The Tamagawa number of a semi-simple simply connected group over a function field is equal to 1, and the space of connections on a principal bundle is contractible. In the 1980s, Professor Harder pointed out to me the intriguing but mysterious relationship between these two statements. A recent proof of Weil's Tamagawa number conjecture in the function field case by Gaitsgory and Lurie using derived geometry sheds new light on this old conundrum. I will report on this and related work.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/7438