In the early 1990's, Elliott conjectured that simple, nuclear C*-algebras are determined up to isomorphism by their K

-theory groups and trace-simplex. Although the strongest interpretation of this conjecture is false, after decades of work on this question by many mathematicians, a positive answer was obtained in the summer of 2015, with the nal steps (in the stably finite case) taken by Elliott, Gong, Lin, and Niu, under the additional hypothesis that all traces on the algebras are quasidiagonal and the algebras have finite nuclear dimension and satisfy the universal coefficient theorem. Shortly after, Tikuisis, White, and Winter showed the quasidiagonality assumption is redundant, and very recently, Castillejos, Evington, Tikuisis, White, and Winter, showed nite nuclear dimension is equivalent to Z

-stability which, in many cases is an easier condition to verify.

These (partial) solutions to the quasidiagonality problem and the Toms-Winter conjecture have shed new light on the classication problem. In joint work with Carrion, Gabe, Tikuisis, and White, building also on my proof of the quasidiagonality theorem and the recent partial solution to the AF-Embedding

problem, we give a simplied proof of the classication conjecture following the same road map laid out by Gong, Lin, and Niu, but using these new techniques to avoid the use of direct limit algebras and tracial approximation structure.

In this series of talks, I will discuss some aspects of the classication theory and these new techniques. I hope to keep the prerequisites to a minimum. The actual topics discussed and the prerequisites will be largely determined by the participants.