Магистратура 2021/2022

Математические методы естествознания

Статус: Курс обязательный (Математика)

Направление: 01.04.01. Математика

Кто читает: Факультет математики

Где читается: Факультет математики

Когда читается: 1-й курс, 1, 2 модуль

Формат изучения: без онлайн-курса

Охват аудитории: для своего кампуса

Преподаватели: Тихомиров Александр Сергеевич

Прогр. обучения: Математика

Язык: английский

Кредиты: 5

Контактные часы: 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

Assessment Elements

• midterm exam
• homework
• final exam

Interim Assessment

• 2021/2022 2nd module
  total 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