Mini-school: "SCHUBERT VARIETIES"
(partially supported by EAGER)
Invited speakers:
Michel Brion (Grenoble)
Anders Skovsted Buch (Aarhus)
Richard Rimanyi (Budapest/Ohio)
Harry Tamvakis (Waltham/Essen)
Time: 17.05.2003 (arrival day) - 23.05.2003 (departure day)
Scientific Program:
The main
goal of the school is to introduce young researchers to different aspects of
Schubert varieties. There will be two courses of five 60 minutes lectures
each:
Michel Brion: Geometry of flag varieties
Flag varieties are
projective algebraic varieties, homogeneous under an action of a linear
algebraic group. They are of importance in algebraic geometry and in
representation theory. By the Bruhat decomposition, each flag variety is
stratified into Schubert cells, isomorphic to affine spaces. Their closures are
the Schubert varieties; these may be singular, but admit nice
desingularizations.
After presenting classical structure results concerning
flag varieties and Schubert varieties, the lectures will discuss some
applications to the geometry of subvarieties of flag varieties, and to
representation theory.
Harry Tamvakis: Gromov-Witten invariants and quantum cohomology of
Grassmannians
The quantum cohomology ring QH^*(X) of a Grassmannian X
encodes the enumerative geometry of rational curves in X, in the form of
Gromov-Witten invariants. The latter are integers which count the number of
rational curves in X of a given degree which are incident to three Schubert
varieties in general position. The ring QH^*(X) is a deformation of the usual
cohomology ring of X which first appeared in the work of string theorists, and
has been studied extensively over the last decade.
The aim of my lectures is
to discuss an approach to the 3-point, genus zero Gromov-Witten invariants on
Grassmannians which uses only basic algebraic geometry, and to apply this to
obtain elementary proofs of the main structure theorems regarding QH^*(X). We
will begin with the usual Grassmannian of linear subspaces in complex affine
space, and then describe the analogue of this theory for isotropic Grassmannians
in the other classical Lie types. Most of these talks are part of a joint
project with Andrew Kresch and Anders Buch.
There will be two cycles of three lectures 45 minutes each:
Anders Buch: Combinatorial K-theory
The Grothendieck ring of vector bundles on an algebraic variety
provides a "generalization" of the usual cohomology or Chow rings,
which gives rise to a more refined intersection theory. I will
describe some results which give explicit descriptions of various
aspects of this intersection theory. This includes a combinatorial
formula for the structure constants which are obtained, when a product
of two Schubert structure sheaves on a Grassmann variety is expressed
as a linear combination of other Schubert structure sheaves. This
formula generalizes the classical Littlewood-Richardson rule, but
replaces semistandard Young tableaux with set-valued tableaux. I will
also describe a formula for the structure sheaf of a general type of
degeneracy locus, which is obtained by putting arbitrary rank
conditions on a sequence of vector bundle maps and their compositions.
Certain quiver coefficients associated with this formula are
conjectured to have signs which alternate with codimension. This
alternation of signs is part of a general K-theory phenomenon, which
for example also occurs in the above mentioned Littlewood-Richardson
rule. If time allows I will discuss a recent proof, with Kresch,
Tamvakis, and Yong, of a special case of this conjecture.
Richard Rimanyi:
Interpolation approach to Schubert and quiver polynomials
We will study numerous geometrically relevant polynomials in a
unified language, the language of "Thom polynomials for group
actions". We will focus on the special cases of double Schubert
polynomials and the polynomials representing quiver cycles
(see [Buch-Fulton]). We will apply the general (interpolation)
theory of Thom polynomials to these cases, concluding in a formula
for the quiver classes in terms of Schubert polynomials (see also
[Knutson-Miller]). The results presented are a joint work with
L. Feher. The talks will be essentially self-contained, with
many examples.
Lecture 1: classifying spaces, characteristic classes; Thom polynomials
for group actions. Lecture 2: double Schubert polynomials as Thom
polynomials. Lecture 3: Thom polynomials for quivers.
Additional lectures will be selected from the participants'
proposals.
Participants:
Katrin Appel (Wuppertal),
Michel Brion (Grenoble),
Anders Skovsted Buch (Aarhus),
Slawomir Cynk (Krakow),
Swiatoslaw Gal (Wroclaw),
Tomasz Elsner (Wroclaw),
Laszlo Feher (Budapest),
Grzegorz Kapustka (Krakow),
Michal Kapustka (Krakow),
Martijn Grooten (Nijmegen),
Gert Moustad Hana (Bergen),
Zbigniew Jelonek (Krakow),
Christian Meyer (Mainz),
Mikkel Oebro (Aarhus),
Damian Osajda (Wroclaw),
Piotr Pragacz (Warszawa),
Slawomir Rams (Krakow),
Erik Reuvers (Nijmegen),
Richard Rimanyi (Budapest/Ohio),
Andrzej Szczepanski (Gdansk),
Marek Szyjewski (Katowice),
Harry Tamvakis (Waltham/Essen),
Alexis Tchoudjem (Grenoble),
Dimitri Timashev (Moscow/Grenoble),
Francisco Leon Trujillo (Roma),
Andrzej Weber (Warszawa),
and Warsaw Mathematicians.
Accommodation:
There are two possibilities:
1. Accomodation at the Institute: price 120 zl=30 euro
per night
2. Accomodation at the University Hotel
which is in 40 minutes walk distance or 15 min by bus:
price 71 zl=18 euro per night
The participants from EU
and associated countries may apply to their home nodes of EAGER network
for financial support.