Year: 2008
Type: dissertation
Abstract: This thesis explores the intersection theory on M g,n , the moduli space of genus g stable curves with n marked points.Our approach will be via hyperbolic geometry and our starting point will be the recent work of Mirzakhani.One of the landmark results concerning the intersection theory on M g,n is Witten's conjecture.Kontsevich was the first to provide a proof, the crux of which is a formula involving combinatorial objects known as ribbon graphs.A subsequent proof, due to Mirzakhani, arises from considering M g,n (L), the moduli space of genus g hyperbolic surfaces with n marked geodesic boundaries whose lengths are prescribed by L = (L 1 , L 2 , . . ., L n ).Through the Weil-Petersson symplectic structure on this space, one can associate to it a volume V g,n (L).Mirzakhani produced a recursion which can be used to effectively calculate these volumes.Furthermore, she proved that V g,n (L) is a polynomial whose coefficients store intersection numbers on M g,n .Her work allows us to adopt the philosophy that any meaningful statement about the volume V g,n (L) gives a meaningful statement about the intersection theory on M g,n , and vice versa.Two new results, known as the generalised string and dilaton equations, are introduced in this thesis.These take the form of relations between the Weil-Petersson volumes V g,n (L) and V g,n+1 (L, L n+1 ).Two distinct proofs are supplied -one arising from algebraic geometry and the other from Mirzakhani's recursion.However, the particular form of the generalised string
Author: Norman Do
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2793766266