Abstract: In this talk, first I will give a brief overview of my current research activities on topological data analysis, certified geometry, optimization and level set method with applications in visualization, graphics, image analysis and simulations.Then, I will present a recent work on multivariate (or multi­field) topology simplification. Topological simplification of scalar and vector fields is well-established as an effective method for analyzing and visualizing complex data sets. For multi­field data, topological analysis requires simultaneous advances both mathematically and computationally. Mathematically, weshow that the projection of the Jacobi Set of multivariate data into the Reeb Space produces a Jacobi Structure that separates the Reeb Space into components. We also show that the dual graph of these components gives rise to a Reeb Skeleton that has properties similar to the scalar contour tree and Reeb Graph, for topologically simple domains. Computationally, we show how to compute Jacobi Structure, Reeb Skeleton in an approximation of the Reeb Space, and that these can be used for visualization in a fashion similar to the contour tree and Reeb Graph.