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If someone told you that mathematics is quite beautiful, you might be surprised. But you should know that some people do mathematics all their lives, and create mathematics, just as a composer creates music. Usually, every time a mathematician solves a problem, this gives rise to many oth­ ers, new and just as beautiful as the one which was solved. Of course, often these problems are quite difficult, and as in other disciplines can be understood only by those who have studied the subject with some depth, and know the subject well. In 1981, Jean Brette, who is responsible for the Mathematics Section of the Palais de la Decouverte (Science Museum) in Paris, invited me to give a conference at the Palais. I had never given such a conference before, to a non-mathematical public. Here was a challenge: could I communicate to such a Saturday afternoon audience what it means to do mathematics, and why one does mathematics? By "mathematics" I mean pure mathematics. This doesn't mean that pure math is better than other types of math, but I and a number of others do pure mathematics, and it's about them that I am now concerned. Math has a bad reputation, stemming from the most elementary levels. The word is in fact used in many different contexts. First, I had to explain briefly these possible contexts, and the one with which I wanted to deal.



What does a mathematician do and why?

Prime numbers:16 May 1981
The conference started with why, for ten minutes. I do mathematics because I like it. We discussed briefly the distinction between pure mathematics and applied mathematics, which actually intermingle in such a way that it is impossible to define the boundary between one and the other precisely; and also the aesthetic side of mathematics. Then we did mathematics together. I started by defining prime numbers, and I recalled Euclid’s proof that there are infinitely many. Then I defined twin primes, (3,5), (5, 7), (11,13), (17,19), etc. which differ by 2. Is there an infinite number of those? No one knows, even though one conjectures that the answer is yes. I gave heuristic arguments describing the expected density of such primes. Why don’t you try to prove it? The question is one of the big unsolved problems of mathematics.
Serge Lang

A lively activity: To do mathematics

Diophantine equations:15 May 1982
Interest in solving equations in integers or rational numbers dates back from antiquity. I tried to show some fundamental problems which are still unsolved. Euclid and Diophantus already solved the equation a2 + b2 = c2, and gave a formula for all the solutions. The next hardest equation like y2 = x3 + ax + b has given rise to very great problems which have been at the center of mathematics since the 19th century. No one knows how to give an effective method for finding all solutions. I described some of the structures which the solutions have, and the context in which one would like to find such a method.
Serge Lang

Great problems of geometry and space

28 May 1983
To do mathematics is to raise great mathematical problems, and try to solve them. Eventually to solve them. This time, we shall treat problems of geometry and space, and we shall classify geometric objects in dimensions 2 and 3. Dimension 2 is classical: it’s the classification of surfaces, which are obtained by attaching handles on spheres. One can also describe surfaces by using the Poincaré −Lobatchevsky upper half plane. What happens in higher dimensions? In dimension ≧5, Smale obtained decisive results in 1960. Last year, Thurston published great results in dimension 3. He conjectured the way such objects can be constructed starting with simple models, and also how one could obtain them from the analogue of the upper half plane in 3 dimensions. He proved a good part of his conjectures. We shall describe Thurston’s vision
Serge Lang
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