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Simon Salamon: Lectures

September 2022 – LECTURE CANCELLED
TITLE: Spinors and instantons

ABSTRACT: Motivated in part by the existence of certain low-rank vector bundles on projective spaces, the talk will focus on the interplay between SU(3) as a global symmetry group, tensors satisfying twistor equations, and reduced holonomy in eight dimensions. It will relate work of Gray–Green 1970, Horrocks 1978, and Atiyah–Witten 2002, on Spin(7) structures, instantons and circle actions, to that of the Collaboration. The approach is based on a joint paper with Udhav Fowdar.

Slides prepared for lecture

September 13, 2019
TITLE: Twistor Spaces and Special Holonomy

ABSTRACT: Adaptations of Penrose’s twistor theory to the Riemannian setting have played a role in the construction and study of metrics with special holonomy. The original idea was to encode conformal structures by objects from complex and algebraic geometry. After explaining this background, the talk will focus on a circle quotient of a well-known metric with holonomy G2 that fibers over the 4-sphere, investigated by Atiyah and Witten in the context of M-theory. The Gibbons-Hawking ansatz reduces the G2 space to R^6 endowed with a singular SU(3) structure invariant by a diagonal action of SO(3). The talk will describe some key features of this reduction (obtained jointly with Acharya and Bryant).

Slides of Lecture

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.

Slides of Lecture

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 G_2 on a hypothetical Fano 7-fold to a known moment mapping.

Slides of lecture

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 G_2 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.

Slides of lecture

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.

Slides of lecture