Equivariant cohomology in algebraic geometry

General data

Course ID:	1000-1M23EK
Erasmus code / ISCED:	11.0 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0540) Mathematics and statistics, not further defined The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title:	Equivariant cohomology in algebraic geometry
Name in Polish:	Ekwiwariantne kohomologie w geometrii algebraicznej
Organizational unit:	Faculty of Mathematics, Informatics, and Mechanics
Course groups:	Elective courses for 2nd stage studies in Mathematics
Course homepage:	https://www.mimuw.edu.pl/~aweber/ekwga
ECTS credit allocation (and other scores):	6.00 Basic information on ECTS credits allocation principles: the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS; the student’s weekly hourly workload is 45 h; 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes; weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS; work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load. view allocation of credits
Language:	English
Main fields of studies for MISMaP:	mathematics physics
Type of course:	elective monographs
Requirements:	Algebraic methods in geometry and topology 1000-135MGT Algebraic topology 1000-135TA
Prerequisites:	Algebraic Geometry 1000-135GEA Differential geometry 1000-135GR
Prerequisites (description):	Singular cohomology theory or de Rham theory, basic differential geometry and algebraic geometry.
Mode:	Classroom
Short description:	"Equivariant Cohomology in Algebraic Geometry" Borel equivariant cohomology theory is introduced topologically and through differential methods. The theory is applied to classical objects of algebraic geometry, such as flag varieties and Grassmannians. Properties of projective manifolds are presented. Applications are based on equivariant formality and localization theorem for torus action. Equivariant Schubert calculus is discussed.
Full description:	Torus action on vector space, weights, characters. Basic information about actions of connected Lie groups on smooth manifolds, Lie algebra actions. Slice theorem, equivariant CW-complexes. Principal bundles, classifying spaces, Stiefel manifolds. Borel equivariant cohomology, computations for homogeneous spaces (Grassmann manifolds, Flag varieties). Differential interpretation of equivariant cohomology. Weil Algebra, connection, Mathai-Quillen twist, de Rham - Cartan theory. Equivariant bundles and equivariant characteristic classes. Equivariant formality in cohomology. Formality of projective varieties. Borel localization theorem for torus action. Atiyah-Bott-Beline-Vergne localization theorem, integration formula, Duistermaat-Heckman formula Hamiltonian actions. GKM spaces, Chang-Skjelbred lemma. Moment map. Application of the localization theorem to computing Eukler characteristic of equivariant bundes. Equivariant Schubert calculus.
Bibliography:	D. Anderson, W. Fulton: Equivariant Cohomology in Algebraic Geometry V. Guillemin, S. Sternberg: Supersymmetry and Equivariant de Rham Theory, Springer 1999
Learning outcomes:	Student learns basic notions of equivariant cohomology theory. Topological construction is understood and compared with the construction based on differential geometry. Student knows application to algebraic geometry, in particular for homogeneous spaces. Student achieves knowledge of the discipline allowing to start independent research.
Assessment methods and assessment criteria:	1/3 solving problems during classes 1/3 essay 1/3 oral exam

Classes in period "Winter semester 2023/24" (past)

Time span:	2023-10-01 - 2024-01-28	Selected timetable range: weekly course term Navigate to timetable MO WYK-MON TU CW W TH FR
Type of class:	Classes, 30 hours more information Monographic lecture, 30 hours more information
Coordinators:	Andrzej Weber
Group instructors:	Andrzej Weber
Students list:	(inaccessible to you)
Examination:	Examination

Course descriptions are protected by copyright.
Copyright by University of Warsaw.