Singularities in birational geometry (Mircea Mustata, Michigan)

Certain classes of singularities (such as log canonical or log
terminal) play a crucial role in the Minimal Model Program. Their
definition involves invariants coming from a resolution of
singularities. However, these notions show up in apparently unrelated
places. One can characterize them via local integrability conditions,
via p-adic integration and via spaces of arcs. Moreover, they are
related to Bernstein-Sato polynomials and with invariants coming from
positive characteristic. We will start from definitions, compute lots
of examples, and look at the various characterizations and the theory
behind each of them. If time permits, we will discuss also the
connection with multiplier ideals and vanishing theorems (for example,
we could look at singularities of theta divisors and to the number of
condition imposed by curve singularities).