A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture". This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8.

Furuta proved the "10/8+2"-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a "10/8+4"-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins and XiaoLin Danny Shi.

First of two talks. Note change of time and room.