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19223811 Masterseminar Topologie "Algebraic K-Theory"

Winter Term 2023/2024

Dozenten: Dr. Georg Lehner, Prof. Dr. Elmar Vogt

  • Time and place:  Thursday,  4pm -- 6pm, SR 009, Arnimallee 6.

  • 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.


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
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.


  • 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