Kioloa workshop on C-projective geometry: March 31 - Apr.3

Wed	Feb. 12	11:00 - 12:00	Bernhard Lamel (University of Vienna)	Symmetries of nonminimal hypersurfaces Symmetries of real hypersurfaces (or, more generally, CR structures) are very restricted and have both interesting regularity properties and a rich structure theory, intimately related to the question of the classification of these objects. In this talk we will survey some recent results on such symmetries in the nonminimal case, draw analogies to the minimal case, and point out the stark differences which occur in this degenerate setting.
Thu	Feb. 27	4:00 - 5:00	Karin Melnick (University of Maryland)	(Colloquium) Normal forms for local flows on parabolic geometries The exponential map in Riemannian geometry conjugates the differential of an isometry at a point with the action of the isometry near the point. It thus provides a linear normal form for all isometries fixing a point. Conformal transformations are not linearizable in general. I will discuss a suite of normal forms theorems in conformal geometry and, more generally, for parabolic geometries, a rich family of geometric structures of which conformal, projective, and CR structures are examples.
Tue	Apr.29	10:30 - 12:30	Mike Eastwood (ANU)	The X-ray transform (part 1 and 2) The simplest x-ray transform is a version of the Radon transform in three dimensions. One starts with a suitably decaying function of three variables and integrates it over the lines in Euclidean three-space obtaining a function on the four-dimensional space of lines. This transform is often named after John who identified its range in 1938. There are many variations on this theme! There is a compactified version, due to Funk in 1913. There is a complex version, due to Bateman in 1904. Nowadays, there are all sorts of x-ray transforms and the purpose of these lectures will be to describe the links between them and to use representation theory and differential geometry to establish their range and kernel in various cases. This is a repeat of talks given at the 34th Czech Winter School in Srni in January this year: they are aimed at a broad geometrically-inclined audience.
Thu	May 1	11:30 - 12:30	Mike Eastwood (ANU)	The X-ray transform (part 3)
Tue	May 20	11:30 - 12:30	Dennis The (ANU)	Homogeneous integrable Lagrangean contact structures in dimension five
Tue	July 8	11:30 - 12:30	Simon Gindikin (Rutgers)	The Cauchy-Fantappie formula and the holomorphic language for analytic cohomology The Cauchy-Fantappie integral formula has interesting possibilities for explicit computations of residues similar to one-dimensional classical receipts. I'll illustrate it on examples from analytic duality on symmetric spaces.
Mon	July 28	11:30 - 12:30	Rupert MacCallum (University of Munster)	TBA
Wed	Aug. 13	11:30 - 12:30	Mike Eastwood (ANU)	Metrisability of three-dimensional path geometries Given a projective structure on a three-dimensional manifold, we find some explicit obstructions to the local existence of a Levi-Civita connection in the projective class (given as projectively invariant tensors algebraically constructed from the projective Weyl curvature). As a separate issue, this raises some purely algebraic questions concerning classical invariant theory. There will be examples. This is joint work with Maciej Dunajski.
Mon	Nov. 17	11:30 - 12:30	Gerd Schmalz (University of New England)	Chern-Moser theory for para-CR-manifolds and degenerate multicontact structures A 3-dimensional non-degenerate para-CR manifold is nothing but the solution manifold of a second order ODE and its geometry has been thoroughly studied under different names and in various contexts. I will present an approach analogous to Chern-Moser normal forms for non-degenerate and degenerate para-CR manifolds. This is joint work with Alessandro Ottazzi (Trento).
Tue	Nov. 18	11:30 - 12:30	Alexandr Medvedev (University of New England)	Fundamental invariants of systems of ODEs of higher order We find the complete set of fundamental invariants for systems of ordinary differential equations of order >=4 under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. To obtain results we use theory of geometric structures on filtered manifold and Cartan connections. This allows us to transfer main difficulty to essentially algebraic computations of special Lie algebra cohomologies associated with the problem.
Wed	Nov. 26	11:30 - 12:30	David Ridout (ANU)	Algebra vs Geometry: A showdown in physics I'd like to pose a challenge to geometers (and/or topologists) relating to our current understanding of certain toy models for string theory called the Wess-Zumino-Witten theories. In essence, there are certain textbook cases that are well understood by using algebras and representations as well as by using geometry and topology. Recent work has clarified the algebraic structure of many more cases. However, a geometric realisation of these cases is completely missing. I will introduce this story with the simplest example, when the string spacetime is a rank 1 Lie group.
Tue	Dec. 2	3:30 - 4:30 (LG15)	Callum Sleigh (University of Auckland)	Tractor calculus, BGG complexes, and cohomology of discrete groups. Let M = \Gamma \ H^n be a compact, orientable, totally geodesic hyperbolic manifold. We will study M using the tools of tractor calculus by viewing the hyperbolic structure as a holonomy reduction of a Cartan geometry. In particular, if F is an irreducible SO(n,1)-module, we give an isomorphism between the algebraic group cohomology H(\Gamma, F) and the cohomology of an appropriate BGG complex on M. Moreover, these tools can be used to prove a nonvanishing theorem for H^1(\Gamma, F) when M contains a compact, orientable, totally geodesic hypersurface. (Joint work with Rod Gover, University of Auckland)