Use an en dash rather than a hyphen between University of Wisconsin and Madison. Spell out on first reference in external publications or publications that will be read widely off campus. UW–Madison (with an en dash, not a hyphen) is acceptable in external publications and in all internal communication for the University of Wisconsin–Madison. Do not use the with UW–Madison (research at UW–Madison).
Also acceptable on second reference are the UW and the university (when the context is clearly UW–Madison). To prevent confusion with other UW System units, however, use the UW as a substitute for UW–Madison only when the context is clearly UW–Madison or the entity is officially named UW instead of University of Wisconsin–Madison, (UW Hospital and Clinics, UW Credit Union, UW Law School). Capitalize The only as a formal title in a formal reference, such as in the headline of a program; generally, though, lowercase the.
UW is acceptable when referring to athletics teams or departments that do not use UW–Madison as part of their official names (UW Health, UW Carbone Cancer Center); or as an abbreviated reference to the University of Wisconsin System as a whole (UW budget, students enrolled at UW institutions).
See also CMS 6.39 and 6.81.