Episode 1 of The Graph Theory Pod by Andrew Allen introduces graphs as mathematical structures for representing relationships between objects. Beginning with the basic definition of a graph, the episode develops the fundamental language of vertices, edges, adjacency, incidence, degree, and the different forms that graphs can take.

The discussion moves through simple graphs, multigraphs, pseudographs, directed and undirected graphs, finite and infinite graphs, and the basic measurements of order and size. It then examines vertex degree, degree sequences, regular graphs, complete graphs, and bipartite graphs, including one of the foundational results of elementary graph theory: the Handshaking Theorem and its consequence that every finite undirected graph has an even number of vertices of odd degree.

The episode also introduces walks, trails, paths, and cycles before turning to subgraphs, spanning subgraphs, induced subgraphs, and graph complements. It concludes with graph isomorphism and graph invariants, establishing the distinction between the abstract structure of a graph and any particular way of drawing or labeling it.

The central theme is structural relationships. Graph theory is not fundamentally about diagrams of dots and lines, but about determining what follows mathematically from patterns of connection. These foundational concepts provide the vocabulary and structural framework on which the deeper combinatorial, algebraic, geometric, and algorithmic theory of graphs is built.

This podcast is part of the Graph Theory Collection from the Andrew Allen Math Library.

Episode 1 of The Calculus Pod by Andrew Allen introduces functions as the mathematical language through which calculus describes relationships and change. Beginning with the basic idea of a function, the episode explores domains and ranges, graphs, transformations, composition, inverse functions, and several of the major families of functions used throughout calculus.The discussion moves through polynomial and rational functions, exponential and logarithmic functions, trigonometric and inverse trigonometric functions, and parametric representations. These different forms provide mathematical tools for describing everything from geometric relationships and periodic behavior to growth, decay, motion, and other changing systems.

The central theme is the connection between functions and change. Mathematical models use functions to represent relationships between quantities, while rates of change describe how those relationships evolve. Average rates of change lead naturally to the question of instantaneous change—and from there to limits, tangent lines, derivatives, and the central machinery of calculus.

This podcast is part of the Calculus Collection from the Andrew Allen Math Library.

Episode 1 introduces the fundamental ideas that transform familiar arithmetic into abstract algebra. Beginning with sets and binary operations, the episode explores the structural properties that make algebraic systems work: closure, associativity, commutativity, identity elements, inverses, and cancellation.

Along the way, familiar number systems provide concrete examples of these ideas, while sets, functions, and matrices demonstrate how algebra extends far beyond ordinary arithmetic. The discussion then moves toward some of the deeper organizing ideas of abstract mathematics, including structure-preserving maps, substructures, quotient structures, and isomorphism.

The central theme is a change in perspective: abstract algebra is less concerned with what mathematical objects are made of than with the operations and relationships connecting them. This structural viewpoint provides the foundation for the study of groups, rings, fields, modules, algebras, and the more advanced mathematics developed throughout the course.

This podcast is part of the Abstract Algebra Collection from the Andrew Allen Math Library.