Topics on Algebraic, Symplectic and Non-commutative Geometry

Programme

10:10-11:00 Eckhard Meinrenken (Toronto)

11:05-11:30 Paul Norbury (Adelaide)

11:35-12:00 Daniel Stevenson (Adelaide)

Lunch at the Eros Greek Restaurant on Rundle Street.
Please let Siye Wu know as soon as possible if you wish to attend.

14:10-15:00 Adam Harris (Melbourne)

15:35-16:00 Siye Wu (Adelaide)

16:05-16:30 Adam Rennie (Adelaide)

Titles and Abstracts

Eckhard Meinrenken
Matrices, moment maps and moduli spaces
Abstract: Moment maps are a mathematical generalization of angular momentum in classical mechanics. The abstract notion of a moment map was introduced in the late 1960's by Souriau, and developed by Guillemin, Kirillov, Kirwan, Kostant, Marsden, Sternberg, Weinstein, and many others. This lecture will be concerned with a "non-linear" theory of moment maps, introduced in 1998 in collaboration with Alekseev and Malkin. We will explain the main properties of non-linear moment maps, and discuss their applications to eigenvalue problems for matrices and to moduli spaces of flat connections over a surface.

Paul Norbury
Boundary algebras of hyperbolic monopoles
Abstract: We define an algebra of projection operators for each hyperbolic monopole. The algebra is applied to settle the open conjecture that the value of the monopole at infinity determines the monopole. Further applications will be described.

Daniel Stevenson
The relationship of bundle gerbes with gerbes
Abstract: Recently there has been interest in higher dimensional notions of p-line bundles on a manifold M which provide a way of geometrically realising classes in H^{p+1}(M;Z). `Gerbes' and `bundle gerbes' provide examples of 2-line bundles. We compare these two objects by constructing a monoidal 2-functor between the monoidal 2-categories of bundle gerbes and gerbes (this is joint work with Michael Murray). We will also show how this construction extends to associate a 2-gerbe to a bundle 2-gerbe.

Adam Harris
Special metrics near isolated complex singularities hypersurfaces
Abstract: Let X be an analytic variety in complex Euclidean space, with isolated singular point at the origin. Intersection of X with a sufficiently small ball punctured at the origin defines a domain whose topology is determined by that of the smooth, compact, real hypersurface corresponding to the outer boundary. In this talk I will discuss the role of boundary-value problems for certain systems of partial differential equations in the existence of special metrics on this punctured domain. Our main result is a criterion for local existence of Ricci-flat Hermitian metrics. Implications for the topology and geometry of singularities and their deformations will also be discussed. (Joint work with Y. Tonegawa of Hokkaido University)

Siye Wu
Geometric quantization of a family
Abstract: Given a family of polarizations of a quantizable symplectic manifold, we calculate the Chern character of the bundle of quantum Hilbert spaces over the parameter space using the family index theorem. We also study the situation when the bundle is projectively flat.

Adam Rennie
Poincare duality in K-Theory; spin^c structures and Morita equivalence
Abstract: For some time it has been known that spin^c structures on a complete manifold are in one-to-one correspondence with unitary equivalence classes of Morita equivalence bimodules between the algebra of functions and the complex (twisted) Clifford algebra. We show that the existence of such bimodules is a consequence only of Poincare Duality in K-theory and a couple of simple algebraic requirements (linearity and irreducibility). In particular, the result continues to hold for spectral triples over noncommutative algebras obeying these conditions.