Fall Semester 2025

Lineare Algebra I D-MAVT/D-MATL Herbst 2025

I am co-organizing a reading group/student seminar on the Schönflies problem. The seminar takes place in room HG D3.3 from 8 to 10.

Week 0, 26/09/2025: Introduction and overall work repartition.
Week 1, 03/10/2025: The Jordan-Schönflies theorem: Part 1. Polygonal curve and a weak version of the Jordan Curve Theorem (JCT). From the beginning of Section 2 to Proposition 2.6 in [6].
Week 2, 10/10/2025: The Jordan-Schönflies theorem: Part 2. End of the proof of the JCT. From Proposition 2.7 until the end of Section 2 in [6].
Week 3, 17/10/2025: The Jordan-Schönflies theorem: Part 3. The Jordan-Schönflies theorem. Section 3 in [6].
Week 4, 24/10/2025: The Jordan Schönflies theorem: Part 4. Triangulation of surfaces and ideas toward the classification of surfaces. Sections 4 and 5 in [6].
Week 5, 31/10/2025: Introduction to the fundamental group. Definition, introduction to the presention of group, Seifert-Van Kampen.
Week 6, 14/11/2025: Alexander horned sphere. Example 2B.2 in [4], see [1] for the original paper.
Week 7, 21/11/2025: Brouwer's fixed point theorem and Brouwer's invariance of domain.Following the proof of Kupka that can be found in [5].
Week 8, 28/11/2025: The generalized Schönflies theorem: Part 1. On locally flat embeddings. Present the results and the proofs in [2].
Week 9, 12/12/2025: The generalized Schönflies theorem: Part 2. The Schönflies theorem for collared spheres. Present the results and the proofs in [3].

References:

[1] J. W. Alexander, " An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected", PNAS, 10 (1), 8–10, 1924.
[2] S. Behrens, "The Disc Embedding Theorem", Oxford University Press, chapter 3, 2021.
[3] M. Brown, "Locally flat imbeddings of topological manifolds", Annals of Math., 75 (2), 331–341, 1962.
[4] M. Brown, "A proof of the generalized Schoenflies theorem", Bulletin of the AMS, 66 (2): 74–76, 1960.
[5] A. Hatcher, "Algebraic topology", https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.
[6] T. Tao, "Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem", Blog.
[7] C. Thomassen, "The Jordan-Schoenflies Theorem and the Classification of Surfaces", American Mathematical Monthly, 99 (2), 116–130, 1992.

I was head TA for the class of Topologie, taught by Dr. Lukas Lewark.

2024-2025, Ricardo Olivo, Morse Theory and the Lefschetz Theorem (Bachelor's thesis).