👉 The Donald IOR, or Donaldson-Ilu-Yu (DII) invariant, is a complex topological invariant used to study the properties of smooth manifolds, particularly in the context of symplectic geometry and low-dimensional topology. It measures the complexity of a manifold's structure by counting certain types of closed, non-simply-connected submanifolds called Lagrangian submanifolds. Specifically, it quantifies how many distinct "twisted" or non-trivial Lagrangian circles can be found within a given symplectic manifold, providing insights into the manifold's topology and symplectic structure. The Donaldson IOR is crucial for understanding phenomena like the existence of exotic smooth structures on manifolds and has applications in areas such as string theory and mirror symmetry.
