We call a non-trivial homology 3-sphere a Kirby-Ramanujam sphere if it bounds a homology plane, an algebraic complex smooth surface with the same homology groups of the complex plane. In this talk, we present several infinite families of Kirby-Ramanujam spheres bounding Mazur type 4-manifolds, compact contractible smooth 4-manifolds built with only 0-, 1-, and 2-handles. Such an interplay between complex surfaces and 4-manifolds was first observed by Ramanujam and Kirby around nineteen-eighties. This is upcoming joint work with Rodolfo Aguilar Aguilar.
The cosmetic surgery conjecture states that no two surgeries on a given knot produce the same 3-manifold (up to orientation preserving diffeomorphism). Floer homology has proved to be a powerful tool for approaching this problem; I will survey partial results that are known and then show that these results can be improved significantly. If a knot in S^3 admits purely cosmetic surgeries, then the surgery slopes are +/- 2 or +/- 1/q, and for any given knot we can give an upper bound for q in terms of the Heegaard Floer thickness. In particular, for any knot there are at most finitely many potential pairs of cosmetic surgery slopes. With the aid of computer computation we show that the conjecture holds for all knots with at most 15 crossings.
Michael Taylor : Anderson-Cheeger limits of smooth Riemannian manifolds, and other Gromov-Hausdorff limits- Geometry and Topology ( 142 Views )
If you take a surface in Euclidean space that is locally the graph of a C2 function, this induces a local coordinate system in which the metric tensor is merely C1. Geodesic flows are well defined when the metric tensor is C2, but there are lots of examples of metric tensors of class C^(2-epsilon) for which geodesics branch. Nevertheless, for the C2 surface mentioned above, the geodesic flow is well defined. This result has been noted several times. It has several proofs. One uses the fact that geodesic flows are well defined whenever the Ricci tensor is bounded. An important class of Gromov-Hausdorff limits of smooth Riemannian manifolds studied by Anderson and Cheeger puts a lower bound on the Ricci tensor (and the injectivity radius), and obtains a limiting manifold whose metric tensor is not quite C1. We will explore the question of whether the geodesic flow is well defined on such a limit, and also look at some other limits of smooth manifolds, with wilder behavior.
I will tell two stories. The first is the story of static spacetimes with black hole boundaries and the attempt to classify them. The second is the story of the Penrose inequality. I will then weave these two stories together in the setting of negative curvature. This last part is a report on joint work-in-progress with A. Neves.
I will discuss a conjecture of Conner and Raymond that any aspherical manifold whose fundamental group has center possesses a circle action, and put it into the context of earlier work and conjectures of Borel and others.
We start by recalling gauge theory and some of its applications in low-dimensional topology. We briefly discuss Donaldson-Thomas program to extend the methods of gauge theory to study higher-dimensional manifolds, specially Calabi-Yau 3-folds and G2-manifolds. Finally, we will see that the study of gauge theory in higher dimensions motivates new ideas and questions in low-dimensional topology.
In this talk, we will study strongly negative amphicheiral knots - a class of knots with symmetry. These knots provide torsion elements in the knot concordance group, which are less understood than infinite-order elements. We will introduce the half-Alexander polynomial, an equivariant version of the Alexander polynomial for strongly negative amphicheiral knots, focusing on its applications to knot concordance. In particular, I will show how it facilitated the construction of the first examples of non-slice amphicheiral knots of determinant one. This talk is based on joint work with Keegan Boyle.