Math and Physics of Seiberg-Witten theory

Math and physics have developed through interactions with each other.
For example, classical mechanics and calculous were born together.
Einstein's theory of gravitation is written in the language of pseudo-Riemann geometry.
Since the late 20th century, physicists centering on Edward Witten have revolutionized modern geometry.
Seiberg-Witten theory is one of such breakthroughs, for both mathematicians and physicists.
In physics it is regarded as a theory describing strong coupling (i.e. low energy) behavior of some supersymmetric gauge theories. It showes confinement (by a mechanism similar to superconductivity) and electric magnetic duality.
Even though this story has not been mathematically justified yet, it is regarded as an important trigger of developments in understanding non perturbative aspects of quantum field theory and string theory, and stimulates broad fields of physics and math.
In math, Seiberg-Witten theory is regarded as a fundamental tool to study 3 and 4-dimensional geometry.
This is based on a PDE called Seiberg-Witten equation, which originates from the "electric magnetic dual description" of monopoles, but people can use it as a tool to study geometry without knowing such a physical origin.
In this talk, developments of Seiberg-Witten theory from both viewpoints will be reviewed and if the time permits, works in math by the speaker and collaborators will be discussed.
The speaker thinks it is unusual for a mathematician to talk about something that has not been mathematically justified yet, but hopes this talk will lead to new interactions between math and physics.