See our celebration of 70th birthday of these unique individuals:
Just to get an idea on the scale of his mathematical inheritence see: Lars Ahlfors Centennial Celebration
Further links to this world can be found through web pages:
In the hydrodynamical setting a corresponding role is carried by the so called rotational Beltrami fields. Beltrami fields are exactly those vector fields that are parallel to their own curl. All such fields are steady solutions to Euler's equations of motion for perfect incompressible fluid. Flows generated by such fields have several noteworthy properties, such as extremization of the energy functional as well as the potential to display the phenomenon sometimes called Lagrangian turbulence, in which a volume-preserving flow has flowlines which fill up regions of space ergodically.
The classic examples of Beltrami fields which exhibit Lagrangian turbulence are the ABC flows generated by a certain family of vector fields on the 3-torus which are eigenfields of the curl operator. Beltrami fields in general are poorly understood. Typically one wants to restrict to spaces like 3-space or 3-torus under the fixed standard metric and volume form. Under these restrictions the analysis becomes very hard since a small perturbation, even within the class of volume-preserving fields almost always destroys the Beltrami property. The difficulty lie especially on global questions about Beltrami flows, such as the existence of closed orbits, the presence of hydrodynamic instability and the minimization of the energy functional. The contact setting provides methods to overcome these difficulties. This point of view has successfully applied in plasma physics and further in molecular biology by H. Gluck, D.DeTurck and M. Teytel (Pennsylvania). In plasma physics Beltrami fields are known as force-free magnetnic fields. They exhibit the familiar 'twistedness' of contact structures present on writhing of DNA.
Beltrami fields arise naturally also in the electromagnetism. Any electric field E can be decomposed into left-handed E-, right-handed E+ and static E_0 parts. The volume forms related to this decomposition catch essentially the same information as the corresponding volume form in the contact setting. Integrating over the underlying manifold gives a quantity called helicity. Natural variational problems can be formulated through this functional. For example one can state an isoperimetric problem to minimize energy among all divergence-free vector fields of given helicity, defined on and tangent to the boundary of all domains of given volume in 3-space.
References:
Further information on finsler geometry can be found throuhg web pages:
Some ideas to this direction can be found from
Shun-Ichi Amari
See also a seminal book in the field:
There are many fashinating connections of information geometry and statistics in neural computing and related methods that could be further enhanced:
Riemann-Finsler geometry
Finsler structures on manifolds provide natural framework to many physical applications.
There one typically has a more general norm (instead of an inner product) or just some
distinguished curves (geodesics)are given. Armed with these structures the finslerian
philosophy gives tools to look the tangent bundle roughly as a riemannian manifold.
This fits nicely to physical setting where one has generic structures on the cotangent bundle.
The duality between tangent and cotangent spaces (legendrian transformation) transmits
the correct metric to the system. The unit tangent bundle and tangent bundle are then the
correct contact and symplectic spaces that are related to the original problem.
Boundary rigidity
Heisenberg Group
The first heisenberg group equipped with a natural sub-riemannian metric is topologically
just the euclidean three space, but it is metrically four dimensional and has a two
dimensional plane distribution that supports planar geometric structures. All this
together gives rise to fractal nature, collapsing phenomena and a rich source of
counterexamples in analysis, geometry and applications. For an overview and some recent achievements
see monograph:
Quasiconformal and Quasiregular mappings
Analytic mappings acting on complex plane have a natural generalization
called quasiregular maps (QR) to higher dimensional euclidean space.
These non-smooth mappings distort the given metric by a bounded amount.
The theory of such mappings is deeply studied since 60's [R1].
The local definition of this mapping class makes sense also for maps between
arbitrary riemannian manifolds M and N of same dimension.
A general classification problem on the existence of nontrivial QR maps
between given riemannian manifolds is highly open. A beautiful
interplay between analysis, geometry and cohomology of the manifold N was
found by M. Bonk and J. Heinonen in the case M is the euclidean n-space (QR-ellipticity).
UQR dynamics
In the UQR setting we study noninjective maps f acting on a manifold M whose
all iterates satisfy distortion condition for fixed K
independently of the number of iterates. Such maps distort the given
metric uniformly by a bounded amount. They are always conformal
with respect to some measurable conformal structure G
that can be constructed from the semigroup of the iterates.
The space of n times n positive definite real matrices G of determinant 1
can be equipped with a natural hyperbolic metric whose distance function
relates the invariant conformal structure G to the distortion constant K of the semigroup elements.
Conversely, for a given
conformal structure G the weak solutions f of a generalized
Beltrami equation
are UQR maps whose iterates satisfy the same equation. Monograph
deals with analysis to this direction.
Bilipschitz mappings
Hyperbolic geometry and Kleinian groups
Meeting point of several different parts of mathematics including Riemann surfaces, complex dynamics and fractals.
Some links:
Ricci flow
Notes and comments on the work related to geometrization
see:
Seifert fibered spaces
Seifert fibered spaces are a nice category of three manifolds that
have a fiber structure that gives in a sense global coordinates.
If you look for nontrivial but concrete examples of manifolds
this class might be worth looking just next after surfaces.
Knot theory
Visually appealing and in the heart of many geometric problems.
Some links:
Information geometry
Information geometry deals with applying differential geometry to families of probability distributions, and to statistical models. So called Kullback-Leibler information, or relative entropy, features as a measure of divergence (not quite a metric, because it's asymmetric), and Fisher information takes the role of curvature. Information geometry gives very strong tools for proving results about statistical models, by considering them as geometrical objects like manifolds and standard structures in differential geometry related to them. To the converse direction statistical models
give new ideas and viewpoints also to the most classical geometry, for instance the dualism of connections on a given manifold and the self dualism of the riemannian connnection.
Teaching in differential geometry
Any of the courses below and much more can be studied independently.
Please contact Kirsi Peltonen for further information.
Last update 1.3.2009