Shifted Symplectic & Poisson Structures and their Quantisations in the context of Derived Algebraic Geometry

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The proposed PhD project is about studying examples of shifted symplectic and Poisson structures and their quantisations in the context of derived algebraic geometry. This is a timely research project connecting to modern developments in the areas of algebraic geometry, quantum algebra and mathematical physics. The techniques used in this project are obtained from pure mathematics. In the first stage of this project, the student will learn about the geometry of commutative differential graded algebras and the definitions of n-forms and derivations on these structures. These tools can then be used to define n-shifted symplectic and Poisson structures. A main example to be explored is how non-degenerate pairings on a Lie algebra and quasi-Lie bialgebra structures can be recovered from this framework. After these primary goals have been achieved, the next step will be to apply the above techniques to investigate more novel and richer examples involving higher algebraic structures, such as differential graded Lie algebras.