Introduction to Order Structures in Groups - Lecture 1

Explore the fundamental concepts of total orders in groups through this comprehensive lecture. Delve into the various types of total orders and their interactions with standard algebraic structures like quotients, free products, and amalgams. Begin with the basics, requiring no prior knowledge of group orderings, and progress to develop essential tools and structures from the ground up. Examine key concepts such as the dynamical realization of an ordering, the Burns-Hale theorem, and Hölder's theorem, along with its relationship to local indicability. Discover how even basic topics in this field naturally lead to open problems, providing ample opportunities for both novice and experienced students to connect with active areas of research in group theory and order structures.