Homological mirror symmetry

Homological mirror symmetry is a mathematical idea suggesting that two complex geometric objects, called Calabi-Yau manifolds, are deeply connected through their underlying algebraic structures. It posits that the "shape" of one manifold (complex geometry) relates to the "shape" of its mirror (symplectic geometry), with their respective categories of mathematical objects being equivalent. This duality bridges different areas of mathematics and physics, offering a unified framework to study geometric and algebraic properties, and has profound implications for string theory and our understanding of the universe's fundamental structure.