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Besides reviewing historical development and prerequisites, we present the main ingredients needed for calculations with spectral sequences for this purpose. These include the Bousfield-Kuhn functor, the Goodwillie tower and so on. We then outline current approaches to running such spectral sequences from cohomology to homotopy, as well as indicate specific questions to investigate.
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This is a sample blog post.
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Reports for reading seminars in Fudan University each semester since Spring 2021.Topics included some basic knowledge in algebraic topology, such as the model category, the May spectral sequence, the setting up of classical Adams spectral sequence and the obstruction theory.
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In this talk we give a brief introduction into the history of the computation of (stable) homotopy groups of spheres. Starting with a review of the Serre spectral sequence, the Eilenberg-Maclean space and the anicent way of computing higher homotopy groups with these tools, we introduce the motivation and the idea behind the construction of the classical Adams spectral sequence as well as its proof. The main reference of this talk is chapter 6 of Homotopical Topology by Fuchs and Fomenko.
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Following the previous talk in SUSTech graduate seminar, we introduce the generalised Adams spectral sequence and show the algorithm of computing $E_2$-pages of classical Adams spectral sequence.(i.e. minimal resolution, Lambda algebra and May spectral sequence) Then, we prove that the ring spectrum structure induces a multiplicative structures on the Adams spectral sequnece as well as show the algorithm of computing it. Finally, we give a brief introduction to the method of computing the differentials in the spectral sequence and the hidden extension, aiming to show the difficulty that we will face in computing stable homotopy groups.
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In this reading seminar, I was asked to give talks on the basics of algebraic K-theory, the Serre-Swan theorem, the Grothendieck-Riemann-Roch theorem, Whitehead torsion and Reidemaster torsion. I divided these into three talks. In the first talk, I introduced the basics of $K_0$ (i.e. the definition and some examples of calculations, relative K-theory). In the second talk, we arrive at the Grothendieck-Riemann-Roch theorem with a geometric approach. The motivation is also discussed from a geometrical point of view. Finally, in the last lecture, I talked about Whitehead’s torsion and its application.
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Besides reviewing historical development and prerequisites, we present the main ingredients needed for calculations with spectral sequences for this purpose. These include the Bousfield-Kuhn functor, the Goodwillie tower and so on. We then outline current approaches to running such spectral sequences from cohomology to homotopy, as well as indicate specific questions to investigate.
Recitation class, SUSTech, 2023
I am a TA of this course. My duties of this course included homework correcting and teaching recitation classes. The supervisor of this course is Prof. Juexian Li.
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Besides reviewing historical development and prerequisites, we present the main ingredients needed for calculations with spectral sequences for this purpose. These include the Bousfield-Kuhn functor, the Goodwillie tower and so on. We then outline current approaches to running such spectral sequences from cohomology to homotopy, as well as indicate specific questions to investigate.