19223811 Masterseminar Topologie "Algebraic K-Theory"
- FU-Students should register via Campus Management.
- Non-FU-students should register via MyCampus/Whiteboard.
Winter Term 2023/2024
Dozenten: Dr. Georg Lehner, Prof. Dr. Elmar Vogt
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Time and place: Thursday, 4pm -- 6pm, SR 009, Arnimallee 6.
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Leistungsnachweis/criteria for proof of performance:
Grade and credit points will be awarded based on a presentation and written summary.
Prerequisites: We assume basic knowledge of topology as taught in Topology I and II. This means concretely some familiarity with CW-complexes and basics of category theory. We also assume some familiarity with basics from commutative algebra, such as rings and projective modules.
Content: This seminar will cover advanced topics from topology and homotopy theory.
Algebraic K-theory is a powerful invariant that appears throughout various areas of mathematics. The K-theory of group rings provides obstructions which appear in the classification of manifolds via the use of surgery theory and is central to geometric topology. The algebraic K-theory of schemes provides a universal cohomology theory whose study was pioneered by Grothendieck. The K-theory of number fields and their rings of integers contains information about the values of the zeta-functions. Assembler K-theory generalizes the Dehn invariant of polyhedra to arbitrary dimensions, which measures when two given polytopes are scissors congruent. In this seminar we will begin with the basics: We will discuss the zero-th and first K-theory groups of a ring and compute them for many different examples. From there on we will discuss negative as well as higher K-theory and work towards the central theorem of K-theory: The Additivity Theorem.
Talks
| Date | Title | Speaker |
|---|---|---|
| 19.10. | First Meeting. Overview of the goals of this seminar. | Georg Lehner/ Elmar Vogt |
| 26.10. | Talk 1: K0, Elementary Examples and Morita-invariance [Milnor] | Maftuna Samatboyeva |
| 02.11. | -- | -- |
| 09.11. | Talk 2: Milnor Squares and some examples of K0’s [Milnor] | Jonas Kahle |
| 16.11. | Talk 3: Wall’s finiteness Criterion [Ferry-Ranicki] | Mark Backhaus |
| 23.11. | Talk 4: K1, Mayer-Vietoris and the LES of an ideal [Milnor, Weibel] | Medha Yelimeli |
| 30.11. | Talk 5: The Fundamental Theorem of Algebraic K-theory for K0 and K1 [Weibel] | Konstantin Roßberg |
| 07.12. | Talk 6: Negative K-theory and triangulated categories [Weibel, Schlichting] | Chris Huggle |
| 14.12. | Talk 07: Higher algebraic K-theory and the +-Construction | N.N. |
| 21.12. | Talk 08: Topological K-theory and the K-theory of finite fields [Weibel, Haines] | Larisa Janko |
| 2024 | ||
| 11.01. | Talk 09: Exact vs. Triangulated vs. Waldhausen categories vs. Stable ∞-categories [Neeman] | Chun Yin Lui |
| 18.01. | Talk 10: The S. Construction [Weibel, Hebestreit | Ruochong Huang |
| 25.01. | Talk 11: The Additivity Theorem | N.N. |
| 01.02. | Talk 12: The Localization Theorem and Applications | N.N. |
| 08.02. | Talk 13: | N.N. |
| 15.02. | N.N. |
Literature:
- Milnor: Introduction to algebraic K-theory.
- Weibel: The K-Book - https://sites.math.rutgers.edu/ weibel/Kbook.html
- Ferry-Ranicki: A survey of Wall’s finiteness obstruction - https://arxiv.org/abs/math/0008070
- Schlichting: Negative K-theory of derived categories - https://www.maths.ed.ac.uk/ v1ranick/papers/schlneg.pdf
- Haines: On the K-theory of finite fields - https://math.mit.edu/ phaine/files/KFF.pdf
- Hebestreit: Algebraic and Hermitian K-Theory - https://www.uni-muenster.de/IVV5WS/WebHop/user/fhebe 01/HigherCats2/KTheory.pdf