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It might be too late to book a flight, but if you already are in the Bay Area, be aware that tomorrow Dan Spielman will deliver the 3rd Motwani Lecture at 11am in the CIS auditorium of the Paul Allen building, which is across the street from Gates.

Dan will talk about sparsification of graphs and matrices.

It has been too long and now there is no point telling stories from the last two days of FOCS, such as the distinguished theoretician who “flew” on the zip-line on Fremont street and then got drunk at Double Down and asked the two big scary guys who were playing pool if they were twins (they said they were).

As soon as the videos of the conference go online, however, I would suggest to everybody who wasn’t there to watch Dan Spielman’s invited talk, which was phenomenal. Dan talked about the problem of solving linear equations when the constraint matrix is the Laplacian of a graph, the result of a long-term collaboration with Shang-hua Teng. Two major parts of this research program have gone on to become their own research area:
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FOCS 2007 started yesterday in Providence with a series of tutorials.

Terry Tao gave a talk similar to the one he gave in Madrid, discussing the duality between pseudorandomness and efficiency which is a way to give a unified view of techniques coming from analysis, combinatorics and ergodic theory.

In typical such results, one has a set $F$ of “simple” functions (for example linear, or low-degree polynomials, or, in conceivable complexity-theoretic applications, functions of low circuit complexity) and one wants to write an arbitrary function $g$ as

$g(x) = g_{pr} (x) + g_{str} (x) + g_{err} (x)$

where $g_{pr}$ is pseudorandom with respect to the “distinguishers” in $F$, $g_{str}$ is a “simple combination” of functions from $\cal F$, and $g_{err}$ accounts for a possible small approximation error. There are a number of ways to instantiate this general template, as can be seen on the accompanying notes, and it is nice to see how even the Szemeredi regularity lemma can be fit into this template. (The “functions” are adjacency matrices of graphs, and the “efficient” functions are complete bipartite subgraphs.)

Dan Boneh spoke on pairing-based cryptography, an idea that has grown into a whole, rich, area, with specialized conferences and, according to Google Scholar, 1,200+ papers published so far. In this setting one has a group $G$ (for example points on an elliptic curve) such that there is a mapping $e: G X G \rightarrow G_T$ that takes pairs of elements of $G$ into an element of another group $G_T$ satisfying a bilinearity condition. (Such a mapping is a “pairing,” hence the name of the area.) Although such mapping can lead to attacks on the discrete log problem in $G$, if the mapping is chosen carefully one may still assume intractability of discrete log in $G$, and the pairing can be very useful in constructing cryptographic protocols and proving their security. In particular, one can get “identity-based encryption,” a type of public key cryptography where a user’s public key can be her own name (or email address, or any deterministically chosen name), which in turn can be used as a primitive in other applications.

Dan Spielman spoke on spectral graph theory, focusing on results and problems that aren’t quite studied enough by theoreticians. He showed some remarkable of example of graph drawings obtained by simply plotting a vertex $i$ to the point $(v(i),w(i))$, where $v$ and $w$ are the second largest and third largest eigenvalues of the laplacian of the adjacency matrix. The sparse cut promised by Cheeger inequality is, in such a drawing, just the cut given by a vertical line across the drawing, and there are nice algebraic explanations for why the drawing looks intuitively “nice” for many graphs but not for all. Spectral partitioning has been very successful for image segmentation problems, but it has some drawbacks and it would be nice to find theoretically justified algorithms that would do better.

Typically, I don’t go to an Italian restaurant in the US unless I have been there before and liked it, a rule that runs into certain circularity problems. I was happy that yesterday I made an exception to go to Al Forno, which proved to be truly exceptional.