Executive Summary : | Skein modules are invariants of 3-manifolds introduced by Turaev and independently by Przytycki generalizing the Jones-Conway, HOMFLYPT, and Kauffman bracket polynomial link invariants. For any surface S, the skein module of the 3-manifold Sx[0,1] admits a natural product structure, and becomes an algebra, known as the skein algebra of S. Skein algebras are related to different areas of mathematics and mathematical physics e.g., topological quantum field theory, SL(2,C) character variety, skein invariants of links in 3-manifolds, Poisson algebras associated to moduli space of a surface, quantum cluster algebras etc. Although the definition of skein algebra is quite elementary, and uses basic topological properties of links in Sx[0,1], the complete presentations of various skein algebras are only known for a few low complexity surfaces. An interesting property of skein algebras of surfaces is that they quantize various Poisson algebras associated to surfaces, e.g. character varieties of S, Poisson algebras of loops on S etc. These Poisson algebras are associated to geometric objects and therefore can be studied using tools from Poisson, symplectic and hyperbolic geometry. A typical approach to study the properties of skein algebra involves the study of topology of links in Sx[0,1] along with general algebraic techniques available to study the specific property. The use of Poisson, symplectic and hyperbolic geometry to study skein algebras via quantization is a very recent topic of research and the existing literature is quite limited. Therefore the objective of the proposed project is twofold: 1) Use quantization to interpret properties of skein algebras in term of properties of Poisson algebras that they quantize. 2) Use tools from Poisson, symplectic and hyperbolic geometry to study the above mentioned properties of Poisson algebras and then use the above interpretation to obtain results about the skein algebras. The possible outcome of the project includes obtaining new results about skein algebras of S as well as obtaining alternative proofs of the existing theorems using tools from Poisson, symplectic and hyperbolic geometry of moduli spaces associated to S. If successfully executed, the proposed project would help us to enhance our understanding of skein algebras and related objects. |