Is or Are

[jack]

The solution to these problems is immensely complex. <--why is "is" not "are"? Is it because "solution" is the subject? Why isn't "problems" the subject? How do i know which one is the subject and verb?

[Casiopea]

The solution to these problems is immensely complex. <--why is "is" not "are"? Is it because "solution" is the subject? Why isn't "problems" the subject? How do i know which one is the subject and verb?

The solution is immensely complex. (Singular subject + Singular verb)
The solutions are immensely complex. (Plural subject + Plural verb)

The phrase 'to these problems' is made up of a preposition (to), a demonstrative (these) and a noun (plural). The entire phrase is called a prepositional phrase and it modifies the word 'solution'. That is, it tells us more information about the word 'solution'.

All the best,

[jack]

"Soaring over the mountains with their snowcapped peaks is always a treat I look forward to." <---Why is "is" not "are"? How do i know?

[Francois]

Soaring over the mountains with their snowcapped peaks is always a treat I look forward to

The subject is "soaring over....", that is, the action of soaring. the part "over the mountains with their snowcapped peaks" just qualifies the action (eg. soaring over a mole-hill would not be as much fun).
So you say "soaring over [blah blah] is always a treat I look forward".
Similarly, you woud say "Driving sport cars is his favorite hobby".

FRC