Abstract: We shall discuss associative binary structures and analyze why such a structure is not a homotopy invariant. This naturally leads to higher associative structures, often called $A_\infinity$ structures, which are prevalent in topology, geometry and mathematical physics. We shall discuss a recognition principle for $A_\infinity$ spaces. We will explain why certain polytopes, which are of great combinatorial interest on its own, govern maps between $A_\infinity$ spaces. The talk will be self-contained.
School of Mathematical Sciences
NISER, PO- Bhimpur-Padanpur, Via- Jatni, District- Khurda, Odisha, India, PIN- 752050