Given a subfactor N of finite index of a factor M, the standard invariant is the direct sum for n from 0 to infinity of the zeroth cohomology of the nth. tensor power of M over N, viewed as an N-N-bimodule. This graded vector space has a lot of structure, in particular that of a ring coming from the simple tensor product. This bears a striking resemblance to the "canonical ring" of algebraic geometry. Indeed it has been shown that the completion of this canonical ring gives a canonical subfactor whose standard invariant is the same as the original one! This structure will be explored in all its detail, especially the planar algbra structure on the standard invariant which is used to describe and analyse the canonical ring. Various combinatorial structures such as Hadamard matrices and Latin squares give rise to these canonical rings but there are serious problems of computational complexity. The large n limit of random nxn matrices also gives such structures. Technically the tensor product over N referred to above is the Connes tensor product and we are looking at how this tensor product may be directly relevant to quantum physics.
Fock space is the mathematical engine of second quantization, a process that, among other things, is necessary to make quantum mechanics compatible with special relativity. There are fermionic and bosonic versions. In our research we are pursuing a more general Fock space based on a subalgebra and an algebra. The small algebra corresponds to the scalars in second quantization and the large one corresponds to the first quantized Hilbert space. One can no longer freely exchange particles as in the fermionic and bosonic situations. But annihilation and creation operators still exist and form a ring analogous to the canonical ring of a projective variety. In our situation the canonical ring/Fock space has much more structure, namely that of a planar algebra. Roughly speaking this gives operations for every map. Imagine the map of a country like the United States. Upon putting an element of the Fock space into each State on the map one would get an output for the whole country, which is another element of the Fock space. To say that our structure is a planar algebra means that such an operation on Fock space exists for every conceivable map that can be drawn on a piece of paper.
