Lectures related to The Degree of a Covering Space

Fundamental Groups

Explores fundamental groups, homotopy classes, and coverings in connected manifolds.

Bar Construction: Homology Groups and Classifying Space

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

Cohomology: Cross Product

Explores cohomology and the cross product, demonstrating its application in group actions like conjugation.

Cohomology: Cup Product

Covers the cup product in cohomology, focusing on examples and computations.

Directed Networks & Hypergraphs

Explores directed networks with asymmetric relationships and hypergraphs that generalize graphs by allowing edges to connect any subset of nodes.

Acyclic Models: Cup Product and Cohomology

Covers the cup product on cohomology, acyclic models, and the universal coefficient theorem.

Irreducible Polynomials: Degree and Roots

Explores irreducible polynomials, focusing on their degree and roots in different fields.

Categorical Perspective: Group Quotients

Explores the categorical sense of constructing a group quotient by a normal subgroup, showing it as a specific example of a more general construction.

Universal Property of Groups

Explores the universal property of groups, defining homomorphisms and exploring subgroup properties.

Lagrange's Theorem: Applications and Homomorphisms

Covers Lagrange's theorem, group homomorphisms, congruence classes, and normal subgroups.

Pushouts in Group Theory: Universal Properties Explained

Covers the construction and universal properties of pushouts in group theory.

Stein Algorithm: Polynomial Identity Testing

Explores the Stein algorithm for polynomial identity testing and the minimization of a cut problem.

Topology of Riemann Surfaces

Covers the topology of Riemann surfaces, focusing on orientation and orientability.

Group Theory Fundamentals

Introduces group theory basics and group action concepts with associated subsets.

Groups: Definitions, Properties, and Homomorphisms

Introduces the basic concepts of groups, including definitions, properties, and homomorphisms, with a focus on subgroup properties and normal subgroups.

Homotopy: Degree

Covers the concept of degree in homotopy theory and the structure of complexes with two simplices.

Cohomology Groups: Hopf Formula

Explores the Hopf formula in cohomology groups, emphasizing the 4-term exact sequence and its implications.

Group Theory Basics: Subgroups and Homomorphisms

Introduces subgroups and homomorphisms in group theory, with examples for illustration.

Elliptic Curves: Theory and Applications

Covers the theory and applications of elliptic curves in cryptography and number theory.

Chinese Remainder Theorem: Rings and Fields

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.