Moduli Space

 
set of isomorphism classes of geometric objects made into a space of some kind(topological space, complex manifold) Example. The moduli space of complex tori $ℂ/Λ$ (≃moduli space of non-singular cubics up to projective transform.) $Λ=⟨w_1,w_2⟩_ℤ=$pairs $(w_1,w_2)/GL(2,ℤ)$ for $w_1/w_2∈ℂ∖ℝ$ moduli space of tori $ℂ/Λ$ $≅ℋ=\{z∈ℂ:\Im(z)>0\}$ $SL(2,ℤ)$ acts by Mobius transform $SL(2,ℤ)=⟨S,T⟩,S=\pmatrix{1&1\\0&1},T=\pmatrix{0&1\\-1&0}$ fundamental domain for 2 special tori: $ℂ/ℤ+iℤ$: $ℤ^4$ symmetry $ℂ/ℤ+e^{πi/3}ℤ$: $ℤ^6$ symmetry $C:x^3+y^3+z^3=0$ What $Λ$ is this $ℂ/Λ$ for? $ℂ/ℤ+e^{πi/3}ℤ$: $ℤ^6$ symmetry $(x,y,z)↦(e^{2πi/3}x,y,z)$ order 3 rotation fixes $(0,1,-1)$