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Regular version of the site
2020/2021

## Abstract Mathematics

Type: Optional course (faculty)
When: 1-4 module
Open to: students of one campus
Language: English
ECTS credits: 3

### Course Syllabus

#### Abstract

Abstract Mathematics is a two-semester course for the third year bachelor’s programme students who selected specialization Economics and Mathematics. The course is based on the Introduction to Abstract Mathematics course of the University of London (UoL) with further expansions into selected topics from algebra, real analysis and topology. The course is taught in English. For the theory itself there are no pre-requisites except for an aptitude for logical reasoning. However, many examples will reference concepts from Calculus, Statistics, Mathematics for Economists and Linear Algebra courses from the 1st and 2nd years of the ICEF bachelor’s programme. The emphasis of the course is on the theory rather than on the method. One central topic of the course is formal mathematical reasoning. The students will practice formulating precise mathematical statements and proving them rigorously. These skills are essential for the current specialization, they often remain in shadows in other math courses where the focus is on solving problems through calculation. The second central topic of the course is the abstract mathematical structures from algebra (groups, fields, etc.), analysis, topology (topological spaces, manifolds) and functional analysis. We will develop some of these theories roughly to the extent of standard 1st and 2nd -year courses of the mathematical departments. The awareness of the theoretical foundations of these classical theories is key in understanding the contemporary theoretical research and the synergies between different areas of mathematics and its applications. #### Learning Objectives

• explain the main mathematical concepts in discrete mathematics, algebra, real analysis, functional analysis and topology;
• to illustrate the concepts by specific examples and counter-examples;
• to teach how to use formal notations correctly and in connection with precise statements in English
• to give definitions, formulate statements of the key theorems and present their proofs
• to critically analyze a proposed proofs of a given statement and make a conclusion on the completeness and accurateness of the proof
• to give a generic understanding of the applications of the discussed classical theories
• to teach how to find and formulate proofs of problems based on the main definitions and theorems #### Expected Learning Outcomes

• be able to find and formulate proofs of problems based on the main definitions and theorems
• be able to critically analyze a proposed proof of a given statement and make a conclusion on the completeness and accurateness of the proof;
• be able to illustrate the concepts in Set Theory by specific examples and counter-examples
• be able to use formal notations correctly and in connection with precise statements in English
• be able to give definitions, formulate statements of the key theorems and present their proofs
• be able to illustrate the concepts of Group Theory by specific examples and counter-examples
• be able to give definitions, formulate statements of the key theorems, such as Lagrange Theorem, Homomorphism Theorem, etc., and present their proofs
• be able to give definitions, formulate statements of the key theorems, such as Chinese Remainder Theorem, The Fundamental Theorem of Arithmetic, etc., and present their proofs;
• be able to illustrate the concepts of Ring Theory and Field Theory by specific examples and counter-examples
• be able to illustrate the concepts of Real Analysis including the axiomatic definition of the set of real numbers; Functional Analysis such as norm, metric, metric spaces; Topology such as a topological space, base of topology by specific examples and counter-examples;
• be able to give the opposite and the contrapositive statements for a given statement #### Course Contents

• Introduction to Set Theory
a. Definitions and notations b. Set operations, maps between sets c. Discussion of the proper foundations of the set theory
• Algebraic Structures: Groups
a. Definition and basic properties of groups b. Subgroups, quotient groups and homomorphisms, cosets & Lagrange’s theorem. c. Homomorphisms, group isomorphism theorems d. Automorphisms and semi-direct products e. Group actions f. Sylow theorems* g. Classification of Abelian groups* h. Introduction to the representation theory of finite groups*
• Algebraic Structures: Rings and Fields
a. Divisibility of integers b. Congruence and modular arithmetic. c. Definition and basic properties of rings d. Ideals, ring homomorphisms e. The Chinese Remainder Theorem f. Definition and basic properties of fields g. Field of complex numbers h. Finite fields i. Rings of polynomials j. Field Theory and Galois theory*
• Elements of Real Analysis, Functional Analysis and Topology
Real numbers as a complete ordered field b. The Archimedian property of real numbers c. Equivalent definitions of completeness in an archimedian field d. Representation of real numbers via decimal fractions e. Topological spaces and operations with them f. Continuity, properties of continuous functions g. Metric spaces h. Norm and normed spaces i. Contracting maps, contracting map theorem for complete metric spaces j. Homotopy groups, homotopy equivalence.* k. Coverings, cell spaces (CW-complexes).* l. Differentiable manifolds, diffeomorphisms* m. Tangent vectors and differentials, differential forms*. n. Lie Groups*
• Introduction to Mathematical Reasoning
• Mathematical Logic
Elements of mathematical logic and formal reasoning Course Review #### Assessment Elements

• Exam-I (midterm)
• Exam-II
• homework and inclass activities
• Exam-III (midterm)
• UoL (or HSE) exam #### Interim Assessment

• Interim assessment (4 module)
0.11 * Exam-I (midterm) + 0.15 * Exam-II + 0.175 * Exam-III (midterm) + 0.115 * homework and inclass activities + 0.45 * UoL (or HSE) exam #### Recommended Core Bibliography

• An introduction to mathematical reasoning : numbers, sets and functions, Eccles, P. J., 2013
• Discrete mathematics, Biggs, N. L., 2004