • Interesting article about the bag men who direct and link the booster ecosystem in NCAA football. A perhaps naive question: this isn’t illegal in terms of law, just in terms of NCAA regulations, right? (Well, except for the tax code.)

  • On human memory. I found this particularly interesting as someone who has very careful notes for a whole range of things—I’ll interrupt conversations with (certain) friends to “consult my journals”, knowing exactly where to find something I can’t remember, especially when it comes to literature.

  • David Auerbach on product management and Marissa Mayer. Also on Yahoo: how can Yahoo be worth less than zero.

  • London’s Great Exodus. Wouldn’t it make sense to tax such outside real estate investment higher?

  • On health care: Why employers will stop offering health insurance. That would be great. Unfortunately, see this recent news: the IRS isn’t letting employers nudge employees to the health exchanges.

  • I must admit I found myself doing the same thing as the author of this article about Tiki Barber’s website that lets fans pay to do various activities with athletes: I started searching for the most obscure and absurd things, e.g., the karaoke.

  • On bicycle laws: why cyclists should be able to roll through stop signs.

  • Why unbundling cable won’t save you much money. Not surprising. Well, you shouldn’t get cable anyways—why not read? (Silicon Valley is great, though.)

  • A somewhat sad but quite interesting article on Tom Lehrer.


  1. At some point I’d like to try to write an exposition of my research geared at readers with only a moderate (say, multivariable calculus) background. We’ll see if I have time. 

Links: Crowds and the Recession, Snowden, Win $1 Million with Circuits, Social Choice Theory, and More

As promised, more substantive posts (including a reflection on my freshman seminar on math, linguistics, and writing) to come soon, once I finish my thesis.1 For now, assorted links accrued over procrastination during thesis-writing.

  • Terry Tao has a really cool idea about using circuit design from electrical engineering to solve the million-dollar Navier-Stokes problem. A good popular writeup here, or, if there are any analysts who read this blog, take a look at the article itself.

  • An excellent article by Adrian Vermeule on social choice theory in The New Republic. I like how Vermeule argues for the relative merit of non-formal analyses over the more common, formal (i.e., mathematical) approaches. From my limited experience on the mathematical side of things,2 I have to agree; indeed, I’m always surprised why there even is that much formal social choice theory—but then I have a mathematician’s bias, since most of the results beyond the basic ones seem3 to be mere elaborations. What’s interesting is where you move beyond the formal. Now if only someone can solve the problem of the irrationality of voting…4

  • Excellent article by Sean Wilentz on Edward Snowden in The New Republic. Each time I get TNR, I skim through the front parts—I want light reading—and then feel guilty as I put off the dense articles in Books and Arts. But that’s where the gems are!

  • This is great: a late note from your train conductor.

  • On teaching philosophies/statements. Uh, yeah, duh… More broadly: there’s a huge challenge in figuring out who is good at things and who isn’t. But the ways of measuring this in job applications are problematic to say the least. And about phds leaving academia: this. Let me add that our country’s immigration policies are really stupid along these lines. I have really smart friends getting their PhDs in math who want to stay in the US and would certainly contribute to the well-being of our country. Why should it be so hard for them to get visas? (Anyone connected to companies hiring math phds that will sponsor visas: I’d be glad to put you in touch with them.)

  • Congrats to Robert Harrison for being named a Chevalier de l’Ordre des Arts et des Lettres. His Entitled Opinions show is highly recommended!

  • Interesting on copyright: Why It’s a Wonderful Life Comes But Once a Year

  • Geeky valetudinarian5 lifehacking software update: If you don’t use f.lux you should; there’s a new version out that’s even better. (For the science why, see this NYT article: tl;dr version is that blue light at night from screens messes up your sleep.) Also, for those readers who aren’t socially confident enough to sport these terribly stylish red sunglasses (these are better worn over glasses) when looking at their smartphone screens late at night (or slaving away in their depressing fluorescent-lit offices into the wee hours of the morning), and who don’t want to jailbreak their phones (to use f.lux on iPhones), finally someone has produced red screen filters. I’ve tried out this one and then this one, which seems more believable because it’s actually red-looking, but seems less good qua screen protector.6

A few readings on finance, an area that doesn’t often attract smart writing.7 First, my standard proviso: finance isn’t inherently evil—it helps our society allocate finite resources in an efficient way—and everyone should learn more about it; demonizing finance means only bad people will go into it, which will make the world a much, much worse place. These links below assume to a varying degree some familiarity with finance, of the sort that a mathy person with an interest in economics and investing can probably acquire by osmosis and a little bit of reading. The basic reading everyone—mathy or not—should do is Burton Malkiel’s A Random Walk Down Wall Street, which introduces the basic ideas of modern finance and gives excellent advice for everyone about how to invest their money.8

  • I just read a really interesting book about the 2008 financial crisis and its antecedents (e.g., the fall of LTCM), Ludwig Chincarini’s The Crisis of Crowding: Quant Copycats, Ugly Models, and the new Crash Normal. This book is several things. Most importantly, I think, it provides a good narrative of developments in modern finance over the past twenty years. Second is its thesis, about how ostensibly safe things—things that are unrisky according to models based on historic correlations—can become very unsafe if these spaces become crowded. While this thesis is only explicit at certain points (e.g., in discussing the quant crisis of August 2007), it is certainly implicit in many areas. Third, it raises very interesting questions about regulation. A real thought-provoker.9

  • Two other readings along these lines (and a bit more accessible) are by Cliff Asness of AQR: The Great Divide over Market Efficiency (with John Liew) and The Five-Percent Solution (with Antti Ilmanen).

  • Also, a general comment: Matt Levine is really good, and quite funny.10 A few months ago, I decided11 I waste too much time online, and so decimated my rss list, removing even the great Matt Yglesias. But I had to add Matt Levine. A good comment he made, hidden in a column generally incredulous at Facebook for its December stock offering, has a good point hidden about how FB, unlike other companies (cf. Twitter, pretty much every other company), priced its IPO according to the demand, which ought to make more sense than leaving money on the table. So why does everyone harp on them?


  1. There’s an actual hard deadline for that, so it’s plausible I’ll actually be able to satisfy such promises in a timely manner. 

  2. Namely an expository undergraduate thesis on applications of topology to social choice theory, which I haven’t gotten around from removing from the internet yet. 
  3. To me—full caveat that I’m not terribly familiar with the literature. There’s deeper formal stuff going on in sister fields like mechanism design. 
  4. Public Service Announcement: This is not a good topic to bring up on dates. Indeed, even if one is fully aware of the prudence of such caution, steering clear of the topic when it’s broached without giving a (false) impression of political apathy (or otherwise being disingenuous) is quite difficult. But I urge you, dear deeply conflicted reader who understands rational choice but nonetheless has a heart and a conscience, try nonetheless. One approach is to wax romantic on Obama as a literary character, focusing especially on his basketball playing, with references to redemption, self-knowledge, Jeremy Lin, and the Princeton offense. (I should write about that some time; I’ve added it to my growing list of posts to write. At this rate, post to appear circa 2017.) 
  5. In the 1940s, the great neurotics like Bellow and Rosenfeld occupied themselves with things like orgone boxes (worth reading, by Christopher Hitchens), so if I try to salve my existential terrors by distracting myself with moderately evidence-based things like eating turmeric, drinking fermented beet-juice while reading about probiotics (exciting essay on the biome coming “soon”), and using f.lux, I guess we should chalk it up as progress for reason and science? 
  6. I had four deep insecurities going into grad school: I didn’t eat vegetables often, I didn’t really like poetry, I was afraid of multivariable calculus, and I didn’t understand physics. (These last two really aren’t so important if you aren’t going to math grad school.) I’ve resolved the first two; thanks, People’s Food Coop of Ann Arbor and John Whittier-Ferguson! The third one still haunts me at times, though luckily researching partial differential equations doesn’t actually involve that much of the annoying parts of multivariable calculus, and Green’s and Stokes’ Theorems made much more sense when presented simply as the generalized Stokes’ Theorem. The fourth one remains a problem. Hence this question: I can’t discern a difference at night with this filter, which on cursory examination is quite clear, but does seem to be blue-ish in the day when reflecting sunlight; does this mean it’s working? Someone who knows physics, please tell me. Anyways, someday, I’m going to read Spivak’s Physics for Mathematicians, whereupon I will have resolved my last remaining insecurity and will be a model of psychological well-being. 
  7. There’s an obvious incentive for people with smart ideas in finance to focus on making money rather than spreading their knowledge. I’ve been reading a lot of finance recently, and much of it is mediocre in content, execrable in writing. (Curiously, people in finance—even people who have smart things to say—seem to be really bad at differentiating between its and it’s. Thought: Is there a correlation between the quality of the writing and the quality of the financial advice? Pretty much all the bad investment advice I read is horribly-written, except in carefully copy-edited publications, and more than half of the good advice is badly-written.) 
  8. The best basic approach is to follow Malkiel’s advice, and invest in diversified, low-cost index funds. An interesting development is Wealthfront, which potentially makes sense for people who (a) aren’t interested in doing things by themselves and/or (b) are in high-enough tax-brackets that the tax-loss-harvesting offsets the lack of flexibility and the .25% extra expenses. Of course, it would be nice if there was something that provided the very straightforward, algorithmic capabilities of Wealthfront without those fees and with fuller flexibility. Also, FYI, Wealthfront, most people don’t have stock options from IPOs waiting to vest… 
  9. I hasten to add that I don’t agree with everything there. I’m of course not an expert in the financial and economic analyses he proffers. My main gripe is with his rhetorical approach in discussing the social implications. I find it a bit odd that someone from an economic perspective adopts the rhetorical technique of blaming the “greed” of home owners (as well as other, more elite groups), rather than casting the blame on a system with misaligned incentives. And while I wholeheartedly agree with his conclusion that our society needs to educate far more people about finance, for him to put affirmative action first on his list of the source of educational maladies leading to this ignorance, as he does on page 297, is not merely tone-deaf but foolish, whatever one’s views are on affirmative action. (The other two bugbears he lists are grade inflation and meaningless degrees.) Far more important is the fact that our educational system doesn’t emphasize mathematical, probabilistic, and economic reasoning, as it should
  10. If my editorial comments in the footnotes are overly snarky, Levine’s malign influence has something to do with it (along with being cooped up doing math without writing other things for too long). 
  11. I’ve decided this many times, of course. 

Quick Update on My Freshman Writing Course

In case anyone is wondering: I haven’t disappeared. My freshman seminar last semester on math, linguistics, and writing went really well, and I’m looking forward to writing an extensive reflection on what I learned from the experience. But I do have a dissertation that needs to be finished, so that reflection is on the back burner until I can finish my dissertation draft.

For now, feel free to take a look at the course website to see what we ended up doing over the semester. Stay tuned for a substantial update, sometime soon I hope!

Links: The Future of the Left, Tech Intellectuals, Self-Serve Gas Stations, Etc

  • Simon Winchester, My First Mistake.

  • An article by Peter Beinart on the future of the left.

  • Paul Berman argues for music lessons.

  • Is anyone else as confused as I am about how it’s possible that a new ligament in the human knee was only just discovered?

  • Interesting interview with literary agent Andrew Wylie.

  • On The Onion

  • Wow, interesting case: a scientist criminally prosecuted and placed under house arrest for misleading statistical interpretation in a press release. [h/t Tyler Neylon]. Some thoughts: (1) we should be vary wary of criminalizing speech; (2) wouldn’t civil prosecution be more appropriate anyways; (3) there certainly are big problems with statistical shenanigans; (4) figuring out the right incentive structures for science + capitalism when it comes to medicine is hard; and (5) this notion that p = 0.05 is the magical threshold is problematic for a lot of reasons, and as they point out perhaps the drug could work for the right population; but (6) later studies seemed much more inconclusive.

  • Henry Farrell on the tech intellectuals

  • Good points by Matt Yglesias on iPhones

  • Mary Beard on the deciphering of Linear B

  • Matt Yglesias on New Jersey’s ban on self-serve gas stations. He fails to mention another drawback: for millenial members of the New Jersey diaspora who don’t own a car and rarely drive (all good things that Matt Yglesias, proponent of public transit and cities, would agree with), NJ’s ban makes it possible never to have pumped one’s own gas. This elicits pangs of anxiety whenever the Zipcar gets close to the 1/4 tank level, and your correspondent (who doesn’t recall if he has ever pumped his own gas, though he did help his NY-to-NJ transplant mother do it in Minnesota once, when she’d forgotten how) starts planning how he’ll have to go on youtube and watch a video explain how to pump his own gas. (Or perhaps he can just rely on the memory those Seinfeld American Express ads.)

  • Now online, the first volume of the Feynman Lectures. Someday, I’ll learn physics.

  • It seems obvious to me that a financial product that could help people hedge against drops in real estate prices would be a good thing. So why doesn’t it exist?

  • On Orwell