The 2016 Nobel Prize in Physics was awarded to David Thouless, John Kosterlitz, and Duncan Haldane for research broadly associated with topology in condensed matter physics. In the first half of the talk I  will describe the two-dimensional XY model which undergoes a Berezinskii-Kosterlitz-Thouless phase transition. This transition, unlike many magnetic transitions, lacks spontaneous symmetry-breaking and is driven by the proliferation of topological excitations, vortices. The analog of of this excitation in three dimensions is the "hedgehog" of the Heisenberg magnet. I will explain what this has to do with quantum phase transitions of the two-dimensional Heisenberg antiferromagnet, explicated by Haldane.