Mirror symmetry is a field in geometry aimed at establishing a relationship between symplectic and algebraic geometry. This was started by exploiting a duality in string theory, but the theory has advanced towards being planted in mathematics itself (albeit still taking inspiration from the physics). That is, given a symplectic manifold M, there should exist a so-called mirror algebraic object W so that the symplectic invariants of M are ‘equivalent’ to the algebraic invariants of W. In the first part of the talk, I will give a survey of how mirror symmetry can be viewed and its various avatars and links between the two fields.
This includes a famous conjecture of Kontsevich that encodes this link in homological algebra and category theory, by stating that the Fukaya category of M is equivalent to the derived category of W. One can aim to prove this conjecture in as many cases as possible, but one can also use it to inspire oneself to find new structure in the derived categories of algebraic spaces. In the second part, I show how Kontsevich’s conjecture inspires theorems for derived categories and the technical tool called an exoflop that helps prove them.
