Schubert Calculus
3-4 hours of lectures per week.
Lecturer
Anders Buch
Content
Schubert calculus is concerned with the study of intersections of general subvarieties in projective space or flag varieties, and has been one of the motivating factors in the development of algebraic geometry and intersection theory. A typical problem is the following: given four randomly chosen lines in a three dimensional space, how many lines are there that intersect all of the chosen lines? More generally one can ask how many subspaces of dimension $m$ intersect $mn$ chosen subspaces of dimension $m$ inside a complex vector space of dimension $m+n$. To answer such questions requires a good understanding of the cohomology ring of a Grassmann variety, and in particular of the combinatorial properties of this ring.
The foundations of Schubert calculus requires some facts from algebraic geometry and cohomology theory. In this course I will mostly grant these facts, and in stead concentrate on the applications and the associated combinatorics. In a similar fashion one can study the (small) quantum cohomology ring of flag varieties, which can be used to compute the number of curves in such varieties that satisfy given conditions. Subject to time constraints we may also explore $K$-theory and the theory of degeneracy loci.
Prerequisites
Algebra 2 and Topology.
Literature
1. William Fulton,
Young tableaux, Cambridge University Press, 1997.
2. L.Manivel,
Symmetric Functions, Schubert Polynomials and Degeneracy Loci, AMS, 2001.
Evaluation
Students who do not intend to take a degree in Mathematics or Statistics from the University of Aarhus, but wish to earn credits for a 2.dels course from the Department of Mathematics, should indicate at the beginning of the course that they wish to be examined.
The form of examination for these students will be active participation together with oral or written contributions.
Credits
10 ECTS
Quarter
Spring 2004