Question

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.

Answer 1

I would upvote the other Grothendieck answer or leave a comment there if I had the reputation. I think the 'crazy genius' tag attached to him is incredibly unfair. Despite the personality cult and the idol-worship, his humanity remains to me as inspiring as the mathematics. I found the following translated quote from Récoltes et Semailles immensely liberating:

"Since then I've had the chance, in the world of mathematics that bid me welcome, to meet quite a number of people, both among my "elders" and among young people in my general age group, who were much more brilliant, much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle -- while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates, almost by sleight of hand, the most forbidding subjects. In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still, from the perspective of 30 or 35 years, I can state that their imprint upon the mathematics of our time has not been very profound. They've all done things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they've remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have had to rediscover in themselves that capability which was their birth-right, as it was mine: the capacity to be alone."

Answer 2

John Baez. "This week's finds in mathematical physics" is a great playground for young mathematicians. I was a graduate student when I first found it, and I really loved the links between various TWFs and the math they discussed. Not only does he show you the breadth of modern math, he also gives you bridges between the various areas.

Answer 3

Alexander Grothendieck. See, for example, his passage about opening a nut. This was very inspiring for me and was one of the key reasons that led me to abandon computer science and start studying math. I also very much like the way he uses geometric intuition in algebraic geometry, it helped me a lot and not only in algebraic geometry.

Answer 4

Does Martin Gardner count, even though he is not a mathematician?

I read all of the "Mathematical Games" columns in Scientific American when I was maybe 12 or 14. And this was a non-trivial task ... I would ride my bicycle to the public library one afternoon a week to read a few more columns (the school library didn't have it). So it took maybe a year to read them all.

Answer 5

Serge Lang's Algebra was my first serious encounter with mathematics, the event was a very singular defining moment in my life.

Back then, I was firmly intent on becoming a poet or, at least, pursuing some kind of literary career. Like most budding poets, I loved books and I liked spending time in the library. I was very curious, I would often wander in a section and pick up a book just to see what that row was about. One day I picked up an old rebound copy of Lang's Algebra. It was dirty purplish grey and it just said Lang: Algebra in half erased white letters. I don't think I had any good reason to pick up that book, it certainly wasn't very attractive, I probably just wondered why one would write such a large tome on algebra. I sat down with the book and read the first page where he defines a monoid and proves the uniqueness of the identity element. I was fascinated. It was so beautiful. I fell in love.

I don't think I read much of Lang's book on that day, I probably only had an hour or less to spare, but I went back to the math section later and I picked up more books. The next one was Willard Van Orman Quine's Set Theory and its Logic, which is probably the worst possible way to get introduced to Set Theory but that's how I eventually became a logician instead of a poet.

Answer 6

Terence Tao. He is one of many who influenced me the most. I don't have to mention how superb his blog and publications are. From his writings I found analysis of PDE as a fascinating subject and I am really happy that I found this topic not too late. It amazes me how much he produces.

Answer 7

Sir Michael Atiyah.

Besides his great technical work (his collected papers are absolutely magnificent!) especially his great interview "Beauty in Mathematics" was very inspiring to me. Another inspiring piece is his "Advice to a Young Mathematician" in the Princeton Companion to Mathematics.

Answer 8

Who: Leonhard Euler.

When: As a highschool student.

Where: On the book "Euler: the master of us all" by William Dunham.

Why: The amount of creativity and genius dispersed among the so-different works of Euler continues to amaze me just now, so it only could have a devastating effect on me 10 years ago. He not only addressed a lot of distinct topics, he layed the foundations of many branches of mathematics and solved with ease many problems that were interesting me at that moment of my life. I learned a lot from him: he really deserves the title of "master of us all".

Answer 9

Gian-Carlo Rota. I really wish I could pinpoint the moment that I came across some of his work, but I can't. And I've only just begun graduate work so it's impossible to be really honest about what sort of impact he has had on me... only time can tell.

But nonetheless, his writings are truly inspiring. It's tough to describe the wonder they have given me. Rota began as a functional analyst (PhD under Jacob Schwartz, of the Dunford & Schwartz fame) and moved over to algebraic combinatorics in the 1960s. One of his first papers that has stuck in my mind is "The Number of Partitions of a Set" in which he applies the techniques of the so-called 'umbral calculus' (which he also worked to rigorously formulate) to beautifully establish some combinatorial results. He's credited as setting the field of algebraic combinatorics on solid ground via his seminal papers On the Foundations of Combinatorial Theory. But it's not just the technical results -- his writing is just plain fun to read.

Given my personal interests, I really appreciate that although Rota did so much work in combinatorics he always seemed to lean back towards his roots in functional analysis & probability. In fact, his goal to find the true nature of classical results in analysis and probability led him to a great deal of good work in combinatorics, e.g. his work on the Rota-Baxter algebra inspired by his ambition to understand "the algebra of indefinite integration", and his work on the foundations of probability with the ambition to understand the middle ground between the discrete and the continuous. Moreso than most people he is willing to put his position out there and speak about mathematics rather than just speak mathematics. A great example of this is his book "Indiscrete Thoughts" -- definitely worth reading. You may not agree with many things he says but it's wonderful to be allowed a glimpse into the mind of a man such as Rota.

Answer 10

Erdos

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.

Answer 11

Thurston. When I was a graduate student, Thurston's work really inspired me to appreciate the role of imagination and visualization in geometry/topology.

A prominent mathematician once remarked to me that Thurston was the most underappreciated mathematician alive today. When I pointed out that Thurston had a Fields medal and innumerable other accolades, he replied that this was not incompatible with his thesis.

Answer 12

Gromov.

Answer 13

Poincaré. Not so much for his mathematical writings (although what I've found in English, or struggled through in French, has been uniformly interesting [if dated, and/or, um, in a language I barely understand]) but for his thoughts on the philosophy and psychology of math. After the already-mentioned John Baez, the first thing I'll implore anyone who bothers to ask to read is "Intuition and Logic in Mathematics," fin-de-siecle thinking and all.

Answer 14

Carl Friedrich Gauss.

The breadth and beauty of his work amazed me when I was a student, and it still inspires me.

He started by building on much less than what many of us take for granted: His doctoral dissertation was the Fundamental Theorem of Algebra. His work covered deep, essential results in many areas, from number theory (quadratic reciprocity, conjecture of prime number theorem) to geometry (Gaussian curvature) to statistics (least squares) to probability (Gaussian distribution). It is difficult to imagine these areas without his fundamental contributions. He also contributed to physics and astronomy.

Even though Gauss explored many areas, he took the time to revisit old results, looking for different and more satisfactory proofs.

Answer 15

G.H. Hardy. Reading "A Mathematician's Apology" in high school really changed the way I think about mathematics.

Answer 16

Otto Forster.
He is the most brilliant expositor I have ever met. I cherish the notes I took a long time ago of courses he gave in Italy and France, in perfect Italian and French. He wrote a wonderful course on Analysis (in three volumes) which has been the reference in German Universities for 30 years, something like Rudin in the States. His book on Riemann surfaces (both compact and non compact) is a masterful blend of Algebra, Topology and Analysis, with tools ranging from cohomology of sheaves to difficult potential theory. He is a brilliant researcher and has made important contributions to complex geometry and also to algebra (Forster-Swan). Working with him was a wonderful experience and he had the generosity of letting me co-sign articles to which my contribution was negligible compared to his. I am very happy of this opportunity to express my gratitude to and admiration for this genuine scholar and real gentleman.

Answer 17

Gödel I was captivated by his belief of a platonic mathematical world and the belief that human can understand such a thing.

Answer 18

Consider looking forward: Grisha Perelman, his strange history wiki G.Perelman. However the first millenium prize awarded. Even note this: Terence Tao said... "well, it's amazing"

Answer 19

John Willard Milnor for his books about "Morse Theory" and the "h-cobordism theorem" (I think it is a crime that it isn't printed anymore) and for writing papers in a way that they are quite self-contained and readable.

Answer 20

JH Conway. He has published work in a diverse set of interesting fields. I first met his name when looking for cool computer programs to write as a kid (ie. the Game of Life) but since then his name kept appearing in mathematics that I found interesting, whether it's Monstrous Moonshine or the properties of finite state automata. He has this incredible knack for turning anything he touches into fun - whether it's knot theory, group theory, quadratic forms, or, more obviously, combinatorial games. As well as working at the frontiers of mathematics he's discovered accessible but surprising and beautiful recreational mathematics, like Conway's soldiers. All in all, an amazing guy. Once of my regrets in life is being too lazy to attend his lectures on finite simple groups when I was an undergraduate.

Answer 21

Arnold Ross. He ran the summer program in Number Theory for high school students at Ohio State University, my first exposure to serious mathematics. His lectures set me on a course from which I've hardly deviated in over 40 years.

Answer 22

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.

Answer 23

Benedict Gross. I saw him lecture a few times on BSD. His enthusiasm and mastery were very inspirational. It reminded me why I want to be a professional mathematician. I had just finished my general exams the previous semester and felt tired from taking so many classes and preparing for exams. It had put a haze over the beauty of mathematics. Professor Gross made it clear again.

Answer 24

Leibniz. Not just for his mathematics (calculus, amazing insights in logic, semantics) but he was just an incredible polymath, with deep work in law, history, linguistics, chemistry, physics, metaphysics, politics, engineering, sociology, he founded 'library science', and on and on.

Answer 25

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.

Answer 26

Riemann

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

Answer 27

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.

Answer 28

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.

Answer 29

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.

Answer 30

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.