The differential geometric analysis of general manifolds began as
the study of curves and surfaces using the methods of calculus. In
time, the notions of curve and surface were generalized to the
notion of a manifold along with other accompanying new notions such
as tensors, forms, connections, Lie-derivatives etc. The motivation
and momentum for this development is mainly found in the need for a
precise geometric framework in which we can effectively analyse the
local and global shapes of - as well as the short time and long
time processes in - higher dimensional configuration spaces.
This course will cover a bouquet of results, examples, and modern
applications concerning this generalized global geometric analysis,
with a focus on general surfaces.
The course aims to give the students a firm background for
acquiring and applying new literature and new results from the
fields of manifold analysis and advanced geometry and
topology.
Læringsmål:
En studerende, der fuldt ud har opfyldt kursets mål, vil kunne:
Recognize a topological paracompact Hausdorff space
Construct atlases for simple but nontrivial manifolds
Find the rank of a given differentiable map and understand its
significance
Prove if a given subset is a submanifold
Explain the definition of a Riemannian manifold
Understand the tangent bundle and its dual
Apply Stoke's theorem to some examples
Compute homotopy and homology groups for simple examples
Be able to work with holomorphic and harmonic maps between
Riemann surfaces
Understand and be able to apply the uniformiztion theorem for
Riemann surfaces
Understand the Ricci flow and its application to
uniformization
Master the basic theory well enough to carry out a self study
of a theoretical extension or a technical
application
Kursusindhold:
Fundamental concepts and results which will be covered in the
course include: Smooth manifolds; Tangent bundle; Integration on
manifolds; Homotopy, and homology groups; Riemann surfaces;
Conformal maps; Riemann mapping theorem; Harmonic maps; Surface
Ricci flow; Circle packings and discrete Ricci flow.
Litteraturhenvisninger:
Riemannian Geometry, af Manfredo do Carmo.
Computational Conformal Geometry, af David Gu og S-T Yau.
Compact Riemann Surfaces, af Jürgen Jost.
Bemærkninger:
The course will be based around Part I of Gu and Yau's book
"Computational conformal geometry". There will be the
opportunity to implement one of the alogirthms in Part II of the
same book as a short project, if desired, however the focus is on
the mathematical theory.
Mulighed for GRØN DYST deltagelse:
Kontakt underviseren for information om hvorvidt dette kursus giver
den studerende mulighed for at lave eller forberede et projekt som
kan deltage i DTUs studenterkonference om bæredygtighed,
klimateknologi og miljø (GRØN DYST). Se mere på http://www.groendyst.dtu.dk
