A real valued function of one variable $\varphi$ is called a metric transform if for every metric space $(X,d)$ the composition $d_\varphi = \varphi\circ d$ is also a metric on $X$. In this talk we present a complete characterization of the class of approximately nondecreasing, unbounded metric transforms $\varphi$ such that the half line $[0,\infty)$ with the transformed Euclidean metric $\varphi(|x-y|)$ is Gromov hyperbolic. As an application we obtain a type of rigidity with respect to metric transformations of roughly geodesic Gromov hyperbolic spaces. This is based on joint work with Andrew Nicas.
I will survey some recent joint work on invariants of smooth 4-manifolds with homology of S^1 x S^3. In our older work with Tom Mrowka and Daniel Ruberman we defined a Seiberg-Witten invariant of such manifolds. I will present a new formula (from our recent work with Jianfeng Lin and Daniel Ruberman) which expresses this invariant in terms of the monopole Floer homology and leads to some interesting topological applications.
We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable. This is joint work with Alix Deruelle.
The standard picture of gravitational collapse leads (using heuristic arguments) to an inequality relating the ADM mass and angular momentum of axisymmetric spacetimes. Such geometric inequalities have been rigorously proved for axisymmetric, asymptotically flat maximal initial data for the vacuum Einstein equations in dimension d=4. After a broad review, I will discuss recent work on extending this class of inequalities to d>4, where a number of qualitative differences arise (e.g. black holes can have non-spherical horizon topology). In particular, I will discuss how a lower bound for the mass, in terms of a (regularized) harmonic energy functional is obtained. The unique minimizer of this energy is then shown to correspond to extreme black hole initial data for fixed angular momenta. This provides a variational characterization of extreme black hole.
We show that if L is an oriented strongly quasipositive link other than the trivial knot or a link whose Alexander polynomial is a positive power of (t-1), or a quasipositive link with non-zero smooth 4-ball genus, then the Alexander polynomial and signature function of L determine aninteger n(L) ≥ 1 such that Σn(L), the n-fold cyclic cover of S3 branched over L, is not an L-space for n > n(L). If K is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that Σn(K) is not an L-space for n ≥ 6, and that the Alexander polynomial of K is a non-trivial product of cyclotomic polynomials if Σn(K) is an L-space for some n = 2,3,4,5. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi- alternating quasipositive links. They also allow us to classify strongly quasipositive alternating links and strongly quasipositive alternating 3-strand pretzel links. This is joint work with Michel Boileau and Cameron Gordon.