Use an en dash rather than a hyphen between University of Wisconsin and Madison. Spell out on first reference in external 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 oficial 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