September 10, 2018
TITLE: Generating equidistant points
ABSTRACT: Work of Gerhard Zauner has led to the belief that, for every n, one can find n^2 points in complex projective space CP^(n-1) that are mutually equidistant in the Fubini-Study (Kaehler) metric. It is also conjectured that one can always find a vector in C^n whose orbit under a finite Heisenberg group generates such a set. I shall give some geometrical background, and new moment map interpretations of the problem.
April 13, 2018
TITLE: Automorphisms of Fano contact manifolds
ABSTRACT: The correspondence between compact symmetric spaces with holonomy in Sp(n)Sp(1) and complex homogeneous contact manifolds was discovered by Joseph Wolf in 1965, yet the possibility of non-homogeneous manifolds subscribing to this model remains open. Recent results of Buczynski-Wisniewski-Weber on torus actions on Fano contact manifolds settle this question for n<5. Aspects of the theory will be explained with reference to an example that relates the action of on a hypothetical Fano 7-fold to a known moment mapping.
September 10, 2017
TITLE: Quotients and hypersurfaces of model metrics
ABSTRACT: I shall outline basic techniques in understanding induced structures on U(1)-quotients and hypersurfaces of metrics with holonomy or Spin(7), following separate joint work with V. Apostolov and D. Conti. I shall discuss some explicit examples that have also been developed by B. Acharya, R. Bryant, and U. Fowdar.
September 9, 2016
TITLE: Manifolds with holonomy Sp(n)Sp(1)
ABSTRACT: This holonomy group from Berger’s list characterizes the class of quaternion-Kähler (or “nearly hyperkähler”) manifolds of real dimension 4n. Each such manifold M carries a parallel 4-form, and is Einstein but is not in general Kähler. In the case of positive scalar curvature, the twistor space (a 2-sphere bundle over M) is Kähler, and it is an open question as to whether the only complete examples are homogeneous. This topic provides fascinating links between complex and Riemannian geometry that illustrate the power of spinor and twistor methods.