Binary quadratic forms are as elementary as they are mysterious—much like prime numbers. In 1997, John Conway introduced topographs, a powerful geometric tool that provides a geometric visualization of binary quadratic forms and their values. These lectures will explore how topographs, combined with telescoping summation techniques, yield elegant formulas — some with intuitive geometric interpretations. For instance, consider the following result:

Let

\(A = \big\{ (x, y) \mid  x,y\in \mathbb{Z}_{\geq 0}^2, \det(x \ \ y) = 1 \big\},\)

the set of pairs of lattice vectors in the first quadrant spanning a parallelogram of oriented area 1. Then,

\(4 \displaystyle\sum_{(x,y) \in A} \frac{1}{|x|^2 \cdot |y|^2 \cdot |x+y|^2} = \pi.\)

Lecture Outline

1. Introduction to Binary Quadratic Forms and Conway’s Topographs
We will begin with the basics of binary quadratic forms and their classification, followed by an introduction to Conway’s topographs—a visual and geometric framework for understanding them.

2. Class Number Formula and Summation over Topographs
Building on the first lecture, we will explore the class number formula and how summation identities arise naturally from the structure of topographs.

3. Evaluation of Lattice Sums via Telescoping over Topographs
The final lecture will focus on telescoping techniques, demonstrating how they can be used to evaluate intricate lattice sums—such as the one above—with geometric meaning.