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)	Eynard-Orantin Theory of the A-Polynomial 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 Eynard-Orantin 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 equi-Poisson sequences A sequence of functions f = f_i (-\infty < i < \infty) on a surface S is said to be \emph{equi-Poisson} if it satisfies the relations df_{i-1}\wedge df_i = df_i\wedge df_{i+1} for all i. We say that f is n-periodic if f_i = f_{i+n} for all i. The n-periodic equi-Poisson 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 far-reaching 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 Lecture notes
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 non-linear PDE A real-analytic 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 non-linear 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 4-folds in dimension 7, the so-called 'coassociative cones' of Harvey and Lawson.
Thu	Nov 7	11:00 - 12:00	Pawel Nurowski (Warsaw)	Ring-type structures on CMB maps (at Mount Stromlo)
Fri	Nov 8	14:00 - 15:00	Pawel Nurowski (Warsaw)	Lagrange solution for 3-body problem and beyond First I briefly review special solutions of the 3-body 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.