By reading the "Advanced" book from cover to cover in the coming months, I'm hoping to bridge this gap by posting on the more commonly encountered grammar points in the text. So, without further adieu, I give you:
あえて： An adverb that expresses the speaker's or writer's desire or will to dare to do something in spite of difficulty, danger or opposition.
Meaning： Daringly; boldly; dare to ～; venture ～ ; force oneself to ～
Related expressions: 思い切って*； 強いて*
The section chief boldly expressed his opposing view to the company president.
He didn't dare object to my plan.
Similar examples, but one uses the affirmative, the other the negative. Pretty simple to grasp!
Notes: In using あえて, the person who "dares to to do something" has to be a person one is familiar with. For example 私 / 父 are acceptable, but あの通行人 is not, because the speaker/writer is not on familiar terms with the "passerby."
C例： 私 / 父 / あの通行人 / は赤信号だったのに、あえて道を渡った。
*Related expression 1: 思い切って
The key difference here is that with あえて, one dares to do something that runs contrary to common sense in spite of difficulty/danger/opposition, but 思い切って is used when one "resolutely" does something difficult to do. あえて doesn't imply strong determination. 思い切って is also used only with an affirmative predicate (ex: 君の言うことにあえて / 思い切って反対しない）. In this case, 思い切って cannot be used.
あえて= doing something against the grain/opposite of common sense.
思い切って= resolving to do something despite its difficulty.
**Related expression 2: 強いて
With 強いて, you are forcing yourself to do something, usually to chose something, but there is no opposition/danger in this. It is often used in the phrase 強いて言えば.