Mon 
Oct 15 
12:00  1:00 
John Ryan (U. Arkansas) 
Dirac type operators and their invariances
We will consider invariances and their consequences for various types of operators associated with Dirac operators. This will be a
continuation of my earlier analysis talk. Operators to be considered include the nonlinear p and infinite Dirac operators
RaritaSchwinger type operators and a square root of the Kohn Laplacian.

Mon 
Oct 22 
12:00  1:00 
Kowshik Bettadapura (ANU) 
An introduction to the StolzTeichner program
The StolzTeichner program began almost a decade ago with the goal of using the framework of field theories to describe cocycles
in certain generalsed cohomology theories, such as TMF. While this goal has not been realised yet, some progress has been made in
the low dimensional cases. In extending Atiyah's classical definition of a topological field theory, by adding notions of
smoothness and supersymmetry, Stolz and Teichner were able to describe cocycles in de Rham cohomology and Ktheory. In this talk I
will discuss these notions and use the (01)dimensional field theories as a case study for modelling cocycles in de Rham
cohomology.

Mon 
Oct 29 
12:00  1:00 
Amitesh Datta (ANU) 
The Rauch Comparison Theorem
The Rauch Comparison Theorem is one of the basic facts in Riemannian geometry. Intuitively, it expresses the plausible fact that
as the curvature grows, lengths shorten. We will begin by briefly recalling some basic properties of Jacobi fields. We will then
prove the index lemma and establish the Rauch Comparison Theorem as a corollary. The proof that we present is an elaboration due to
several mathematicians in the fifties of Rauch's original 1951 proof. If time permits, we will discuss some standard applications.

Mon 
Nov 5 
12:00  1:00 
Mike Eastwood (ANU) 
Conformally Fedosov manifolds
Symplectic and projective structures may be compatibly combined. The resulting structure closely resembles conformal geometry and
a manifold endowed with such a structure is called conformally Fedosov. This talk will present the basic theory of conformally
Fedosov geometry and, in particular, construct a Cartan connection for them. This is joint work with Jan Slovak.

Mon 
Nov 12 
11:30  12:30 
Dennis The (ANU) 
The gap phenomenon in parabolic geometries (revisited)
For Cartan geometries of a given type, the maximal amount of symmetry is realized by the flat model. However, if the geometry is
not (locally) flat, how much symmetry can it have? This lecture is concerned with this "gap" between maximal and submaximal
symmetry dimensions in the case of parabolic geometries. In my previous talk on this topic, I showed how Tanaka prolongation and
Kostant's BottBorelWeil theorem play a key role, and results were obtained for conformal geometry, systems of 2nd order ODE, etc.
In this talk, I'll describe a new Dynkin diagram recipe and derive consequences for the gap problem in general parabolic
geometries. (Joint work with Boris Kruglikov.)

Mon 
Nov 19 
11:30  12:30 
Arman TaghaviChabert (Masaryk University) 
Pure spinors and curvature in higher dimensions
We give a generalisation of the PenrosePetrov classification of the Weyl tensor from four to higher dimensions for pseudoRiemannian manifolds of complex or split signature, based on the concept of principal pure spinors. We classify the various degrees of integrability of the totally null distribution associated to a pure spinor field. The relation between these classifications and solutions to spinorial differential equations such as the twistor equation is discussed. Finally, we apply these results to characterise the Riemannian extension of a projective structure.

WED 
Nov 21 
11:00  12:00 
Jeanne Clelland (UC Boulder) 
Equivalence of geometric structures in control theory via moving frames
The method of equivalence was introduced by Elie Cartan in 1908 as a procedure for finding invariants of geometric structures under the actions of pseudogroups, which most commonly appear as
subgroups of the diffeomorphism group of a manifold preserving some underlying geometric structure. The method was further developed and systematized over the course of the 20th century, and it
has been applied in a great variety of contexts.
After briefly demonstrating Cartan's method in the familiar context of Riemannian geometry, we will describe applications of the method of equivalence to some geometric structures related to
control theory:
1) SubRiemannian geometry (Hughen, Moseley) and subFinsler geometry (C, Moseley, and Wilkens), both of which involve metrics which are defined only on a distribution on a manifold (i.e.,
on a subbundle of the tangent bundle of the manifold). Such metrics appear naturally in the context of optimal control of "driftless" kinematic systems.
2) Affine distributions (C, Moseley, and Wilkens), which appear naturally in the contexts of kinematic control systems with drift and dynamic control systems.
3) Dynamic equivalence of control systems (Stackpole). Dynamic equivalence generalizes the more thoroughly understood notion of static equivalence for control systems, which is addressed by the
methods mentioned above. In his justcompleted Ph.D. thesis, Stackpole shows how the method of equivalence may be used to address this more general notion of equivalencewhich is defined in
terms of submersions rather than diffeomorphismsby extending to infinite prolongations.

TUE 
Nov 27 
11:30  12:30 
Jeanne Clelland (UC Boulder) 
A Tale of Two Arc Lengths
In Euclidean geometry, all metric notions (arc length for curves, the first fundamental form for surfaces, etc.) are derived from the Euclidean inner product on tangent vectors, and this inner
product is preserved by the symmetry group of Euclidean space (translations, rotations, and reflections).
In equiaffine geometry there is no invariant notion of inner product on tangent vectors that is preserved by the full symmetry group of affine space. Nevertheless, it is possible to define an
invariant notion of arc length for "nondegenerate" curves, and an invariant first fundamental form for "nondegenerate" surfaces in affine space. This leads to two possible notions of arc length
for a curve contained in a surface, and these two arc length functions do not necessarily agree! In this talk we will explain all this, derive necessary and sufficient conditions under which
the two arc length functions DO agree, and illustrate with lots of examples. (This is joint work with a group of independent study students.)

Mon 
Dec 3 
11:30  12:30 
TBA 
NO SEMINAR
