How to count curves: from nineteenth-century problems to twenty-first-century solutions

Area of Science:

• Enumerative geometry
• Algebraic geometry
• Theoretical physics

Background:

• The problem involves counting curves on a plane, a fundamental concept in enumerative geometry.
• The sequence, representing the number of curves through a given number of points, remained unsolved for decades.
• Early attempts to solve the problem highlighted its complexity and the need for novel mathematical approaches.

Purpose of the Study:

• To describe the mathematical problem of finding the next term in the sequence 1, 1, 12, 620, 87304.
• To explain the connection between this sequence and curve-counting in enumerative geometry.
• To detail the unexpected mathematical and physical concepts utilized in deriving the solution.

Main Methods:

• The study likely involved advanced techniques from algebraic geometry and theoretical physics.
• The solution required developing new mathematical tools to handle complex curve-counting problems.
• The article explains the derivation of a formula to calculate any term in the series.

Main Results:

• A closed-form formula was discovered for the sequence, enabling the calculation of any term.
• The solution connected seemingly disparate fields of mathematics and physics.
• The problem's resolution marked a significant advancement in enumerative geometry.

Conclusions:

• The solution to the curve-counting problem demonstrates the power of interdisciplinary mathematical and physical approaches.
• The discovered formula provides a definitive method for calculating terms in the sequence.
• This work has implications for further research in algebraic geometry and related fields.