9704359 Szabo The main theme of this project is the study of simply-connected smooth closed 4-manifolds and their Seiberg-Witten invariants. The project has three parts. The first part is related to the classification problem for these 4-manifolds. While the recent failure of the Minimal Conjecture shows that smooth 4-manifolds tend to defy classification schemes, there are other related conjectures like the 3/2 Conjecture and the Simple-type Conjecture, and the investigator will try to disprove these conjectures as well. In the second part, the investigator will continue his joint work with John Morgan to study how Seiberg-Witten invariants change along an h-cobordism. This problem is related to the Simple-type Conjecture and also to the third part of the project, in which the investigator will study the relation between the combinatorial presentations (Kirby calculus pictures) and Seiberg-Witten invariants of simply-connected smooth closed 4-manifolds. A long-term goal in this direction is to find generalizations of the Seiberg-Witten invariants. The study of smooth structures of simply-connected 4-dimensional manifolds was initiated in 1984 by Simon Donaldson. In his ground breaking work, Donaldson applied the Yang-Mills equation and the corresponding Yang-Mills moduli spaces over the 4-manifolds. The corresponding Donaldson invariants were used to settle various problems in 4-dimensional topology. The Yang-Mills equation is a partial differential equation that has special importance in theoretical physics, and its use in the classification of smooth 4-manifolds revealed a surprising link between topology and theoretical physics. This link has been underlined by Seiberg and Witten in 1994, when, using ideas coming from theoretical physics, they discovered a pair of new partial differential equations and the corresponding Seiberg-Witten invariants of smooth 4-manifolds. ***
