A recent paper of myself and some collaborators studies ways to extend the tensor product of vector spaces to a classes of infinite dimensional vector spaces. An interesting feature of the construction is that there are certain choices of this extension. It seems to be possible to organize all those choices into a mathematical structure known as an operad. Operads originated in the 70s in algebraic topology and since then have found many applications in different areas of mathematics. For the first class of infinite dimensional vector spaces the operad obtained from the construction is the well-known associative operad. The goal of this project is to obtain a basic understanding of operads and tensor products and explore the operads that arise from the second and higher classes of those infinite dimensional vector spaces. These could be examples of new operads in the mathematical literature and we have reasons to believe that they have a geometric description, which would be very interesting to understand.
Math 416 and Math 417 (or Math 427)