Monday

This talk will describe some recent constructions of exotic smooth closed 4-manifolds with definite intersection form and fundamental group Z_2. It is a joint work with Andras Stipsicz.

The global version of the Donaldson-Scaduto conjecture predicts the existence of a pair of pants special Lagrangian in the Calabi-Yau 3-fold K3 x R^2, and the local one, in (A2-type hyperkahler 4-manifold) x R^2. This conjecture is expected to play a crucial role in gauge theories in higher dimensions. After reviewing the background and motivations for this conjecture, we prove the local version of the Donaldson-Scaduto conjecture, which also implies the global version for a non-empty open subset of hyperkahler K3 surfaces. This talk is based on a joint work with Yang Li (MIT).

The following question was asked by Atiyah in the early days of instanton Floer theory: Is there a Milnor fiber description of the Floer homology of the links of singularities? In the late 1990s, I proved a closed form formula for the instanton Floer homology of Brieskorn homology spheres, which could be viewed as an answer to Atiyah's question for the links of Brieskorn singularities. In this talk, we will revisit that formula in light of the progress in gauge theory of the past twenty years. We obtain several new formulas for the instanton Floer homology of Brieskorn homology spheres, including one exclusively in terms of the monodromy of the Milnor fiber. This is a joint project with Kyoung-Seog Lee and Anatoly Libgober.

Does every four-manifold admit either no smooth structure or infinitely many of them? I'll report on recent work, joint with A. Stipsicz and Z. Szabó, addressing this question for oriented four-manifolds with finite cyclic fundamental groups. A bonus discussion may feature fake projective planes in the context of yet another dichotomy.