Primes and Knots - Akshay Venkatesh

Explore the fascinating connections between number theory and three-dimensional geometry in this 51-minute public lecture by Akshay Venkatesh, Robert and Luisa Fernholz Professor at the Institute for Advanced Study's School of Mathematics. Delve into the surprising parallels between problems in prime number theory and questions in knot theory, understanding how these seemingly disparate fields inform each other. Learn about the fundamental metaphor linking these areas, the production of knots and prime numbers, and the intriguing statistics of prime factors. Discover the concept of modular arithmetic and its relation to knot linking, and grasp the significance of the law of quadratic reciprocity. Gain insights into recent developments in this interdisciplinary field of mathematics, showcasing how ancient questions continue to drive cutting-edge research.

Intro
The fundamental metaphor
Producing many knots
Prime numbers and the factorization problem
Some examples
Statistics of prime factors
Producing tangles (collections of knots)
Linking of knots
Modular arithmetic
The analogy between square roots and linking
The law of quadratic reciprocity
Back to statistics
Recent developments