So, I want to title a talk.
Which of these is the right usage and why?
"Web Development Paradigms and Djangoic approach to solve them"
"Web Development Paradigms and Djangoic approach to solving them"
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The construction is typically "approach to [noun]". The gerund can be used in place of a noun here, as it can in many other situations.
The infinitive should not be used. It is something of a coincidence that the infinitive has the prepended word "to", and that "to" is the appropriate preposition to use after "approach" here.
So to be clear, the following is correct:
For me, the first (""Web Development Paradigms and Djangoic approach to solve them") is not possible, for the reason Noldorin gave. Also, it reads oddly without an article before "Djangoic".
They are equivalently correct. But if you want to sound expensive and impressive, use the gerund form.