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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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    $\begingroup$ It's always your advisor(s) that influence you the most, aren't they? $\endgroup$
    – Zsbán Ambrus
    Commented Jul 5, 2010 at 22:00
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    $\begingroup$ @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! $\endgroup$
    – Jose Brox
    Commented Jul 5, 2010 at 23:12
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    $\begingroup$ 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. $\endgroup$
    – Thierry Zell
    Commented Aug 14, 2010 at 1:12
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    $\begingroup$ This is a nice list, but perhaps it is long enough. I vote to close. $\endgroup$
    – user9072
    Commented Sep 2, 2011 at 18:17
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    $\begingroup$ I agree with quid, and am voting likewise $\endgroup$
    – Yemon Choi
    Commented Sep 2, 2011 at 20:44

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Who: H. S. M. Coxeter.

When: When I was an undergraduate.

Why: Not only was he a prince among mathematicians, but he was also a gentleman of the first rank. Several of his books also inspired me. Moreover, by transitivity, he was (for me) clearly the most influential.

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Vladimir Igorevich Arnold.

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    $\begingroup$ You've answered the who, but not the when, where, or why, of the posted question. $\endgroup$
    – Gerry Myerson
    Commented Oct 20, 2010 at 2:17
  • $\begingroup$ V.I. Arnold influenced me the most as well, on many occasions, although I never met him personally. When I was in high school I start reading "Mathematical Methods of Classical Mechanics", and looked up in other books the things I didn't quite understand. Then as an undergrad I kept reading the core of MMCM and a couple of Appendices, as well as his ODE and Catastrophe Theory. Much later his articles and general expository papers were very illuminating as well. For me his general expository papers are on par with Atiyah's. $\endgroup$
    – Michael
    Commented Dec 12, 2013 at 18:05
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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: 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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Taking into account the butterfly effect, I guess Roger Penrose would have influenced me the most. At first I was into physics and taught myself some calculus to understand it better; but it was more a tool than an end in itself. Then, at about 14, I read The Emperor's New Mind and was totally blown away by the ideas and proofs around Gödel's and Turing's work. Previously I had no idea the human mind could be so powerful!

It definitely pushed me into mathematics, and to this day I am very logically and discretely inclined.

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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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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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    $\begingroup$ The Eisenstein Ideal paper is one of the masterpieces of mathematics; it ushered in a new approach to number theory. $\endgroup$
    – Emerton
    Commented Apr 20, 2012 at 0:05
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Walter Rudin: His texts in Analysis are the ones which got me into mathematics.

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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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Andrew Gleason. An inspiring teacher in Math 55, the 2nd-year advanced calculus course at Harvard, and Math 213, the graduate complex variables course. He had a knack for getting at the essence of anything he lectured about. I have tried (with considerably less success) to do that in my teaching and my writing.

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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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    $\begingroup$ 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! $\endgroup$
    – Jose Brox
    Commented Jul 5, 2010 at 21:43
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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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Herbert Federer: his work on geometric measure theorey radiacally changed my view on differential geometry. Besides that, it is extremely practical when studying geometric flows.

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John Milnor too many great books.

Friedhelm Waldhausen for his papers on three manifold topology.

William Thurston for his lecture notes on hyperbolic three manifolds.

Fathi, Laudenbach and Poeneru for "Travaux de Thurston"

Atiyah and Bott, for "Yang-Mills on Riemann Surfaces."

Kobayashi for "Differential Geometry of Complex Vector Bundles"

Bill Meeks for his lecture notes on Minimal Surfaces.

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    $\begingroup$ Please, one mathematician per post! $\endgroup$
    – Jose Brox
    Commented Jan 17, 2011 at 7:44
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I should say three of them:

a. Philippe Flajolet

His work on analytic combinatorics inspired me enough to decide to study mathematics further after having majored in theoretical computer science. He wrote a book along with Sedgewick called analytic combinatorics, not to mention lots of papers on analysis of algorithms using the techniques he developes, he's a Cauchy of modern combinatorics.

b. Lucjan Jacak

Mathematician & quantum physicist, his lectures from quantum physics have inspired me to study this field for over two years. His most famous work concers quantum dots.

c. Bollobas, Kozma, et. al

And their work on non-constructive, probabilistic methods in graphs, also neuropercolation theory etc. Somewhat a revolutionary idea.

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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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Albert H Beiler. When I was in high school, someone gave me his book, Recreations in the Theory of Numbers. So different from any mathematics I had seen before, and so much fun! From there, it was just a short step to the Ross program....

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Silvanus P. Thompson: "Calculus Made Easy" (old simian proverb... "what one fool can do, so can another").

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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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  • $\begingroup$ A nice relationship, indeed! :) $\endgroup$
    – Jose Brox
    Commented Jul 6, 2010 at 9:47
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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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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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Henry Ernest Dudeney, author of Amusements in Mathematics, another book that set me, in my early teenage years, on the path to mathematics.

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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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  • $\begingroup$ Well, the voting serves from a statistical viewpoint! (In the sense that a collection of personal views transforms on a set of tendencies when we blur the names out and look from the distance). As I stated on the question, one of the things I wanted to see, on the whole, was which mathematicians have or have had more impact on our community, nowadays. As you mentioned some of them that have appeared on the list before, I suggest you upvote them to contribute to the statistics! Thank you a lot. $\endgroup$
    – Jose Brox
    Commented Jul 5, 2010 at 21:49
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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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  • 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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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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    $\begingroup$ So, is this a vote for Galois, or for Leopold Infeld? $\endgroup$
    – Gerry Myerson
    Commented Apr 26, 2011 at 12:50
  • $\begingroup$ If you can recommend better references, say it. $\endgroup$
    – Vicfred
    Commented Apr 27, 2011 at 7:00
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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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Charles Sanders Peirce, his Collected Papers, first encountered in the less-traveled Library of Congress from B to BD corner of the math library my freshman year, and compelling me to the prodigal expense of $35.00 in late 1960-ish dollars to buy Volumes 3 & 4 bound as 1. Every year that goes by is a year I add to the number of years his thought was ahead of his time.

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    $\begingroup$ Might you want to say what he has contributed, for those of us who might not recognize the name? (Is he the same as in the Peirce Decomposition for a ring in terms of its idempotents?) $\endgroup$
    – alekzander
    Commented Nov 26, 2009 at 6:28
  • $\begingroup$ I linked to the MacTutor Bio that should provide basic info on the man. I haven't looked deeply enough into the history, but my first guess would be that Peirce Decomposition is due to Benjamin, his father, who is credited with coining the terms idempotent, nilpotent, and others of that ilk. But C. did edit and supply addenda to some of B.'s papers. There's a link to B. Peirce's bio and a page from his Linear Associative Algebra here. $\endgroup$
    – Jon Awbrey
    Commented Dec 18, 2009 at 15:20
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Lou van den Dries

Frankly speaking.

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    $\begingroup$ Maybe give us some explanation 'why'? $\endgroup$
    – Michal Kotowski
    Commented Jul 5, 2010 at 12:34
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    $\begingroup$ Yes, and maybe also 'when' and 'where' (in which work, if it applies). Thanks! $\endgroup$
    – Jose Brox
    Commented Jul 5, 2010 at 19:18
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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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  • $\begingroup$ Sounds like a quite interesting book! I'll have a look at it :) $\endgroup$
    – Jose Brox
    Commented Jul 6, 2010 at 9:44
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