Abstract
There are mathematicians able to solve incredibly difficult problems devising amazingly new ideas. There are mathematicians with a sure grasp of entire subjects, able to single out the more promising research directions. There are mathematicians with a crystal clear vision, able to explain and clarify any subject they talk about. And then there is John Milnor. He is all three: he solves, understands and explains, at an exceptional level on all counts. And he is a nice guy too.
In this short note, after a brief bibliographical sketch, I shall try and describe one of the most famous theorems proved by Milnor. We shall not go very much beyond explaining what the statement means, but I hope it would be enough for giving at least an idea of the brilliance of Milnor’s mathematics.
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Notes
- 1.
To be more precise, two surfaces are homeomorphic if there is a continuous bijection between them having a continuous inverse; but allow me to use a slightly imprecise language here for the sake of clarity.
- 2.
Again, the technical definition is more complicated than this: two smooth surfaces are diffeomorphic if there is a differentiable bijection between them having a differentiable inverse.
- 3.
Of course, this is not a proof; however, a formal proof is not that difficult, and can be understood by third-year math undergraduate students.
References
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Abate, M. (2013). Exotic Spheres and John Milnor. In: Emmer, M. (eds) Imagine Math 2. Springer, Milano. https://doi.org/10.1007/978-88-470-2889-0_24
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