Black Holes, Branes, and Knots in String Theory
String theory has proven to be fertile ground for interactions between physical and mathematical ideas. This dissertation develops several new points of contact where emerging mathematical ideas can be applied to stringy physics, and conversely where stringy insights suggest new mathematical structures. In the first half, I explain how the formula of Kontsevich and Soibelman (KS) can be used to compute the spectrum of stable BPS particles in Calabi-Yau compactifications and I show that dimer models can be used to prove the KS formula for walls of the second kind in toric Calabi-Yau manifolds. I also explain how dimer models give a way of associating integrable systems to Calabi-Yau threefolds and show that this map agrees with an existing gauge-theoretic map. The second half of this dissertation focuses on a background in M-theory that defines the refined topological string. I explain how orientifolds can be introduced into this background, leading to new integrality conditions on amplitudes and to new invariants for torus knots. Finally, I introduce a new duality that relates the refined counting of supersymmetric balck hole entropy and refined topological string theory.