Step-by-step Solution:

Let's analyze each sentence for subject-verb agreement: I. 'Neither the principal investigator nor the associates was...' is incorrect. In a 'neither... nor' structure, the verb must agree with the subject closest to it. Since 'associates' is plural, the verb should be 'were'. II. 'A series of workshops... are' is incorrect. The subject is 'A series', which is singular. Thus, the verb should be 'is'. III. 'The majority of the manuscript has been revised' is correct. Here, 'majority' refers to a portion of a singular, uncountable noun ('the manuscript'), so it correctly takes the singular verb 'has'. IV. 'Each of the participants were...' is incorrect. 'Each' is a singular indefinite pronoun, so it requires the singular verb 'was'. Therefore, only sentence III is correct.