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Mathematics for non-mathematicians: I recommend a talk by Dan Freed, "The Hodge Conjecture". You'll get a glimpse on some central ideas of geometry from this talk, even if you're not interested in the Hodge Conjecture.
content on Konrad Voelkel's Blog – Mathemagically is licensed under CC-BY-NC 3.0, if not indicated otherwise.
http://blog.konradvoelkel.de/2010/01/introduction-to-some-important-themes-in-geometry/
