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:

[math]\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}[/math]

where [math]\phi[/math] is the golden ratio [math]\phi = \dfrac{1+\sqrt{5}}{2}[/math] (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 [math]\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)[/math]

Pure gold.

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.

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.(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

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.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,

likethe, theFields Medal, or theAbel Prize, to come up with a partial list.Cole PrizeWhereas 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.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:

That is probably me, for the following reasons:

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

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

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:

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.But if I were to still answer it then I would go with

Endre Szemeredi, simply because I have known him in person and haveseenhim 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.

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

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

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)

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

44) Srinivasa Varadhan

45) Jacques Tits

46) Stephen Smale

47) Lotfi A. Zadeh

48) Louis Nirenberg

49) Yakov Sinai

50) John Griggs Thompson

52) Lennart Carleson

53) Isadore Singer

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

62) Vladimir Voevodsky

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

72) Vladimir Drinfeld

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

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

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

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

JOHN HORTON CONWAYGrigori Perelman for solving the Poincaré conjecture.

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

Isadore Singer — 44,300

Peter Lax — 118,000

Lennart Carleson — 47,500

Srinivasa Varadhan — 15,800

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.

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