turkmath.org

I will address several seemingly unrelated problems, such as the 64 lines in Schur’s quartic x(x^3-y^3)=z(z^3-w^3), 72 tritangents to the plane sextic curve x^6+y^6+z^6=10(x^3y^3+y^3z^3+z^3x^3), and 405 two-spaces in Fermat’s cubic four-fold x_0^3+x_1^3+\ldots+x_5^3=0. The first problem is classical, whereas the two others are relatively new. I will state that the figures indicated are indeed the maxima for the respective problems, and then I will outline the proof (for the last two problems) using a reduction to the so-called Niemeier lattices.