Have any significant/notable academic advancements (like new theorems or proofs in mathematics, or new scientific concepts, etc.) emerged from discussions on the Stack Exchange network?
MathOverflow, the question and answer site for professional mathematicians in the Stack Exchange network, has a topic on their Meta about this: Best of MathOverflow, or papers inspired by MathOverflow
This is a place to collect MathOverflow success stories!
Was some of your research inspired by something on MathOverflow? Do you know questions & answers that led to interesting research? MathOverflow citations? Open problems solved on MathOverflow?
As Glorfindel said, Math Overflow has led to a number of published papers, but it's not the only site in which this has happened. Mathematics has inspired or directly lead to several papers over the years. Other sites have had posts cited by academic papers (e.g. Theoretical Computer Science), and I would be very surprised if there aren't many instances of this across the network.
Also see answers to Academic papers using Stack Exchange data.