#
Algebraic systems with only real solutions

October 18 to October 22, 2010
at the

American Institute of Mathematics,
San Jose, California

organized by

Evgeny Mukhin,
Natasha Rozhkovskaya,
and Vitaly Tarasov

## Original Announcement

This workshop 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

- 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?
- 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?
- 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.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Wronskians, cyclic group actions, and ribbon tableaux

by Kevin Purbhoo,

*Trans. Amer. Math. Soc. 365 (2013), no. 4, 1977-2030 * MR3009651Discriminants, symmetrized graph monomials, and sums of squares

by Per Alexandersson and Boris Shapiro,

*Exp. Math. 21 (2012), no. 4, 353-361 * MR3004251