A typical FOCS day divides up the same way as a day in a cricket test match: pre-lunch session, afternoon session and post-tea session. And approximately the same fraction of the audience is awake and/or paying attention?
I set myself the goal of attending the first talk of the day, which naturally meant that I stumbled in around halfway through Manindra Agrawal's talk on "Arithmetic Circuits: A Chasm at Depth Four" (Agrawal and Vinay). This is one of my favorite papers in the conference: the result is surprising, easy to state and not hard to prove. The authors show that an exponential lower bound on arithmetic circuit size for depth-4 circuits actually implies an exponential lower bound on circuit size for general arithmetic circuits. There are precedents for this kind of depth reduction for arithmetic circuits (unlike for Boolean circuits), eg. the Berkowitz-Skyum-Valiant-Rackoff result that polynomial size circuits can be simulated by polylogarithmic-depth circuits, but the nice thing about this result is that it gives an explanation for why most recent progress on arithmetic circuit lower bounds has been in the very low depth regime. Manindra's talk was marred a little by technical glitches, but he has an unparalleled ability to stay cool in a crisis.
Then came an excellent talk on Madhur Tulsiani on "Dense Subsets of Pseudorandom Sets" (Reingold, Trevisan, Tulsiani and Vadhan). This is a topic that Luca has said a lot about in his blog, and I certainly can't hope to better that. The final talk in the session was by Timothy Chow on "Almost-natural Proofs", about how the natural proofs framework of Razborov and Rudich ceases to be a barrier to proving circuit lower bounds if it is relaxed slightly. The talk was good, but I was even more intrigued by the speaker. He is unusually wide-ranging in his choice of problems to work on - he comments quite often on the FOM mailing list, and I know he's worked among other things on logics for polynomial time. It's good for our heavily algorithms-and-complexity centered community to have people from outside the mainstream working on our problems.
You'll notice that most of the talks discussed so far are complexity-related - old habits die hard. But I did try compensating by going to Mihai Patrascu's talk on "Dynamic Connectivity: Connecting to Networks and Geometry" (Chan, Patrascu and Roditty), about data structures for dynamic connectivity in graphs where both vertex and edge updates are allowed. There did seem to be some elegant ideas involved, but the talk was geared for an audience which knew something already about dynamic data structures, and I didn't follow the details. This reminds me of why I don't tend to go to talks too far afield of areas I work in - presenters do tend to take certain things for granted in their audience, and it makes sense for them to do so, since most members of the audience probably have a reasonable background in the subject of the talk. Also, it wasn't the most enthusiastic of talks, but I am looking forward to Mihai's talks on "Succincter" and "(Data) Structures", both of which concern very natural questions that are interesting even to non-specialists in data structures.
Next for me was another algorithmic talk: Chris Umans on "Fast Modular Composition in any Characteristic" (Kedlaya and Umans), which showed that the composition of two polynomials f and g of degree n modulo a third polynomial of the same degree can be computed very efficiently, in time that's nearly linear in n. The proof goes through an earlier reduction by Chris of the problem to simultaneous evaluation of multilinear polynomials on several points; the new contribution is a clever solution to this simultaneous evaluation problem through recursively applying Chinese remaindering. The talk, as is usual with Chris, was crisp and well-organized.
That concluded my pre-lunch session – I wouldn't have been alert enough to do justice to the remaining talks. Stay tuned for a report on the rest of the day's play.