Steenrod Operations: From Classical to Motivic in 2022
Steenrod operations are natural transformations of cohomology groups of spaces being compatible with suspensions, which play an important role especially in homotopy theory, for example, the Hopf invariant one problem and Adams spectral sequence. In this course, we introduce the Steenrod operations for both singular cohomologies and motivic cohomologies, focusing on constructions and basic properties. In particular, we prove the Adem relations and give the Hopf algebra (algebroid) structure of Steenrod algebra and its dual.
Lecturer
Date
15th September, 2022 ~ 5th January, 2023
Website
Prerequisite
Algebraic topology and algebraic geometry
Syllabus
1. Classical Steenrod operations
2. Classical Steenrod algebra and its dual
3. Equidimensional cycles
4. Motivic Steenrod operations
5. Motivic Steenrod algebra and its dual
2. Classical Steenrod algebra and its dual
3. Equidimensional cycles
4. Motivic Steenrod operations
5. Motivic Steenrod algebra and its dual
Reference
1. O. Pushin, Steenrod operations in motivic cohomology, thesis
2. G. Powell, Steenrod operations in motivic cohomology, Luminy lecture notes
J. Riou, Opérations de Steenrod motiviques, arXiv:1207.3121v1
C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006).
A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, Annals of Mathematical Studies, vol. 143, Princeton University Press, 2000
A. Hatcher, Algebraic topology
2. G. Powell, Steenrod operations in motivic cohomology, Luminy lecture notes
J. Riou, Opérations de Steenrod motiviques, arXiv:1207.3121v1
C. Mazza, V. Voevodsky, C. Weibel, Lecture notes on motivic cohomology, American Mathematical Society, Providence, RI, for the Clay Mathematics Institute, Cambridge, MA (2006).
A. Suslin and V. Voevodsky, Relative cycles and Chow sheaves, Annals of Mathematical Studies, vol. 143, Princeton University Press, 2000
A. Hatcher, Algebraic topology
Audience
Graduate
Video Public
No
Notes Public
No
Language
Chinese
Lecturer Intro
Nanjun Yang got his doctor and master degree in University of Grenoble-Alpes, advised by Jean Fasel, and bachelor degree in Beihang University. Then he became a postdoc in YMSC. Currently he is a assistant professor in BIMSA. His research interest is the Chow-Witt group of algebraic varieties, with publications on journals such as Camb. J. Math and Ann. K-Theory.