I’m Jordan Ellenberg, a math professor at the University of Wisconsin-Madison. I study number theory, algebraic geometry and topology, which basically means I study very old questions about numbers using very new methods developed in the last few decades. I’m also a writer. My new book, Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else, is all about geometry, interpreted very very broadly. It’s not just about triangles! (Though there are some triangles in the book.) We are thinking geometrically every time we talk about where things are and what they look like and how close things are to other things. In the book I write about the end of human dominance in checkers, the geometry of legislative redistricting, social networks and the spread of pandemics, what a neural network is and how modern machine learning works, Abraham Lincoln’s love of Euclid, mathematicians with poetry envy, poets with math envy, the end of human dominance in checkers, the sometimes-stultifying way we teach math in school, the map of all words, why you are your own negative-first cousin, and a whole lot more.

The book has been reviewed in the New York Times https://www.nytimes.com/2021/05/18/books/review-shape-geometry-jordan-ellenberg.html

and excerpts have appeared in the Wall Street Journal https://www.wsj.com/articles/what-honest-abe-learned-from-geometry-11621656062

and Slate https://slate.com/technology/2021/05/covid-models-cubic-fit-rinderpest.html

And I’m on Twitter, talking about geometry and other stuff, at https://twitter.com/JSEllenberg

Proof (see what I did there?): https://i.redd.it/6ulpirabmo271.jpg

Comments: 98 • Responses: 31  • Date: 

psu_1312 karma

If you could add one class to and remove one class from the current 6th to 12th grade sequence in the U.S. what would you remove and what would you add?

Jordan-Ellenberg18 karma

The obvious answer for "what to add" is a course in data analysis and statistics, but it needs to be said that to some extent this has already been added! Statistics is taught in a substantial proportion of high schools now, and the stat AP is more popular than the calc BC AP and almost as popular as calc AB. And having two kids on their way through K-12, I can tell you that ideas about data, visualization, and statistical analysis are now present throughout the curriculum, starting in kindergarten. BUT: I do think it would be good for a year to be devoted to that topic and for it to be marked as part of the sequence, not as an elective.

But what to take out? I'm going to cheat a little, because I don't think any existing course is fully expendable. I think I would make both pre-calculus and calculus shorter by reducing the emphasis on games with trig identities in pre-calc and the emphasis on integration tricks in calculus. (But again I'm still cheating because the integration tricks are in calc BC which is really not part of the 6th-12th sequence per se!)

pulkitism10 karma

What are some of your favorite mathematics documentaries?

Jordan-Ellenberg39 karma

Mean Girls

Jordan-Ellenberg18 karma

OK serious answer: after all these years, the class of the field remains the NOVA episode "The Proof," about Wiles and Taylor's proof of Fermat's so-called Last Theorem. It captures the emotional side of doing research in mathematics as no other documentary has done. https://www.pbs.org/wgbh/nova/proof/

I've heard great things about the movie about Maryam Mirzakhani but I haven't seen it yet! http://www.zalafilms.com/secrets/

And I can't omit the NOVA episode I'm in and where through the miracle of special effects they make it look like I can shoot a three-pointer over my shoulder. https://www.pbs.org/wgbh/nova/video/prediction-by-the-numbers/

JoshuaZ18 karma

Preamble: I just finished reading the book yesterday. I've previously read How Not To Be Wrong. I've taught voting theory courses before, and the bit about slime molds voting I had never seen before, so that was neat.

Anyways, four questions:

First:: shape seemed in some respects to be more overtly political than How Not to Be Wrong. How difficult was it to write a nuanced take on the more substantially political issues which were intertwined with the mathematics?

Second: A lot of Shape was about issues surrounding COVID and pandemics. Obviously, some of the general pandemic material was going to be there already. How much were you intending pandemics and diseases to be discussed in early drafts? Was there anything that you took out so you could fit in more of that?

Third, I was surprised in reading the pandemic sections that you didn't mention Bernoulli's early work (or if you did I somehow missed it). Is there a reason you decided not to include it?

Fourth, to me at least the failure of monotonicity makes Instant-Run Off seem like a fundamentally flawed system, in that this seems to my intuition to be one of the most basic things a voting system should do if it resembles consensus in any way. But IRV is in practice the most common non-plurality voting system proposed in the US. Why do you think of the many different non-plurality systems, IRV gets so much attention?

Jordan-Ellenberg10 karma

  1. I don't know if it's more political! Of course there is the second on gerrymandering, where the subject itself is the interaction of mathematics and politics; so unavoidably that part is political (though I strive to be nonpartisan.) I wrote the book during what was (is?) obviously a very politically strained time in the U.S. and I actually found it kind of a respite to mostly stay at arms length from that stuff in this book.
  2. Oh I had no idea I was going to write any of that in advance. Like a lot of mathematicians I suddenly got very interested in pandemic modeling and mathematical epidemiology in early 2020. But because everything in math is connected, it all ended up very much in tune with the stuff about random walks and networks I already knew I wanted to write about, the use of differential equations to study the natural world brought it in contact with Poincare, etc. I didn't have to take anything out because I don't write to a strict word limit. I suppose if I told Penguin I wanted to do 800 pages they might start to get itchy but by and large their philosophy is that I should write the book I want to write and that determines the length.
  3. I didn't mention it because I don't know about it! Part of talking about the book, once it's out, is finding out about all the research tunnels I didn't go down and stuff that could have been in the book but isn't. So tell me!
  4. There's no non-flawed system. My inclination is certainly to see IRV as an improvement over what we do now, though I'll be watching carefully what happens in Maine and NYC and other places that have taken it on to see if that changes my assessment. I don't have a good answer as to why IRV is getting traction while, e.g. approval voting isn't. It may just be psychological -- when people vote they like to be able to put their favorite candidate first. IRV feels like less of an emotional departure from the system Americans are used to, in that respect.

Jordan-Ellenberg6 karma

OK, everybody, gotta go pick up a kid from school! Thanks for all your questions -- as always, I learned a lot!

draxus995 karma

What does Mathematical Research look like these days, in your experience. What can I do as someone with a programming background and a fascination with 'graphing' my thoughts, to contribute to math research? Are there any online communities that you might recommend? Thanks :)

Jordan-Ellenberg7 karma

I think the newer the field (or the newer the question within the field), the more possible it is for people with ideas to enter and make meaningful contributions. There was a wonderful paper a few years ago by Aubrey de Grey, who is not a professional mathematician, about the chromatic number of the plane:



kevinwangg4 karma

I've been thinking casually about that anti-isosceles set of lattice points problem in the past week -- has anyone came to you with any good leads? Do you have a conjecture?

Jordan-Ellenberg6 karma

I truly don't, which is one of the reason I wanted to blast the question out to the public! I have no idea whether the right answer is closer to N^2 or to N^1, and this ignorance reminds me of the cap-set problem, one of my very favorites. But I'll say this -- on Twitter, after I posted the problem on 538, some people showed me computer searches they did which suggest that the optimal solutions have their points concentrated very near the edges of the N x N square. So that suggests you're closer to N than N^2, but it also suggests that maybe it's the wrong way to frame the problem? One feels that if the optimum is on the boundary you maybe didn't choose the maximally interesting objective to optimize? Obviously, my thoughts on this are still in flux.

Pattastic3 karma

Hey Jordan, I'm a Wisconsin alumnus, I enjoyed your first book. Did you write any cryptic messages in your book similar to the one you referenced in the Torah?

Also, where do you like to get your news? Is there a source that you think that looks at data well from the perspective of your book?

Jordan-Ellenberg8 karma


The cryptic messages in the book are all lines that reference lyrics to songs I like. You don't have to count every 573rd letter, you just have to have been into guitar-based indie rock in the 90s.

Pattastic1 karma

Oh my god you did not disappoint

Jordan-Ellenberg1 karma

Off the top of my head, I know where there are Guided By Voices, Magnetic Fields, and Brian Eno references. Can you find all three? (There are also non-hidden Eno references but that's not what I'm talking about.)

Jordan-Ellenberg4 karma

As for news: the level of attention to data in major news sources has absolutely skyrocketed in the last decade. (Nate Silver's success, showing tradition-minded newspapers and magazines that readers are actually hungry for this stuff, is a big part of that.) The New York Times Upshot is consistently good. So is 538 itself.

The truth is, a lot of my news is served to me by Twitter; that's good in some ways (e.g. by following state government reporters for the Wisconsin State Journal and the Milwaukee Journal-Sentinel I can get very fine-grained and rapid updates on what's happening in my state government) but bad in other ways (because Twitter is Twitter and you can bias your information intake towards what you expect to want to hear.) After many years without, we started getting paper newspapers delivered, and honestly, the print newspaper is a miraculously great information delivery device and I feel silly for having gone so long without it. It's a great ways to get news you didn't mean to get.

EmmanuelSarkozy3 karma

What were some books that really shifted or enhanced the way you look at mathematics?

Jordan-Ellenberg10 karma

As a kid, for sure, Douglas Hofstadter's Godel Escher Bach. I loved about it that I couldn't understand all of it but there were parts I could understand, and then as I re-read it as I got older, I could follow more and more of it. I try to make my own books work like that; you can read them with a pencil and paper and really work through the parts where it gets technical, or you can skim those parts because you're never far from getting back to stories about humans. Hofstadter's books, of course, are also really insistent on mathematics as a human activity that's inextricably wound around our other mental activities -- talking, art-making, philosophizing, etc. -- that has affected my view of what mathematics is and certainly it has affected what I choose to write about.

Jordan-Ellenberg10 karma

And as a researcher, Joe Silverman's book The Arithmetic of Elliptic Curves (a lot of number theorists would say the same thing) which just absolutely sells the hell out of the idea that the road to understanding the most classical, simple-to-state problems inevitably runs through abstractions developed only in the 20th century, and that the path doesn't have to be brambly and terrible but is in fact really fun to follow.

CharlesDeGaulle3 karma

During your research on gerrymandering, what districts did you find to be the most absurd? On a state level, which Wisconsin districts were the worst? Thanks!

Jordan-Ellenberg7 karma

The badness of a gerrymander is in part about the badness of certain districts, in part a global thing about the badness of the map as a whole. But I'd point to the districts that span the Waukesha county - Milwaukee county border; legally, state legislative districts aren't supposed to cross county borders unless population equality makes that necessary, which it clearly doesn't here, and those districts are very openly designed to mix rather different political communities together in proportions intended to deliver them all to one party (if you live in Wisconsin or if you've read that part of the book, you know which one.)

kitten-choir3 karma

why is math and statistics so daunting for so many people in your experience/opinion, and how could we improve this?

ps. I'm a huge fan, HNTBW was the decisive factor for me to go for it and study statistics :)

Jordan-Ellenberg6 karma

I feel a lot of responsibility now and I hope you're happy with your decision!! (But as the child of two statisticians I of course think it's a great line of work.)

Why so daunting? I think one reason is that it's one of the places in school where you're most liable to be told you're wrong, and that's naturally uncomfortable for most people. So I think we have to strive for a classroom where people can be as comfortable as possible being wrong sometimes! Ironic, I know, for a person who wrote a book about How Not To Be Wrong to say this, but in a classroom setting, being wrong a lot of the time is how you know you're working on the skills you need to learn.

HarryPotter57773 karma

What are your thoughts on spending one's time thinking about natural elementary problems that seem captivating for their own sake (isoceles triangles on an NxN grid seems like one such), versus "serious" mathematical research of the sort that takes a graduate course or two to even understand why someone would care about the problem?

I get the sense that problems of the former sort are often looked down in mathematical research - that being able to explain to a bright 5th grader what the problem is and why it's cool is a mark against the quality of a problem, rather than a sign of its value. At minimum, it seems like relatively few mathematicians actually spend much of their time thinking about such problems, even for the many such which are pretty tractable. This twitter thread on John Conway gets at some of the same thing.

(Of course, if you genuinely love pondering the homology of quasi-monoidal infinity categories or whatever the latest hot topic in mathematical research is, great! But sometimes the thing you're passionate about has practically no widespread interest in "real" mathematics, even if the questions are very natural to ask.)

Are people like John Conway (or their less-talented admirers) doing the "wrong" sort of math? Are the Fields medalists?

Jordan-Ellenberg7 karma

I write a lot about Conway in the book actually! And I actually am not totally on board with that thread; it seems weird to say Conway got no recognition when he was a tenured full professor at Princeton for decades! He didn't get the Fields Medal -- well, so didn't a lot of people!

At any rate, which questions are deemed "interesting" is of course a very interesting mix of mathematics and sociology, and can't be understood through either lens alone. All I can say is that I've worked on very mainstream steep-learning-curve problems like modularity of Galois representations, and I've worked on very "olympiad-style" math like the cap set problem, and I truly don't believe that in my own life one was taken more seriously than the other. And I mean look at this whole zone of extremal combinatorics and algebraic methods, of which the isosceles triangle problem is a part -- I mean, look at people who are working in that area, you have Lisa Sauermann and Yufei Zhao and Larry Guth at MIT, you have Jacob Fox at Stanford, you have Terry Tao -- this is obviously not work that's looked down on!

ban_mat_karna_BC2 karma

What do you think about Arnold v/s Bourbaki ?

Jordan-Ellenberg7 karma

I can't help being naturally attracted to the very austere French style of exposition, while at the same time not really *thinking* of mathematics in that way -- does anyone? I touch on this issue in the book, though not directly in terms of Arnold and Bourbaki; Poincare writes very compellingly about the fact that you need *both* the rigidly formal and the intuitive in order to get things done; he compares the intuition to the sponge and the formalism to the skeleton it builds around itself. The sponge needs the skeleton to keep itself from collapsing, while the skeleton makes no sense unless you know about the sponge.

HeegaardFloer2 karma

I'm acquainted with one of your students, Silas Johnson (from the days at Stanford). I am interested in math education in general, and was wondering what how you think the pandemic has affected math education. Here are some concrete questions in that direction: What are the main things that need to be improved with remote learning? What things can we take away from remote learning as we slowly transition to in-person classes? What things are very inconvenient, and can be improved by technology?

Jordan-Ellenberg5 karma

I don't know anybody who thinks teaching on the screen is *as good* as teaching in person, but I do think I got much better at it over the course of the year and I really hope the skills we've built up aren't wasted. For instance; now that we've been forced to learn how to deliver a course this way, can we do better at making college courses available for HS students who are ready to take them but don't live near a college campus?

ban_mat_karna_BC2 karma

I am a professional applied mathematician (PDEs/Dynamical systems), but far from number theory and algebraic geometry. If had to read one text-book (other than SHAPE which I will buy anyway) on learning what you do, what would be your recommendation ?

Jordan-Ellenberg6 karma

I might try J.P. Serre's *A Course In Arithmetic*. The style is very different from mine, though -- crystal clear and terse rather than talky like mine. But it is very short and incredibly rich in important ideas and requires no more than undergraduate algebra.

spanthis2 karma

Has your experience writing pop-math books with broader appeal influenced the way you write math papers? Or are these two completely different kinds of writing?

Jordan-Ellenberg7 karma

It hasn't influenced the way I write math papers but I think it's influenced my classroom teaching. From writing the books and seeing what people respond do, I think I understand better how to approach math teaching as storytelling. That came naturally to me when writing prose on a page because I used to want to be a novelist. But I knew less about how to do it in front of people. I'm getting better at it.

dnlwng2 karma

To what extent do you think your early interest/talent in mathematics was fostered by your parents/teachers, as opposed to innate ability? Were your parents mathematical people?

Bonus question: are your children interested in math at all? Would you be sad if they weren’t?

Jordan-Ellenberg3 karma

My parents are both statisticians and no question, that meant I had an easier path into mathematics than somebody else without those resources. My kids are both interested in math -- at the moment, I wouldn't say they're so interested that they want to make a life out of it! If they really were entirely cold to the subject, I suppose I'd be sad, but not a serious sad, more like "A favorite movie of mine doesn't speak to them." Serious sad would be if I had kids with no sense of humor or who rooted for the Yankees.

manuelestavillo2 karma

Is math discovered or invented?

Jordan-Ellenberg2 karma

We discover it by inventing it.

manuelestavillo1 karma

I’ll admit that I don’t know exactly what you mean by this lol.

Jordan-Ellenberg3 karma

I mean that we simultaneously feel like we are creating things, but then, having created them, feel that they were already there. I don't want to police people's feelings; I feel that both those feelings are right!

whitesoxs1412 karma

Do you have any thoughts on about how mathematically oriented grad students should weigh a career in academia versus industry?

Jordan-Ellenberg5 karma

This is hard and there's no one-size-fits-all answer; the only way through this is to know yourself well enough to know what kind of workplace would fit you best. Of course one hopes the faculty in your graduate program understand that their role is to help you figure that out, not to create clones of themselves. (At Wisconsin I think we have a pretty good culture around this issue.) One thing I hear people talk about a lot -- the timescale of industry and academia are really different. Industry has much faster turnaround expectations, which can be stressful, but you also get much faster feedback, the lack of which can be stressful! Talk to people, talk to people, talk to people, people who are now living the lives you might lead in the future.

BaroqueIsMyJam2 karma

Who is the mathematician with the wildest life who didn't make it into Shape or How Not To Be Wrong?

Jordan-Ellenberg8 karma

Well, Alexander Grothendieck barely sneaks in, coming in for a cameo right at the end, and he basically created the version of geometry my own research is about, and he had quite a life (which ended with him rejecting the mathematical community and going off to live alone in a small mountain village.) It's bizarre that there's not a best-selling book and feature film about his life. But to do a good job writing that book, you would have to be fluent in French, which I am not.

BaroqueIsMyJam1 karma

Alexander Grothendieck

I have done a brief google, and he seems absolutely amazing. I sadly am also not fluent in French, so someone else will need to take that project on. Thank you!

Jordan-Ellenberg4 karma

Here's a start: Scharlau has written a whole book in German, much of which has now been translated -- maybe aimed at more of a mathematical than a popular audience but very good reading (I of course speak only of the English version because I can't read German either!)


le_coque_grande1 karma

Hey Jordan! I loved your book “How not to be wrong”. What I have is maybe less a question, but more a remark...I hope that’s ok. If I recall correctly, there is a point in the book where you mention that sometimes the sequence 0,1,0,1,... is considered to converge to 1/2, and I was wondering in which area of math this happens? From an analysis perspective, this sequence would be a non-converging, alternating sequence. I guess I’m asking this because there are “fake proofs” that use this convergence to prove 1+2+3...=-1/12, which is clearly not true (I think one could get the same result by taking the analytic extension of the Riemann zeta function and plug in the value for -1, but the mistake would be that the analytic extension doesn’t have “look” like the Riemann zeta function if the input isn’t great than 1).

Jordan-Ellenberg7 karma

I really like Hardy's book Divergent Series for this stuff. I don't think of the statement


as either "true" or "not true" because I don't think the left hand side refers to anything in particular until you make some decision about what it refers to. If that decision is that it's shorthand for zeta(-1), which is not a completely unreasonable choice, well, then the assertion is true. If on the other hand you don't think the LHS has any meaning at all, but that the whole string of symbols

"1 + 2 + 3+ .... ="

is shorthand for some statement about a limit, then the assertion is false. But you gotta decide!

0xE4-0x20-0xE61 karma

In the past, it seems like universities provided students a well-rounded education, teaching equally as many courses from the humanities as from the sciences. Now however, students are expected to specialize in one field, and thus not learn much outside of what they’ve chosen. Do you think this is good? Bad? If bad, how do you think we can change this?

Jordan-Ellenberg3 karma

I think this varies a lot country by country. In the US, universities by and large are still in the liberal arts tradition, and in most places, students have distribution requirements which requires them to venture outside their pre-professional studies. But you're certainly right that after high school students are generally much more free, and even encouraged, to heavily weight a particular area. I think that's OK! College students are adults who are starting to know which parts of the intellectual universe really sing to them. I think it's good that they don't all take the same portfolio of courses, and I think it's good that some spend more time learning math and some spend more time learning business and some spend more time learning dairy science (at least here at Wisconsin.) That said, I'm pretty in favor of distribution requirements. Frankly, anyone you talk to who's a college professor has thrived in the existing higher ed system so of course we're biased to think it's great! (Cue picture of Abraham Wald's bullet-riddled plane.)

PaboBormot1 karma

Hey I’m from the /r/math subreddit! How useful is mathematical theory for a machine learning practitioner? I’m thinking stuff like VC dimension and concentration inequalities. Does it actually help you do better work, or is it more useful to just use heuristics?

Jordan-Ellenberg1 karma

Stopping in late to answer this. One of my students now does ML full time and I asked him, "do you feel like all the time you spent learning algebraic geometry and number theory was wasted?" and he said absolutely not -- not because he does AG and NT in machine learning, but because pure math training creates different mores and habits of mind than engineering training, and he felt he had something rather unusual to bring to the problems of ML which is in short supply among current practitioners.

guillemot_221 karma

Do you think a basic knowledge of number theory should be required in high school?

Jordan-Ellenberg7 karma

Despite being a number theorist, I'd say no. But I think we can and should work a little more of it into the curriculum. (Ideas like prime factorization are already there.) Not because students need it for any particular "next thing," but because for some students that will be the exact thing that excites them about math, just as for some students plane geometry is the exact thing that excites them about math. You want everyone to get the chance to experience that excitement and people are really different in what they respond to!

crownofperception1 karma

As a professional researching mathematician, who also writes books, what does a typical day look like? And how do you find the time?

Jordan-Ellenberg5 karma

There's no typical day. (Except that every day involves a lot of answering email.) Some days I'm in the classroom a lot, some days I'm 1-on-1 with my Ph.D. students a lot, some days I'm alone with a preprint writing on it in pen as I try to get it into final form.

One important factor for "finding the time" is that US academia has a sabbatical system, where every seventh year you don't teach or serve on committees with the idea that you are being granted a set span of time to work on a project that really needs your entire attention. And that is why my books come out seven years apart.

ApocalypseSpokesman0 karma

Do you know anything about Eric Weinstein's Geometric Unity?

Jordan-Ellenberg3 karma

You're not the first person to ask me that! You're the second. But the answer remains no.