Vladimir Voevodsky is a mathematics professor at the IAS in Princeton, most famous for his proof of the Bloch-Kato conjecture, work which won him a Fields Medal in 2002. This conjecture relates the K-theory of fields and their étale cohomology (note that there are other, different, Bloch-Kato conjectures on special values of L-functions). For a description of Voevodsky’s ideas from 2002, see this by Soulé. The proof of Bloch-Kato was only finished later, including work by other people, for more about this see Weibel’s lectures on the proof, or Voevodsky’s talk at the IHES conference honoring Grothendieck. For a popular talk by Voevodsky, see “An Intuitive Introduction to Motivic Homotopy Theory”, video here, write-up here.
Voevodsky has had a somewhat unusual career, for an interview from 2002 where he discusses his early years in Moscow and at Harvard, see here. A recent interview with him by Roman Mikhailov in two parts has appeared (in Russian, I’m relying on Google Translate to get the gist of it) here and here. He describes what appear to be various delusional episodes, especially during a period in 2006 and 2007 when he was unable to work.
In recent years he has moved away from his work on K-theory, towards topics in applied math (for a while he was investigating population genetics) and foundations of mathematics. This year the IAS will run a year-long program he is organizing on what he calls Univalent Foundations of Mathematics. Back in 2010 he gave a popular talk at the IAS, entitled What if Current Foundations of Mathematics are Inconsistent?
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