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On Thursday Shou-wu Zhang concluded his lecture series by talking about the higher genus case , and in particular focusing on some recent work of his which is related to the effective Mordell conjecture and the abc conjecture. The higher genus case is substantially more difficult than the genus 0 or genus 1 cases, and one often needs to use techniques from many different areas of mathematics (together with one or two unproven conjectures) to get somewhere.
This is perhaps the most technical of all the talks, but also the closest to recent developments, in particular the modern attacks on the abc conjecture. (Shou-wu made the point that one sometimes needs to move away from naive formulations of problems to obtain deeper formulations which are more difficult to understand, but can be easier to prove due to the availability of tools, structures, and intuition that were difficult to access in a naive setting, as well as the ability to precisely formulate and quantify what would otherwise be very fuzzy analogies.)