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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Interview with Vladimir Voevodsky in Russian contain some mystical stuff

http://baaltii1.livejournal.com/200269.html

Very interesting…

Anyone interested in Voedvosky’s foundational work, or who has viewed the video of his talk in 2010, should also take a look at the discussions on the FOM email list in May of last year, which included much discussion of this talk and his foundational views. One can start with this post, if one likes: http://www.cs.nyu.edu/pipermail/fom/2011-May/015407.html

What’s the advantage of basing the foundations on homotopy theory?

I think it is more type theory itself which has applications in automated theorem proving. The main general theorem proving environment is Coq, I think, based on type theory, and which can interface well with human-style proofs, because of the ease of making definitions.

I also think the connection to homotopy theory is not very useful in applications yet, it is more something interesting to explore mathematically. And it seems to provide a natural framework for the connection between logic and algebraic geometry/topology. So perhaps we can say it is useful psychologically, it clarifies our ideas of what is possible.

This is a quote from Awodey’s survey on the subject:

“The homotopy interpretation of Martin-Löf type theory into Quillen model categories, and the related results on type-theoretic construc- tions of higher groupoids, are analogous to the basic results inter- preting extensional type theory and higher-order logic in (1-) toposes. They clearly indicate that the logic of higher toposes—i.e., the logic of homotopy—is, rather remarkably, a form of intensional type theory.”

(I only comment this because nobody replied. If someone can make a real answer instead, please do.)

The mystical bent of the two part interview with Mikhailov is somewhat shocking. For those who might be curious, Voevodsky himself takes part in the extensive discussion attached to the interviews under the screen name vividha. On, the other hand, from what I can tell, the mystical and mathematical parts appear to be disjoint for Voevodsky himself.

http://www.wired.com/business/2012/08/ff_wallstreet_trading/all/

Deserves a separate note.

P. asked: “What’s the advantage of basing the foundations on homotopy theory?”

From the point of view of formal logic it makes the whole theory more coherent to regard also the notion equality “constructively”, hence to consider intensional identity types and thus homotopy type theory. See

http://ncatlab.org/nlab/show/homotopy%20type%20theory

for more. What is remarkable, and this was Voevodsky’s insight, is that with this assumed the formal logic is automatically one that describes homotopy theory in its modern guise of “infinity-category theory”. This is remarkable since the latter is seen more and more to be of fundamental relevance in many areas of mathematics.

And of fundamental physics. Adding one extra axiom to Voevodsky’s homotopy type theory, that of “cohesion”

http://ncatlab.org/nlab/show/cohesive%20homotopy%20type%20theory

makes a large chunk of aspects of quantum field theory appear pretty straightforwardly from the formal logic itself, see

http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos .

Some more dedicated discussion of the relation between _Quantum gauge field theory in Cohesive homotopy type theory_ is now here

http://ncatlab.org/schreiber/show/Quantum+gauge+field+theory+in+Cohesive+homotopy+type+theory