Is there any possible to extend the currnet infinite-dimensioal Morse theory, which can be applied to the nonlinear elliptic equations? Almost every paper I read reflects that this version of the theory is (too) well-developed. Is there still any hope to find something new?
This topic has close connection with supersymetric QM,
You usually have some nilpotent elements, the Qi expressed in terms of these, and a hamiltonian H in terms of the Qi.
When you say infinite dimensional you mean an infinite number of Qi ?
Or of what?
For example Witten has the nilpotent elements d and d* differentialoperators of the Laplace teorem in the topic of p forms.
Alternatively you can use simple 2 by 2 matrices with just 1 at a lower left or upper right corner.
Its mostly just fancy algebra, adapted to whatever problem you think about,
I think it has flexibility depending on what and where it is used.
Quantum Field theory with creation and destruction operators, used with Bosons or Fermions uses a lot of this structure.
Of course you may have to understand some Physics.
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