Wed 
Sep 25 
15:00  16:00 
Lenka Zalabova (University of South Bohemia) 
Symmetries of parabolic geometries
We present various generalizations of symmetric spaces and use them as the motivation for the definition of symmetric parabolic geometries. We describe symmetries of flat models in
detail. We describe some properties of symmetries on general parabolic geometries.

Mon 
Oct 14 
14:30  15:30 
Callum Sleigh (University of Auckland) 
EynardOrantin Theory of the APolynomial
In the early 2000s, the physicists B. Eynard and N. Orantin introduced a
family of invariants associated to an algebraic curve. These invariants
are related to several difficult and interesting problems in enumerative
geometry and mirror symmetry; however  from the point of view of a
mathematician  they remain quite mysterious. I will discuss an attempt
to understand some of the geometry underlying the theory of
EynardOrantin invariants and will apply these ideas to the A polynomial
of a knot. This is conjectured to have important repercussions in
quantum topology.

Tue 
Oct 15 
15:30  16:30 
Mike Eastwood (ANU) 
The moduli space of genus 2 Riemann surfaces (Algebra & Topology seminar)
An exploration of this moduli space using only nineteenth century mathematics and a small computer. We shall discover the most bizarre Riemann surface of them all.

Fri 
Oct 18 
14:00  15:00 
Robert Bryant (Duke University) 
The geometry of periodic equiPoisson sequences
A sequence of functions f = f_i (\infty < i < \infty) on a surface S is said to be \emph{equiPoisson} if it
satisfies the relations df_{i1}\wedge df_i = df_i\wedge df_{i+1} for all i. We say that f is nperiodic if f_i = f_{i+n} for all i. The nperiodic equiPoisson sequences for low values of n
turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras. In this talk, I will explain what is known about the classification
(up to a natural notion of equivalence) of such periodic sequences and its surprising relationships with differential geometry and the theory of overdetermined differential equations.

Mon 
Oct 21 
14:30  15:30 
Rod Gover (University of Auckland) 
Nearly Kaehler Geometry and the (2,3,5)distribution
We look at a beautiful convergence of these two geometries facilitated by projective geometry. This is joint work with Roberto Panai and Travis Willse.

Thu 
Oct 24 
16:00  17:00 
Robert Bryant (Duke) 
The geometry of calibrations (Colloquium)
It has now been more than 30 years since Harvey and Lawson's fundamental paper "Calibrated Geometries" appeared, and its
influence in geometry and mathematical physics continues to grow. Introduced as a farreaching generalization of the ideas behind
Wirtinger's Theorem in complex geometry, the idea of calibrations as tools for proving that various geometric configurations are
actually optimal (rather than just 'stationary'), calibrations have now found their way into many different aspects of the calculus
of variations.
In this talk, I'll begin by describing the geometric motivation behind calibrations, what Caratheodory famously called "the
royal road to the calculus of variations", give the basic definitions, introduce some of the important examples, and report on some
recent results in geometry and topology that follow from applications of calibrations.
The focus will be on basic examples, and most of the talk should be accessible to those with no more than a basic
familiarity with the calculus of variations.

Fri 
Oct 25 
14:00  15:00 
Pawel Nurowski (Warsaw) 
On exceptional contact geometries

Thu 
Oct 31 
17:30  19:30 
Pawel Nurowski (Warsaw) 
Gravitational
slingshot and space mission design 
Mon 
Nov 4 
14:30  15:30 
Gerd Schmalz (University of New England) 
Rigid Spheres and a nonlinear PDE
A realanalytic real hypersurface M in C^2 is called a rigid sphere if it is locally holomorphically equivalent to
the Heisenberg sphere Im w= z^2 and it's defining equation has rigid normal form.
The problem to find all rigid spheres has been raised by Nancy Stanton in 1991 in relation with her rigid normal form construction.
The original motivation was to identify the different realisations of the sphere in rigid normal form. On the other hand it is of
its own interest to classify rigid hypersurfaces that are equivalent to the sphere.
In her paper Stanton has presented a family of rigid spheres that depends on 4 real parameters. Though Stanton's list turned out to
be incomplete, we show that the complete list can be obtained from Stanton's family by an algebraic trick.
This also solves the vanishing Cartan curvature equation for rigid hypersurfaces, which (after some trivial reduction) is a
overdetermined 3rd order nonlinear PDE.
This is joint work with Vladimir Ezhov (Flinders University).

Mon 
Nov 4 
15:30  16:30 
Emma Carberry (University of Sydney) 
Harmonic maps, Toda frames and extended Dynkin diagrams
I shall discuss harmonic maps from surfaces into homogeneous spaces $G/T$ where $G$ is any simple real Lie group (not necessarily compact) and $T$ is a Cartan subgroup. All immersions of a
genus one surface into $G/T$ possessing a Toda frame can be constructed by integrating a pair of commuting vector fields on a finite dimensional subspace of a loop algebra. I will provide
necessary and sufficient conditions for the existence of a Toda frame and describe those $G/T$ to which the theory applies in terms of involutions of extended Dynkin diagrams. Applications will
be given to harmonic maps into de Sitter spaces and to Willmore tori in $S^3$.

Tue 
Nov 5 
11:00  12:00 
Boris Doubrov (University of Minsk) 
Fundamental invariants of systems of ODEs
This is a recent joint work with Alexander Medvedev, where we compute fundamental invariants for systems of ODEs of order greater
or equal to 4. This follows earlier results of Boris Doubrov on the case of contact invariants of a single ODE, results of Mark
Fels on systems of 2nd order and results of Alexander Medvedev on invariants of systems of 3rd order. The main idea is to construct
a normal Cartan connection associated with a given system of ODEs and to compute the harmonic part of its curvature tensor. This is
similar to parabolic geometries, but requires special techniques for cohomology computation. 
Tue 
Nov 5 
12:00  13:00 
Robert Bryant (Duke University) 
The exceptional geometry of binary sextics (Algebra & Topology seminar)
A very classical problem in algebra is the study of the invariants of binary forms of degree $n$ under the action of $SL(2)$. When $n=2$, the basic invariant of a binary quadratic $Q = a\,x^2 + 2b\,xy + c\,y^2$
under unimodular changes of $x$ and $y$ is the discriminant $\Delta(Q) = b^2  ac$. When $n=3$, the basic invariant of a binary cubic $C = a\,x^3 + 3b\,x^2y + c\,xy^2 + d\,y^3$ is a polynomial $D(C)$ of degree~$4$ in
the coefficients, and so on. The algebra of these invariants on these representations of $SL(2)$ become quite complicated as $n$ increases, but it turns out that the case of sextics (i.e., binary forms of degree $6$)
is exceptional in a number of ways.
In this talk, I will describe some features that make the case of binary sextics particularly interesting, beginning with its connection with the exceptional group $G_2$. I will explain how certain features of the
algebra of invariants in this case turn out to be the key to understanding the differential equations that govern the existence of and explicit formulas for certain absolutely minimizing 4folds in dimension 7, the
socalled 'coassociative cones' of Harvey and Lawson.

Thu 
Nov 7 
11:00  12:00 
Pawel Nurowski (Warsaw) 
Ringtype structures on CMB maps (at Mount
Stromlo) 
Fri 
Nov 8 
14:00  15:00 
Pawel Nurowski (Warsaw) 
Lagrange solution for 3body problem and beyond
First I briefly review special solutions of the 3body problem obtained
in 18th century by Euler and Lagrange. Then I will present a few
generalizations of these solutions. Two of them will concern with a
system of 3 and 4 charges obeying the Coulomb law.
