In my essay for the Part III course in Cambridge I studied breakthrough papers on interlacing polynomials by Marcus, Spielman and Srivastava. Using this innovative approach they proved the long standing Kadison-Singer conjecture from operator algebra theory, and proved the existence of bipartite Ramanujan graphs of arbitrary degree.
In my master thesis I worked on quantum spin glasses. The main novel finding was a phase transition between a semicircular and Gaussian density of states.
In my PhD thesis I worked on three challenging problems in random matrix theory: (1) Universality of random matrices with correlated entries, (2) Pearcey-kernel universality at cusp points, and (3) entrywise linear statistics and Young tableaux.
