Counting surfaces

CRM Aisenstadt Chair lectures

Autor*innen

• Bertrand Eynard

Parameter

• Mathematik

Mehr zum Buch

The problem of enumerating maps, defined as sets of polygonal "countries" on various topologies, is significant in mathematics and physics, with applications in statistical physics, geometry, particle physics, telecommunications, and biology. This issue has attracted attention from researchers in combinatorics, probability, and physics. Since 1978, physicists have developed "matrix models" to tackle this problem, yielding numerous results. Another crucial mathematical and physical challenge, particularly in string theory, involves counting Riemann surfaces, which are parametrized by a finite set of real parameters (moduli). The volume of the moduli space, a compact manifold or orbifold with complex topology, represents the number of Riemann surfaces. Additionally, algebraic geometry seeks to characterize moduli spaces by calculating their volumes and intersection numbers. Witten's conjecture, first proved by Kontsevich, posits that Riemann surfaces can be derived from limits of polygonal surfaces, suggesting that the number of maps in a specific limit correlates with the intersection numbers of moduli spaces. This book elucidates the "matrix model" method, presents key results, and contrasts it with combinatorial and algebraic geometry techniques. Aimed at graduate students, it offers comprehensive proofs, examples, and a general formula for enumerating maps on surfaces of any topology, concluding with a discussion linking the

• Sprache: Englisch
• Erscheinungsdatum: 2016