In this course, we will introduce several elementary topics in pure math. We will discuss some basic techniques and notions of abstract algebra, topology, analysis and geometry. There will be a focus on broad themes in mathematics and we will discuss some recent math research. There will be optional reading and exercises handed out at the end of every class.

Topology is the study of shape and space. Traditionally, topological study begins with the study of geometric spaces that we can easily picture, such as 3-dimensional flat euclidean space (our perceived world). It then covers more abstract geometric and set-theoretic topics, like connectedness, the number of holes, and finiteness, before arriving at a theory of an abstract topological space. In this class, we take topology in a totally different direction: motivated by questions in computer science, we will build a 'topological' understanding of logic via the theory of semi-decidability, ultimately arriving at a geometric understanding of logic and computation. For us, we will see that a topology will capture an essential computational notion: "open sets are semi decidable properties". The theory, developed from this perspective, will produce results applicable to understanding traditional abstract topology, domain theory, first order logic, decidability theory in computer science and modern algebraic geometry. No previous exposure to domain theory, topology or formal logic needed.

In this course, we will introduce several elementary topics in pure math. We will discuss the basic techniques and notions of analysis, topology, abstract algebra and geometry. There will be a focus on broad themes in mathematics and we will discuss recent math research. There will be optional reading and exercises handed out at the end of every class.

Want to build machines that understand the world? Algebra at your service...