**Souvik Goswami**

TIFR

March 31, 2016

**Business of Height Pairing**:
In algebraic geometry there is the notion of a height pairing of algebraic
cycles, which lies at the confluence of arithmetic, Hodge theory and
topology. After explaining the basic notions, we introduce new directions in this subject. Specifically, we address an extension of Beilinson's height pairing to the (still conjectural) graded pieces of a Bloch-Beilinson filtration. Along the way, we use machineries from arithmetic intersection theory to provide an example computation.