We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N = 2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.

Let g be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra U(g) has intermediate growth and thus infinite Gelfand–Kirillov (GK-) dimension. We prove that the GK-dimension of U(g) is just infinite in the sense that any proper quotient of U(g) has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra S(g) by proper Poisson ideals. In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.

We study the topology of a complete asymptotically hyperbolic Einstein manifold of which its conformal boundary has positive Yamabe invariant. We prove that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

We relate previously defined quantum characteristic classes to Morse theoretic aspects of the Hofer length functional on Ham (M, ω). As an application we prove a theorem which can be interpreted as stating that this functional is “virtually” a perfect Morse-Bott functional. This can be applied to study the topology and Hofer geometry of Ham(M, ω). We also use this to give a prediction for the index of some geodesics for this functional, which was recently partially verified by Yael Karshon and Jennifer Slimowitz[5].

We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.