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The Sylvester-Gallai Theorem

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The Sylvester-Gallai Theorem states that, given any set P of n points in the plane, not all on one line, there is a line passing through precisely two of them (and “ordinary line”). I will discuss the history of this theorem and a couple of proofs of it. After that I will hint at some more recent work which establishes that there must in fact be at least n/2 ordinary lines for all sufficiently large enough n. I’ll also discuss some examples of sets with few ordinary lines, which involve some quite interesting constructions involving elliptic curves. The talk will be accessible to Part IA students.

This talk is part of the Trinity Mathematical Society series.

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