For centuries mathematicians have generalised statements like “there is a unique line through any 2 points”, but with increasing technical difficulties. It was not until the late 1990s that new ideas from mathematics and string theory allowed rigorous definitions to be made of these “curve counting problems”.

I will outline two different ways to count curves, assuming only a bit of undergraduate complex analysis. The famous “MNOP conjecture” is that the two definitions give equivalent information. Its recent proof by Pandharipande and Pixton has enabled the solution of various counting problems, such as the “KKV conjecture” from string theory, expressing all curve counting problems on “K3 surfaces” in terms of modular forms.

If time allows I will also outline some other enumerative theories suggested by physics and implemented in algebraic geometry, such as Vafa-Witten theory.

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