Galois groups of Schubert problems (and "The Shape of Space" at the Math Club)Algebra & Discrete Mathematics
|Speaker:||Frank Sottile, Texas A&M|
|Start time:||Wed, Apr 17 2013, 12:10PM|
Work of Jordan from 1870 showed how Galois theory can be applied to enumerative geometry. Hermite later showed that a showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field extension, and in 1979 Harris applied this to study the Galois groups of many enumerative problems. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group. White and I have given a new proof of this based on 2-transitivity.
My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.
Note, Professor Sottile is also speaking about "The Shape of Space" in the Math Club meeting at about 5:30pm. (Bring a belt if you have one!)