Abstract: 
A cork is a contractible piece of a smooth four-manifold. Cutting it out and gluing it back by a symmetry of its boundary can change the smooth structure while preserving the underlying topological manifold. 

How does the exterior of the cork distinguish the two gluings? We will introduce a framework based on Heegaard Floer homology, and inspired by lasagna modules construction, that compares the two fillings of the same boundary and tracks which differences remain detectable as the contractible piece is enlarged. We will use it to distinguish smooth structures on small rational surfaces and, by passing to a double cover, on the definite four-manifolds of Stipsicz and Szabó. It also gives a test for a proposed exotic smoothing of the projective plane blown up just once.

Finally, we will describe two embeddings of the same contractible piece in a standard four-manifold. A topological isotopy carries one embedding to the other, yet their exteriors remain nondiffeomorphic after any number of common blowups.

Bio-Sketch:

Eaman Eftekhary heads the School of Mathematics at the Institute for Research in Fundamental Sciences (IPM) in Tehran, Iran. He received his PhD from Princeton University in 2004 and was a Benjamin Peirce Assistant Professor of Mathematics at Harvard University before joining the IPM faculty in 2007. He is a member of the Academy of Sciences of Iran and a former president of the Iran National Science Foundation. His research interests include Heegaard Floer theory, low-dimensional topology, and knot theory.