On Twitter I posed the following question:

I got a great repsonse. Here is the complete—unedited—list of names (in alphabetical order).

- Micheal Atiyah
- Art Benjamin
- Andrea Bertozzi
- Manjul Bhargava
- Joan Birman
- Bela Bollobas
- Stephen Boyd
- María Chudnovsky
- Fan Chung
- John H. Conway
- Ingrid Daubechies
- Keith Devlin
- Marcus du Sautoy
- Jordan Ellenberg
- Joe Gallian
- Rob Ghrist
- Tim Gowers
- Judith Grabiner
- Brian Greene
- Benedict Gross
- Dusa McDuff
- Danica McKeller
- Curt McMullen
- Maryam Mirzakhani
- John Allen Paulos
- Mary Rees
- Jean-Pierre Serre
- Michael Sipser
- Ian Stewart
- Steven Strogatz
- Daina Taimina
- Terence Tao
- Eva Tardos
- Ravi Vakil
- Robin Wilson

If you have other suggestions, add them in the comments.

John Milnor!

Living mathematician and expositor Ivars Peterson’s distinguished career at Science News easily qualifies him for this list.

Even without generalizing the meaning of “expositor” to include great leaps in typesetting technology, I think one could also make a pretty strong argument for including Donald Knuth.

I thought of Knuth, too, because of his book

Surreal Numbers. I came to add Paul Lockhart. I recently finished readingMeasurementand loved it.If you include people who may not qualify as mathematicians, but as great popularizers of math, you could add Raymond Smullyan and Marilyn Burns.

Cédric Villani qualifies as both, I think.

David Spiegelhalter, if you’ll have a statistician, and John Baez has a good reputation in this area. And don’t we hear good things about Cédric Villani?

John Baez

William Dunham.

Andrew Wiles

Donald Knuth is not known for being a mathematician but he has quite a bit of nice work under his belt. And he is certainly one of the best expositors around. I’m not thinking of Surreal Numbers as someone else mentioned, but rather his famous books and his collected papers. He explains things so plainly and yet so eloquently, with just the right number of diagrams and with a voice that’s authoritative yet familiar at the same time.

I would certainly add, having met him personally, Stephen Hawking. He is a brilliant mathematician and the universe would be a much bigger mystery without his contribution to clear expression.

Surprised not to find Persi Diaconis on the list.

Edward Witten

You guys didn’t mind the factor ‘expositor’. All of people such as Wiles or Witten are great enough but I don’t agree they are good expositors. If you don’t agree with me, prove it.

Valid point. “Best fit” probably doesn’t mean “maximize the following sum of two real variables multiplied by a boolean.” Actually Dave didn’t even pose it that way, it’s just a sum, so I guess if all we were doing was maximizing, somebody like Gauss (er, I mean Gauss) would get in.

…except the title does imply the boolean has to be 1, so I think Dave really meant to write “*” instead of “+” in both places where it appears.

Ha ha, you guys crack me up. An interesting list is a good outcome. But when I wrote the question I was asking for someone who embodied all three characteristics. High marks in one category can’t overcome low marks in another. Clearly some of the people on this list should not be there.

http://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz

Dave, which names from this list do you think best fit the description? What about anyone not on the list?

I don’t know. It is a difficult question to answer. I was hoping for names like John Conway, Steve Strogatz, etc.—people who have a solid track record of high quality research and also can deliver an excellent talk to people who don’t know any math.

Some of the names on the list fall into one camp, but not the other. Many (most of them) are clearly in one camp, and I just don’t know if they fall into the other. For example, Keith Devlin, Marcus du Sautoy, and John Allen Paulos are great spokespeople for mathematics, but I am unfamiliar with their scholarly work—I don’t know if they are top mathematical researchers. On the flip side, Terrence Tao and Tim Gowers are outstanding scholars, and I know they can write well for a mathematical audience, but I don’t know what they would be like if they were asked to write/speak for a nonmathematical audience.

Raymond Smullyan’s papers on logic aren’t as well known as his puzzle books, but I concur with suevanhattum that he should be here.

This is a great idea, Dave. It held surprises: Atiyah and Serre still living and having written or spoken popularizations? I didn’t know either.

I’d be curious to see how many of these find their way into the “greatest mathematicians of all times” lists that one can find at places like

http://fabpedigree.com/james/mathmen.htm

Or how many mathematicians make your list after you relax “still living.”

Teaching History of Mathematics, I find it hard to bring the history all the way up to the 21st century. This list is a good starting point for thinking about that.

I saw McMullen give his lecture on Oct 11, 2006 on “Geometry of 3-manifolds” at Harvard – I first thought I was in the wrong place bc the place was packed and he goes to lectures on such onscure topics?. Then after Lisa Randall introduces him, he give the best math lecture I have ever seen http://athome.harvard.edu/threemanifolds/watch.html

The guy had props, elan – it was fantastic

vi hart

vi hart is a great expositor, but not a mathematician.

Depends on your definition of mathematician. I do not have a phD (just a masters). Until 5 years ago, I would not have called myself a mathematician; now I do. Great mathematician? Nah. Vi Hart has probably done nothing new as a mathematician, so she probably can’t score on the first criteria either.

Dave, why did you choose great mathematician as one of the criteria for your list? What does this list do for you? (My personal interest is in great math expositors and popularizers, so I might play around with this idea in my own way later.)

Alexandre Grothendieck, simple as that.

Marsden

Mario Livio’s book “Is God a Mathematician” is a clearly-written book with a great treatment of some of the giants of mathematics. It deals with a pretty tough subject, and I learned a lot from it.

I am in the midst of reviewing the Selected Works of George Andrews for MAA Reviews, and I would suggest adding him to the list. He may not have done as much exposition for truly general audiences, but some really good stuff.

Don Zagier (http://people.mpim-bonn.mpg.de/zagier/) deserves to be included as a contender, and it’s possible that I would choose him as my favorite if I knew enough about the other mathematicians listed to form a judgment.

—

MATHEMATICAL WORK

Some of his striking discoveries are:

•His finding with Hirzebruch about intersection numbers of curves on Hilbert modular surfaces being the coefficients of a modular form.

>(with F. Hirzebruch) Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus

My impression is that this marked the birth of the rich theory of special cycles on Shimura varieties.

•His conjecture on expressing special values of Dedekind Zeta functions in terms of polylogarithms.

>Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields in Arithmetic Algebraic Geometry (G. v.d. Geer, F. Oort, J. Steenbrink, eds.), Prog. in Math. 89, Birkhäuser, Boston (1990) 391-430 (pdf)

•His work with Benedict Gross giving a formula for norms of differences of singular moduli

>(with B. Gross) Singular moduli J. reine Angew. Math. 355 (1985) 191-220 (pdf)

and his later work showing that traces of singular moduli are the coefficients of a modular form

>Traces of singular moduli In “Motives, Polylogarithms and Hodge Theory (Eds. F. Bogomolov, L. Katzarkov), Lecture Series 3, International Press, Somerville (2002), 209-244 (pdf)

•The Gross-Zagier formula

>(with B. Gross) Heegner points and derivative of L-series

Invent. Math. 85 (1986) 225-320 (pdf)

•The computation of the (orbifold) Euler characteristic of the moduli space of curves of genus g with n marked points

>(with J. Harer) The Euler characteristic of the moduli space of curves

Invent. Math. 85 (1986) 457-485 (pdf)

EXPOSITORY WORK

I’ve found his papers to be a great pleasure to read.

For those who have a solid undergraduate background (including complex analysis), I would strongly recommend his notes on Elliptic Modular Forms and Their Applications (http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/978-3-540-74119-0_1/fulltext.pdf).

Great J. Milnor, M. Goromov, A. Avila…..No one seen??

Having hosted his public talk/film screening and his topology seminar here last week, we’d add Ettiene Ghys to your list.

What about Tom Apostol?

Andrew Granville and Don Knuth.