The Gauss-Bonnet and Hopf theorems are beautiful results in global geometry. There are a few combinatorial versions of the classical Gauss-Bonnet theorem. The project is to work out a combinatorial version of the Hopf theorem on CW-complexes, or on simplcial complexes, or on graphs.
Learn basic parts of simplical and CW-complex topology, graph theory, and algebraic topology to understand the problem by reading book chapters and giving reports in weekly visit. Search literature to find out possible previous work done by others. Read and understand the previous related work. Work with the professor on the combinatorial version on graphs first, then on simplcial complexes, and possibly on CW-complexes. Write joint research papers if possible.
Build up solid knowledge on simplicial complex topology, CW-complex topology, algebraic topology, differential geometry, and differential topology, polyhedral geometry, etc. Gain research experience from academic courses to up-to-date front of interested research areas. Prepare themselves for future postgraduate study.