Mirrors of Curves and Their Fukaya Categories I

Explore homological mirror symmetry for curves in this comprehensive lecture by Denis Auroux from Harvard University. Delve into the symplectic geometry of curve mirrors, comparing coherent sheaves on curves to suitable Fukaya categories of their mirrors. Examine the construction of Landau-Ginzburg models mirror to curves in (C*) or toric surfaces, and understand the concept of fiberwise wrapped Fukaya category. Discover computations and verifications of HMS in this framework, based on joint work with Mohammed Abouzaid. Investigate the geometric relationship between smooth and singular fibers of Landau-Ginzburg models, their total space, and corresponding functors between Fukaya categories. Learn about the application of these results to hypersurfaces in higher-dimensional toric varieties, abelian varieties, and complete intersections. Explore speculative views of mirror symplectic geometry from lower-dimensional perspectives, including "tropical Lagrangians" and geometry inside the critical locus. Gain insights into a new flavor of Lagrangian Floer theory in trivalent configurations of Riemann surfaces and its relation to curve geometry.

Mirrors of Curves and their Fukaya Categories I