This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Auroux-Donaldson-Katzarkov discovered a startlingly simple picture of smooth non-negative-definite 4-manifolds, generalizing Donaldson's interpretation of symplectic 4-manifolds as Lefschetz pencils. They view a 4-manifold, after blowing it up, as the total space of a broken Lefschetz fibration (BLF). In the simplest picture available, the blown-up 4-manifold X is the union of two 4-manifolds X_1 and X_2, each bounding a fibered 3-manifold Y, where X_1 is a Lefschetz fibration over a disc, and X_2 is essentially standard. This proposal focuses on computational and qualitative consequences of this escription. It concerns the Seiberg-Witten theory of broken Lefschetz fibrations and, more particularly, their "Lagrangian matching invariants" - gadgets developed by the P.I. using symplectic geometry associated with BLFs which conjecturally recapture the Seiberg-Witten invariants. The P.I. will use conceptual tools from symplectic topology to compute Floer-theoretic invariants for the two parts of the 4-manifold: X_1 (combinatorially complicated but symplectic) and X_2 (simple but non-symplectic). These computations are directed at making inroads into some major open problems in 4-dimensional topology: existence of symplectic structures, Seiberg-Witten simple type, and algorithmic computation of Seiberg-Witten invariants. They aim to shed light on the algebraic structures of 3- and 4-dimensional gauge theory.
Mathematicians regard 4 as the most mysterious dimension - more so than 2, 3, 5 or 1000. It is also the dimension of physical space- time, and equations devised by physicists have led to techniques that probe the topological structure of 4-dimensional spaces and show that they are governed by more complicated rules than anything in higher dimensions. So far, we have a very limited knowledge of what those rules are, and understanding them better is the focus for this project. Recent developments have shown that we can build all 4- dimensional spaces (technically, smooth, compact 4-dimensional manifolds) from simple but highly structured building blocks. This project will study how the known characteristics (invariants) for 4- dimensional manifolds can be understood in terms of those building blocks by invoking methods from another part of geometry that evolved from physics, symplectic topology. One aim is to elucidate what geometric information the invariants capture. Another is to seek genuinely new invariants. One can hope to do so by using the known invariants as a template; but that will require a deep understanding of how those invariants arise from the building blocks.
