Singularities of Flips and Pseudo-Effectivities of Second Chern Classes and for Terminal Threefolds

This project is to study singularities of ips and pseudo-e?ectivities of secondChern classes in dimension 3. Pseudo-E?ectivities of second Chern classes for ter-minal threefold were proved when the numberical dimension of anti-canonical di-visor ?(KX) 6= 2. Then, Xie gave some partial results for the remaining case?(KX) 6= 2. We would like to try to ?nish this program by studying the di?er-ence of singularities via the birational map in minimal model program.