Topics in Analysis for undergraduate level essay project

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As part of my real analysis 1 course, we are being asked to write a short (3-5 page) paper on a topic connected to analysis. The professor has provided ideas such as optimal transport theory, information theory, probability theory etc. However upon looking up these topics, they don't feel quite accessible given the half semester's worth of background in analysis I have. I was wondering if anyone could suggest some references to topics or problems in real analysis, preferably those with some history and open problems remaining.