I consider the algebraic realization problem for G vector bundles with my adviser Prof. Heiner Dovermann. In 1973 Tognoli proved Nash’s conjecture that closed manifolds are algebraically realized. He reduced the question to a bordism problem that had previously been solved by Milnor. Various authors (Dovermann, Masuda, et al.) extended the study to closed manifolds with compact Lie group actions, which include the realization of equivariant vector bundles. I investigate the algebraic realization conjecture for equivariant vector bundles over a 2-sphere. I am also interested in doing future work in applied algebraic topology, more specifically applications to image processing and neuroscience.
Why do I care about vector bundle isomorphism classes?
In 1973, Tognoli proved Nash’s conjecture using cobordism techniques developed by Milnor. In the 1980’s, Akubulut and King, along with Benedetti extended the algebraic realization problem to vector bundles over closed manifolds.
In the 1990’s, others, including Dovermann and Masuda started to consider vector bundles whose base spaces are closed manifolds with group actions on them. We’d like to know when these structures can be approximated by a nonsingular algebraic set.
Note that compact Lie groups acting on representatives of the same vector bundle isomorphism class act identically (assuming that the bundles are equivariant).