Dr Asilata Bapat
The Australian National University
Triangulations, rigid motions, and applications to representation theory
The course will begin with a brief survey of the theory of triangulations of a convex n-gon, which appear in several different places in mathematics. This course will focus on one such, possibly unexpected, appearance, namely in the theory of rigid motions of points in the plane. With this perspective, we will move to non-convex arrangements of points, and discuss the appropriate replacement of a triangulation. Finally, we will say a word about how these constructions are related to representation theory, via configuration spaces of points in the plane and a certain category of quiver representations. I will also mention some open questions in this direction.
Pre-requisites
- You should be very comfortable with linear algebra, including inner products and orthogonal decompositions.
- Think about the following questions if you don’t know the answers already:
- how many triangulations are there of a convex n-gon in the plane?
- How many edges does any given triangulation have?
- Would you need to modify the definition of a triangulation if you arrange n points in a non-convex arrangement? Why or why not?
Dr Asilata Bapat
The Australian National University
Asilata Bapat is a Lecturer at the ANU working on categorical methods in representation theory, with connections to algebraic geometry, topology, and combinatorics. Asilata finished her PhD in 2016 at the University of Chicago, was a postdoctoral researcher at the University of Georgia, and has been at ANU since 2018.
Asilata enjoys working on problems that bring together different areas of mathematics, and in which lots of examples can be written down by hand or by computer. Asilata often works with triangulated categories, which serve as links between representation theory and geometry/topology. Sometimes, these categories are equipped with interesting group actions that give them extra structure. In particular, she thinks about Bridgeland stability conditions on these categories, which give interesting ways to measure the size of objects in the category.
In her free time, Asilata can be found baking sourdough bread, solving puzzle hunts, playing board and video games, or tinkering with her emacs configuration.