Abstract: We will discuss the logarithmic minimal model program (mostly) for log surfaces (X,D). For D=0 the theory is a new way of saying old things. But
if D is nonzero things get complicated because there may exist inner extremal rays (represented by curves contained in D). Their effective
characterisation is a basic obstacle to apply the logMMP to problems concerning quasiprojective varieties by analyzing their completions. For a
reduced D the theory was developed by Fujita, Miyanishi, Tsunoda, Sakai et al., leading in particular to the solution of the Zariski cancellation
conjecture in dimension 2. We will state the definitions and results in modern terms. This will be a preparation for the next lecture where we
discuss our results concerning D with rational coefficients and their applications to some open problems.