Lectures related to Excision: An Example

Explores the homology of Riemann surfaces, including singular homology and the standard n-simplex.

Bar Construction: Homology Groups and Classifying Space

Covers the bar construction method, homology groups, classifying space, and the Hopf formula.

Explores the topological Künneth Theorem, emphasizing commutativity and homotopy equivalence in chain complexes.

Explores the comparison of long exact sequences for vibrations and the relationship between homotopy and homology groups.

Group Cohomology

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.

Topology Seminar: Tower Sequences and Homomorphisms

Explores tower sequences, homomorphisms, and their applications in topology, including the computation of homology and the construction of telescopes.

Homotopy Theory of Chain Complexes

Explores the homotopy theory of chain complexes, including path object construction and fibrations.

Topology: Classification of Surfaces and Fundamental Groups

Discusses the classification of surfaces and their fundamental groups using the Seifert-van Kampen theorem and polygonal presentations.

Shape of Data: Algebraic Topology and Shape Representation

Covers algebraic topology, Betti numbers, and shape representation methods for efficient data shape measurement and analysis.

Active Learning: Functors and Geometric Realization

Covers the computation of nerves and geometric realization in simplicial sets, along with functors into and out of the category of simplicial sets.

Higher-order Interactions in Brain Networks

Explores higher-order interactions in brain networks using simplicial complexes and information theory, analyzing data from fMRI, financial time-series, and infectious diseases.

Simplicial and Singular Homology Equivalence Log in to Mediaspace to watch this video

Demonstrates the equivalence between simplicial and singular homology, proving isomorphisms for finite s-complexes and discussing long exact sequences.

Singular Homology: First Properties Log in to Mediaspace to watch this video

Covers the first properties of singular homology and the preservation of decomposition and path-connected components in topological spaces.

Simplicial Homology Revisited Log in to Mediaspace to watch this video

Covers the concept of simplicial homology, focusing on finite complexes and induced maps.

Homology Groups: Basics Log in to Mediaspace to watch this video

Introduces reduced homology groups and explains their properties and applications in topology.

Simplicial Homology: Structure and Complexes Log in to Mediaspace to watch this video

Covers the structure of topological spaces with A-complexes and chain complexes.

Homology Theorem Log in to Mediaspace to watch this video

Covers the proof of Theorem A, discussing homology, quotients, and isomorphisms.

Homology: Introduction and Applications Log in to Mediaspace to watch this video

Introduces homology as a tool to distinguish spaces in all dimensions and provides insights into its construction and applications.

Homology and the fundamental group Log in to Mediaspace to watch this video

Explores simplicial and singular homology, reduced homology groups, and their connection to the fundamental group.

Zig Zag Lemma Log in to Mediaspace to watch this video

Covers the Zig Zag Lemma and the long exact sequence of relative homology.