Oberseminar Algebra und Geometrie
Winter term 2024/25: Mixed and non-mixed topics
Chaired by I. Halupczok, H. Kammeyer and B. Klopsch.
All talks take place on Fridays at 14:30 in 25.22.03.73.
This term, the oberseminar is divided in two parts. In the first part we will learn about local class field theory and in the second part we will have mixed topics, mainly with guest speakers. Please get it touch if you have a suggestion on whom to invite. Note also the changed time of the seminar.
If you want get announcements about the seminar, please get in touch with I. Halupczok so that your address is added to the mailing list.
Infos für Studierende
Im SoSe 2024 wird das Seminar (hauptsächlich?) aus in sich abgeschlossenen Einzel-Vorträgen bestehen, größtenteils von externen Gästen, und größtenteils über aktulelle Forschungsthemen. Das Oberseminar richtet sich an alle, die einen Einblick in aktuelle Forschung erhalten möchten, ist aber tendenziell eher für fortgeschrittene Studierende geeignet (ab Master). Besonders empfohlen wird die Teilnahme an Oberseminaren, wenn Sie sich vorstellen können zu promovieren.Wenn Sie intessiert sind, können Sie sich einfach (ohne Anmeldung) ins Seminar reinsetzen - gerne auch nur zu einzelnen Vorträgen, die Sie interessieren.
Die Vorträge in diesem Oberseminar sind auf englisch. Üblicherweise nehmen an Oberseminaren auch viele Doktorand:innen, Postdocs und Professor:innen teil. Wahrscheinlich werden Sie nicht alles verstehen; das passiert aber auch den fortgeschritteneren Teilnehmer:innen, und auch, wenn man nicht alles verstanden hat, hat man doch am Ende oft einen interessanten Einblick in ein neues Thema erhalten.
Local class field theory
Understanding the absolute Galois group of an arbitrary base field has turned out to be a very hard problem. The case of
Schedule
- 18.10.: Doris Grothusmann: Group cohomology 1
Define the cohomology groups
as in Definition (2.1). Mention for as well as the difference to conventional cohomology in degrees 0 and . State Theorem 3.2 (no proof required). Give explicit descriptions of , , , and (Theorem 3.19.). (Source: Part I Sections 2 and 3.) - 25.10.: Fabian Rodatz: Group cohomology 2
Define the cup product as in Definition (5.1) and state its important properties (excluding those referring to inflation and restriction). Describe the cohomology of cyclic groups and define the Herbrand quotient. Prove its given properties. (Source: Part I Sections 5 and 6.)
- 1.11.: No Seminar (holiday)
- 8.11.: Clotilde Gauthier: Group cohomology 3
Define inflation, restriction and corestriction and prove their basic properties (Theorem 4.6, Theorem 4.14). Sketch the proof of Tate's Theorem (Theorem 7.3). (Source: Part I Sections 4 and 7.)
- 15.11.: Jan Hennig: Abstract class field theory
Give the definition of a class field formation (Defintion 1.3). Explain, why it fulfils the requirements of Tate's Theorem (Theorem 1.7) and deduce Theorem 1.9. Sketch the proof of Theorem 1.15. (Source: Part II Section 1.)
- 22.11.: Giada Serafini: Local class field theory in the unramified case
Recount Theorem 2.2 and Theorem 2.3 (no proof required, if you are interested, this will most likely be shown in this semesters special lecture). Show that
is a class formation for the maximal unramified extension of a local field (Theorem 4.6). Recount definitions related to local fields if necessary. (Source: Part II Sections 2, 3, and 4.) - 29.11.: Benjamin Klopsch: Local lass field theory in the general case
Show that, even in the ramified case,
is a class formation for an algebraic closure of a local field (Theorem 5.6). Deduce Theorem 5.7 and Theorem 5.9. State Theorem 5.13 (no proof required). (Source: Part II Section 5.)
References:
[N] J. Neukirch, Class Field Theory, Spinger-Verlag, Berlin, 2013. link
[G] G. Gras, Class field theory, Spirnger-Verlag, Belin, 2003. link
[M] J. S. Milne, Class field theory, Course notes, 2020. link
Mixed Topics
- 6.12.: tba
- 13.12.: tba
- 10.1.: Francesco Fournier-Facio (Cambridge): tba
Abstract: tba
- 17.1.: tba
- 24.1.: Amir Weiss-Behar (Jerusalem): tba
Abstract: tba
- 31.1.: Jaro Eichler (Frankfurt): tba
Abstract: tba
Archive
SS 2024: Mixed topics
WS 2023/24: Central Simple Algebras
SS 2023: Knot theory and quandles
WS 2022/23: Combinatorics and Commutative Algebra
SS 2022: Discrete Groups, Expanding Graphs and Invariant Measures
WS 2021/22: Superrigidity
SS 2021: Group cohomology
SS 2020 and WS 20/21: cancelled due to pandemic
WS 2019/20: Intersection theory
SS 2019: Knots and primes
WS 2018/19: The Grothendieck group of varieties and stacks
SS 2018: Arithmetic Groups - Basics and Selected Applications
WS 2017/18: Algebraic K-theory
SS 2017: Berkovich spaces
WS 16/17: Resolution of singularities and alterations
SS 2016: Modular Representation Theory
WS 15/16: The Milnor Conjectures
SS 2015: Rationality
WS 14/15: Essential Dimension
SS 2014: Varieties of Representations
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