Geometry III (Winter 2021/2022)

Discrete differential geometry: Discrete confocal quadrics and related topics

We study (smooth and discrete) confocal quadrics and the corresponding confocal coordinate systems. Focussing on this basic example we discuss (smooth and discrete) orthogonal coordinate systems, special parametrizations and diagonally related nets on surfaces, and the classification of pencils of quadrics.

This is an advanced course of the Berlin Mathematical School (BMS) and will be held in English.

References

• Techter. Discrete confocal quadrics and checkerboard incircular nets. PhD thesis, 2020
• Bobenko, Schief, Suris, Techter. On a discretization of confocal quadrics. II. A geometric approach to general parametrizations International Mathematics Research Notices, 2018 (preprint on arxiv)
• Akopyan, Bobenko, Incircular nets and confocal conics, Trans. AMS 370:4, 2018 (preprint on arxiv)
• Akopyan, Bobenko, Schief, Techter, On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs, Discrete and Comp. Geometry, 2020 (preprint on arxiv)
• Casas-Alvero, Analytic Projective Geometry, EMS, 2010
• Bobenko, Suris. Discrete Differential Geometry: Integrable Structure, Graduate Studies in Mathematics, Vol. 98, AMS, 2008

Lecture Notes

• Geometry 3 - Confocal quadrics

Additional Notes

• October 27 - residues, circle points
• November 03 - confocal coordinates, focal conics
• November 10 - spheres
• November 17 - Jacobi elliptic functions
• November 24 - Lame coefficients, Euler-Poisson-Darboux system
• December 01 - curvature line parametrization, focal nets, parallel nets
• December 08 - 1st and 2nd fundamental forms of confocal coordinates
• January 05 - diagonally related nets, pencils of conics
• January 19 - spheres and circles, circular cross sections of ellipsoids
• January 26 - polynomial matrices, Smith normal form
• February 02 - minor divisors, invariant factors, elementary divisors
• February 09 - Segre symbols for polynomial matrices of degree 1

News

[2022, January 10]

No lecture on Tuesday, January 11.
Instead this weeks lecture will be on Wednesday, January 12.

[2021, October 25]

Zoom link for Tuesdays: https://tu-berlin.zoom.us/j/62532060765?pwd=UUtNdG8waHBLNUdQUW44YWtzWjFsdz09
Zoom link for Wednesdays: https://tu-berlin.zoom.us/j/64250003173?pwd=TzBsdlBVQ0hwOS9FcUQvb255THNydz09

[2021, October 19]

The first lecture will be on Tuesday, October 26.

[2021, October 18]

First organisational meeting on Tuesday, October 19, 12:15 on jitsi:
https://meet.innocampus.tu-berlin.de/Geometry3
Backup zoom link:
https://tu-berlin.zoom.us/j/68515161428?pwd=WG1Ga0VqQjdkV3FoSHpkSTEyMU4vQT09

[2021, September 23]

The course will probably be an online course.
We will decide on the exact format and the actual dates of the lectures at the first online meeting on Tuesday, October 19.
If you are interested in this course please write a mail.
In particular let me know if you can't make it to the first meeting.