For my PhD thesis I investigated what conditions turn a particular Gerstenhaber Algebra into a Batalin-Vilkovisky (BV) algebra. More precisely, let $\mathcal{A}=\oplus_i \mathcal{A}\^i$ be a Gerstenhaber algebra generated by $\mathcal{A}^0$ and $\mathcal{A}^1$ as an associative suber-algebra. Given a degree $-1$ operator $D$ on $\mathcal{A}^0\oplus\mathcal{A}^1$, I investigated a condition on $D$ that makes $\mathcal{A}$ a BV-algebra.

When applied to the Gerstenhaber or BV algebra associated to a Lie algebroid, this result gives a global proof of the correspondence between BV-generators and flat $A$-connections obtained Ping Xu.

Currently, I am working on quantization of a special type of classical dynamical r-matrices, called a {\it Classical Dynamical $r$-matrix Coupled with a Poisson Manifold}. In general, classical dynamical $r$-matrices are the solutions of so-called Classical Dynamical Yang-Baxter Equation (CDYBE), which is a classical counterpart of Quantum Dynamical Yang-Baxter Equation (QDYBE). A fundamental question is whether a classical dynamical $r$-matrix can always be quantized. In the non-dynamical case, the positive answer was found by Etingof and Kazhdan. In the dynamical case, quantization has been established for many special classes of a classical dynamical $r$-matrix, but the general case still remains open.