  In this paper, we develop new techniques for understanding surfaces in
$\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to
smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue
to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan
subsequently developed the theory of bridge trisections for smoothly embedded
surfaces in 4-manifolds. The main application of these techniques is a new
proof of the Thom conjecture, which posits that algebraic curves in
$\mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented
surfaces in their homology class. This new proof is notable as it completely
avoids any gauge theory or pseudoholomorphic curve techniques.
