Nets Katz

Nets Katz was born in Arlington, Texas on August 19, 1972. His father was a theoretical physicist who left academia to become an aerospace engineer and entrepreneur, before eventually returning to academia as the Cudsworth Professor of Aerospace engineering at the University of Alabama. His mother opted to be a stay-at-home mother. He has one older sister.

He skipped three grades and attended Rice University at the age of 15. At age 20, he completed his PhD in pure math at the University of Pennsylvania, successfully passing his thesis defense in May 1993.

He received a Guggenheim Fellowship in 2012. In 2015, he was awarded the Clay Research Award with Larry Guth. He currently is IBM Professor of Mathematics at the California Institute of Technology, where he has been since 2013.

He has a new book Calculus for Cranks, published by Yale University Press, coming out in January 2021.

(The following interview was conducted via video chat on December 14, 2020.)

I’m not sure my parents ever noticed anything unusual about me.  (Laughs.)  We were immigrants, so we were speaking Hebrew at home.  I was learning English from watching TV, so I’d watch Sesame Street.  I guess I was a little bit hyperlexic, so I learned how to read that way, and I was reading relatively fluently when I was 4 or 5.  I was okay at math.  I could do some addition and subtraction.  I thought about numbers a lot; I thought about counting a lot.  But no one thought this was particularly odd or unusual.

When I got to first grade – this was in Grand Prairie, Texas, in 1978 – there was tracking in the school.  There was a set of multiple choice tests that they gave you in the first week for deciding which track you’re supposed to be in.  I don’t know why, but I was placed in the second track, not in the top track, and my parents were a little bit offended by this, because they said, “He can read.”

So my father organized a demonstration for the principal and some teachers.  He demonstrated that I could read, and I could so some math, and it didn’t make any sense for me to be in the second track.  Most of the teachers reacted to this by saying okay and putting me in the first track.  The one teacher who didn’t was my math teacher, Mrs. Owens, who said that instead, she really wanted to teach me.

So I was sitting in Mrs. Owens’ class, and her method of teaching was pretty good; at least it was pretty good for me.  She had an astonishing collection of workbooks filled with rote problems, and she said, “Here are a bunch of workbooks.  Just work through them.”  I was already pretty good at math, but it wasn’t really formal, so there were things I didn’t know.  For instance, when I subtracted, I didn’t subtract by borrowing exactly; I had a rearrangement algorithm for subtracting that I came up with ad hoc.

By the end of the year, I knew all of elementary school arithmetic, or at least as it was understood in that school.  And then, for the next two years, my teachers just didn’t know what to do, so basically I did no math for two years.

By the time I got to fourth grade, an ambitious high school teacher heard about me.  Mr. Hart was coach of the local high school math team.  It wasn’t competing in quality math competitions.  There’s something called the Math Olympiad, which is the world standard.  We were definitely not doing that.  We were in Texas, and Texas has its own state league for public schools for various kinds of competitions.  They have math contests, and at the time, there were two types of tests; one was called Number Sense, and one was called Calculator.  Mr. Hart believed in both of these.

Calculator was essentially a manual dexterity test.  There were these ridiculous problems that were called number crunchers, where you multiplied and divided numbers with eight significant digits, and you did this on these old TI calculators, and the trick was to be able to move your fingers fast enough.  It’s like typing.  Anyway, I was terrible at that.  There was no possibility that I would ever do that.  It took him forever to believe this.

But Number Sense was more interesting.  It was mental arithmetic.  So you would be handed a sheet of paper with 80 problems on it.  You had 10 minutes to do them.  And the only thing you were allowed to write on the sheet of paper was the answers.  My parents weren’t very thrilled about this idea, partly because they were a little behind the times.  They’d say things like, “Math?  That’s not a contest.  Why should there be a contest in math?”

So what Mr. Hart did to try to get me into the system was that he offered to teach me some math.  I hadn’t been learning any new math in second or third grade, and here I was in fourth grade, and he said, “Why don’t you come to the high school for one period a day, and I’ll teach you some math.”  That sounded good, and everyone was in favor of this.  It was a little bit crazy because it was all independent reading, and it wasn’t arranged in any very clever order, so it started with probability, and then it was like, algebra I, geometry, algebra II, and so on.  Over a couple of years, I was making progress, and I had a respectable working knowledge of algebra.

Then, when I was 11 and starting sixth grade, he really wanted to get me to do these contests, and I said, “Okay, I can do Number Sense.  I can do this mental arithmetic thing somewhat.”  Every weekend, there would be an invitational contest.  Some high school in the Dallas-Fort Worth area would be holding a contest, and everyone else could go over there.  He took me to the first one of these contests, and I didn’t do so well; it was my first time, and so nothing happened.

He took me to the second contest, and I did really much better there.  Everything was arranged by grade level, and it was a high school contest, so the lowest grade level was ninth, so even though I was a sixth grader, I signed up as a ninth grader.  And I got second place.  Some kids talked to me during the breaks, and they heard that I wasn’t really a ninth grader, and this got around, and the result that I was disqualified.  (Laughs.)  I didn’t get awarded the trophy at the award ceremony.

For Mr. Hart, this was like the worst thing that could happen.  So the next week, I received a battlefield promotion, and I was promoted to ninth grade – instantaneously.  This was a little bit of a shock.  In math and science, it was just great.  I got to take physical science, which was a huge amount of fun.  I was still doing the reading with Hart’s method of learning math, but the following year I would take calculus as a class. In the humanities I was a little bit behind. I was this kind of patchy high school student who was pretty good in math and science and not so great at anything else, but I was eleven.

So that’s how I got through the public schools quickly.  It was a complete accident.

Academically, how did you find university life?

I was very happy.  It was intellectually a lot more stimulating.  I went to Rice as an undergraduate, and it was a more stimulating atmosphere than high school.  There were a lot more people who were really interested in things.

Well, I was a bit of an outcast socially, but I think I’ve always been a bit of an outcast socially.  Maybe I still am.  It wasn’t worse.  In high school, when you’re an outcast socially, it means that no one really wants to talk to you, and to a certain extent, you get bullied and things like that.  In college, there was nothing really negative, and lots of people were very friendly and talked to me, but I wasn’t really on the inside of groups.

Did your sister also start university early?

She did, sort of, but it was by a different system.  We got to Grand Prairie before I was born, and when we got there, she started middle school, and she absolutely hated it.  She had had actually in some ways a much better education than I did, in stronger schools in various places where they’d lived before.  Something about this middle school was not okay with her, and she got my father to homeschool her.  This was probably a bad idea, because this was much before homeschooling was widespread and accepted.  It was a kind of halfway legal thing, and it also didn’t work entirely well, because some of his expectations were excessive about what he could do and what she would learn in homeschooling.

When she was 16, almost on a whim, they decided one day to sign her up for junior college.  They didn’t start at a university because she couldn’t get admitted to university.  There wasn’t a pathway from being homeschooled.  It’s different now, but that’s how it was then, at least in Texas.  So she went to junior college for one year and transferred into university.

You finished your bachelor’s degree in three years and then finished your PhD in three years.  That’s very impressive!  How were you able to finish so quickly?

For the undergraduate, I have to admit it was partly a financial thing.  We were a little bit more strapped than usual when I was going to college, and I had a lot of AP credits, nearly a year’s worth of advanced placement credit.  It was possible without too much stress to finish in three years, and my parents really wanted me to do that.

For the PhD, it’s completely different.  I was 18.  I had an NSF Graduate Fellowship, so I was financially independent.  There was no particular pressure to finish quickly.  Getting a PhD means you get a new result, and I basically had a new result by the end of the summer after the first year. That was more or less when I started research, and I was immediately getting stuff.  I’m not sure it was really well thought out either to graduate that quickly, but there wasn’t any reason not to.

One thing that was missing in all of this was that I didn’t have particularly good career advice.  So it wasn’t that I was thinking about what do I need to do to be sure I can have a job later, or to make it the best job it can be.  I was just letting things happen to me.  That was kind of the way everything worked through my first postdoc.

What drew you to math in particular?

It was something that I found myself thinking about a lot.  So, the fact that it is something one can just choose to think about and doesn’t need any direction made it attractive.  I was sufficiently good at it that it became sort of natural.

Could you describe some of your research interests in layman’s terms?

I’m interested in analysis.  The way I would define analysis is that it’s the part of mathematics that’s about inequalities.  So you try to prove that something is smaller than something else, or that the number of ways of doing something is smaller than something.  I particularly like analysis that is connected to geometric combinatorics, so I’m interested when the thing you’re trying to get an upper bound on is some sort of a count, and it’s a count of things that have geometrical meaning.

I can give an example that will allow me to state my most famous theorem.  There’s a very old problem that asks that, given $n$ points in a plane, how many different distances can there be between them?  For instance, if $n$ is 3, then the number is 1, since the 3 points can be in an equilateral triangle.  But if $n$ is 4, then the number has to be at least 2; if the points are in a square, there’s the length of the side and the length of the diagonal.

What one is really interested in is the asymptotics.  When $n$ gets large, you’d like to know how can you arrange the points so that they have the fewest possible distances.  For instance, you could arrange the points in a lattice.  If $n$ were a perfect square, you could have a $\sqrt{n}\times\sqrt{n}$ lattice of equally spaced points, and the number of distances is the number of sum of squares of two numbers up to $\sqrt{n}$.  To know how many of these there are, you have to know a little bit of number theory.  Not every number is the sum of two squares, but it’s known what is the rough density of numbers that are sums of two squares.  You get order of $n/\sqrt{\log{n}}$ distances, and that’s in fact the best example that’s known.

For a long time people have been working on showing that you have to have that many distances.  It’s been pretty hard, and it’s been a long road, so there are various results that people got over the years.  In 1946, Erdős got that there have to be at least order of $\sqrt{n}$ distances.  Gradually people have been raising the power, $n^{2/3}$, $n^{4/5}$, $n^{6/7}$.  With Larry Guth, I was able to show there have to be at least constant times $n/\log n$ distances, which has the right power.  It’s not quite as good as the lattice example, but it’s only different in the logs.  That’s actually the theorem I’m best known for.

What do you find most rewarding about what you do?

I’m allowed to think about whatever I want to, and when I get results, they’re permanent.  I’ve really found out something that is true that wasn’t known was true before, and it’s always going to be true.  So it feels kind of good.

Do you feel getting a PhD early has helped you?

Not necessarily, no.  I don’t know that I wouldn’t have done better career-wise if I had known what I was doing, but it’s just difficult to plan.

How does your life today compare with what you imagined when you were young?

I think what I imagined when I was young was a little bit too science-fictiony.  I should be colonizing other planets or something.  It’s a complete failure.  (Laughs.)  But I’m not sure that’s a fair thing to compare it to.

What three words do you think your friends or colleagues would use to describe you?

I have no idea.  You should ask them.  (Laughs.)

It’s a textbook for our introductory course at Caltech.  Our introductory math class is somewhat unusual in that we teach rigorous calculus, meaning epsilon-delta based calculus, calculus that’s based on really using the definition of the limit.  This is something that at almost all other places is only done for math majors, but we require this course of all students.  I think it’s a really good idea, but the difficulty is that the existing books for this were written in the 1960s, and it was kind of a different era.  So it was, this is math, no apologies.  It was also shaped like a calculus book.  We have standard calculus books, which are these horrendous things, where each section is full of examples, and every problem that you’re going to do is basically a copy of one of the examples with some of the numbers changed.  It’s not a thing that you would read from cover to cover.

The calculus books that can be used in this kind of rigorous course are shaped like calculus books, except they’re full of proofs.  The most notable examples are Spivak’s Calculus and Apostol’s Calculus.  Those are great books, but they’ve been found by our students to be hard to read.  They just aren’t speaking to them.  I wrote this book to try to motivate the course better, to try to show why somebody would want to be using the definition of the limit, why ingredients in showing that limits exist are important beyond just wanting to know that the limit exists.  You don’t just want to know that the limit exists.  You want to know what is the delta that goes with each epsilon.  You want to know how close you have to get to the value at which you’re taking the limit to be sufficiently close to the limit.

The book is written from this viewpoint.  It’s addressed not to math majors particularly but to future scientists and engineers.  It’s also pretty thin.  It is a book that one can read from cover to cover.  People complain a little bit about not having enough examples, about not being able to just copy their homework from an example, but I think that’s important too, because what we’re really trying to do is teach students how to think for themselves.  That’s the idea behind the book.

You mentioned before the interview that you had two children.  How have your childhood experiences affected how you approach parenting?

I have to say I’m afraid that they haven’t.  Both of my children are on the spectrum.  It’s not that I know for a fact that I’m not, but I certainly don’t have quite their symptoms.  Unfortunately they need help, and a lot of what I learned from my childhood hasn’t helped.

What do you do in your spare time?

Do I have spare time?  I watch TV.  I play chess.  I read.

Before the interview, you mentioned you had an interest in cults.

I do.  You know, the pandemic has been terrible, and it makes us watch really obnoxious TV.  There’s been a lot of TV lately about cults.  There have been these documentaries about NXIVM.  What sort of dawns on me when I watch these is that maybe math is a little bit like a cult.  (Smiles.)  Maybe we should be learning from the successes that cults have how to organize our own, so that more people can join.  But I think the secret of cults is that you really have a lot of people that are interested in doing the same thing.  It’s unfortunate that in most cases it’s a useless or destructive thing, but it doesn’t have to be.  That’s a thought I’ve been playing with.

What goals do you still have for yourself?

Well, I have research problems that I’d really like to solve. We’ll see how that goes.  I have a few books that I would still like to write as a continuation of the calculus franchise.