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Differential geometry and applications

People

Pekka Alestalo Matias Dahl Anton Isopoussu
Riikka Kangaslampi Tony Liimatainen Kirsi Peltonen
Juha-Matti Perkkiö Tri Quach Antti Rasila

Brief description

One goal is to develop differential geometric methods towards applications and physically motivated questions in areas like electromagnetism and inverse problems. Research to this direction see:

  • M. Dahl, Contact geometry in electromagnetism, Progress In Electromagnetics Research, Vol. 46, pp. 77-104, 2004.
  • M. Dahl, Electromagnetic fields from contact forms,Preprint

Beyond smooth mappings, the categories of mappings that distort the metric by a bounded amount are under research. One goal is to study approximation and extension of bilipschitz and quasisymmetric mappings. See:

  • P. Alestalo, D.A.Trotsenko: Plane sets allowing bilipschitz extensions. Math. Scand. 2008. 

Related to various geometries, we study dynamics and classification of a surprising class of mappings generalizing planar rational functions into higher dimensional(sub)riemannian compact manifolds. See:

  • Kangaslampi, R: Uniformly quasiregular mappings on elliptic riemannian manifolds Ann. Acad. Sci. Fenn. Math. Diss. 151 (2008)
  • Martin, G., Mayer, V., Peltonen, K.: The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms, Proc. Amer. Math. Soc. Vol 134, No 7, 2006.
  • Martin, G., Peltonen, K.: Stoilow factorization for quasiregular mappings in all dimensions. PAMS 138(1), 2010.  

See recent publications by our group.

Collaborators and related people

  • Laura Astola did her Phil. Lic. Thesis on UQR mappings in the group. In her PhD Thesis she applied Finsler geometry to imaging problems.
  • Zoltan Balogh is working with the group on problems related to the Heisenberg group.
  • Yasha Eliashberg has been the greatest guide to the wonderlands of contact and symplectic geometry. Kindness, encouragement, wisdom and good sense of humor- what else do you need?
  • Richard Kenyon Tiling problems are in the heart of many constructions in UQR dynamics.
  • Matti Lassas has shown in practise that geometry does matter. Recent highlights see:
    • Greenleaf, A., Kurylev, Y., Lassas M. and Uhlmann G: Invisibility and Inverse Problems, BAMS, Vol 46, nr 1, 2009, 55-97.
  • Gaven Martin Some people find their collaborators as far away as possible. Great preprints on exciting examples of UQR maps are to be finished in the very near future. We insist fast but ecological connections between Finland and New Zealand!
  • Ari Sihvola is the most important connection between electromagnetism and geometry. We are grateful for all support and encouragement in various levels starting from basic teaching up to the latest research.
  • Erkki Somersalo is a tireless source of support towards geometrization of physics and dear connection to inverse problems. On the most geometrical aspects see:
    • Kurylev, Y., Lassas, M., Somersalo E.: Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems, Journal de Mathematique Pures et Appliquees 86 (2006), no. 3, 237-270.
    • Ola P., Päivärinta L, Somersalo: Inverse problems in time-harmonic electrodynamics. Inside Out: Inverse Problems and Applications. Mathematical Sciences Research Institute Publications 47. Cambridge University Press 2003.
  • Alina Vdovina works on geometry and analysis on groups, buildings, knot theory, noncommutative geometry and geometry on surfaces.

 

Mathematical fathers and constant sources of inspiration and support:

See our celebration of 70th birthday of these unique individuals:

 

Those who have past away but left so strong fingerprints that you meet them every day:

  • Juha Heinonen was the one who insisted us to organize the celebration for Lars Ahlfors and finally, when everything was done, he was not able to join us. See Juha and think what this world has lost when he was forced to leave us so soon.

 

Topics of interest

We hope the list below is useful especially for students looking for interesting mathematical problems on any level. Please do not hesitate to contact any of the group members and you will be kindly guided to a fascinating world.

Contact and symplectic geometry

Contact topology was born over two centuries ago in the work of Huygens, Hamilton and Jacobi on geometric optics. Later it has been studied by many great mathematicians, such as Sophus Lie, Elie Cartan and Darboux. Only recently it has moved into the foreground of mathematics. The last decade has witnessed many remarkable breakthroughs in contact topology, resulting in beautiful theory with many potential applications. More specifically-though sketchy-picture of contact topology has been developed, a surprisingly subtle relationship arose between contact structures and low dimensional topology. In addition, the applications of contact topology have extended far beyond geometric optics to include non-holonomic dynamics, thermodynamics and more recently Hamiltonian dynamics and hydrodynamics.

Further links to this world can be found through web pages:


Geometrization of physics

In the series of articles J. Etnyre and R. Ghrist have built a connection between contact topology and hydrodynamics. In the contact setting the so called Reeb vector fields have a central role. For a fixed contact form the Reeb vectorfield characterizes the transversal directions for the contact distribution.

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 magnetic 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 Eo 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:


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.

Further information on finsler geometry can be found through web pages:

 

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:

  • Luca Capogna, Donatella Danielli, Scott D. Pauls, Jeremy T. Tyson: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Springer, 2007

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). 

  • Bonk, Mario, Heinonen, Juha: Quasiregular mappings and cohomology, Acta Math. 186 (2001), 219-238.
  • Rickman, Seppo: Quasiregular mappings, Springer, 1993.
  • Astala, Kari and Iwaniec, Tadeusz and Martin Gaven: Elliptic Partial differential equations and quasiconformal mappings in the plane, Princeton, 2009

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. The following monograph deals with analysis in this direction:

  • Iwaniec, Tadeusz and Martin, Gaven: Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, 2001

 

 

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 connection.

Some ideas to this direction can be found from Shun-Ichi Amari.

See also a seminal book in the field:

  • Shun-Ichi Amari, Hiroshi Nagaoka: Methods of Information Geometry, Translations of Mathematical Monographs

There are many fascinating connections of information geometry and statistics in neural computing and related methods that could be further enhanced:


Teaching in differential geometry

Any of the courses below and much more can be studied independently. Please contact Kirsi Peltonen for further information.