McShane identities for finite-area convex real projective surfaces
Although Teichmueller theory began as the study of Riemann surface structures, one popular modern approach is via hyperbolic surfaces. Every point in the Teichmueller space T(S) describes a different possible hyperbolic structure on the topological surface S. Hyperbolic geometry allows us to define geometrically meaningful coordinates for T(S), such as length and twist coordinates, which explicitly describe the underlying hyperbolic structures on S. One major success story in this direction, is that of McShane discovering geometric identities which are valid for all cusped hyperbolic surfaces and Mirzakhani's later generalization and application of these identities to prove Witten's conjecture and to study the growth rates of the number of non-self-intersecting closed geodesics on hyperbolic surfaces.
Another popular approach to Teichmueller theory is more algebraic: the hyperbolic structure on a surface S may be encoded as a SL(2,R) representation of the fundamental group of S. This approach lends itself to natural generalizations of Teichmueller theory where we increase the rank of SL(2,R) to SL(n,R). For n=3, there is a geometric interpretation of this higher (rank) Teichm\"uller theory as the theory of strictly convex real projective structures on S. We show that there is a generalization of McShane's identity to this context: a type of infinite-sum trigonometric identity which holds for all cusped convex real projective surfaces. This is work in collaboration with Zhe Sun (YMSC).
Dr Yi Huang, Tsinghua University