We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is at most polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of polyloglog(n). To prove this, we show that any k-edge-connected graph (for k=Omega(log(n))) has a polylog(k)/k thin tree. In this talk I will highlight the main ideas of our proof. This is a joint work with Nima Anari.