## Saturday, October 16, 2004

### Some Dagstuhl Presentations

At Dagstuhl Manindra Agrawal presented recent work of his students Neeraj Kayal and Nitin Saxena (the trio that showed a polynomial-time algorithm for primality testing) on rings given by a matrix describing the actions on the base elements. They show a randomized reduction from graph isomorphism to ring isomorphism and from factoring to #RI, counting the number of ring isomorphisms. They also show a polynomial-time algorithm for determining if there are any non-trivial automorphisms of a ring and that #RI is computable with an oracle for AM∩co-AM. Agrawal conjectured that #RI is computable in polynomial time, a conjecture that would imply factoring and graph isomorphism have efficient algorithms.

We also saw a series of presentations by Andris Ambainis, Robert Špalek and Mario Szegedy. Ambainis described his improved method for showing lower bounds for quantum algorithms that provably beats the degree method. Špalek talked about his work with Szegedy showing that Ambainis techniques as well as different tools developed by Zhang, Laplante and Magniez, and Barnum, Saks and Szegedy all gave the same bounds. Szegedy, in his presentation, called this measure the div complexity and showed that the size of a Boolean formula computing a function f is at least the square of the div complexity of f.