Fall Semester 2025

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

I am head TA for the class of Lineare Algebra I D-MAVT/D-MATL, taught by Prof. Dr. Norbert Hungerbühler.

Student seminar on the Schönflies problem

I am co-organizing a reading group/student seminar on the Schönflies problem. It is possible that the plan will suffer some change but the next lines should serve as a good basis.

• Week 0: Introduction and overall work repartition.
• Week 1: The Jordan-Schönflies theorem: Part 1. Polygonal curve and a weak version of the JCT. From the beginning of Section 2 to Proposition 2.6 in [6].
• Week 2: 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: The Jordan-Schönflies theorem: Part 3. The Jordan-Schönflies theorem. Section 3 in [6].
• Week 4: The Jordan Schönflies theorem: Part 4. Triangulation of surfaces and ideas toward the classification of surfaces. Sections 4 and 5 in [6].
• Week 5: Introduction to the fundamental group. Definition, introduction to the presention of group, Seifert-Van Kampen.
• Week 6: Alexander horned sphere. Example 2B.2 in [4], see [1] for the original paper.
• Week 7: Brouwer's fixed point theorem and Brouwer's invariance of domain.Following the proof of Kupka that can be found in [5].
• Week 8: The generalized Schönflies theorem: Part 1. On locally flat embeddings. Present the results and the proofs in [2].
• Week 9: The generalized Schönflies theorem: Part 2. The Schönflies theorem for collared spheres. Present the results and the proofs in [3].
• Week 10: A bit of knot theory. Definition, locally flat knots are PL and wild knots.

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] M. Brown, "Locally flat imbeddings of topological manifolds", Annals of Math., 75 (2), 331–341, 1962.
• [3] M. Brown, "A proof of the generalized Schoenflies theorem", Bulletin of the AMS, 66 (2): 74–76, 1960.
• [4] A. Hatcher, "Algebraic topology", https://pi.math.cornell.edu/~hatcher/AT/AT.pdf.
• [5] T. Tao, "Brouwer’s fixed point and invariance of domain theorems, and Hilbert’s fifth problem", Blog.
• [6] C. Thomassen, "The Jordan-Schoenflies Theorem and the Classification of Surfaces", American Mathematical Monthly, 99 (2), 116–130, 1992.

Spring Semester 2025

Topologie Frühlings 2025

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