Units Unit 0: Introduction Unit 1: Gauss Bonnet Unit 2: Poincare Hopf Unit 3: Index expectation Unit 4: Euler's Gem

Appendices Appendix 0: Graphs Appendix 1: Sets Appendix 2: Complexes Appendix 3: Linear algebra Appendix 4: Topology

Code

This is a 1-liner that I had submitted to the WTC2023 competition. It computes the simplex generating function of an arbitrary graph using the Gauss-Bonnet theorem (134 bytes):

f[s_,x_]:=Module[{v=VertexList[s]},n=Length[v];1+Integrate[Sum[w=v[[k]];
  f[VertexDelete[NeighborhoodGraph[s,w],w],t],{k,n}],{t,0,x}]];
f[RandomGraph[{100,1000}],x]