http://en.wikipedia.org/wiki/Weil_conjectures is something that Grothendieck contributed to, which seems amazing and whose statement can actually be comprehended. Does it have any applications, or does it just seem amazing? Well, I actually am aware of a few applications of the Weil conjectures, but I think most or all of those applications were later also accomplished in far simpler ways without needing to go anywhere near said conjectures.

Hello, I met once Pierre Cartier in Montréal for a conference he gave. I also have known some mathematicians for years that were very fond of the Bourbaki style. Pierre Bouchard, André Joyal, Gilbert Labelle, Also, my own directors of masters thesis were experts in category theory, the theory of species. I had a friend also named Pierre Bouchard, a very strict Bourbaki mathematician. Myself about that style : I think I can say I am the complete opposite. I did not say anything yet because I wanted to see what is the rest of the world is thinking about 'Le Monde' saying that he was the greatest mathematician of the XX'th century. This is about abstraction and the means to get to an answer using logic, topos, category theory. For years there was those conferences in Montréal about The theory of species and combinatorics. I even participated to the making of the book on Species they made (Pierre Leroux, Labelle and Bergeron). Now, what do I think of Grothendieck : nothing. For me that type of mathematics is way too dry and formal. I was very curious to see what Pierre Cartier had to say when he came to Montréal for that conference. I am afraid that the abstraction is too high. The only explanation I could give about this to some friends recently is that : Suppose you want to justify the use of algebra for the solution of the n'th degree equation. I am talking here of the ordinary, one variable equation. As we know it, the use of algebra is justified, for degree 2, 3, 4 and 5. For the fifth degree, we need Galois theory to explain why some equations can or cannot be solved. Every mathematician will agree that this abstraction is necessary to get to the answer. The problem of Bourbaki mathematics is that they use algebra (of algebra (of algebra of some 'things' ))) to justify in 22 symbols what is the purpose of the empty set. This is authentic : the definition of the empty set in Bourbaki's theory of sets uses 22 symbols. This is insanely abstract for nothing in my opinion. I remember when I stumbled on that definition a long time ago. I was shocked (and amused). I also saw what type of justification was necessary to justify with the theory of species what are the Bernoulli Numbers in terms of combinatorics with that theory : Horribly complicated and useless. When I came in with the EIS project in 1990 with Neil Sloane, they laughed at me because I was using numerical recipes, numerical tricks to get the generating function of sequences. After a visit of Neil Sloane in Montreal, he convinced Pierre Leroux and Gilbert Labelle that this idea of numerics was good and they accepted that I do my master thesis on that subject. This was the complete opposite of the current projects they had with generating functions and the theory of Species. They had this huge program called Darwin programmed in Lisp, a white elephant, completely formal and dry, useless. It never worked actually. We call it 'une usine a gaz' a gaz factory in french. Grothendieck is the perfect example of Bourbaki style, formal, very abstract and dry. My friend Paul Simon (not the singer), an amateur mathematician asked me : what is or are the greatest mathematicians in the past century : In my opinion, John Conway, Martin Gardner, Ingrid Daubechies. Because there is a lot of imagination and richness in what they did and also because many of Conway's discoveries can be explained to a child. I made the test myself about the sequence 1, 11, 21, 1211, 1112211, many kids knows that sequence in schools in Canada and France. This is an amusing example of something not trivial, interesting, and that can be explained in a few words in plain language. I was giving the example of Grothendieck's work : I cannot find any example, profound phrase or only one simple idea that came out of all this. Maybe it is very profound but in my opinion : too formal. Best regards, Simon Plouffe