42nd Autumn School in Algebraic Geometry

Homogeneous Spaces, Characteristic Classes

Lukecin, Poland, September 9 - September 13, 2019

Teachers: Luis Sola Conde (Universita di Trento), Andrzej Weber (University of Warsaw)

Lecture series:

  • Luis Sola Conde: Geometry of rational homogeneous spaces.
    During these lectures we will look at rational homogeneous spaces from the point of view of projective geometry. We will start by going through their classification and the description of their Mori cones and contractions. We will then study how their cohomology rings can be written in terms of certain features of their groups of automorphisms (roots, Weyl groups), introducing Schubert cycles and -their standard desingularizations- Bott-Samelson varieties. Along the different lectures we will describe projectively many examples, of classical (Grassmannians, flags, quadrics,...) and exceptional type.
  • Andrzej Weber: Characteristic classes of singular varieties in equivariant theories.
    Spaces with torus action, GKM-spaces and the moment map (main examples toric varieties, homogeneous spaces and their subvarieties). Equivariant cohomology theories: Borel theory and K-theory, localization theorems, formula for cohomology/K-theory of GKM-spaces. Equivariant cohomology and K-theory of homogeneous spaces, Demazure operations, Schubert varieties and their fundamental classes, inductive definition of Grothendieck polynomials. Equivariant characteristic classes of singular varieties in equivariant cohomology (CSM-classes) and K-theory (motivic Chern classes). Equivariant characteristic classes of toric varieties and Schubert varieties in G/B. Action of the Hecke algebra on characteristic classes.

Prerequisites: Basic knowledge of algebraic geometry, e.g. at the level of Harstshorne's book, ch. I-III, or equivalent; basic knowledge oc characteristic classes, e.g. at the level of Milnor-Stasheff book, or equivalent.

Readings:

Program of the school: Two lectures each morning, 90 min each, followed by two 60 min excercise sessions in the afternoon.

Organizers: Aleksandra Borowka (Institute of Mathematics, Polish Academy of Sciences) and Jakub Koncki, Eleonora Romano, Jaroslaw Wisniewski (Institute of Mathematics, the University of Warsaw).

The school is supported by: Institute of Mathematics, Faculty of Mathematics, Informatics and Mechanics, the University of Warsaw and by research projects Algebraic geometry: varieties and structure (2013/08/A/ST1/00804) and Complex contact manifolds and geometry of secants (2017/26/E/ST1/00231)

Venue: The school will take place in a Warsaw University pension in Lukecin, on Western part of Polish Baltic sea shore, see the map. The accommodation (full board, shared double room) will cost about 120 zloty (PLN) a day (1 Euro is approx. 4.3 PLN, but the exchange rate is not fixed). A fee of 50 PLN will apply.

Graduate students and young researchers with inadequate support from their home institutions are encouraged to apply for accommodation cost waiver (we may ask you to provide additional letter of recommendation). The organizers will not pay for participants' travel.

Travel to Lukecin: The closest airport is in Szczecin (Poland), you may consider also traveling through Berlin (Germany) and then taking a train to Szczecin (it takes about 2hrs), see the German railway page, or a bus, see busses from Berlin airports to Szczecin and Koszalin The organizers will provide a bus from Szczecin Glowny train station to Lukecin on the day of arrival, Sunday evening.


All inquiries should be directed to Eleonora Romano, elrom AT mimuw.edu.pl.


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