Abstract. Complexity in computer science is both a challenge and a resource: we can often solve complex problems using complex mathematical objects. Many of our best algorithms rely on pseudorandom constructions — deterministic objects which sample a space of possible solutions as well, or even better, than random sampling can do, with respect to some class of functions. Building these objects has forced computer scientists to wrestle with some of the deepest and most beautiful branches of mathematics (which, of course, is a good thing in itself). It has also engendered new work in mathematics, challenging mathematicians to go from existence proofs to efficient algorithms and explicit constructions.
Polynomials over finite fields, Fourier analysis, and group theory gave us many important such constructions, including efficient error-correcting codes, secret-sharing schemes, small-bias and k-wise spaces, and explicit expander graphs. The next mathematical frontier to touch computer science is algebraic geometry — the interplay between algebraic structures like groups, matrices, and tensors, and geometric structures like polynomials and curves.
This working group will bring together an intimate group of people building bridges along this frontier. We will have 1-2 talks per day, of arbitrary length and with an arbitrary amount of interruptions, ranging from basic pedagogy on algebraic geometry to current work.