Algebraic systems with only real solutions

October 18 to October 22, 2010

at the

American Institute of Mathematics, Palo Alto, California

organized by

Evgeny Mukhin, Natasha Rozhkovskaya, and Vitaly Tarasov

This workshop, sponsored by AIM and the NSF, will be devoted to the study of different versions of the B. and M. Shapiro Conjecture and related questions in real algebraic geometry. The recent progress in settling several cases of the Conjecture has been achieved by a number of surprisingly different approaches including techniques of Algebraic Geometry, Complex Analysis, Representation Theory and Integrable Systems. Many of these proofs seem to be adhoc, and the reasons why they have worked are not clear. The general aim of the workshop is to attempt to solve this mistery and do determine the real power and limitations of the methods used in the area.

The main topics for the workshop are

  1. Can a combination of the existing methods solve any other cases of the B. and M. Shapiro Conjecture? For example, can we handle the B. and M. Shapiro Conjecture for gl(N) flag manifolds also known as the Monotone Conjecture? Can we solve the cases of Orthogonal and Lagrangian flag manifolds?
  2. Currently, there are several generalizations of the B. and M. Shapiro Conjecture motivated by the existing methods and by some extensive numerical experiments. Can we formulate general principles which encompass these generalizations of the B. and M. Shapiro Conjecture?
  3. We plan to discuss subjects sharpening the original version of the B. and M. Shapiro Conjecture -- such as transversality of intersections, monodromy problems, connection matrices and the corresponding combinatorics.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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