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Should "mathematics" be capitalized in the following two cases:

a Ph.D. program in (mathematics or Mathematics)

the Ph.D. program in (mathematics or Mathematics). (referred to the Ph.D. program in the Department of Mathematics in a university, mentioned earlier)

Thanks!

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Short answer: Yes for both examples, if they are being used in a formal context.

The Chicago Manual of Style (CMS) recommends writing academic degrees in lower case, except when using the degree as a title. (Grammerbook.com)

Capitalize abbreviations (B.A., M.S., M.B.A., Ph.D., etc.) and use periods. Capitalize complete names of degrees but do not capitalize when degrees are referred to informally. (pdx.edu)

When mathematics is used in a sentence without specifically referring to the program of study of a university or a title, it is not capitalized.

Supervision for MPhil/PhD research is available in computational learning theory, financial mathematics, stochastic control, search theory, and in the applications of mathematics in areas such as telecommunications and economics.

Here are some links that use both uppercase and lowercase versions of mathematics.

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