From the very start in the main stream, deformation problems of sub-objects in algebraic geometry and those in differential/symplectic/calibrated geometry are treated very differently. A very noticeable example comes from mirror symmetry that relates the Fukaya(-Seidel) category of Lagrangian submanifolds (with flat bundles) on a Calabi-Yau manifold and the derived category of coherent sheaves on a mirror Calabi-Yau manifold. How a differential/symplectic geometer thinks about deformations of a Lagrangian submanifold (possibly with a bundle supported thereupon) in a symplectic manifold, what issues to address, and techniques to address these issues and how an algebraic geometer thinks about deformations of (complexes of) coherent sheaves on a variety/scheme, what issues to address, and techniques to address these issues are usually very remote from each other. And yet, motivated by D-branes in string theory, the objects (or the category they form) on each side are somehow mysteriously and closely linked. In this lecture I'll explain how the early notions from synthetic differential geometry --- initiated around 1967 through lectures of F. William Lawvere with intent to merge Grothendieck's theory of schemes and topos into differential geometry, promoted by several others in the 1970s and 1980s, and recast in 2010 by Dominic Joyce as C^∞ algebraic geometry --- and a recent progress on understanding D-branes as fundamental objects in string theory together motivate a new look to calibrated objects in symplectic/calibrated geometry.

 

Reference

Joyce: "Algebraic geometry over C^∞-rings", 1001,0023 [math.AG].

Chien-Hao Liu and Shing-Tung Yau, D(11.1) (arXiv:1406.0929 [math.DG]).

Chien-Hao Liu and Shing-Tung Yau, D(12.1) (arXiv:1504.01841 [math.SG]).