UCI TOPOLOGY SEMINAR 1997-1998 Math 298 Course Code 44915 This seminar meets in PS1 314 from 3-4 PM

Tuesday March 3 Speaker: Xio-Sin Lin, UC Riversde Title: Invariants of homology 3-spheres Abstract: The Casson invariant of homology 3-spheres is known to be the first non-trivial perturbative SU(2) quantum invariant. Higher order perturbative invariants are generalizations of the Casson invariants. We establish surgery formulae for these higher order invariants. We will also discuss some applications of higher order invariants and their relationships with the Casson invariant.

Tuesday January 20 Speaker: Marc Lackenby, Berkeley Title: Dehn surgery and negatively curved 3-manifolds Abstract: Dehn surgery is perhaps the most common way of constructing 3-manifolds. The procedure is to start with a given 3-manifold (usually $S^3$), then to remove a knotted solid torus from this 3-manifold, and then to glue in a new solid torus in a different way. It is a classical result that any closed orientable 3-manifold can be obtained from $S^3$ by iterating this procedure. In this talk, I will examine 3-manifolds $M$ obtained by Dehn surgery along a knot $K$ in $S^3$. It is known that for a given 3-manifold $M$, there may be several different knots $K$, each of which yield $M$ on Dehn surgery. But it is a long-standing open problem whether, for a given 3-manifold $M$, there can be an infinite number of such knots $K$. I will give a theorem (which is joint work with Daryl Cooper) that goes some way in answering this and other questions. The techniques of the proof involve hyperbolic structures and (more generally) negatively curved metrics on 3-manifolds.

Tuesday November 18 Speaker: John Etnyre, Stanford Title: Tight Contact Structures on Lens Spaces Abstract: At present, tight contact structures on 3-manifolds remain quite mysterious. The question of existence of such structures remains open (in spite of recent progress) and even less is know about uniqueness. In this talk, after recalling some basic facts from contact geometry, I will discuss recent progress on both these question as they apply to lens spaces, focusing on the uniqueness of tight contact structures.

Tuesday October 14 Speaker: Bruno deOliveira, UCI Visiting Assistant Professor Title: Stein deformations of strictly pseudoconvex surfaces Abstract: In this talk we prove that every strictly pseudoconvex (s.p.c) surface has a small deformation which is a Stein surface. We also show that every s.p.c. surface is a union of two Stein. The proof of the first result relies on the analysis of the behaviour of analytic cycles on deformations. We also apply this result to the study of contact structures on 3-dimensional manifolds.

Thursday October 9 Speaker: Terry Fuller, UCI Visiting Assistant Professor Title: "Lefschetz Fibrations and 3-fold Branched Covers II.

Tuesday October 7 Speaker: Terry Fuller, UCI Visiting Assistant Professor Title: "Lefschetz Fibrations and 3-fold Branched Covers, I. Abstract: We discuss the relationshiip between two current threads of research in smooth 4-manifold topology. Firstly, one may ask which smooth 4-manifolds are branched covers of a fixed smooth 4-manifold, such as $S^4$ or $S^2 \times S^2.$ Secondly, recent work has shown that every symplectic 4-manifold admits a "canonical" Lefschetz pencil, and conversely that every such pencil admits a symplectic structure. Thus the existence of Lefschetz pencils provides a topological characterization of symplectic 4-manifolds. We unite these two themes by proving that most smooth Lefschetz fibrations may be obtained as a 3-fold branched cover of $S^2 \times S^2,$ branched over an embedded surface.

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