Explore the intricate relationship between physics and mathematics in this 41-minute lecture by Gregory Moore at the Institute for Advanced Study. Delve into the concept of "pre-established harmony" between these disciplines, as observed by Gottfried Leibniz. Examine how mathematical ideas have been crucial for physicists in discovering nature's fundamental laws, and how physics has reciprocally provided new insights into advanced mathematics. Investigate the potential for a unified theory encompassing all universal laws. Journey through various dimensions, from one to four, and their associated challenges in physical mathematics. Learn about topological invariants, gravity, nuclear forces, and electromagnetic interactions. Discover the significance of commutative versus noncommutative mathematics and explore groundbreaking concepts like Donaldson invariants and Seiberg-Witten equations. Gain insights into topological field theory, quantum field theory, and the importance of topology in understanding the universe. Witness the excitement of scientific breakthroughs and their impact on long-standing debates in the field.
Overview
Syllabus
Intro
Physical Mathematics
Two Basic Questions
One Dimension
A One-Dimensional Topological Invariant
Two Dimensions
Three Dimensions: Things Get Harder
More Than Three Dimensions?
Four Dimensions Is The Hardest !
Gravity
Nuclear Force
Electric & Magnetic Forces
Commutative vs. Noncommutative
Meanwhile, back at the ranch
Cartoon Of Donaldson Invariants
Using this new (1984) topological invariant Donaldson could prove dramatic new results
Witten's Answer
Topological Field Theory
What does matter is topology!
Quantum Field Theory
Seiberg-Witten equations have soliton-like solutions: vortices in a superconductor
Mad Dash After A Breakthrough
Settling A Debate
Taught by
Institute for Advanced Study
