Paul Hacking (UMass Math): What is Mirror Symmetry

Mirror symmetry is a duality in string theory: one physical theory has two very different mathematical incarnations. Roughly, from a mathematical point of view: there are pairs X and Y of Calabi--Yau manifolds such that the symplectic geometry of X is equivalent to the complex geometry of Y and vice versa. This is made precise via the homological mirror symmetry conjecture of Kontsevich (1994). The Strominger--Yau--Zaslow conjecture (1996) asserts that X and Y admit dual Lagrangian torus fibrations over a common base; then the physical duality reduces to a simpler duality called T-duality on the torus fibers. I will give a gentle introduction to the mathematics of mirror symmetry, including some brief remarks on the relation to physics (which I hope can be expanded upon on another occasion by someone more qualified than myself). This is a wide-ranging and challenging topic, spanning complex geometry, symplectic geometry, differential geometry, and homological algebra, but I will attempt to convey the essence of the mirror correspondence as mathematicians currently understand it, with a minimum of prerequisites and technical details. I will teach a graduate class at UMass in Fall 2025 giving a more extensive introduction to the mathematics of mirror symmetry. 

References: Kontsevich: Homological algebra of mirror symmetry; Strominger--Yau--Zaslow: Mirror symmetry is T-duality