Course Identification
Analysis on Manifolds, Local and Global -Preliminary Program
Lecturers and Teaching Assistants
Prof. Sergei Yakovenko
Course Schedule and Location
Second Semester
Monday, 09:15 - 11:00
22/03/2021
Field of Study, Course Type and Credit Points
Mathematics and Computer Science: Lecture; Elective; Regular; 2.00 points
Prerequisites
Familiarity with the multidimensional calculus and firm knowledge of the linear algebra.
Attendance and participation
Grade Breakdown (in %)
The exam will be given at 9.7.21 and you will have one month.
Estimated Weekly Independent Workload (in hours)
Syllabus
- Notion of a manifold: topological manifolds and smooth manifolds.
One-dimensional and two-dimensional manifolds: classification.
Other examples: spheres, real projective spaces, Grassmanian varieties,
Lie groups.
- Notion of a submanifold. Examples of submanifolds in R^n. Implicit function theorem. Real plane curves. Real surfaces in R^3.
- Example: smooth complex analytic (and/or algebraic) varieties in a
complex projective space CP^n as smooth manifolds.
- Homology groups: intuitive definition. Examples: homology of 2-dimensional oriented manifolds. Intersection in homology. Pairing in middle homology. Possible examples: vanishing cycles; curves on complex surfaces.
- Infinitesimal theory: Tangent space to a manifold. Vector bundles on manifolds. Tangent and cotangent bundle. Their sections: vector fields and 1-forms. k-forms on a manifold.
Lie derivative on functions, vector fields and 1-dorms.
- Integration: change of variable formula suggests that the natural object to integrate are n-forms. Stokes theorem.
- de Rham cohomology: Poincare lemma and de Rham complex. Global sections of the de Rham complex and definition of the de Rham cohomology group.
Computation of de Rham cohomology of a circle and of a torus.
Learning Outcomes
Upon successful completion of this course students will be able to:
- Demonstrate knowledge of the invariant language of analysis in the coordinate-free form that is used in modern mathematics from differential geometry to mathematical physics.