Master
2021/2022
Mathematics of Science
Type:
Compulsory course (Mathematics)
Area of studies:
Mathematics
Delivered by:
Faculty of Mathematics
Where:
Faculty of Mathematics
When:
1 year, 1, 2 module
Mode of studies:
offline
Open to:
students of one campus
Instructors:
Alexander S. Tikhomirov
Master’s programme:
Mathematics
Language:
English
ECTS credits:
5
Contact hours:
60
Course Syllabus
Abstract
The first part of the course will be cover basic notions and results in calculus on manifolds with applications to differential equations and classical mechanics. In particular, we will review integration of differential forms, the Stokes theorem, vector fields, Noether’s Theorem. Time-permitting we will discuss vector bundles, connections, curvature, the de Rham cohomology and the Chern – Weil theory.
Learning Objectives
- To bring together and to shape systematically the basic notions of differential geometry
Expected Learning Outcomes
- Ready to study more advanced courses including Differential geometry, Morse theory, Differential (and some chapters of algebraic) topology and probably some more
- Show and teach the application of modern methods of mathematics to the problems of modern natural Sciences
Course Contents
- Manifolds.
- Riemannian Manifolds.
- Affine Connections.
- Vector Bundles.
- Connections on a Vector Bundle
- Connection, curvature, and torsion forms
- Geodesics.
- Exponential maps
- Distance and volume.
- Operations on vector bundles
- Vector-valued forms.
- Connections and curvature again
- Characteristic classes
- Pontrjagin classes. The Euler class and Chern classes.
- Some applications of characteristic classes
- Principal bundles
- Connections on a principal bundle
- Curvature on a principal bundle
- Characteristic classes of principal bundles
- Smooth manifolds
- Vector bundles
- Vector fields
- Differential forms
- Integration of differential forms
Interim Assessment
- 2021/2022 2nd moduletotal grade = 0,3(grade for homework)+0,2(grade for midterm exam)+ 0,5(grade for final exam)
Bibliography
Recommended Core Bibliography
- Spivak, M. (1998). Calculus On Manifolds : A Modern Approach To Classical Theorems Of Advanced Calculus. New York: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421137
Recommended Additional Bibliography
- Reed, M. (1972). Methods of Modern Mathematical Physics : Functional Analysis. Oxford: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=567963