Elisa Prato, Univ de Nice, France

Symplectic quasifolds and non-rational convex polytopes

A quasifold is a space $X$ that is locally modelled by the quotient of ${\bf R}^m$ by the linear action of a discrete group; manifolds and orbifolds are examples of quasifolds. It is possible to endow quasifolds with smooth functions, vector fields, differential forms and symplectic structures, just as one does for manifolds and orbifolds. Symplectic quasifolds arise naturally when taking the symplectic quotient with respect to the action of a non-compact Lie group on an ordinary symplectic manifold. The analogue of a torus in this setting is a quasitorus, $D$, which is the quotient of ${\bf R}^n$ by a quasilattice. One can define Hamiltonian quasitorus actions and moment maps on symplectic quasifolds, and (under reasonable compactness assumptions) extend the Atiyah, Guillemin-Sternberg convexity theorem to show that the image of the moment map is a convex polytope. The novelty with respect to the manifold case is that the polytope is no longer necessarily rational.

Weiping Zhang, Nankai Institute of Mathematics, Tianjin, China

An analytic approach to the Guillemin-Sternberg geometric quantization
conjecture

We explain the joint work with Youliang Tian (Courant Institute, New York) which includes an analytic proof of the geometric quantization conjecture of Guillemin-Sternberg as well as an extension of this conjecture to cases of the manifolds with boundary.

Adam Harris (University of Adelaide)

Extension techniques for holomorphic vector bundles

For every unitary vector bundle on a complex manifold there is a one-to-one correspondence between unitary connections with curvature form of type $(1,1)$, and compatible holomorphic structures. Since the celebrated removable singularities theorem of Uhlenbeck, criteria for the extendibility of unitary gauge fields across subsets of Euclidean space have evolved from $L^2$ ``finite energy'' assumptions, but as we hope to indicate in this talk, another line of investigation is opened by considering the associated holomorphic structures. In particular, the interplay between unitary connections of curvature type $(1,1)$ and holomorphic connections will be seen to provide an alternative (though possibly related) class of removable singularities theorems.

Siye Wu (University of Adelaide)

On the instanton complex of holomorphic Morse theory

We consider holomorphic group actions on vector bundles over complex manifolds. When the manifold is K\"ahler and the group is a circle, Morse-type inequalities which strengthen the Atiyah-Bott fixed-point formula were obtained by Witten from physical considerations. They were proved analytically by Mathai, Zhang and myself and were generalized to cases with torus and non-Abelian group actions. Furthermore, unlike the fixed-point formula, the holomorphic Morse inequalities could fail without the K\"ahler condition. In this talk, I show that there is a holomorphic analog of the instanton complex for any meromorphic group action on a complex manifold, if the set of connected components of the fixed-point set is partially ordered in a suitable way under the group action; these assumptions are much weaker than the K\"ahler condition. This holomorphic instanton complex, constructed via a spectral sequence, not only gives an algebraic-geometric proof of the holomorphic Morse inequalities in a more general setting, but also completely determines the Dolbeault cohomology groups using the combinatorial data of the group action.

Varghese Mathai (University of Adelaide)

On the nonexistence of positive scalar curvature metrics on certain
symplectic manifolds

This is intended to be a survey type talk on the Gromov-Lawson-Rosenberg method to study obstructions to the existence of metrics of positive scalar curvature, and then to apply it to certain symplectic manifolds. I also hope to mention some of my work on this problem.

Weiping Zhang

Higher spectral flow and Toeplitz families

In this joint work with Xianzhe Dai (University of Southern California, Los Angeles) we extend the spectral flow of Atiyah-Patodi-Singer to the case of fibered manifolds, using the concept of spectral section of Melrose and Piazza. We also extend the definition of the Toeplitz operator to the fibration cases and relate its index to the higher spectral flow.

Mike Eastwood (University of Adelaide)

Some normal forms in geometry

I shall discuss the use of normal forms in differential geometry and, in particular, an affine normal form currently being used in joint work with Vladimir Ezhov.

Vamanamurthy, Univ of Auckland, New Zealand

Generalised elliptic Integrals and modular equations

Elliptic functions and integrals arise from the conformal mapping of a rectangle onto a half-plane. If we replace the rectangle by an arbitrary parallelogram, we get Generalised Elliptic functions and integrals. These functions are also related to modular equations in analytic number theory. We study some monotone and convexity properties of these functions and obtain sharp inequalities. This is a recent joint work with Glen Anderson, (Michigan State University), Qiu Songliang (HIEE, Hangzhou, China) and Matti Vuorinen (University of Helsinki, Finland)