Geometry II: Discrete Differential Geometry (Summer 2018)

This is a basic course of the Berlin Mathematical School and lectures will be held in English. The tutorials will be held in English or German, depending on the students attending each one.

News

[28.9.]
          Oral exams will be offered 9-12h on Thu. 8 Nov.

[23.7.]
         One source for the Delaunay triangulation algorithms is these lecture notes from ETH Zürich; see Chapters 6 and 7.

[2.7.]
          There will be no lecture on Wednesday, 4.7.
          Isabella's office hours this week are by appointment only (not on Thursday as usual), please write an email.

          Here are the references mentioned in today's tutorial:

          Medkova, "The Laplace Equation" (Smooth theory of Dirchlet boundary value problems, Hölder continuity for domains explained in 1.17),

          Pinkall, Polthier, "Computing discrete minimal surfaces and their conjugates" (Reference [cotan-minimal] for the simplicial minimal surface algoithm in the script)
          and
          Wardetzky et al., "Discrete Laplace operators: No free lunch"

          We will talk about aspects of the second reference in next week's tutorial!

          Isabella is taking questions for the last tutorial on July 16th. Please send an email with topics you would like to have covered or questions about any of the topics from this semester.

[26.6.]
          Here are some further references mentioned in the tutorial (not relevant for the exam):
            Izmestiev et al., "There is no triangulation of the torus with vertex degrees 5,6,... "
            and
            Günther et al., "Smooth polyhedral surfaces" (proof spherical area of Gauss map equals discrete Gaussian curvature)

[7.6.]
          You can find Conway's Zip Proof here.

[29.5.]
          You are allowed to hand in the following exercises after the due dates:
          Ex. 5.1 -> June 4th
          Ex. 6.2 -> June 11th
          We will cover the topic "elastic rods" in the tutorial on Monday.

[23.5.]
          Isabella's office hours tomorrow will be from 10:15-12.

[15.5.]

[11.4.]

There is no lecture on Wed 18 Apr; the first lecture will be on Thursday 19 April.

There are no tutorials on Mon 16 Apr; tutorials and office hours will start in the second week.

Contents

Classical differential geometry studies smooth curves and surfaces, for instance in terms of their curvatures. This course will consider analogous notions and results for discrete (usually polygonal) curves and discrete (usually polyhedral) surfaces. Specific topics will include:

• Curvatures of discrete curves, discrete elastic curves
• Curvatures of discrete surfaces, special parametrizations and classes of surfaces
• Discrete conformal and harmonic maps, the discrete Laplace operator, Delaunay triangulations, circle packings, discrete conformal equivalence, discrete Riemann surfaces, variational principles

References

• Alexander I. Bobenko, Lecture notes from the Summer 2014 version of this course.
• John M. Sullivan, Curves of Finite Total Curvature, in Discrete Differential Geometry, Birkhäuser, 2008, pp. 137–161.
• John M. Sullivan, Curvatures of Smooth and Discrete Surfaces, in Discrete Differential Geometry, Birkhäuser, 2008, pp. 175–188.

Exercise sheets

• Sheet 1, due 30 April
• Sheet 2, due 7 May
• Sheet 3, due 14 May
• Sheet 4, due 23 May
• Sheet 5, due 28 May
• Sheet 6, due 4 June
• Sheet 7, due 11 June
• Sheet 8, due 18 June
• Sheet 9, due 25 June
• Sheet 10, due 2 July
• Sheet 11, due 9 July

Homework policy and exam information

• To get a certificate for the tutorial and thus qualify to take the oral exam, you need to satisfactorily complete 60% of the homework assignments.
• The exercises are to be done in groups of two people and handed in at one of the tutorials each Monday.
• Oral exams will be offered 10-12h on 25 July and 27 July. It will also be possible to take the exam in late September or in early November.

Office Hours