Семинар по геометрия на МЦМН
2025-04-30 16:00 - 17:30 (Europe/Sofia) | Институт по математика и информатика – БАН, Block 8, 1113 БАН IV км., София, Bulgaria
Следващата сбирка на Семинара по геометрия на МЦМН
ще се проведе в сряда, 30 април 2025 г. от 16:00 ч. в зала 403 и онлайн в Zoom:
Доклад на тема
Binary Quadratic Forms and Conway’s Topographs (Lecture 1 of 3)

ще изнесе Nikita Kalinin, Guangdong Technion Israel Institute of Technology.

Следващите две лекции ще бъдат на 07.05.2025 и 14.05.2025 от 16:00 ч.

Abstract: 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,yin 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 displaystylesum_{(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.

Zoom link:
https://us02web.zoom.us/j/86186281353?pwd=[REDACTED]

Семинар по геометрия на МЦМН
2025-05-07 16:00 - 17:30
Следващата сбирка на Семинара по геометрия на МЦМН
ще се проведе в сряда, 7 май 2025 г. от 16:00 ч. в зала 403 и онлайн в Zoom:
Доклад на тема
Binary Quadratic Forms and Conway’s Topographs (Lecture 2 of 3)

ще изнесе Nikita Kalinin, Guangdong Technion Israel Institute of Technology.

Следващата лекция ще бъде на 14.05.2025 от 16:00 ч.

Abstract: 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,yin 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 displaystylesum_{(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.

Zoom link:
https://us02web.zoom.us/j/86186281353?pwd=[REDACTED]

Видеозапис и презентация на първата лекция:

https://youtu.be/z7Pz33JyCuA?si=rQ2L4yXJHMdiDkpe
https://kilin-math.github.io/assets/numbers/telescopic_presentation1.pdf
