In this talk I will present the paper Multi-Dimensional Morse Index Theorems and a Symplectic view of Elliptic Boundary Value Problems by J. Deng and C.K.R.T. Jones (Trans. Amer. Math. Soc. v363 (2011) pp1487-1508). The paper discusses how symplectic geometry can be used to gain understanding of the structure of the solution space of a class of elliptic boundary value problems, and relates the multidimensional case to well-known results from the theory of Ordinary Differential Equations. If time, I will talk about my own efforts on extending this result, as well as some new unpublished advancements that have been brought to my attention.