I will discuss a particularly elegant approach to parallel matching which was proposed by Mulmuley, Vazirani, and Vazirani in their 1987 paper ‘Matching is as Easy as Matrix Inversion’. Their paper introduced the celebrated isolation lemma and used it to prove that searching for perfect matchings in bipartite graphs can be done in RNC.
I will also discuss approaches to de-randomise this algorithm, particularly those proposed by Datta, Kulkarni, and Roy in "Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs" and by Tewari and Vinodchandran in “Green’s Theorem and Isolation in Planar Graphs”.