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While reading The Princeton companion to Mathematics Timothy Gowers choose not to list alive mathematicians in the last part but I think it's important to know about them. Recently I've read about Alexander Grothendieck, Jean Pierre Serre, Sir Michael Atiyah, Vladimir Arnold and Robert Wisbauer. It was very interesting and enlightening, I am reading La Clef des Songes by Alexander Grothendieck, watched this lecture by Vladimir Arnold, read an advice to a young mathematician by Sir Michael Atiyah on the Princeton companion to Mathematics, and I usually read Terence Tao's personal blog.

I'd like you to point me to more contemporary mathematicians and readings to know their work, as Abel said one should learn from the masters and being a beginner mathematician it's not clear what one should read to achieve that.

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Even if I'm not a big fan of prizes, I would say one should know the Abel prize winners as well as the Fields medalists. – t.b. Oct 3 '11 at 11:35
I would hardly consider the current list of Abel prize winners as contemporary. Beside the fact that many of them are likely to be eligible to appear in the next edition of the Princeton Companion, being Abel prize laureates almost guarantees that the illustrious books on their lives' works have been ready to be closed for quite a while (which is not to say that they are all done writing yet). That said, you should certainly know about them. – Willie Wong Oct 3 '11 at 12:32
The ten greatest contemporary mathematicians are Arturo Magidin, Qiaochu Yuan, Bill Dubuque, Ross Millikan, joriki, Aryabhata, Andre Nicolas, Pete L Clark, Matt E, and t.b., in that order. You will find links to their recent work at – Gerry Myerson Oct 4 '11 at 5:49
Depends on what is meant by contemporary. Personally I enjoy Barry Mazur's expository writings a lot. Following link could be a start: Mathematicians born from 1940 to the present – user18325 Dec 18 '11 at 6:09
Saharon Shelah. – Asaf Karagila Apr 24 '12 at 6:28
up vote 7 down vote accepted

Wikipedia has a list of 21st-century mathematicians while the book Mathematicians : an outer view of the inner world has the following list in portrait form with their autobiography:

Adebisi Agboola -- Michael Artin -- Michael Francis Atiyah -- Manjul Bhargava -- Bryan John Birch -- Joan S. Birman -- David Harold Blackwell -- Enrico Bombieri -- Richard Ewen Borcherds -- Andrew Browder -- Felix E. Browder -- William Browder -- Lennart Axel Edvard Carleson -- Henri Cartan -- Sun-Yung Alice Chang -- Alain Connes -- John Horton Conway -- Kevin David Corlette -- Ingrid Chantal Daubechies -- Pierre Deligne -- Persi Warren Diaconis -- Simon Donaldson -- Noam D. Elkies -- Gerd Faltings -- Charles Louis Fefferman -- Robert Fefferman -- Michale Freedman -- Israel Moiseevich Gelfand -- William Timothy Gowers -- Phillip Griffiths -- Mikhael Leonidovich Gromov -- Benedict H. Gross -- Robert Clifford Gunning -- Eriko Hironaka -- Heisuke Hironaka -- Friedrich E. Hirzebruch -- Vaughan Frederick Randal Jones -- Nicholas Micahel Katz -- Robion Kirby -- Frances Kirwan -- Joseph John Kohn -- János Kollár -- Bertram Kostant -- Harold William Kuhn -- Robert Phelan Langlands -- Peter David Lax -- Robert D. MacPherson -- Paul Malliavin -- Benoit Mandelbrot -- William Alfred Massey -- John N. Mather -- Barry Mazur -- Margaret Dusa McDuff -- Curtis McMullen -- John Willard Milnor -- Maryam Mirzakhani -- Cathleen Synge Morawetz -- David Mumford -- John Forbes Nash, Jr. -- Edward Nelson -- Louis Nirenberg -- George Olatokunbo Okikiolu -- Kate Adebola Okikiolu -- Andrei Okounkov -- Roger Penrose -- Arlie Petters -- Marina Ratner -- Kenneth Ribet -- Peter Clive Sarnak -- Marcus du Sautoy -- Jean-Pierre Serre -- James Harris Simons -- Yakov Grigorevich Sinai -- Isadore Manual Singer -- Yum-Tong Siu -- Stephen Smale -- Elias Menachem Stein -- Dennis Parnell Sullivan -- Terence Chi-Shen Tao -- Robert Endre Tarjan -- John T. Tate -- William Paul Thurston -- Gang Tian -- Burt Totaro -- Karen Keskulla Uhlenbeck -- Sathamangalam Rangaiyengar Srinivasa Varandhan -- Michèle Vergne -- Marie-France Vigneras -- Avi Wigderson -- Andrew John Wiles -- Shing-Tung Yau -- Don Zagier.

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AA Razborov is a notable absentee from this list – Mark Bennet Nov 14 '14 at 21:58

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