Are there any semantic or grammatical between "past year" vs "last year" vs "previous year'? Are they similar from the definite article point of view? (last year vs the last year). I know that when used with "the" it means the last year before the time of the sentence (relative time) but when used as without "the" it means the previous calendar year (absolute time)
The expressions the past year and the last year both generally refer to the previous 12 months although they are sometimes used in different contexts.
While the past year always means (roughly) the 12 months up until now, the last year can be used in constructions such as:
The last year of their lives/marriage etc
Last year, without the article, refers to the calendar year previous to the current year.
However, when reporting what people have said, last year would refer to the year prior to that in which the speech was made:
Speaking in 2010, the president said: Last year our economy grew.
Last year too is often used broadly rather than exactly, and to refer to a period during the year rather than the whole year.
To say that she was ill last year would be taken to mean at some point or duration during the year and not for all 365 days.
The previous year (or the year before) refers to the calendar year prior to the year being discussed.