New math revives geometry's oldest problems
14 hours ago
- #Motivic Homotopy Theory
- #Enumerative Geometry
- #Mathematical Revival
- Apollonius of Perga's problem about circles touching three given circles was solved after 1,800 years, with the answer being eight.
- Enumerative geometry, which counts solutions to geometric conditions, was popular among ancient Greeks and continued to interest mathematicians.
- Mathematicians like Sheldon Katz note that these problems are easy to understand but hard to solve.
- Enumerative geometry evolved into its own field but lost interest by the mid-20th century, except for a brief resurgence in the 1990s.
- A new approach using motivic homotopy theory has revived enumerative geometry, connecting it to algebra, topology, and number theory.
- Kirsten Wickelgren and Jesse Kass realized that enumerative geometry could provide deep insights, leading to new methods for solving old problems.
- Their work involves rewriting enumerative geometry problems in terms of spaces of equations and applying motivic homotopy theory to compute quadratic forms.
- This method provides answers in various number systems, including complex, real, and finite systems like clock arithmetic.
- The technique has been applied to famous problems, such as the 27 lines on a cubic surface, and has opened new avenues in string theory.
- The new approach is concrete and accessible, attracting young mathematicians and providing a bridge between high abstraction and tangible results.