To join this seminar via Zoom please click here.

If you would like to join the seminar and are not currently affiliated with ANU, please contact Edmund Heng at edmund.heng@anu.edu.au

Filtrations of mathematical objects are ubiquitous in many parts of mathematics. This talk will focus on filtrations in abelian and triangulated categories, with the aim of introducing the dual notions of distinguished triangles and filtrations — as (actual) oriented triangles and oriented triangulated polygons. These dual notions were introduced in the work of Dyckerhoff-Kapranov.

Although the aim is to do things in triangulated categories, no prior knowledge of triangulated categories will be needed. In fact, we will only end up with drawing triangles and polygons in this setting — that’s the goal. I will however assume some familiarity with vector spaces and modules, as the talk will start with filtrations in the abelian category setting (read: vector spaces, modules). I will present a simple operation involving concatenation of filtrations, which relies on the first and third isomorphism theorems. I will show that the same type of operation can be done on filtrations in triangulated categories, using the octahedral axiom instead. When interpreted dually, this will end up being a flip of a triangulated polygon, switching from one triangulation to another.