Differential Geometry Seminar  2015 

Thu  Feb.19  4:005:00  Matthias Hammerl (Greifswald)  Holonomy Reductions of Parabolic Geometries and Curved Orbit Decompositions
An approach via holonomy groups is often a fruitful way to understand geometric structures, and there has thus been a long established theoretical pursuit to explore the geometric implications
of reduced holonomy and to understand the possible holonomy groups for a given geometric structure. While in particular the holonomy groups of affine connections and (pseudo)Riemannian metrics
have been intensively studied, the appropriate notion of an holonomy reduction for general Cartan geometries has long remained elusive. For in this case, which includes the class of parabolic
geometries and in particular projective and conformal structures, it is no longer geometrically evident how to interpret an holonomy group in terms of underlying geometric data. In this talk I
am going to discuss a general holonomy reduction method for Cartan geometries developed in joint work with A. Cap (Univ. of Vienna) and A. R. Gover (Univ. of Auckland). The main result is the
curved orbit decomposition theorem: It is shown that an holonomy reduction of a Cartan geometry gives rise to a natural decomposition of the underlying manifold into initial submanifolds, each
of which carries an induced geometric structure and corresponds to a grouporbit on an homogeneous model. In particular, this provides an algebraic/geometric explanation of the singularity sets
that are typically observed for parabolic holonomy reductions. The results are applied to study solutions of geometric overdetermined PDEs on parabolic geometries.

Wed  Mar. 4  10:4511:45  Ilya Kossovskiy (University of Vienna)  On Poincare's "Probleme local"
In 1907 Poincare formulated his "Problem local": for given germs of
realanalytic hypersurfaces in complex twospace, find all local
biholomorphic maps between them. This problem can be interpreted in the
framework of Cartan (equivalence of $G$structures).
In addition, it has important application to Several complex Variables,
since the study of mappings between domain in complex space can be
reduced to that of local maps between their boundaries. Poincare's
question naturaly splits to the equivalence problem for two given
germs, and to the problem of describing local automorphisms of
realanalytic hypersurfaces. Poincare did a substantial progress in
solving both problems, by showing first that two germs in general
position are inequivalent, and second by showing that the dimension of
the symmetry group of a germ in the Levinondegenerate case does not
exceed 8. More detailed results in the Levinondegenerate case were
obtained in further work of Cartan, Tanaka, Chern and Moser, and
Beloshapka. However, for hypersurfaces with Levi degeneracies the
question on possible automorphism groups remained unsolved. In the
finite type case (i.e., when a real hypersurface does not contain
germs of complex hypersurfaces) the problem was solved independently
by Beloshapka, Ejov and Kolar. They showed that the dimension of the
group does not exceed 4. However, their method (e.g., the method of
"polynomial models") is not applicable to the infinite type case,
i.e., when a real hypersurfaces contains a complex germ. It was a
longstanding problem to obtain the description in the infinite type
case.
In our work with Shafikov, we developed a method of solving this
problem by using connections between real hypersurfaces and second
order complex differential equations. The infinite type case
corresponds in this way to ODEs with an isolated singularity. By
studying symmetries of an appropriate class of second order singular
ODEs, we were able to classify all hypersurfaces with groups of
dimension 4 and higher. It turns out that there is a gap for the
dimensions which looks as $dim=\infty,8,5,4,3,2,1,0$ (this gap was
conjectured by Beloshapka and known as the Dimension Conjecture).

Thu  Mar. 5  4:005:00  Ian Anderson (Utah State)  Googling Einstein and wikipeding Cartan
Recent software advances in computer algebra systems and in laptop and desktop computing power have now given mathematicians effective tools for research in computational intensive fields such
as differential geometry and its applications. In this talk, I will discuss new ways in which we can use computer algebra systems, ways which go wellbeyond the use of such systems for long,
complex computation.
First, I want to show how Maple can be used to create dynamic, interactive databases of mathematical or mathematical physics knowledge and how this knowledge can be made accessible to a broader
audience. As a case study, we will look at the subject of exact solutions to the Einstein equations of general relativity. I will describe our efforts: [1] to create a comprehensive database of
known solutions; [2] to verify the correctness of these solutions; [3] to calculate an extensive set of properties of these solutions; [4] and to develop an easytouse search engine to access
this database.
Second, I want to give a brief demonstration of how Maple can be used to create rich interactive documents for teaching advanced mathematics. Here we will look at the structure theory of simple
Lie algebras. This classification was begun by W. Killing, completed by E. Cartan in his PhD thesis, and cast into its current form by E. Dynkin. This material is covered in many text books and
is readily available on the web. I'll show how one can use Maple to present this same material in a dynamical new way which makes the material much easier (and more fun!) to learn.

Tue  Mar.10  12:1013:45 (in COP G027)  Ian Anderson (Utah State)  Introduction to DifferentialGeometry with Maple
In this workshop we will:
[i] review a a few basics of Maple;
[ii] learn to create vector fields, differential forms and tensors;
[iii] learn about the basic differential operators of Lie bracket, exterior derivative, Lie derivative and covariant derivatives;
[iv] calculate the curvature tensor;
[v] solve the Einstein equations for some simple metrics.
If time permits, we will learn about Killing vectors and discuss ways to analyze the structure of Lie algebras.
A second workshop will be given if there is sufficient interest.

Wed  Mar.11  10:4511:45  Ian Anderson (Utah State)  An Introduction to the "8 thru 11" Variables Paper of E. Cartan (1911)
It is a remarkable fact that Cartan's 1910 paper on the geometry of rank 2 distributions in 5 dimensions is still the basis for much current research in the field of geometric methods for
differential equations. This paper, often referred to as the "five variables paper", is widely cited for its amazing solution to the equivalence problem for rank 2 distributions in 5 dimensions.
Unfortunately, Cartan's original goal of integrating 2nd order partial differential equations in 1 dependent and 2 dependent variables, has been largely forgotten. I'll begin this talk with a
review of the geometric theory of these PDE.
Cartan's 1911 paper deals with systems of 2nd order partial differential equations in 1 dependent and 3 independent variables. In many ways, this paper is even more astonishing than the 1910
paper and surely contains a wealth of interesting and largely untouched research topics. I'll give a brief synposis ofthis paper and describe some of the recent related work of K. Yamaguchi and
N. Sitton.
The seminar will also contain some demonstrations of the DifferentialGeometry software package.

Thu  Mar.12  11:4512:45  Thomas Leistner (University of Adelaide)  Explicit ambient metrics and holonomy
I will report on recent work with I. Anderson and P. Nurowski in which we present three classes of conformal structures for which the equations for the FeffermanGraham ambient metric to be
Ricciflat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal ppwaves
and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit FeffermanGraham ambient metrics
provide a class of metrics with holonomy equal to the exceptional noncompact Lie group G_2 as well as ambient metrics with holonomy contained in Spin(4,3).

Tue  Mar.31  11:3012:30  Katja Sagerschnig (University of Vienna)  Almost Einstein (2,3,5)distributions
I will discuss some aspects of conformal structures that are determined by
(2,3,5) distributions and admit almost Einstein scales. This is joint work
in progress with Travis Willse.

Tue  Apr.7  11:0012:00  Jeanne Clelland (UC Boulder)  Isometric embedding via strongly symmetric positive systems
(Joint work with GuiQiang Chen, Marshall Slemrod, Dehua Wang, and Deane Yang)
In this talk, I will give an outline of our new proof for the local existence of a smooth isometric embedding of a smooth
3dimensional Riemannian manifold with nonzero Riemannian curvature tensor into 6dimensional Euclidean space. Our proof avoids the
sophisticated microlocal analysis used in earlier proofs by BryantGriffithsYang and NakamuraMaeda; instead, it is based on a new
local existence theorem for a class of nonlinear, firstorder PDE systems that we call "strongly symmetric positive." These are a
subclass of the symmetric positive systems, which were introduced by Friedrichs in order to study certain PDE systems that do not
fall under one of the standard types (elliptic, hyperbolic, and parabolic).
As in earlier proofs, we construct solutions via the NashMoser implicit function theorem, which requires showing that the
linearization of the isometric embedding PDE system near an approximate embedding has a smooth solution that satisfies "smooth tame
estimates." We accomplish this in two steps:
(1) Show that the approximate embedding can be chosen so that the reduced linearized system becomes strongly symmetric positive
after a carefully chosen change of variables.
(2) Show that any such system has local solutions that satisfy smooth tame estimates.
The main advantage of our approach is that step (2) is much more straightforward than similar results for other classes of PDE
systems used in prior proofs, while step (1) requires only linear algebra.
The talk will focus on the main ideas of the proof; technical details will be kept to a minimum.

Thu  Apr.9  14:3015:30  Sean Curry (University of Auckland)  Constructing
Invariants of CR Embeddings Using Tractors
In 1907 Poincare showed that two real hypersurfaces in C^2 are
generically not locally biholomorphically equivalent and hence
inequivalent as CauchyRiemann (or CR) geometries. Applying this
to the boundaries of domains in C^2 showed that the Riemann Open
Mapping Theorem breaks down in higher dimensions. In the early
1930s Elie Cartan resolved the equivalence problem for CR
hypersurfaces by constructing their basic geometric invariants.
Cartan's methods have been extended in the modern CR tractor
calculus, in an effort to better understand CR invariants (and to
find them all). When considering holomorphic mappings between
domains in complex spaces of different dimension the notions of CR
mappings and CR embeddings arise. In this talk we show that the tractor
calculus also provides us with a rich invariant calculus in the setting
of CR embeddings.

Interested in giving talk? Please contact Dennis The (Dennis.The AT anu.edu.au) or Travis Willse (Travis.Willse AT anu.edu.au)
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