Colloquium DATE 2017-02-21¡@16:10-17:00 PLACE ¼Æ¾ÇÀ]3174±Ð«Ç SPEAKER ¿à«C·ç ±Ð±Â¡]¥xÆW¤j¾Ç¼Æ¾Ç¨t¡^ TITLE On Canonical Maps of Projective Varieties ABSTRACT Our aim to study the geometry of projective varieties. In the classification problem, the dominating class is the collection of varieties of general type, i.e. varieties such that the canonical divisor K_X is big. In dimension one, these are all genus $g\geq2$ compact Riemann surface. According to the Minimal Model Conjecture, up to birational isomorphism, one can put an extra assumption on varieties of general type, namely K_X being nef (i.e. X is a minimal model). In this situation, I will explain two projects concerning the volume $K_X^n$ and the map defined by the linear system $|K_X|$: 1. Surface of maximal canonical degrees (joint w/ Sai-Kee Yeung) 2. Higher dimensional Noether inequality (joint w/ Jungkai Chen) Both topics exploit the geometry of the canonical system and certain inequalities involving $K_X^n$ and $p_g(X)$. If time permits, I will also discuss a work in progress on the anti-canonical volume $(-K_X)^3$ of Fano threefolds.