# Who are the best living mathematicians?

Apart from some of the obvious answers like Terrence Tao, Andrew Wiles and Grigori Perelman which other people have mentioned, I would like to add some less-obvious names:

Manjul Bhargava: He’s a field medalist, and has made fundamental contributions in number theory (Geometry of numbers), representation theory, p-adic analysis and theory of elliptic curves (analyzing the BSD conjecture). Check out his paper on Average rank of elliptic curves, the fact that he explains such a strenuous topic so easily and efficiently makes me admire his mathematical skills.

Ravi Vakil: He is an algebraic geometer and his research work spans on various topics like Gromov-Witten theory. He is one of the very few mathematicians, to have worked on topics like Schubert calculus. He has solved several old problems in Schubert calculus. Among other results, he proved that all Schubert problems are enumerative over the real numbers, a result that resolves an issue mathematicians have worked on for at least two decades. Ravi Vakil (with other mathematicians) has also provided solutions of previously asked Putnam Questions. You can find them here. Some of his solutions are so elegant and unique, ‘mind-boggling’ is the word which comes to my mind to describe it.

Anyhow, all these mathematicians (Tao, Wiles, Grigori, Manjul, Tate, Scholze etc.) are still popular among other mathematicians or aspiring mathematicians. These mathematicians are outstanding in their respective fields (or many other fields).

However, I would like to mention two other unheard names here. These two other people are not the best number theorists or algebraic geometers or topologists. I don’t even know if they’ve studied these topics extensively or not. I don’t even know if they’re officially mathematicians or not. I just know that they’re one of the best ‘integrators’ in the world.

First: The mysterious User Cleo (MSE guys will know). People refer to her as ‘the second goddess of Namagiri’ as her style is just like Ramanujan’s. Loads of results in integrals, series and astounding relations to the zeta function. Polylogarithms, Hypergeometric functions, Trigamma, Huwitz Zeta, Dirichlet eta functions, you name it and she has mastered them all. She is ‘mysterious’ as she has provided closed forms of such integrals which neither Mathemamica, Wolfram nor Maple could find. One of my personal favorites of her results is:

$\displaystyle \int_{-1}^{1} \dfrac{1}{x}\sqrt{\dfrac{1+x}{1-x}}\ln \left(\dfrac{2x^2+2x+1}{2x^2-2x+1}\right) = 4\pi\cot^{-1}\sqrt{\phi}$

where $\phi$ is the golden ratio $\phi = \dfrac{1+\sqrt{5}}{2}$ (Prove this, I dare you, I double dare you!)

She also conjectured a few strange integrals with Airy Functions and their relations with the zeta function which have now surprisingly been proved true. I believe she is outstanding with special integral transforms (like Jacobi and Gegenbauer transforms) as well.

Second: Cornel loan Valean. Have you ever solved an American Mathematical Monthly problem? Most of those integrals/series/analysis/functional equation problems are proposed by this guy. One of the most brilliant applied mathematicians I’ve ever seen! Check out his recently published paper of cubic harmonic series: A master theorem of series and an evaluation of a cubic harmonic series, it’s worth appreciating.

His problems might look scary, but are extremely delightful and fruitful to solve, one of them being this:

I recently proved one of his famous results on Quora which was also asked on AMM recently, and I’m deeply honored to have received appreciation from him 🙂 ! Here is that problem:

Prove that $\displaystyle \int_{0}^{\infty} \int_{0}^{\infty} \dfrac{\sin x \sin y \sin (x+y)}{xy(x+y)} \mathrm{d}x\,\mathrm{d}y = \zeta(2)$

Pure gold.

## 19 Replies to “Who are the best living mathematicians?”

1. Tomáš Kania says:

It's a travesty that no one so far has mentioned John Tate:

In roughly chronological order, here are some of his groundbreaking achievements, all of which revolutionized number theory and continue to have ramifications (no pun intended) to this day.

• His 1950 PhD thesis (under the supervision of Emil Artin) reproved Hecke's theorem that the L-functions associated to Grossencharacters have an analytic continuation and a functional equation. (These simultaneously generalize the Riemann zeta-function and Dirichlet's L-functions, which come up in his proof of the theorem on primes in arithmetic progressions.) More important than the result was the method: he introduced the technique of obtaining arithmetic results by doing harmonic analysis on adelic algebraic groups, arguably initiating (in embryonic form) what eventually became the famous Langlands program.
• With his advisor Artin, Tate reworked the foundations of classfield theory using group cohomology. In addition to introducing important techniques in homological algebra, his work greatly clarified (in my opinion) the proofs of the main results of early 20th century number theorists. The Artin-Tate notes on the subject (from the early 1950s) remain a definitive reference in the field.
• He invented the field of p-adic analytic ("rigid analytic") geometry, with the so-called Tate parametrization of an elliptic curve. It was a remarkable leap of imagination to realize that although the p-adic numbers (and algebraic varieties defined over the p-adic numbers) form a totally disconnected topological space (a Cantor set), one can still carry over the techniques of complex analysis to this realm; in particular, a notion of "analytic continuation" makes sense, despite being nonsensical from a naive perspective.
• His paper on p-divisible groups from the mid-1960s arguably began the subject of p-adic Hodge theory, another instance of porting results from geometry over the complex numbers to the p-adic realm. This subject remains vibrant and is a crucial input into the deep results on the Langlands program from the last 20-30 years, in particular into "modularity lifting theorems" such as the one Wiles proved in the course of proving Fermat's last theorem.
• With Lubin, Tate conducted some important early research into formal groups. In addition to applications to classfield theory (the content of the first famous Lubin-Tate paper), their study of moduli of formal groups has led to important (and I think surprising) applications in homotopy theory, a branch of algebraic topology seemingly far-removed from the initial number theoretic context.
• With Honda, Tate made a definitive study of abelian varieties (e.g., elliptic curves) over finite fields, and especially their endomorphism algebras.
• All of the above is really only the tip of the iceberg. For a survey of his work, illustrating its awe-inspiring breadth and impact in both number theory and algebraic geometry (as well as abstract algebra, more generally), I refer you to Milne's writeup: The Work of John Tate

Another striking omission from the current list is Pierre Deligne:

In addition to the foundational work he did on algebraic geometry (in the SGA seminars of Grothendieck in the 1960s), especially crucial work on etale cohomology…

• He proved the (hardest part of) the Weil conjectures, completing a program initiated by Grothendieck. At the same time, he proved the Ramanujan-Petersson conjecture on the growth rate of the Fourier coefficients of modular forms.
• He revolutionized Hodge theory in a famous series of papers, in particular introducing the "yoga of weights".
• With Mumford, he introduced the idea of algebraic stacks, a generalization of algebraic varieties which is now the lingua franca for studying "moduli problems" in algebraic geometry.
• He has a number of deep results on algebraic geometry in characteristic p (and its relation to characteristic 0 algebraic geometry), e.g. on K3 surfaces, and on the "Hodge-de Rham spectral sequence".
• With Lusztig he made major contributions to the study of finite groups; especially, they gave a geometric construction for the irreducible representations of a finite group of Lie type (which comprise "most" finite simple groups).

(While this list is shorter than the one for Tate above, that's mostly because of my own lack of expertise on Deligne's work. In particular, it is difficult to overstate how influential all of the items above have been.)

The Mathematician who solved a 358 year old problem
Andrew Wiles , who proved the age old Fermat's Last Theorem, which had its name even in Guinness Book of World Records for being the "most difficult mathematical problems" and had troubled generations of mathematicians.

The importance of this theorem can be judged from the fact that it led to development of algebraic number theory in 19th century and modularity theorem in 20th century.

The proof itself is over 150 pages long and consumed seven years of Wiles' research time. John Coates (who happened to be his PhD supervisor and a leading mathematician of the time ) described the proof as one of the highest achievements of number theory( though he initially told Andrew Wiles that this theorem can't be proved ), and John Conway called it the proof of the century. For solving Fermat's Last Theorem, he was knighted, and received other honors.

And by the way Manjul Bhargava, Field's Medal awardee of 2014 was his doctoral student.

More on him :
Andrew Wiles
Wiles' proof of Fermat's Last Theorem

3. Daniel McLaury says:

In no particular order:

Terence Tao: Harmonic analysis, partial differential equations, additive combinatorics, ergodic theory, random matrix theory, and analytic number theory.

Noga Alon: Combinatorics.

Endre Szemeredi: Combinatorics.

John Milnor: Differential topology, K-theory and dynamical systems.

Jean-Pierre Serre: Algebraic geometry, number theory, and topology.

4. Tomáš Kania says:

Essentially the same answer as the I one I gave to Who according to you is the best mathematician? a little while ago.

Since I am allowed to mention more than one this time, I may look from among the winners of coveted awards, like the Fields Medal, the Abel Prize, or the Cole Prize, to come up with a partial list. Whereas there can be no doubt whatsoever about the quality of these mathematicians, no list can ever be complete, specially when the question by itself is not well-defined.

5. Allan Steinhardt says:

I don't think that question has any real meaning today, because different people will give different answers (as other posters have demonstrated), based on their own particular tastes and prejudices, and there is no objective way of choosing between them.

The fact is that modern-day mathematics (meaning mathematics since the early 20th century) is a vast discipline; it is both very broad, with around 200 sub-branches (depending on how you divide it up), and very deep. This means that professional mathematicians are forced to specialize in a small number of related fields if they are to have any hope of making an original contribution and earning a reputation for themselves. Today, it is literally impossible for a single mathematician to learn and become proficient in all the different branches of mathematics, because there are simply too many of them, and there is not enough time in a human lifespan for the required study.

Many people believe that the last truly 'universal' mathematician was the frenchman Henri Poincare (1854 – 1912) (http://en.wikipedia.org/wiki/Hen…), as it was believed that he had mastered every field of mathematics that existed at the end of the 19th century, and he made significant contributions to many. He was also a theoretical physicist, and it is thought  that he was very close to developing his own theory of relativity, before Einstein beat him to it.

Since Poincare's time, the corpus of mathematical knowledge has grown exponentially and, as a consequence, the age of the mathematical universalist is over,  and all mathematicians today are narrow specialists to one degree or another.

That being said, mathematicians can still be judged by their brilliance in their own narrow field, and by the impact their work has had on mathematics as a whole (and some branches of mathematics, being more central and fundamental, and less highly specialized, are better-positioned to have a wide-ranging impact on other branches in this respect). By these criteria, three names that immediately spring to my mind are:

• Ed Witten ( http://en.wikipedia.org/wiki/Edw… ) – possibly the world's greatest living physicist, and also a brilliant and prolific mathematician, who has made significant contributions to string theory, M-theory and related areas of mathematics.
• Alexander Grothendieck ( http://en.wikipedia.org/wiki/Ale… ) – one of the visionary geniuses behind the 'Nicolas Bourbaki' group, who made significant contributions to category theory, functional analysis, homological algebra and algebraic geometry.
• John Horton Conway ( http://en.wikipedia.org/wiki/Joh… ), who made significant contributions to the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also invented the concept of cellular automata and The Game of Life.
6. Ricky Kwok says:

That is probably me, for the following reasons:

• Every article that I got published so far won me a Fields medal.
• I won a million dollars for every Millenium Problem that I solved.
• My Erdős number is $\infty$

I cannot name any professional mathematician, who can claim that (but I can find many non-mathematicians 😉 )

7. Alon Amit says:

Assuming he's still alive, which nobody knows for sure:

8. Anurag Bishnoi says:

grigori perelman  who has made landmark contributions to Riemannian geometry and geometric topology

He solved the poincare conjecture somewhat recently. He is supposedly a recluse and turns down all awards.

Him refusing the fields medal:

9. Dan Vanderkam says:

I am not sure what this question means. All such questions might also be considered unfair to the many giants mathematicians on whose shoulders the ones who would be quoted here stood.

One might even say that in a strict sense there are no mathematicians left today except the very few. In what sense? I'd quote the great Norbert Weiner, while this might seem to be off topic, one might easily think that "Mathematician" is too broad a characterization. If one refines the question to ask – Who is the best Graph Theorist? Who is the best Algebraic Geometer etc? Then we are talking.

Since Leibniz there has perhaps been no  man who has had a full command of all the intellectual activity of his  day. Since that time, science has been increasingly the task of  specialists, in fields which show a tendency to grow progressively  narrower. A century ago there may have been no Leibniz, but there was a  Gauss, a Faraday, and a Darwin. Today there are few scholars who can  call themselves mathematicians or physicists or biologists without  restriction.

A man may be a topologist or an  acoustician or a coleopterist. He will be filled with the jargon of his  field, and will know all its literature and all its ramifications, but,  more frequently than not, he will regard the next subject as something  belonging to his colleague three doors down the corridor, and will  consider any interest in it on his own part as an unwarrantable breach  of privacy.

But if I were to still answer it then I would go with Endre Szemeredi, simply because I have known him in person and have seen him do mathematics and am reasonably familiar with some of his very deep results. I can't pretend to understand someone as great as say Mikhail Gromov based on wikipedia and popular articles. Someone's work might be very beautiful and important but might never get enough attention till many decades later. Hence "best" for me is very fuzzy.

However, if "best" relates to more "deep results" then Grothendieck and Terence Tao also get my vote.

10. Tomáš Kania says:

On a more "discrete" side, I would mention Laszlo Lovasz.

11. Dan Vanderkam says:

Some candidates:

* Tim Gowers
* Curt McMullen
* Terence Tao
* Andrew Wiles
* Ed Witten
* Shing-Tung Yau
* Joe Harris
* Stephen Smale
* Bill Thurston
* Mikhael Gromov
* Barry Mazur
* Noam Elkies

12. Tushant Jha says:

Alongside those mentioned, I'd like to add some from my particular subfield (machine learning/topological data analysis) who deserve recognition for their accomplishments and aren't well-known to the general public:

Gunnar Carlsson (founder of Ayasdi and co-developer of the Mapper algorithm)
Larry Wasserman (who has pioneered statistical inference in TDA)
Jerome Friedman (the father of gradient boosting)
Emil Saucan (Ricci curvature and geometry on networks)
Jurgen Jost (Ricci curvature and geometry on networks)
Melanie Weber (a recent student of Saucan and Jost who is carving out a new field of study)
Hemant Ishwaran (father of random survival forests and machine-learning-based survival models in general)

13. Mathieu Dutour Sikiric says:

A list of great living mathematicians:

1) John Tate
2) Pierre Deligne
3) Endre Szemeredi
4) Laszlo Lovasz
5) Grigori Perelman
6) Terence Tao
7) Noga Alon
8) John Milnor
9) Jean-Pierre Serre
10) Tim Gowers
11) Curt McMullen
12) Andrew Wiles
13) Ed Witten
14) Shing-Tung Yau
15) Stephen Smale
16) Barry Mazur
17) Noam Elkies
18) John Conway
19) Michael Atiyah
20) Simon Donaldson
21) Peter Sarnak
22) James Harris Simons
23) Manjul Bhargava
24) Stanislav Smirnov
25) Charles Terence Clegg "Terry" Wall
26) John Ball
27) Ingrid Daubechies
28) Robert Langlands
29) Ben J. Green
30) Joseph B. Keller
31) Brian D. Ripley
32) Frank Kelly
33) Mikhail Gromov
34) Bernard Silverman
35) Wendelin Werner
36) Elon Lindenstrauss
37) Yurij Manin
38) Christopher Zeeman
39) Roger Penrose
40) John Baez
41) Donald Knuth
42) Peter Lax
43) Lennart Carleson
45) Jacques Tits
46) Stephen Smale
48) Louis Nirenberg
49) Yakov Sinai
50) John Griggs Thompson
52) Lennart Carleson
54) Shinichi Mochizuki
55) Martin Hairer
56) Maryam Mirzakhani
57) Artur Avila
58) Cédric Villani
59) Stanislav Smirnov
60) Ngô Bảo Châu
61) Andrei Okounkov
63) Richard Borcherds
64) David Mumford
65) Charles Fefferman
66) Grigory Margulis
67) Alain Connes
68) Shing-Tung Yau
69) Simon Donaldson
70) Gerd Faltings
71) Michael Freedman
73) Vaughan Jones
74) Shigefumi Mori
75) Jean Bourgain
76) Pierre-Louis Lions
77) Jean-Christophe Yoccoz
78) Efim Isaakovich Zelmanov
79) Maxim Kontsevich
80) Laurent Lafforgue
81) Andrei Okounkov
82) Enrico Bombieri
83) Sergei Novikov
84) Heisuke Hironaka
85) Alan Baker
86) Klaus Roth

14. Ishan Dutta says:

I'll throw in votes for Serre, Grothendieck (still alive, though not mathematically active), and John Tate.

15. Anurag Bishnoi says:

Difficult to choose one over another but here I choose
Terence Tao
not because of his long list of achievements but because of the age at which he achieved those

16. Daniel McLaury says:

There is one guy you just cannot skip. Probably a living legend.

JOHN HORTON CONWAY

17. Kiyoto Tamura says:

Grigori Perelman for solving the Poincaré conjecture.

18. Kiyoto Tamura says:

The Greatest Mathematician Alive

When the Abel Prize was announced in 2001, I got very excited and started wondering who would be the first person to get it. I asked my friends and colleagues who they thought was the greatest mathematician alive. I got the same answer from every person I asked: Alexander Grothendieck. Well, Alexander Grothendieck is not the easiest kind of person to give a prize to, since he rejected the mathematical community and lives in seclusion.

Years later I told this story to my friend Ingrid Daubechies. She pointed out to me that my spontaneous poll was extremely biased. Indeed, I was asking only Russian mathematicians living abroad who belonged to “Gelfand’s school.” Even so, the unanimity of those responses continues to amaze me.

Now several years have passed and it does not seem that Alexander Grothendieck will be awarded the Prize. Sadly, my advisor Israel Gelfand died without getting the Prize either. I am sure I am biased with respect to Gelfand. He was extremely famous in Soviet Russia, although less well-known outside, which may have affected the decision of the Abel’s committee.

I decided to assign some non-subjective numbers to the fame of Gelfand and Grothendieck. On Pi Day, March 14, 2010, I checked the number of Google hits for these two men. All the Google hits in the rest of this essay were obtained on the same day, using only the full names inside quotation marks.

Alexander Grothendieck — 95,600

Israel Gelfand — 47,900

Google hits do not give us a scientific measurement. If the name is very common, the results will be inflated because they will include hits on other people. On the other hand, if a person has different spellings of their name, the results may be diminished. Also, people who worked in countries with a different alphabet are at a big disadvantage. I tried the Google hits for the complete Russian spelling of Gelfand: “Израиль Моисеевич Гельфанд” and got an impressive 137,000.

Now I want to compare these numbers to the Abel Prize winners’ hits. Here we have another problem. As soon as a person gets a prize, s/he becomes more famous and the number of hits increases. It would be interesting to collect the hits before the prize winner is announced and then to compare that number to the results after the prize announcement and see how much it increases. For this endeavor, the researcher needs to know who the winner is in advance or to collect the data for all the likely candidates.

Jean-Pierre Serre — 63,400

Michael Atiyah — 34,200

Peter Lax — 118,000

Lennart Carleson — 47,500

John Thompson — 1,610,000

Jacques Tits — 90,900

Mikhail Gromov — 61,900

John Thompson is way beyond everyone else’s range because he shares his name with a famous basketball coach. But my point is that Gelfand and Grothendieck could have been perfect additions to this list.

I have this fun book at home written by Clifford Pickover and titled Wonders of Numbers: Adventures in Mathematics, Mind, and Meaning. It was published before the first Abel Prize was awarded. Chapter 38 of this book is called “A Ranking of the 10 Most Influential Mathematicians Alive Today.” The chapter is based on surveys and interviews with mathematicians.

The most puzzling thing about this list is that there is no overlap with the Abel Prize winners. Here is the list with the corresponding Google hits.

Andrew Wiles — 64,900

Donald Coxeter — 25,200

Roger Penrose — 214,000

Edward Witten — 45,700

William Thurston — 96,000

Stephen Smale — 151,000

Robert Langlands — 48,700

Michael Freedman — 46,200

John Conway — 203,000

Alexander Grothendieck — 95,600

Since there are other great mathematicians with a lot of hits, I started trying random names. In the list below, I didn’t include mathematicians who had someone else appear on the first results page of my search. For example, there exists a film director named Richard Stanley. So here are my relatively “clean” results.

Martin Gardner — 292,000

Ingrid Daubechies — 76,900

Timothy Gowers — 90,500

Persi Diaconis — 84,700

Michael Sipser — 103,000

James Harris Simons — 107,000

Elliott Lieb — 86,100

If prizes were awarded by hits, even when the search is polluted by other people with the same name, then the first five to receive them would have been:

John Thompson — 1,610,000

Martin Gardner — 292,000

Roger Penrose — 214,000

John Conway — 203,000

Stephen Smale — 151,000

If we had included other languages, then Gelfand might have made the top five with his 48,000 English-language hits plus 137,000 Russian hits.

This may not be the most scientific way to select the greatest living mathematician. That’s why I’m asking you to tell me, in the comments section, who you would vote for.

Source:tumblr articles and some edits are mine.

I hope that downs your question.

19. Kiyoto Tamura says:

He may not be a popular choice, but Grigori Perelman should be in the running.