Research

My broad area of research is algebraic topology. More specifically, I am working in homological algebra and homotopy theory. I am also interested in differential topology & geometry, algebraic geometry.

Suppose $X$ be a topological space with a binary operation i.e., there is a map from $X\times X\to X$ taking $(a,b)\mapsto ab$. Now if the binary operation is not associative, we have two non-identical cannonical maps from $X\times X\times X\to X$ namely $f_{1,2}:(x_1,x_2,x_3)\mapsto x_1(x_2x_3)$ and $f_{2,1}:(x_1,x_2,x_3)\mapsto (x_1x_2)x_3$.
  $f_{1,2}$ and $f_{2,1}$ are not equal but we ask whether they are homotopic. If homotopy associative (called weakly associative), then we have a map $m_3:K_3\times X^3\to X$ defined through the homotopy between $m\circ (m\times 1)$ and $m\circ (1\times m)$, where $K_3$ is an interval. Similarly, we can define five different maps from $X^4\to X$ using $m$, and between any two such maps, there are two different homotopies (using the chosen homotopy associativity). If those two homotopies are homotopic themselves, then this defines a map $m_4:K_4\times X^4\to X$, where $K_4$ is a filled pentagon. If we continue this process, we obtain a map $m_n:K_n\times X^n\to X$ for $n\geq 2$. These complexes $K_n$, called associahedra, are our main objects of interest.

Publications and Preprints

  1. Equivalence Between Four Models of Associahedra (joint with Dr Somnath Basu), 2022
    arXiv, Preprint

    Abstract: We present a combinatorial isomorphism between Stasheff associahedra and an inductive cone construction of those complexes given by Loday. We give an alternate description of certain polytopes, known as multiplihedra, which arise in the study of $A_\infty$ maps. We also provide new combinatorial isomorphisms between Stasheff associahedra, collapsed multiplihedra, and graph cubeahedra for path graphs.