New Math Revives Geometry’s Oldest Problems Overview A relatively young mathematical theory has led to revival in the ancient field of enumerative geometry—the study of counting the number of geometric objects that satisfy certain conditions, such as how many circles touch three others at exactly one point each. This article by Joseph Howlett explores how mathematicians have applied modern techniques to long-standing questions, offering not just answers but deeper insights into various numerical systems. --- Historical Context Ancient Greeks like Apollonius of Perga studied problems on counting geometric objects (e.g., up to 8 circles tangent to 3 given circles). Enumerative geometry flourished but lost interest by the mid-20th century as mathematicians focused more on abstract theory. The field had a brief revival due to string theory applications in the 1990s but faded again afterwards. Resolving enumerative problems in different number systems (like integers or real numbers) remained a challenge. --- The Modern Revival Breakthrough Insights Mathematicians Kirsten Wickelgren and Jesse Kass applied motivic homotopy theory to enumerative geometry, stemming from an insight based on a 1977 paper by Harold Levine and David Eisenbud. Levine and Eisenbud’s work involved quadratic forms (polynomials with terms of degree 2) which encode solution counts through something called a signature (difference between counts of positive and negative terms). Motivic homotopy theory views solutions as mathematical spaces and connects these with quadratic forms, offering a powerful framework applicable to any number system. Methodology Instead of directly solving equations, they translate enumerative problems into questions about spaces of functions related to these equations. Applying motivic theory yields a quadratic form summarizing the counting problem. For complex numbers, counting variables in the quadratic form equates to the number of solutions. For real numbers, the signature of the quadratic form provides a lower bound on the number of solutions. For exotic number systems—such as finite fields or modular arithmetic—properties like the determinant of the quadratic form's matrix reveal proportions of solutions with certain geometric characteristics. Key Results Demonstrated for classical problems, e.g., a cubic surface always has 27 lines over complex numbers; their method confirmed this and provided new information in other numeric contexts. Provided a unified theory that encompasses all number systems previously treated separately. --- Significance and Impact Opens new avenues for research into how geometric problems behave over different number systems, enriching understanding of the mathematical landscape. Offers concrete, accessible ways to explore highly abstract concepts, attracting young mathematicians. Still many mysteries remain regarding the full interpretation of quadratic forms in various contexts. Renewed interest in old enumerative problems fueled by new conceptual tools and connections to other fields like algebra, topology, and number theory. --- Related Topics and Further Reading Connections with string theory revisited in new number systems. Articles on algebraic geometry, number theory, and related mathematical areas. "What Are Sheaves?" and "A Mathematician Dancing Between Algebra and Geometry" from Quanta Magazine. --- About the Author Joseph Howlett is a staff writer for Quanta Magazine. --- Visuals & Media Images of mathematicians like Sheldon Katz, Kirsten Wickelgren, Jesse Kass. Diagrams illustrating geometric problems (e.g., circles and lines). Video demonstrating the classical theorem about 27 lines on cubic surfaces. --- Summary This article highlights a transformative mathematical development where modern tools, particularly motivic homotopy theory and quadratic forms, revive