You mention an interest in symplectic geometry and Fukaya categories. A very natural problem in the field is given, say, a Liouville domain (a very reasonable class of exact symplectic manifolds) what are the diffeomorphism types of exact Lagrangian submanifolds. The strongest results on these sorts of problems are all obtained with a heavy dose of homological or homotopical algebra.
In the simplest case of a cotangent bundle of a compact manifold, you can examine Abouzaid's works a series of papers culminating in:
http://arxiv.org/abs/1005.0358
In later work with Smith, they used similar homological algebra (as well as some very interesting geometry) to classify Lagrangians in certain plumbings of cotangent bundles:
http://arxiv.org/abs/1107.0129
Seidel has also done some very spectacular work, by studying a homotopical analogue of a dilating $\mathbb{C}^*$ action on an $A_{\infty}$-category.
http://arxiv.org/abs/1202.1955 http://arxiv.org/abs/1307.4819 http://arxiv.org/abs/1403.7571
Finally there is work by Fukaya (studying a weaker condition than exact) focused on Lagrangains inside of $\mathbb{C}^n$ in a paper called "Applications of Floer Homology of Lagrangian Submanifolds to Symplectic Topology". This uses ideas from operads and string topology.