Solution: Infirmary

SOLUTION:

Each of the 21 pictures represents the lining of an academic hood worn in a Ph.D. or Masters ceremony. Hood linings represent an academic institution (each institution has a distinctive hood lining - MIT, for instance, is red with a silver-grey chevron).

The flavortext relevant to the puzzle theme is clued by "Doc" (cluing the doctorate and academia) and "my head is cold" (cluing a hood).

Lists or pictures of hood linings are, surprisingly, not present online, thus making the puzzle a little more difficult. An example page of what IS online is here.

Thus the reference materials I used were books. Thanks to Jill Bogard, Director of Library & Information Service of the American Council on Education, and UC Davis librarian Daniel Goldstein.

"Academic Heraldry in America" by Kevin Sheard, Northern Michigan College Press, 1962 (LB2389.S539, available in the MIT Humanities library reference collection, which is good because it can't be checked out; this was my primary reference); (UCD LB2389.S5)
"Scholars on Parade" by David Lockmiller, 1969 (LA226.L816, available in the MIT Humanities library stacks);
"The Degrees And Hoods of the World's Universities and Colleges" by Frank Haycraft, W.E. Baxter, Lewes, Sussex, England, 1972 - this one is unique - it's individually numbered and made for libraries only, and does not seem to be in the MIT libraries.

(Alternatively, I would expect that even in the absence of books, it would not be difficult to start calling up universities and asking what their hood colors are once the universities are narrowed down.)

The next part of the puzzle is to narrow down the list of universities since, for any particular color combination, there are often several universities that correspond to the particular color. (The actual colors used in the 21 pictures have mostly been replicated by visiting the institution's web site and using their university colors.) For example, picture 15 (orange with black chevron) is most often associated with Princeton, but in this case is not Princeton.

The list of potential universities can be narrowed down to the land grant institutions. Each state was given a land grant in exchange for that state establishing a university. About half of these universities have "State" in the name and concentrate on agriculture and engineering. The list I used is here.

The land grant universities were chartered by the Morrill Act of 1862. Googling "1862 colleges" or "1862 universities" has this as the first link:

"1862 Land Grant Colleges and Universities"

The land grant schools are clued by "'62" (1862) and "more ill" (Morrill). Googling "moreill 1862" gets you a "Do you mean ..." that gets you to the right place.

The linings chosen for the puzzle are unique among the land grant schools, which have a handful of conflicts. (Florida/Auburn, for example, and Kentucky/Utah State.) The list is below:

University of Arizona
Louisiana State
University of New Hampshire
Purdue
Kansas State
University of Vermont
University of Wyoming
University of Minnesota
University of Nebraska
University of Wisconsin
University of Georgia
University of Maryland
University of Illinois
University of Tennessee
Oklahoma State
Oregon State
Mississippi State
Cornell
Virginia Tech
Michigan State
University of California (Berkeley)

The final piece of the clue is to use the locations of the schools, clued by "I don't know in what city I am", and to read diagonally down the city names, clued by "slanted". This made the puzzle much easier to write given the letters available and also avoided the difficulty of how to represent a university - "University of Arizona"? "Arizona"? "Arizona University"?


Tucson
bAtonrouge
duRham
wesTlafayette
manhAttan
burliNgton
Laramie
mInneapolis
liNcoln
madIson
atheNs
colleGepark
Urbanachampaign
kNoxville
stIllwater
corVallis
Starkville
iThaca
blAcksburg
easTlansing
berkEley

TARTAN LINING UNIV STATE

The only school with a tartan hood lining in the United States is Carnegie Mellon University, which lines its hoods with Andrew Carnegie's ancestral tartan.

CMU is located in PENNSYLVANIA, the answer to the puzzle.