By John W. Morgan

The hot creation of the Seiberg-Witten invariants of delicate four-manifolds has revolutionized the research of these manifolds. The invariants are gauge-theoretic in nature and are shut cousins of the much-studied SU(2)-invariants outlined over fifteen years in the past by way of Donaldson. On a pragmatic point, the recent invariants have proved to be extra robust and feature resulted in an enormous generalization of previous effects. This ebook is an advent to the Seiberg-Witten invariants.The paintings starts off with a evaluate of the classical fabric on Spin c buildings and their linked Dirac operators. subsequent comes a dialogue of the Seiberg-Witten equations, that is set within the context of nonlinear elliptic operators on a suitable endless dimensional house of configurations. it really is confirmed that the gap of options to those equations, referred to as the Seiberg-Witten moduli area, is finite dimensional, and its measurement is then computed. not like the SU(2)-case, the Seiberg-Witten moduli areas are proven to be compact. The Seiberg-Witten invariant is then basically the homology category within the house of configurations represented by means of the Seiberg-Witten moduli area. The final bankruptcy provides a style for the purposes of those new invariants through computing the invariants for many Kahler surfaces after which deriving a few easy toological outcomes for those surfaces.

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The mixing of both object types is not considered here. Hence we restrict to cs ∈ int(s). We can easily determine which objects are scalable and which are not. An object s is said to be strongly star-shaped if there is a point ts ∈ int(s) such that for any point p ∈ s the straight line segment ts p is contained in s, but does not contain any point of bd(s), except possibly p. 3 An object s is scalable if and only if it is strongly starshaped. Proof: Suppose that s is scalable and has scaling point cs .

6, we think of -separated as being a slightly more general notion, since the property of being -separated is invariant under a scaling of the space. 32 Chapter 4. Geometric Intersection Graphs and Their Representation We now show that any intersection graph of scalable objects has an separated representation. 7 For a family A of closed scalable objects, any A-intersection graph has an -separated representation for some > 0. Proof: Let G be an A-intersection graph and S any representation of G. We prove that S can be turned into an -separated representation of G.

5 can be proved for intersection graphs of other scalable objects. In particular, we conjecture that similar techniques apply to intersection graphs of (unit) regular hexagons. Finally, observe that for the results in this section it does not matter if the disks or squares are open or closed. 2 From Separation to Representation The above theorems were quite specific to the object type. We can prove that the converse holds in a more general setting. In the following, let zs denote the size of an object s.