Bachelor
2023/2024

# Abstract Mathematics

Type:
Compulsory course (International Programme in Economics and Finance)

Area of studies:
Economics

Delivered by:
International College of Economics and Finance

When:
3 year, 1-4 module

Mode of studies:
offline

Open to:
students of one campus

Instructors:
Dmitri Piontkovski

Language:
English

ECTS credits:
8

Contact hours:
120

### 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 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 mathematical logic. 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

- to 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 critically analyze a proposed proof of a given statement and make a conclusion on the completeness and accurateness of the proof;
- be able to find and formulate proofs of problems based on the main definitions and theorems
- be able to give definitions, formulate statements of the key theorems 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 give definitions, formulate statements of the key theorems, such as Lagrange Theorem, Homomorphism Theorem, etc., and present their proofs
- be able to give the opposite and the contrapositive statements for a given statement
- be able to illustrate the concepts in Set Theory by specific examples and counter-examples
- be able to illustrate the concepts of Group 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 illustrate the concepts of Ring Theory and Field Theory by specific examples and counter-examples
- be able to use formal notations correctly and in connection with precise statements in English

#### Course Contents

- Introduction to Set Theory
- Algebraic Structures: Groups
- Algebraic Structures: Rings and Fields
- Elements of Real Analysis, Functional Analysis and Topology
- Introduction to Mathematical Reasoning
- Mathematical Logic

#### Assessment Elements

- Final exam
- Homework and in class activities
- Exam-III (midterm)
- Exam-II
- Exam I (midterm)