As an author, you are the default copyright owner of work that you create unless you have previously entered an agreement indicating that your work belongs to someone else or you subsequently transfer your copyright to a publisher or other third party.
Employment or enrollment at Stanford does not, in most situations, impact your ownership of copyright in scholarly work you create. According to Stanford's copyright policy, "all rights in copyright shall remain with the creator unless the work is a work-for-hire (and copyright vests in the University under copyright law), is supported by a direct allocation of funds through the University for the pursuit of a specific project, is commissioned by the University, makes significant use of University resources or personnel, or is otherwise subject to contractual obligations.” The policy specifies that, following academic tradition "Stanford does not claim ownership to pedagogical, scholarly, or artistic works, regardless of their form of expression."
Works do not need to be registered with the U.S. Copyright Office in order to receive copyright protection. Copyright protection applies automatically regardless of whether a work is registered or the creator displays a copyright symbol on it.
Registration does enable the copyright owner to sue for copyright infringement and creates a public record of ownership. If a scholarly work is potentially of high monetary value or the subject of an anticipated dispute, registration is recommended. However, for most academic works, registration does not usually yield practical benefits.