Speaker: 秦雨轩（重庆理工大学）
Time: 9:30-11:00, Jan. 8 2025
Online: Zoom: 811 5638 1768（Password: msrc）
Title: The basic geometry of hyperbolic space
Description:
we will begin with the abstract properties of spectral sequences and their relation to the double complexes,We point out the similarities and the differences whenever appropriate. In particular, there is a very brief discussion of the Kunneth formula and the universal coefficient theorems in this new setting. Thereafter we apply the spectral sequences to the path fibration of Serre and compute the cohomology of the loop space of a sphere. The short review of homotopy theory that follows includes a digression into Morse theory, where we sketch a proof that compact manifolds are CW complexes,we also discuss the Hopf invariant and the linking number and explore the rather subtle aspects of Poincare duality concerned with the boundary of a submanifold. Returning to the spectral sequences, we compute the cohomology of certain Eilenberg-MacLane spaces.
Speaker: 张时铭（重庆师范大学）
Time: 9:30-11:00, Dec. 25 2024
Online: 腾讯会议：146412019
Title: A generalization of the Künneth theorem
Abstract:
This discussion focuses on the applications of spectral sequences in algebra. Firstly, by introducing the Künneth theorem, the content related to the Tor functor and its generalization to the case of differential graded modules are expounded. Secondly, theorems are given to compare related homologies. Finally, through the generalization of the Künneth theorem, double complexes and total complexes are introduced, leading to the method of constructing two spectral sequences in the process of calculating the total complex. Our discussion this time provides a theoretical basis for subsequent research on homological algebra and related content of spectral sequences.
Speaker: 秦雨轩（重庆理工大学）
Time: 9:30-11:00, Dec. 18 2024
Online: Zoom: 811 5638 1768（Password: msrc）
Title: Chern-weil theory
Description:
we will begin with the abstract properties of spectral sequences and their relation to the double complexes,We point out the similarities and the differences whenever appropriate. In particular, there is a very brief discussion of the Kunneth formula and the universal coefficient theorems in this new setting. Thereafter we apply the spectral sequences to the path fibration of Serre and compute the cohomology of the loop space of a sphere. The short review of homotopy theory that follows includes a digression into Morse theory, where we sketch a proof that compact manifolds are CW complexes,we also discuss the Hopf invariant and the linking number and explore the rather subtle aspects of Poincare duality concerned with the boundary of a submanifold. Returning to the spectral sequences, we compute the cohomology of certain Eilenberg-MacLane spaces.
Speaker: 张时铭（重庆师范大学）
Time: 9:30-11:00, Dec. 11 2024
Online: 腾讯会议：639924446
Title: Construction of Spectral Sequences of Exact Couples
Abstract:
In this discussion, we first put forward a proposition and a theorem in the construction process of spectral sequences of exact couples. Secondly, we got to know the exact sequences formed. Finally, we expounded on the relationship between the collapse of spectral sequences and the convergence of spectral sequences. In this work, we are grateful to Teacher Li Ming for the relevant guidance and help provided.
Speaker: 秦雨轩（重庆理工大学）
Time: 9:30-11:00, Dec. 4 2024
Online: Zoom: 811 5638 1768（Password: msrc）
Title: Pontryagin classes
Description:
we will begin with the abstract properties of spectral sequences and their relation to the double complexes,We point out the similarities and the differences whenever appropriate. In particular, there is a very brief discussion of the Kunneth formula and the universal coefficient theorems in this new setting. Thereafter we apply the spectral sequences to the path fibration of Serre and compute the cohomology of the loop space of a sphere. The short review of homotopy theory that follows includes a digression into Morse theory, where we sketch a proof that compact manifolds are CW complexes,we also discuss the Hopf invariant and the linking number and explore the rather subtle aspects of Poincare duality concerned with the boundary of a submanifold. Returning to the spectral sequences, we compute the cohomology of certain Eilenberg-MacLane spaces.
Speaker: 张时铭（重庆师范大学）
Time: 9:30-11:00, Nov. 27 2024
Online: 腾讯会议：145722968
Title: Filtered Differential Modules and Exact Couples
Abstract:
This discussion centers around spectral sequences. Firstly, it expounds on its definition and fundamental properties, which involve the constitution of differential bigraded modules and spectral sequences, as well as the related concepts of the submodule tower. Subsequently, it presents the generation methods, such as filtered differential modules and exact couples, and also discusses the relevant proofs and applications.
Speaker: 秦雨轩（重庆理工大学）
Time: 9:30-11:00, Nov. 20 2024
Online: Zoom: 811 5638 1768（Password: msrc）
Title: Stiefel-Whitney classes
Description:
we will begin with the abstract properties of spectral sequences and their relation to the double complexes,We point out the similarities and the differences whenever appropriate. In particular, there is a very brief discussion of the Kunneth formula and the universal coefficient theorems in this new setting. Thereafter we apply the spectral sequences to the path fibration of Serre and compute the cohomology of the loop space of a sphere. The short review of homotopy theory that follows includes a digression into Morse theory, where we sketch a proof that compact manifolds are CW complexes,we also discuss the Hopf invariant and the linking number and explore the rather subtle aspects of Poincare duality concerned with the boundary of a submanifold. Returning to the spectral sequences, we compute the cohomology of certain Eilenberg-MacLane spaces.
Speaker: 张时铭（重庆师范大学）
Time: 9:30-11:00, Nov. 13 2024
Online: 腾讯会议：861985068
Title: The action of Graded Algebras on Spectral Sequences and the introduction of Graded Commutative Algebras.
Abstract:
This part of the content focuses on spectral sequences and encompasses numerous significant aspects.
Firstly, starting from the connection between differentials and the dimension of vector spaces, the magnitude rules of the Poincaré series of each term of the spectral sequence are derived, and methods for determining the collapse of the spectral sequence are presented. These methods include two approaches: proving the equality of dimensions and using partial sums to prove the equality of relevant Euler characteristics.
Secondly, in the application domain, on the one hand, the role of spectral sequences in the process of calculating the Euler characteristic of manifolds is demonstrated. On the other hand, the action of graded algebras on spectral sequences is introduced. This covers the definition of the action, conditions, and relevant convergence definitions, and its significance is highlighted through examples.
Finally, based on specific assumptions of algebraic spectral sequences, the relevant concepts of graded - commutative algebras are introduced. Through examples, the idea of reversely deducing other information based on known partial information is elaborated, and at the same time, the relevant rules and properties of differentials are mentioned.
Speaker: 秦雨轩（重庆理工大学）
Time: 9:30-11:00, Oct. 30 2024
Online: Zoom: 811 5638 1768（Password: msrc）
Title: The spectral sequence of a fiber bundle and Some applications
Description:
we will begin with the abstract properties of spectral sequences and their relation to the double complexes,We point out the similarities and the differences whenever appropriate. In particular, there is a very brief discussion of the Kunneth formula and the universal coefficient theorems in this new setting. Thereafter we apply the spectral sequences to the path fibration of Serre and compute the cohomology of the loop space of a sphere. The short review of homotopy theory that follows includes a digression into Morse theory, where we sketch a proof that compact manifolds are CW complexes,we also discuss the Hopf invariant and the linking number and explore the rather subtle aspects of Poincare duality concerned with the boundary of a submanifold. Returning to the spectral sequences, we compute the cohomology of certain Eilenberg-MacLane spaces.
Speaker: 张时铭（重庆师范大学）
Time: 9:30-11:00, Oct. 23 2024
Online: 腾讯会议：329815039
Title: Spectral Sequences of Algebra and Euler Characteristic
Abstract:
We will mainly discuss how to utilize spectral sequences to study algebraic structures, especially the structures of graded algebras. Firstly, graded algebras and bigraded algebras, as well as differential graded algebras and differential bigraded algebras, are defined. Secondly, the definition of the algebraic spectral sequence is provided, and the filtration is restricted regarding the product mapping ϕ, so that the spectral sequence can restore a graded algebra H*. Finally, the Euler characteristic, a useful arithmetic invariant of the graded vector space, is introduced.
