New Directions in Exterior Differential Systems
A conference in honor of Robert Bryant’s 60th birthday
July 15–19, 2013
Titles and Abstracts
Workshop Mini-Courses
Andreas Cap, Michael Eastwood & J.M. Landsberg. Invariant differential
operators and exterior differential systems.
Tuesday, 3:30 - 4:30; Wednesday, 4:00 - 5:00; Thursday, 11:30 - 12:30.
This course will present new developments in parabolic geometries, generalized BernsteinGelfand-Gelfand sequences and representation theory, and illuminate their relationship with EDS and Cartan geometry. The emphasis will be on explicit examples (for
example, the prolongation of the Killing and conformal Killing operators) and open
questions.
Mark Green & Phillip Griffiths. Linear PDE systems and discrete series representations, including their limits.
Monday & Tuesday, 11:30 - 12:30 & 2:00 - 3:00.
As an outgrowth of recent work relating Hodge theory and representation theory
through complex geometry, certain linear PDE systems have appeared whose solutions include the Harish-Chandra modules associated to the representations in the
title. Their construction depends on the complex geometry of flag domains and the
realization of the Harish-Chandra modules as cohomology groups associated to homogeneous line bundles. For very regular Harish-Chandra parameters, i.e. those far from
the walls of their Weyl chamber, these PDE systems turn out to be closely related
to classical Spencer resolutions of involutive over determined differential operators.
For singular Harish-Chandra parameters, a seemingly new type of linear PDE system
emerges, one whose structure has not been worked out. These lectures will give an
overview of this theory, and we hope they will provide a complement to those of CapEastwood-Landsberg. The commutative algebra background of tableaux of exterior
differential systems, characteristic modules and varieties, and generic initial ideals will
be covered in the interests of making the lectures more complete and user-friendly.
The outline of the talks is:
(1) Complex geometry and representation theory–introduction of PDE system.
(2) EDS theory, tableaux, characteristic modules and characteristic varieties, Spencer
cohomology, Koszul cohomology.
(3) Relation between discrete series with very regular infinitesimal character and
Spencer sequences, geometry of the characteristic varieties, general case including limits of discrete series, examples.
(4) Initial ideals, generic initial ideals, interpretation of the Cartan coefficients si
and involutivity in terms of generic initial ideals, scheme structure of characteristic varieties.
Chuu-Lian Terng. Moving frames, geometric curve flows, and integrable systems.
Wednesday, 2:30 - 3:30; Thursday, 9:00 - 10:00; Friday, 11:30 - 12:30.
In this mini-course, we plan to use some concrete and simple examples to give an
elementary introduction on the following topics:
(i) Construct soliton equations from simple Lie algebras and finite order automorphisms.
(ii) Natural moving frames for curves on homogenous spaces.
(iii) Geometric curve flows on homogeneous spaces whose geometric invariants flows
are soliton.
Conference Talks
Jeanne Clelland. The geometry of lightlike surfaces in Minkowski space (joint with
Brian Carlsen). Thursday, 3:15 - 4:00.
We investigate the geometric properties of lightlike surfaces in the Minkowski space
R2,1 , using Cartan’s method of moving frames to compute a complete set of local
invariants for such surfaces. Using these invariants, we give a complete local classification of lightlike surfaces of constant type in R2,1 and construct new examples of
such surfaces. (This work is based on Brian Carlsen’s undergraduate honors thesis.)
Michael Eastwood. Conformal geometry in four variables and a deformation complex in five. Tuesday, 9:00 - 10:00.
Starting with a neutral signature 4-dimensional conformal structure, one can build a
5-dimensional bundle over it equipped with a 2-plane distribution. Generically, this
is a (2,3,5)-geometry in the sense of Cartan’s five variables paper, an aspect that was
recently pursued by Daniel An and Pawel Nurowski (finding new examples concerning
the geometry of rolling bodies where the (2,3,5)-geometry has G2-symmetry). I shall
explain how to understand some elementary aspects of this “twistor construction”
using “BGG-machinery.” This is joint work with Katja Sagerschnig and Dennis The.
C. Robin Graham. Conformal BGG Sequences via Poincaré Metrics. Friday, 9:00
- 10:00.
A new derivation of BGG sequences on conformal manifolds will be outlined. The
BGG operators arise as obstructions to solvability of differential equations on a
Poincaré space having the given conformal manifold as conformal infinity. This is
joint work in progress with Olivier Biquard.
Marianty Ionel. Austere Submanifolds in Complex Projective Space. Thursday,
10:15 - 11:00.
A submanifold in Rn is austere if all odd-degree symmetric polynomials in the eigenvalues of the second fundamental form, in any normal direction, vanish. Harvey and
Lawson showed that this condition is necessary and sufficient for the normal bundle
of Rn to be special Lagrangian in the tangent bundle of Rn . In joint work with Tom
Ivey, we investigate the conditions under which the normal bundle of a submanifold
in CPn is special Lagrangian with respect to the Stenzel metric on the tangent bundle
of the complex projective space, including some examples and classification results.
Thomas Ivey. Austere Submanifolds in Euclidean Space. Tuesday, 10:15 - 11:00.
A submanifold in Euclidean space is austere if for any normal vector ν all the odddegree symmetric polynomials in the eigenvalues of the second fundamental form
in direction ν vanish. In their foundational calibrations paper of 1982, Harvey and
Lawson showed that the conormal bundle of M ⊂ Rn is special Lagrangian in T ∗ Rn ∼
=
Cn if and only if M is austere. The next significant work in this area was Robert
Bryant’s 1991 paper “Some remarks on the geometry of austere manifolds”, in which
austere 3-folds are classified, and all pointwise models for the second fundamental
forms of austere 4-folds are also determined.
In this talk I’ll discuss joint work with Marianty Ionel (Universidade Federal do
Rio de Janeiro) in classifying austere 4-folds. While this problem is still unsolved, we
have made progress in two areas: the classification of austere 4-folds that are maximal
(i.e., where the second fundamental form spans one of the maximal austere subspaces
on Bryant’s list) and classifying those that are ruled by 2-planes. In particular, in
the first case we find that not all maximal subspaces can be realized (even locally) by
austere submanifolds. In the second case we find new examples of families of ruled
real Kähler 4-folds associated to holomorphic maps into certain k-symmetric spaces.
Blaine Lawson. A geometric perspective on nonlinear partial differential equations.
Monday, 9:00 - 10:00.
I will discuss a geometric approach to the study of fully nonlinear pde’s on manifolds.
Basic concepts of affine jet equivalence and monotonicity will be introduced. The discussion will include existence and uniqueness for the Dirichlet problem, a geometric
characterization of the maximum principle, and removable singularity results. Applications to calibrated geometry and to potential theory on almost complex manifolds
will also be discussed.
Joseph M. Landsberg. Two things I learned from Robert and how they have helped
me. Monday, 4:30 - 5:15.
I will discuss the arts of exploiting (i) symmetry and (ii) the sacred equation d2 = 0.
These arts have been useful in most of my work. To the extent time permits, I will give
an overview of applications of these arts to algebraic geometry (secant varieties and
rigidity), representation theory (geometric proof of the Cartan-Killing classification
of complex simple Lie algebras), and theoretical computer science (the complexity of
matrix multiplication).
Benjamin McKay. Holomorphic exterior differential systems. Friday, 10:15 - 11:00.
A landscape painting of various types of geometric structures and exterior differential
systems on smooth complex projective varieties.
Christopher Moseley. Finsler and Sub-Finsler Geometries. Monday, 3:30 - 4:15.
Finsler geometry, in the words of Shiing-Shen Chern, is “Riemannian geometry without the quadratic restriction.” This talk will review some of the basic notions and
important results in Finsler geometry, with special attention to Robert Bryant’s contributions. I will then present some results in sub-Finsler geometry from joint work
with Jeanne Clelland and George Wilkens.
Colleen Robles. Homological rigidity of Schubert classes. Thursday, 4:15 - 5:00.
Robert Bryant (and Maria Walters, independently) showed that the varieties homologous to a multiple of a Schubert class in a compact Hermitian symmetric space are
characterized by an exterior differential system. Since then the rigidity and flexibility
of this system has been studied by several people. I will present a survey of this work
with an emphasis on two of the dominant themes of the conference: the application
of EDS and representation theory to geometric problems with symmetry.
Abraham D. Smith. Hydrodynamic Integrability from the EDS perspective. Thursday, 2:00 - 2:45.
Integrability is a somewhat vague term that different people use in different ways for
different types of PDE. In more than 2 independent variables, hydrodynamic integrability is a particularly geometric notion of integrability arising from semi-Hamiltonian
systems of conservation laws. In this talk, I’ll explain how this notion of integrability
can be defined, studied, and generalized in an invariant manner using EDS and the
characteristic variety. If time permits, I’ll discuss the implementation of EDS tools
in the Sage mathematics software system, and also some hypotheses about the application of these integrable EDS to Bryant’s generalization of Cartan’s realization
theorem for Lie pseudogroups.
Sung Ho Wang. Conservation laws for CMC surfaces in three dimensional space
forms. Monday, 10:15 - 11:00.
The second order elliptic Monge-Ampere equation for CMC surfaces in a three dimensional space form is one of the prototypes of an integrable equation. It possesses many
characteristic properties which are widely used in the theory of integrable PDE’s and
differential geometry. We determine and give a recursive formulae for the infinite
sequence of conservation laws making use in particular of an enhanced prolongation
and a nonlocal symmetry called spectral symmetry.
We also introduce the corresponding secondary spectral conservation law, and
indicate the possible application of these structures to the theory of high genus CMC
surfaces.
Deane Yang. Affine integral geometry from a differentiable viewpoint. Friday, 3:30
- 4:15.
We will describe how to define both known and new affine geometric invariants of a
convex body using the homogeneous contour integral, which is an n-dimensional real
analogue of the contour integral for a meromorphic function of one complex variable.