Case theory explains how noun phrases receive morphological case (nominative, accusative, dative, etc.) through syntactic position and grammatical relations. In Minimalism, case is checked via feature agreement between functional heads and their dependents; failure to check case results in ungrammaticality.
Examine languages with rich case systems (Finnish, Japanese, German) to see how case tracks argument roles; study case-marking patterns in complex structures like passives and ditransitives.
Case is not merely morphology; abstract case exists even in languages with no morphological case realization and governs syntactic well-formedness.
From your study of morphological structure, you know that case endings on nouns mark grammatical roles — in Latin, *puella* (nominative, subject) vs. *puellam* (accusative, object). From your study of the Minimalist Program, you know that grammar is driven by feature checking: syntactic operations apply to satisfy formal requirements on lexical items. Case theory brings these two threads together: case features are a core mechanism through which noun phrases are licensed to appear in particular syntactic positions.
The key insight of abstract case is that case is not just a surface morphological phenomenon but a syntactic requirement that holds even when there is no visible marking. In English, nouns don't inflect for case on their surface form — "the dog" is the same whether subject or object. But English still obeys case constraints. Consider: "I expect him to win" (grammatical) vs. *"I expect he to win" (ungrammatical). The pronoun following "expect" must be accusative (*him*, not *he*), even though nothing visible on "him" tells you the syntactic structure. Abstract case theory says that every overt noun phrase must receive case, and that certain positions assign case while others don't. A noun phrase without case — unvalued case feature in Minimalist terms — causes the derivation to crash.
How does case get assigned? In the Minimalist Program, case is valued through Agree: a functional head like T (Tense) or v (little-v, the verb phrase–introducing head) has a case feature that probes downward, finds a goal (a noun phrase with unvalued case), and values it. Nominative case is assigned to the subject by T in its specifier position; accusative case is assigned to the object by v or by V itself. Dative case is assigned by a different functional head, often P (preposition) or by a ditransitive verb structure. In languages with richer morphology — German, Finnish, Russian, Japanese — these abstract assignments surface visibly, giving you direct evidence for the underlying structure. Languages like English "hide" the system but still follow its logic.
One of the most instructive case environments is passivization. In an active sentence "Mary kicked John," John receives accusative case from the verb. In the passive "John was kicked (by Mary)," the verb is morphologically changed — its accusative case-assigning capacity is absorbed by the passive morphology. John now must receive nominative case from T, which forces movement to the subject position. Abstract case theory thus explains movement: noun phrases move not randomly, but to reach positions where they can get their case valued. This same logic explains a range of cross-linguistic patterns, including why subjects of infinitival complements must sometimes appear as accusatives (they can't get nominative from the infinitival T, which lacks tense and therefore can't assign it) and why there are restrictions on what can appear in subject position in certain clause types.