You might want to break up the ofs with the word "the". The department of Science of the University of Oregon. Better yet, change one. The department of Science at the University of Oregon. Having several instances of the same word in a sentence is redundant and doesn't sound very nice.
An institution is a noun describing an organization, school/university, or business. An example of a sentence using institution: Harvard University is a very prestigious institution that only accepts the finest students.
Example sentence - I want to attend an Ivy League university.
If you can get into Yale, Harvard or Stamford for example, you can be assured that you will get the best education available. If not, or if you are looking at a specific geographical area, I recommend that you check out collegexpress.com or topuniversities.com and check out their rankings. Don't worry, the little voices in my head tell me that I'm not crazy.
Example sentence - The department of education attempted to educate all children equally.
Example sentence - The meeting was mandatory for everyone in the sales department.
The state of the Department's security was truly deplorable.
One can enroll in a business school that is associated with a university. For example, Harvard University has a business school, as does the University of British Columbia and Queen's University in Canada.
Example sentence - He loves his chosen profession as a nurse in the neonatal department.
At the beginning of a sentence and when it forms part of the proper noun. Example: Harvard Law School
Example sentence - The proper forms must be completed in order for the finance department to disperse the funds.
Example sentence - He was directed to the human resources department to file his complaint at work.
The word university is a common noun so it should not be capitalized. If the word university comes in as a proper name, for example Harvard University, then university should be capitalized as it is part of a proper name.