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There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, which can inspire the desire, even the need for doing mathematics, or can make one to confront some kind of problems, dedicate his life to a branch of math, or choose an specific research topic.

I think that this kind of force must not be underestimated; on the contrary, we have the duty to take advantage of it in order to improve the mathematical education of those who may come after us, using the work of those gifted mathematicians (and even their own words) to inspire them as they inspired ourselves.

So, I'm interested on knowing who (the mathematician), when (in which moment of your career), where (which specific work) and why this person had an impact on your way of looking at math. Like this, we will have an statistic about which mathematicians are more akin to appeal to our students at any moment of their development. Please, keep one mathematician for post, so that votes are really representative.

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closed as off topic by quid, Yemon Choi, Andres Caicedo, Ryan Budney, S. Carnahan Sep 5 '11 at 14:08

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It's always your advisor(s) that influence you the most, aren't they? –  Zsbán Ambrus Jul 5 '10 at 22:00
@Zsbán: I'm not so sure about that! I have the feeling that many mathematicians are most influenced by some others, or some works, or some open problems, or even some teachers BEFORE getting to have an advisor at all! –  Jose Brox Jul 5 '10 at 23:12
Interesting question. I noticed that all the stronger mathematicians I know (or know of) have other mathematicians that they look up to (sometimes long gone mathematicians who only communicate with us through their writings). So that the most influential may also be the most influenced (insert "shoulders of giants" Newton quote here). You would expect some self-made geniuses out there, people who feel they owe their success mostly to themselves, but I have yet to come across one. –  Thierry Zell Aug 14 '10 at 1:12
This is a nice list, but perhaps it is long enough. I vote to close. –  quid Sep 2 '11 at 18:17
I agree with quid, and am voting likewise –  Yemon Choi Sep 2 '11 at 20:44

62 Answers 62

Who: G. H. Hardy

When: As a high school student

Where: The book "Pure Mathematics" -- from which I learned real analysis.

Who: Serge Lang

When: As a college student

Where: At Columbia, Serge Lang was my mathematical mentor. I took Math I C/II C from him (which I'd describe as freshman mathematics for prospective Ph.D.'s -- it was pretty much an undergraduate Abstract Algebra, plus Real Analysis plus more in two semesters). His energy and love of mathematics was inspiring. I know that nobody who met him felt neutral about him. He was incredibly dedicated to his students. If he liked you he would move mountains.

Who: Lipman Bers

When: As a college student

Where: At Columbia, Lipman Bers was my other inspiration. I took Math III C/IV C from him -- sophomore mathematics for prospective Ph.D.'s. Besides being a very lucid lecturer, with fantastic geometric intuition, he was sophisticated and kind -- a prince among men! By example he showed how one could live a mathematical life (at perhaps a bit less than the frenetic pace of Serge Lang).

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As a graduate student, it's hard to say who's influenced me the most. Certainly my advisor seems to be a strong candidate, though others mentioned above have also influenced me. Still, there is an individual who has influenced my mathematical development at several different times in my career so far and who deserves a mention. From my talks with other grad students, I know I am not alone in being grateful for this person's work and his clear way of thinking and writing about mathematics.

Who: Keith Conrad

Which work: his body of expository papers at http://www.math.uconn.edu/~kconrad/blurbs/

When/Why: First, sophomore year of undergrad, in an elementary number theory course. This course and Professor Conrad's writings helped convince me to go to grad school. Then also junior year when I saw him give a talk at a conference and later at my own college. And more recently in the first year of grad school when I learned about tensor products, modules, exterior algebras, Galois theory, and several other topics.

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Simon Donaldson. His proofs involve (to quote wikipedia) a creative use of analysis. I loved his proof of the theorem of Narasimhan and Seshadri.

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Barry Mazur. The Eisenstein ideal paper, the one on towers of abelian varieties, as well as his beautiful expositions on visibility, Galois deformations, Kolyvagin systems etc continue to inspire me everyday.

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The Eisenstein Ideal paper is one of the masterpieces of mathematics; it ushered in a new approach to number theory. –  Emerton Apr 20 '12 at 0:05

Curtis McMullen. If you have ever seen him give a talk, you'll know what I'm talking about. He has a knack for delivering seemingly complicated ideas with clarity and charm. He is also a brilliant expositor. See Milnor's article on his work here.

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Colin Adams Knot theory was the first topic I was really excited about as an undergraduate from reading "The Knot Book." I did an summer program with Colin Adams and got my first glimpse of research, even at an undergraduate level and realized it's what I wanted to do for the rest of my life.

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Walter Rudin: His texts in Analysis are the ones which got me into mathematics.

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The first inspiration was Gauss's solution of sum of first n natural numbers when i was in high school...I went on to learn his notion of congruence etc which were really breath taking at that time.

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I was an (computer systems) engineering student, I decided tu study Mathematics after reading "Whom the gods love" it's a book about the life of Évariste Galois. I was thinking about that but reading that book gave me the courage. I also feel that mathematics is not very different from the topics I like about computer science.

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So, is this a vote for Galois, or for Leopold Infeld? –  Gerry Myerson Apr 26 '11 at 12:50

Nigel Hitchin has an amazing ability to find a new mathematical structure out of every physical context. His articles and papers are always clear, concise and provide the necessary intuition for the reader to grasp the concept/application while reading the definitions. I have always felt that many mathematics papers ignore the reader and focus on presenting things in such a concise matter that the true meaning is obfuscated. Hitchin never seems to do that and almost holds the reader's hand as he guides him/her through the wonders of mathematical physics.

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Who: Manin, Parshin, Serre, Tate.

When: When I was an undergraduate.

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The number 1 personality behind Bourbaki. Even though he was famous for taking the most extreme positions and was widely dismissed as a radical, his vision of mathematics is one that has largely been adopted by almost all mathematicians everywhere. Reading any piece of mathematical work he wrote, it his hard not to feel the respect and passion he felt for mathematics as a subject.

Dan Kan

Singlehandedly developed categorical homotopy theory into a full-fledged replacement for the homotopy theory of spaces (Kan complexes, combinatorial homotopy groups, subdivision, $Ex^\infty$, among many other things) as well as a large part of the foundations of homological algebra (Dold-Kan correspondence), category theory (adjoint functors, Kan extensions), and the modern theory of simplicial localization (with Dwyer) among numerous other achievements.

It's said that Kan's breakthrough paper on adjoint functors convinced Eilenberg and Mac Lane that pure category theory was not only a viable mathematical discipline (rather than a language), but also a deep and rich one.

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A really nice answer to my question. Thanks! –  Jose Brox Jan 28 '11 at 10:57
@Harry Gindi: Thanks for the classification. The elaboration you give is close to things I heard before. It seems my confusion arose from the fact 'personality' has a slightly different meaning for you an me. –  quid Jan 28 '11 at 16:38
  • Paul Cohen & Kurt Gödel
    • They gave us the tools to construct models of set theory.
  • Kenneth Kunen
    • His book "Set theory: An Introduction To Independence Results" was the book that got me interested in the field I would later call my home.
  • Saharon Shelah
    • His work on forcing, and singular cardinals keep me asking questions, and open up the possibility for questions I didn't even know could be asked.
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Richard Courant. Several years before I started studying mathematics in earnest, I spent a summer working through his calculus texts. Only recently, on re-reading them, have I come to realize how much my understanding of calculus, linear algebra, and, more generally, of the unity of all mathematics and, to use Hilbert's words, the importance of "finding that special case which contains all the germs of generality," have been directly inspired by Courant's writings.

From the preface to the first German edition of his Differential and Integral Calculus:

My aim is to exhibit the close connexion between analysis and its applications and, without loss of rigour and precision, to give due credit to intuition as the source of mathematical truth. The presentation of analysis as a closed system of truths without reference to their origin and purpose has, it is true, an aesthetic charm and satisfies a deep philosophical need. But the attitude of those who consider analysis solely as an abstractly logical, introverted science is not only highly unsuitable for beginners but endangers the future of the subject; for to pursue mathematical analysis while at the same time turning one's back on its applications and on intuition is to condemn it to hopeless atrophy. To me it seems extremely important that the student should be warned from the very beginning against a smug and presumptuous purism; this is not the least of my purposes in writing this book.

Another example: while not a "linear algebra book" per se, I have yet to find a better introduction to "abstract linear algbera" than the first volume of Courant's Methods of Mathematical Physics ("Courant-Hilbert"; so named because much of the material was drawn from Hilbert's lectures and writings on the subject). His one-line explanation of "abstract finite-dimensional vector spaces" is classic: "for n > 3, geometrical visualization is no longer possible but geometrical terminology remains suitable."

Lest one be misled into thinking Courant saw "abstract" vector spaces as "$\mathbb{R}^n$ in a cheap tuxedo," he introduces function spaces in the second chapter ("series expansions of arbitrary functions"), and most of the book is about quadratic eigenvalue problems, or, as Courant saw it, "the problem of transforming a quadratic form in infinitely many variables to principal axes."

As a final example: Courant's expository What is Mathematics? is perhaps best described as an unparalleled collection of articles carefully crafted to serve as an object at which one can point and say "this is." Moreover, while written as a "popularization," its introduction to constrained extrema problems is, without question, a far, far better introduction than any textbook I've ever seen.

I should also mention Felix Klein, not only because Klein's views on "calculus reform" so clearly influenced both the style and substance of Courant's texts, but since a number of Klein's lectures have had an equally significant influence on my own perspective. For those unfamiliar with the breadth of Klein's interests, I'm tempted to say "his Erlangen lecture, least of all" (not that there's anything wrong with it).

Lest my comments be mistaken for a sort of wistful "remembrance of things past," I'd easily place Terence Tao's writings on par with Courant's, for many of the same reasons: clear and concise without being terse, straightforward yet not oversimplified, and, most importantly, animated by a sort of — je ne sais quoi — whatever it is, it seems to involve, in roughly equal proportions: mastery of one's own craft, a genuine desire to pass it on, and the considerable expository skills required to actually do so.

Finally, I can't help but mention Richard Feynman in this context, and to plug his Nobel lecture in particular. While not a mathematician per se, Feynman surely ranks among the twentieth century's best examples of a "mathematical physicist" in the finest sense of the term, not merely satisfied by a purely mathematical "interpretation" of physical phenomena, but surprised, excited, and, dare I say, delighted by the prospect! Moreover, he was equally excited about mathematics in general, see, e.g., the "algebra" chapter in the Feynman Lectures on Physics.

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Cliched perhaps, but my fellow graduate students when I was in grad school. They're the ones that answered my questions when I got lost, shared their half-baked ideas and listened to mine, showed me just how many interesting fields of math there are and how many different perspectives people can have on the same subject, and cheered me on when things were difficult.

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Joseph-Louis Lagrange For his modesty as a human being, and his great mathematical work, on almost every field of mathematics, but especially in mechanics.

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I would have to say equal parts Godel and Raymond Smullyan. When I first started caring about math I picked up both Newman and Nagel's book on the Incompleteness Theorems and Smullyan's "First Order Logic". I then bought as many of the Smullyan puzzle books I could find. I also read Smullyan's "The Tao is Silent", which influenced me as a person.

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Vladimir Igorevich Arnold.

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You've answered the who, but not the when, where, or why, of the posted question. –  Gerry Myerson Oct 20 '10 at 2:17

Bernhard Riemann.

The idea of the Riemann Surface and manifolds stroke me when I was a high school student.

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Sophomore year when I decided that I didn't like physics classes I just happened to be reading "The Man Who Loved Only Numbers" by Hoffman. Between this and "How to Read and Do Proofs" by Solow, I saw mathematics as something much more beautiful. This combined with reading about Erdos style of mathematics made me really attracted to research and led to my first REU experience. It was all downhill from there.

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My research has been based on fields which he has had great influence. –  Kristal Cantwell Nov 14 '09 at 19:41

Dedekind, whose championing of concepts (vs. calculation) left a longstanding impression on the way that I conceive mathematics - even long after I first started reading the masters as a student. Back then I had to grovel through the bowels of the MIT libraries but now, with many important historical works easily accessible online, there is no excuse not to read the masters.

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The graduate advisor at Queens College of the City University Of New York, Nick Metas, was and continues to be my greatest influence.

I first had a conversation on the phone with Nick over 15 years ago when I was a young chemistry major taking calculus and just becoming interested in mathematics. We spoke for over 3 hours and we were friends from that moment on.

It was Nick who indocrinated me into the ways of true rigor through his courses and countless conversations,and the equal cardinality of the stories he's told. Nick is a true scholar and my enormous knowledge of the textbook literature and research papers from the 1960's onward,I learned from Nick.My learned capacity for self-learning got me through the lean years at CUNY during my illnesses,when there wasn't much of a mathematics department there.

In relation to the reference to Gian-Can Rota above,I am Rota's mathematical grandson through Nick. Nick loved Rota and his eyes light up when he speaks of his dissertation advisor and friend from his student days at MIT. I hope someday there's someone famous I can feel that way about. But no one's influenced me more then Nick.

Nick's has been my friend and advisor for all things mathematical and he celebrated his 74th birthday yesterday quietly in his usual office hour,with dozens of students asking him for advice or just listening to his wonderful stories and jokes. Regardless of what happens,it will be Nick who's influence on me as a mathematician, student and mentor who's shaped me the most.

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Louis Comtet, through his book "Analyse Combinatoire vol 1 and 2", now republished in english translation with additions and corrections as "Advanced Combinatorics".

When ? My first year in Paris University while I was attending boring courses in Analysis and Linear Algebra that were very inferior to what I have been exposed in high school the year before.

These two little pocket books were relatively easy and cheap to find and gave a wealth of packed information and links to the existing litterature on combinatorics. Combinatorial Mathematics were not in fashion in France in the 1970s, neither in the 1980s. Among many things I liked were the fancy notations, the diagrams, the density of results, the careful index, the intersection with so many other mathematical theories such as set theory, differential equations, topology, group theory. And it was also my first contact with a slightly formalized graph theory, Eulerian numbers, integer partitions, multiple summation, etc.

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(I think that for a question like this with the answers being entirely personal, the voting is of little or no significance.)

For me there are so many that I hardly know where to begin. Initially, Martin Gardner. Among those I knew personally: my undergrad profs (espcially I.M. Singer) who taught me what math is. Then Bill Thurston, with whom I shared an office in grad school. Stephen Smale, my de facto co-thesis advisor.

Notably Gauss, Riemann, Klein, Poincaré, Milnor.

Above all, my thesis advisor, Morris Hirsch, with whom I've had a continuing connection since 1970.

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In terms of style of math, I'm not quite sure yet. However, I think that it is certainly true for me, and no doubt for countless others, one's advisors role is one of the most crucial influences one may have.

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I have quite a few on my list.

Newton and Leibniz since the day I learned they were 22/19 (respectively) when they invented the calculus, Riemann as well (one of my teaching assistants was mad about him as well... it caught on)

Gödel after I'd took a course focusing on completeness and incompleteness, as well after you read his biographies.

Saharon Shelah, after one of my professors that did his Ph.D. under Shelah told me a lot about him. Finished his master degree in one year, Ph.D. in two. Invented so much... he's a real inspiration for me.

Grothendieck is a personal inspiration from another end. Not as a mathematician but as a human being. The fact he was able to get up and leave everything. That is amazing for me.

And while we're at it, Albert Einstein since I was 21 and read the book Ideas and Opinions.

What matters is less the work, but rather the ability to express with clarity a new idea that no one had before. That's what makes a great mathematician in my eyes... at least from where I stand today.

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Thank you for your answer! Since you mentioned several people already named in posts above, you may upvote them in order to maintain the statistics! Thanks! –  Jose Brox Jul 5 '10 at 21:43

For me its Aryanhatta who estimated the value of pi and proved that it is irrational way back in 400 B.C.

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Proved? Do you have a reference for that? –  Faisal Jul 6 '10 at 12:04
Proved? Not at all. He just gave an approximate value (circle of diameter 20000 has circumference 62832, which corresponds to 3.1416) and said "in this way, [the value] can be approached", and the last word has been speculated on and exaggerated to "proved". (Also, 500 AD, not 400 BC.) –  shreevatsa Jul 7 '10 at 19:46

who: Maxim Kontsevich

when: when I was a PhD student, and onwards.

why: probably because of my main research interest when I was a PhD student, namely deformation quantization. Also because before moving to (many) other subjects Kontsevich has formulated a lot of very reasonnable conjectures and guessed a lot of possible developpments in the field. Some of them I have been following. Even now, I am still thinking quite often about a few questions he raised .

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Lou van den Dries

Frankly speaking.

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Maybe give us some explanation 'why'? –  Michal Kotowski Jul 5 '10 at 12:34
Yes, and maybe also 'when' and 'where' (in which work, if it applies). Thanks! –  Jose Brox Jul 5 '10 at 19:18

Raymond Smullyan, in elementary school. His book "Alice in Puzzleland" was a childhood favorite of mine and is what and inspired a life long interest in math.

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