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Next Tuesday, May 18, at noon California time, I will speak (in Italian) in a free `webcast’ organized by Oilproject on the impact of new collaboration and communication tools in mathematical research. (A recording will be available after the event.)

Those who watch it live can ask questions, and I haven’t planned very specifically what I will talk about, hoping that the questions will drive the discussion. Two things that I want to talk about are:

• the story of Polymath’s combinatorial proof of the Density Hales–Jewett theorem, demonstrating the viability of a “massively collaborative” approach to mathematical research, and what it means for mathematics, both “philosophically” and practically.
• the way mathematical blogs have become an effective way to disseminate the kind of mathematical lore (the insights, the concrete ways of visualizing very abstract constructions, the facts that are “well known” to experts and “implicit” in classic papers, but impossible to see for the non-experts, etc.) that cannot be found in monographs and research articles, and that, previously, was exclusively handed out from advisors to advisees and from colleague to colleague. This will make it much easier for the brilliant students who don’t happen to be in the top schools to master their research area and make new breakthroughs.

This will be part of a series of webcasts on how new communication technologies affect the economy, news, technologies, science, etc., with some notable speakers. Last week Stefano Andreoli spoke about spinoza.it, which is roughly The Onion of Italy.

Counting Problems

Today we describe counting problems and the class ${{\bf \#P}}$ that they define, and we show that every counting problem ${{\bf \#P}}$ can be approximately solved in randomized polynomial given access to an ${{\bf NP}}$ oracle.