### On smooth complex surfaces of general type with minimal holomorphic Euler characteristic

The talks focus on complex smooth surfaces of general type with minimal holomorphic Euler characteristic (namely $\chi=1$). A complete classification of such surfaces is still missing.

The first talk is introductory. I will introduce inequalities in the theory of fibrations on surfaces and Beauville's result which shows surfaces with $\chi=1$ have invariants $p_g=q\leq 4$. Then I report the complete classification on surfaces with $p_g=q=3$, due to Catanese-Ciliberto-Mendes Lopes, Pirola and Hacon-Pardini. Finally, I will talk about the surfaces with $p_g=q\leq 2$, mostly concerning those with geometric genus zero.

In the second talk, I will talk about several families of smooth surfaces of general type with invariants $p_g=0$ and $K^2=7$. One of which is recently constructed by Y. Shin and myself. Then I will discuss their bicanonical maps, automorphism groups, moduli spaces and Bloch conjecture for these surfaces.