Ricci solitons are solutions to the Ricci flow modulo a diffeomorphism, and arise as singularity models for the Ricci flow. In this talk, we will outline the proof of the uniqueness of the conjugate heat kernel on non-compact shrinking Ricci solitons, without any a priori curvature or volume assumptions. We then use the uniqueness of the conjugate heat kernel to prove that Perelman's celebrated Entropy functional has a unique minimiser on shrinking Ricci solitons. Consequently, this shows that the Riemann tensor is bounded with quadratic decay and that the injectivity radius is bounded from below. These geometric conditions imply shrinking solitons have a unique tangent cone at infinity.