**Pre-seminar**

Title: Notions of hyperbolicity for complex algebraic varieties

Abstract: Riemann surfaces can be classified as either parabolic, elliptic, or hyperbolic. In this pre-seminar, we motivate and introduce several notions of hyperbolicity for complex algebraic varieties of any dimension, e.g. Brody hyperbolicity and log algebraic hyperbolicity, and discuss some examples and non-examples.

**Seminar**

Title: Hyperbolicity of the complement of generic quartic plane curves

Abstract: A complex algebraic variety is said to be Brody hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. It is conjectured that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. The case of quartic plane curves has drawn interest for decades. In two collaborations (with X. Chen and E. Riedl, and with K. Ascher and A. Turchet), we prove an algebraic version of this conjecture for the complement of a very general quartic plane curve with <=2 components. Moreover, we completely characterize the exceptional locus for the complement of a very general irreducible quartic plane curve, identifying it as the union of its flex and bitangent lines.