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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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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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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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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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In this short talk, I have summarized the main content of ‘The Chow t-structure on the $\infty$-category of motivic spectra’ and organized its logical flow to facilitate the audience’s understanding of the technical details discussed in the subsequent reports. A mind map has also been drawn to help the audience understand the connections between different subsections of this paper.
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.
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This is a read note of ‘The special fiber of the motivic deformation of the stable homotopy category is algebraic’. In this note, I give a summary of the theoretical part of this paper: according to the equivalence between some parts of the $E_2$-pages of two spectral sequences the authors found by computation, an equivalence between two $\infty$-categories which induces the equivalence between two spectral sequences is found. This result shows us a new way of computing the differentials in the Adams spectral sequences.
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This is a read note of ‘The Chow t-structure on the $\infty$-category of motivic spectra’.I summarized the main content of this paper and organized its logical flow to facilitate the audience’s understanding of the technical details discussed in the subsequent reports.