This is an intensive short course on enumerative geometry. The course covers an introduction to intersection theory, and applies the acquired techniques to some classical problems. We will introduce the basics of intersection theory: Chow ring, Chern classes, and basics of Schubert calculus. The theoretical tools which are developed will be applied to the enumerative geometry of some Grassmannian problem and to the Thom-Porteous formula for the calculation of the degree of determinantal varieties. If time permits, we will draw connections to the representation theory of the general linear group.
It will be held by Fulvio Gesmundo and it will be online.
A preliminary version of the notes is available here.
You may want to test yourself with some exercises!
January 11, 2021: 16:30 - 18:30 CET
January 13, 2021: 16:30 - 18:30 CET
January 15, 2021: 16:30 - 18:30 CET
January 18, 2021: 16:30 - 18:30 CET
January 20, 2021: 16:30 - 18:30 CET
January 22, 2021: 16:30 - 18:30 CET
Lecture 1, 2: Basics of intersection theory. Chow ring. Grassmannians.
Lecture 3, 4: Chern classes. Schubert calculus. Enumerative problems.
Lecture 5, 6: Thom-Porteous’s Formula. Representation Theory.
A first course in algebraic geometry is recommended but not strictly required. Familiarity with the notion of algebraic variety and the Zariski topology in affine and projective space is assumed. Some familiarity with commutative algebra or algebraic topology will be helpful but not necessary.
The main reference is D. Eisenbud, J. Harris 3264 and All That (Cambridge 2016).
Lecture notes adapted from this reference will be provided (in non-final version).
Other references:
E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris Geometry of Algebraic Curves, Vol. I (Springer 1985)
L. Manivel Symmetric Functions, Schubert Polynomials, and Degeneracy Loci (SMF/AMS 1998)
W. Fulton, J. Harris Representation Theory: A First Course (Springer 1991)
Attendance is free but registration is required. To register or for any other information contact Fulvio Gesmundo (gesmundo@mis.mpg.de) or Chiara Meroni (meroni@mis.mpg.de).