How can we find the most efficient route for delivering mail in a community? How can we use internal emails at Enron to discover the individuals working on illegal activities inside the company? In the 1700s, Euler asked a simple question about how to cross a set of bridges exactly once in Königsberg, Prussia (now Kaliningrad, Russia). His study introduced the use of graphs to solve problems. Today, we use graphs to model everything from social networks to road networks. This action-packed course will apply the basics of graph theory, linear algebra, and algebraic topology to study the data and networks all around us. Outside the class, you are encouraged to discuss and think about the human aspect on the topics we will discuss. Would you, say, implement more efficient routes for street-sweeping trucks in your city if doing so leads to truck drivers losing their jobs?
Prerequisites. Algebra II
Learning Outcomes:
- Applying course techniques to model real life.
- Exposure to career paths and research areas in the sciences.
Course Structure:
You will watch course videos and do exercises before class. In class, we will go over the exercises and spend most of the time on group activities.
Syllabus
- Graphs, degree of vertices, and Euler’s formula
- Simplicial complexes and the Euler characteristic
- Systems of linear equations and the matrix notation
- Matrices and adjacency matrices
- Diagonalization of matrices
- The vector space, R^n, and its subspaces
- Spanning sets and the dimension of a vector space
- Matrix transformations and the rank-nullity theorem
- Betti numbers of a simplicial complex
