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Mathematics Colloquium

Date:
Time:
4:00 pm – 4:50 pm
Avery Hall Room: 115
1144 T St
 Lincoln NE 68508
Additional Info: AVH
Contact:
Steve Cohn, (402) 472-7223, scohn1@math.unl.edu
Speaker: Nick Hein
Affiliation: University of Nebraska-Kearney
Local Host: Judy Walker
Title: Frontiers in Schubert calculus

Additional Public Info:

<b>Abstract:</b> Let V be an n-dimensional space. There are combinatorial criteria which determine how many k-dimensional subspaces (k-planes) intersect fixed but general subspaces with the intersections having prescribed dimensions. It is a problem in Schubert calculus to find these k-planes, and it is called a Schubert problem if there are finitely many k-planes.

In 1993, B. and M. Shapiro conjectured that if the fixed subspaces defining a Schubert problem osculate a rational normal curve at real points, then the solution k-planes are all real subspaces. This was first thought too good to be true. Computations of Sottile gave strong support of the conjecture, which was eventually proved by Mukhin, Tarasov, and Varchenko. The original proof used ideas from mathematical physics, and the theorem is related to the pole placement problem.

Large computational investigations have uncovered many compelling generalizations and variants of the Shapiro conjecture, some of which are now proven. Computation has also played a big role in determining monodromy groups for Schubert problems. Conversely, the need to solve Schubert problems computationally has inspired new theorems in pure mathematics. We describe the intertwining of computation as a means of discovery with recent conjectures and theorems in Schubert calculus.

Refreshments will be served in 348 Avery, 3:30-4:00 PM.
This talk is free and open to the public.

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This event originated in Math Colloquia.