Fri |
Jan 25 |
2:00 - 3:00 |
Jan Slovak (Masaryk University) |
Schwachhoeffer's construction and its consequences
Schwachhoeffer's construction of all special holonomies of symplectic connections starts from the homogeneous models of
contact parabolic geometries. The talk will try to link this phenomenon to conformally Fedosov structures.
|
Tue |
Jan 29 |
11:30 - 12:30 |
Lashi Bandara (ANU) |
Square roots of perturbed subelliptic operators on Lie groups
Subelliptic operators can be defined on Lie groups through an
algebraic basis with an associated sub-Laplacian being a fundamental
object. Interesting "divergence-form" operators arise when perturbing
this sub-Laplacian with bounded, complex, measurable coefficients. We
solve the Kato square root problem for such operators on connected Lie
groups. In the general setting we deduce inhomogeneous estimates and
when the group is nilpotent, we prove stronger homogeneous estimates.
Furthermore, we prove Lipschitz stability of the estimates under small
perturbations of the coefficients. This is joint work with Tom ter Elst
(Auckland) and Alan McIntosh (ANU).
|
Tue |
Feb 5 |
11:30 - 12:30 |
Mat Langford (ANU) |
A proof of the Lawson Conjecture
In 1970, Lawson conjectured that any embedded minimal torus in the
three-sphere is congruent to the `obvious' one: the Clifford torus,
$Cliff :=\{z\in S^3\subset R^4 : z_1^2+z_2^2=z_3^2+z_4^2=1/2\}$. This
conjecture was recently proved in the positive by Brendle. The proof I
will present is similar in spirit to Brendle's. It is motivated by a
geometric technique for parabolic equations known as non-collapsing.
This technique transfers to the elliptic minimal surface problem since
minimal surfaces are stationary solutions of the mean curvature flow.
|
WED |
Mar 13 |
11:00 - 12:00 |
Vladimir Matveev (Jena) |
Obata-Tanno equation and parallel objects for cone structures
One of two main steps in the proof of the classical Lichnerowicz-Obata and Obata-Yano conjectures was to understand the existence of a solution of the so-called Obata-Tanno equations. These are
certain systems of geometric PDE that naturally and independently appeared in different branches of mathematics; I will start my talk with historical overview.
In the mathematical part of my talk I will show that these equations are closely related to the existence of parallel tensors on the cone over the manifold and use this observation to show that
the existence of a nontrivial solution on a closed manifold implies that this manifold is a sphere or a complex projective space with the standard metric. This part of my talk is based on the
joint results with A. Fedorova, V. Kiosak, P. Mounoud and S. Rosemann.
I will also discuss in what sense the equations are projectively and h-projectively invariant. This part of my talk is an ongoing project with R. Gover.
|
Wed |
Mar 20 |
11:00 - 12:00 |
Konrad Schoebel |
Separation of Variables and Moduli Spaces of Stable Curves
Integrable Killing tensors are used to classify orthogonal coordinates
in which the classical Hamilton-Jacobi equation can be solved by a
separation of variables. We explicitly describe the projective variety
of integrable Killing tensors on the 3-sphere and relate its algebraic
geometric properties to the differential geometric properties of the
corresponding Killing tensors. This leads to an isomorphism between the
moduli space of separation coordinates on the n-sphere and a well known
object in algebraic geometry: the moduli space of stable curves of genus
zero with n+2 marked points.
|
Wed |
Mar 27 |
11:00 - 12:00 |
David Calderbank (University of Bath) |
What is... a parabolic building?
TBA
|
Fri |
May 24 |
13:30 - 14:30 |
Igor Zelenko (Texas A&M) |
Wilczynski type invariants in the geometry of distributions
The geometry of vector distributions can be studied very often via the path geometry of distinguished integral curves called abnormal extremal trajectories. This approach was developed by Boris
Doubrov and myself mainly for the uniform treatment of distributions of any rank and arbitrary high corank regardless of their Tanaka symbol and the main applications are outside of the scope of
the parabolic geometries. However, even for the classical equivalence problem of rank 2 distributions (fields of planes) in R^5 the approach gives a new insight on the classical Cartan tensor
of such distributions. In this case the approach gives the passage from one G2-geometry to other ones (via the natural double fibration) and the Cartan tensor can be interpreted as the
Wilczynski invariant of self-dual curves in projective spaces. In particular, this gives an alternative way of computing the Cartan tensor via the Hamiltonian formalism. In this talk I would
like to concentrate on this computational aspect. If the time will permit the case of rank 3 distributions in R^6 will be discussed in this line as well.
|
Tues |
May 28 |
10:00 - 11:00 |
Matthew Randall (ANU) |
Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor
A projective surface is a 2-dimensional manifold equipped with a projective structure i.e. a class of torsion-free affine that have the same geodesics as unparameterised curves. Given any
projective surface we can ask whether it admits a torsion-free affine connection (in its projective class) that has skew-symmetric Ricci tensor. This is equivalent to solving a particular
semi-linear overdetermined partial differential equation. It turns out that there are local obstructions to solving the PDE in two dimensions. These obstructions are constructed out of local
invariants of the projective structure.
|
Wed |
May 29 |
11:00 - 12:00 |
Amitesh Datta (ANU) |
The Classification of Symplectic Toric Manifolds
A 2n-dimensional symplectic toric manifold is a compact connected symplectic manifold equipped with an effective hamiltonian action of an n-torus and with a corresponding moment map. Delzant
proved that symplectic toric manifolds are classified by Delzant polytopes - convex polytopes in Euclidean space satisfying certain conditions. Delzant's theorem is important, in part, due to
the various connections that exist between toric geometry and other branches of mathematics. In this talk, we will briefly sketch some fundamentals of symplectic geometry such as the
Marsden-Weinstein-Meyer theorem and the Atiyah-Guillemin-Sternberg theorem and explain how these results lead to a proof of Delzant's theorem.
|
Tue |
July 9 |
12:00 - 13:00 |
Travis Willse (ANU) |
Doubrov and Govorov's exceptional (2, 3, 5)-distribution
In his so called Five Variables paper, Cartan solved the equivalence problem for (2, 3, 5)-distributions. In particular, he claimed to classify up to local equivalence such distributions with
infinitesimal symmetry algebra of rank 6, and gave an ostensible quasi-normal form for such distributions whose harmonic curvature satisfies a natural degenerancy condition, which in particular
implied that all distributions in this class have solvable symmetry algebra. A few months ago, Govorov and Doubrov upended this classification by constructing such a distribution with
nonsolvable symmetry algebra, and have apparently shown that it is the only distribution Cartan missed, at least that satisfies a transitivity condition. (This counterexample was nearly found by
Strazzullo in his 2008 thesis.) This distribution enjoys several unusual features. For one, unlike all of the distributions that satisfy Cartan's normal form, the conformal structure it induces
via Nurowski's construction is not almost Einstein. Despite this, one can still find (with some effort) an ambient metric for this conformal structure and with some computer assistance show that
its holonomy is equal to (split, real) G_2, hence furnishing examples of two uncommon phenomena.
|
Tue |
Oct 15 |
TBA |
Mike Eastwood (ANU) |
TBA (in algebra/topology seminar)
|
Tue |
Oct 17 |
TBA |
Pawel Nurowski (Warsaw) |
TBA - alumni talk on space mission design
|
Wed |
Jan 30 |
11:00 - 12:30 |
Mike Eastwood (ANU) |
Projective structures and mobility
It has been known for at least 2,500 years that different Riemannian
metrics can have the same geodesics (as unparameterised curves).
However, this phenomenon is quite rare: the generic metric has little
"mobility." More generally, one could try to fit a metric to a system of
curves that one would like to be its geodesics but generically this is
hopeless: no "mobility" at all! I'll explain some of this classical area
(Beltrami 1865, Liouville 1889) from a modern perspective (tractor
bundles and the like).
|
Wed |
Feb 6 |
11:00 - 12:30 |
Katharina Neusser (ANU) |
H-projective
geometry - part 3
TBA
|
Wed |
Feb 13 |
11:00 - 12:30 |
Katharina Neusser (ANU) |
H-projective
geometry - part 4
|
Wed |
Feb 20 |
11:00 - 12:30 |
Mike Eastwood (ANU) |
H-projective
geometry - part 5
This talk will discuss the representation theory entailed in breaking up all those annoying tensors in H-projective
geometry into their irreducible parts. As part of the discussion, we'll revisit complexification and also see how it applies to manifolds. To maximise confusion (but also utility) we'll see what
happens if we complexify complex manifolds! Mostly this talk will be about algebra and will be independent of parts 1-4 (already delivered by Katharina Neusser).
|
FRI |
Feb 22 |
1:30 - 2:30 |
Mike Eastwood (ANU) |
H-projective geometry - part 6
|
FRI |
Mar 1 |
1:00 - 2:30 |
Katharina Neusser (ANU) |
H-projective geometry - part 7
|
Tue |
May 7 |
12:00 - 1:30 |
Mike Eastwood (ANU) |
Prolongation 101
Prolongation is a process for augmenting a system of partial differential equations with further equations in order to
produce an equivalent but more congenial system. This talk will be a beginner's guide starting with examples from
differential geometry, e.g. the Killing equation from Riemannian geometry. Prolongation is often regarded as more of an
art form than a science but, with the help of Lie algebra cohomology, some theory can be constructed.
|
Tue |
May 14 |
12:00 - 1:30 |
Dennis The (ANU) |
On a new normalization for tractor covariant derivatives
Last week, Mike discussed prolongation and motivated the use of tractor bundles in this study. More generally, in
studying the prolongation of curved BGG operators, in order to get a 1-1 correspondence between solutions of the BGG
operator and parallel sections of the corresponding tractor bundle, one should modify the notion of "parallel", i.e. one
should take a modified tractor connection instead of the canonical one induced from the Cartan connection associated to
the given geometry. I'll go through some of these ideas, as first introduced in a 2010 article by Hammerl, Somberg, Soucek, and Silhan.
|
Tue |
May 21 |
12:00 - 1:30 |
Travis Willse (ANU) |
On a new normalization for tractor covariant derivatives - part 2
|