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# Introduction to one-parameter semigroups of operators and linear evolution equations

2021/2022
ENG
Instruction in English
7
ECTS credits

### Course Syllabus

#### Abstract

SUMMARY OF THE COURSE The course is dedicated to one-parameter strongly continuous semigroups of linear bounded operators (also called C_0-semigroups) and their applications to linear evolution partial differential equations (examples: heat equation, Schroedinger equation, and their analogues with derivatives of higher order and variable coefficients). It will be explained how solution to initial value problem (also called Cauchy problem) for linear evolution equation is represented with the help of C_0-semigroup. Also we will discuss methods of approximation of C_0-semigroups (hence approximations of the solutions to Cauchy problem) that arise from the Chernoff product formula, including methods proposed by the lecturer in his papers. The course has an introductory level and consists of three parts: 1) reminder on real analysis, 2) reminder on functional analysis, 3) introduction to C_0-semigroups and evolution equations. LEVEL OF KNOWLEDGE EXPECTED FROM STUDENTS WHO WISH TO ATTEND THE COURSE The course is self-contained hence theoretically it is accessible for strong students who fulfilled one year of Bachelor program (Mathematics, Physics, Chemistry, IT, Economics etc), who had calculus (differentiation + integration for real-valued functions of one real variable) and linear algebra (vectors spaces and linear operators). It is a plus to have some background in ordinary and partial differential equations. We hope the course will be also interesting to a wide range of students with different background, including Master students and Ph.D. students. The motivation for our hope is that during the two <<reminder>> parts classical notions of real and functional analysis will be sometimes considered from a different perspective, and also we will include some notions (e.g. unbounded linear operators) that are not often included in basic courses of functional analysis. Students who already had courses in linear algebra, real+functional+complex analysis, ordinary+partial differential equations will feel most comfortable during the course. This corresponds to the level that usually have first year students of Master in Mathematics educational program. AFTER THE COURSE We expect that after successfully passed course students can read professional textbooks on C_0-semigroups without fear and misunderstanding. To become real expert in C_0-semigroup theory it is not enough to pass this introductory level course, but it is not possible to become such an expert without understanding of all the notions touched in the course. Hope this first-step course will help young mathematicians in their professional development.

#### Learning Objectives

• The course aims to prepare students to work with mathematical problems in which linear evolution plays the key role.

#### Expected Learning Outcomes

• Students have understanding of basic facts of the theory of C0-semigroups and their connection with linear evolution partial differential equations (two important examples are heat equation and Shcrodinger equation).
• Students understand basic facts of linear functional analysis: linear spaces, linear operators and their properties

#### Course Contents

• Overview of the basic facts of functional analysis
• One-parameter semigroups, their applications and methods of approximation
• Applications to linear evolution partial differential equations
• Feynman formulas and their analogues
• Representation of solutions of evolution equations via semigroups

#### Assessment Elements

• Итоговый устный опрос
• Итоговый устный экзамен
Экзамен проводится в устной форме (опрос по материалам курса). Экзамен проводится на платформе Zoom (https://zoom.us). К экзамену необходимо подключиться согласно расписанию ответов, высланному преподавателем на корпоративные почты студентов накануне экзамена. Компьютер студента должен удовлетворять требованиям: наличие рабочей камеры и микрофона, поддержка Zoom. Для участия в экзамене студент обязан: поставить на аватар свою фотографию, явиться на экзамен согласно точному расписанию, при ответе включить камеру и микрофон. Во время экзамена студентам запрещено: выключать камеру, пользоваться конспектами и подсказками. Кратковременным нарушением связи во время экзамена считается нарушение связи до 5 минут. Долговременным нарушением связи во время экзамена считается нарушение 5 минут и более. При долговременном нарушении связи студент не может продолжить участие в экзамене. Процедура пересдачи аналогична процедуре сдачи.
• Коллоквиум
• Домашняя работа

#### Interim Assessment

• 2021/2022 1st module
• 2021/2022 2nd module
• 2021/2022 3rd module
0.25 * Итоговый устный опрос + 0.5 * Домашняя работа + 0.25 * Коллоквиум

#### Recommended Core Bibliography

• Aydın Aytuna, Reinhold Meise, Tosun Terzioğlu, & Dietmar Vogt. (2011). Functional Analysis and Complex Analysis. [N.p.]: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=974875
• Krantz, S. G. (2013). A Guide to Functional Analysis. [Washington, D.C.]: Mathematical Association of America. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=561154