Aditya De Saha

Research

Geometric group theory, large-scale geometry, and persistent (co)homology.

My research lies at the intersection of geometric group theory, metric geometry, and algebraic topology. Below is a description of the broad areas I am interested in.

Geometric Group Theory

Very broadly, I am interested in the large-scale behaviour of (mostly discrete) groups, when interpreted as metric spaces.

  • Dimension Theory for Metric Spaces: Cohomological dimension, asymptotic dimension, Lusternik-Schnirelmann Category, etc. These are starkly different notions, but the relationships between them bring forth a lot of very interesting mathematics.
  • Coarse Geometry: I am interested in large-scale analogues of classical topological invariants like coarse homotopy groups, coarse LS-category, etc. These numerical invariants provide tools to attack hard problems like the coarse Baum-Connes conjecture and Novikov conjecture.

Persistent Homology Theory

Persistent homology is the study of how homological features (connected components, voids, etc.) appear and disappear as one "thickens" or varies a scale parameter in a collection of metric spaces. This is the primary useful tool in the field of Topological Data Analysis (TDA).

  • Persistent Equivariant Cohomology: Many spaces naturally come with a group acting on them, and it is natural to study the equivariant homology instead of the ordinary homology. Persistent Equivariant Cohomology is the theory of persistence in this equivariant setting. One interesting example is the natural action of $S^1$ on the Vietoris-Rips complex of the circle $VR(S^1, r)$, as we vary the scale parameter $r$.