# 2011 Niven Lectures Abstracts

Speaker: Denis Auroux

Undergraduate talk: Seeing into the fourth dimension.

May 24th, 4pm in 100 Willamette Hall.

Reception to follow in Willamette Hall atrium.

Abstract: Low-dimensional topology studies the shape of closed geometric spaces (“manifolds”). In two dimensions, these are the familiar closed surfaces (sphere, torus, and so on), which can be drawn inside usual 3-space. But what about three and four dimensional spaces? Topologists have developed a variety of mathematical tools to represent these spaces in terms of diagrams that can be drawn on a two-dimensional sheet of paper. The fundamental idea behind those diagrams is that of handle decompositions, which explain how a space is built out of smaller pieces. In this talk, we will illustrate how this general idea underlies various classical representations (Heegaard diagrams of 3-manifolds, Kirby diagrams of 4-manifolds) and some less classical ones as well (surface diagrams of broken fibrations on 4-manifolds).

Colloquium talk: Building 3-manifold invariants by composing correspondences.

May 25th, 4pm in 100 Willamette Hall.

Tea at 3pm in Willamette Hall atrium.

Dinner to follow. ContactÂ Ben Webster if you are interested.

Abstract: Heegaard-Floer homology, introduced by Ozsvath and Szabo about 10 years ago, provides versatile and computable topological invariants of 3- and 4-dimensional smooth manifolds. This picture has been extended by Robert Lipshitz, Peter Ozsvath and Dylan Thurston in 2008, whose “bordered Heegaard-Floer homology” makes these invariants fit into an extended topological field theory, associating algebras to surfaces and modules over these algebras to 3-manifolds with boundary. Recent work of Yanki Lekili and Tim Perutz further suggests a natural geometric interpretation of bordered Heegaard-Floer homology in terms of Lagrangian correspondences. In this talk, we will try to present this circle of ideas from the perspective of the symplectic geometry of symmetric products of surfaces; our goal will be to show how sophisticated tools such as Fukaya categories can be used to explain the remarkable structure behind Heegaard-Floer theory.