"A lot of ideas" is or are?

Solution 1:

Logically you are correct; "a lot" is the subject of the sentence. Colloquially, "a lot" can also mean "many", which takes a plural verb.

If you were talking about a lot more directly as a lot, you would use the singular:

This lot of wares is being shipped to (address).
Up next at the auction is lot 6124.

For cases where I really mean "many" I try to say either "lots of" or "many" instead of "a lot of X are".

Solution 2:

The original meaning of lot as used here is "a number of units of an article, a single article, or a parcel of articles offered as one item" (Merriam-Webster) and is thus clearly singular (e.g., The driver delivered a lot of trees to the parking lot in time for Christmas).

Early on (evidently 16th century), this concrete sense began to be used in a figurative or transferred sense (e.g., 1575: "a lot of wasps" --OED) and in variations such as "a lot," "lots of," "the lot," etc.

With time and frequent use, the phrase "a lot of" has come to be considered plural. Though lot will forever remain grammatically singular, it will also always be considered plural by most readers and writers.

This presents the careful writer with a dilemma: to be grammatically correct or to blend in with the herd? Should one write "In the background of this picture there are a lot of trees" or ". . . there is a lot of trees"?

In defense of plural construction I have actually encountered the arguments: "the grammar of English has changed" and "a lot of" has become a "syntactic unit" (or a "lexical unit"), both of which I find spurious.

No, dilemmas such as this (some of which have already been noted in this thread) abound in English, and, regrettably? there can be no resolution; each writer must decide for him or herself.