# Course Descriptions

## Mathematics and Statistics

## Mathematics Course Descriptions & Information

Be sure to read the University Course Calendar entry in your course planning to ensure you have all the pre-requisites.

Not all courses are listed here.

# First Year Courses

#### MATH-1101(6) Introduction to Calculus

Introductory calculus is mainly concerned with differentiation and integration of continuous functions. Each of these operations has a geometric interpretation: differentiation is the art of finding slopes of tangent lines while integration involves computing the area under a curve.

Perhaps the most widely studied fact in all of university science, the Fundamental Theorem of Calculus establishes a surprising connection between the operations of differentiation and integration. This tool is the centrepiece of Introduction to Calculus.

The course begins by covering background material on functions, limits and continuity. Then the theories of differentiation and integration are built from basic principles, with an emphasis on skillful computations and a variety of applications. After investigating the Fundamental Theorem, this tool is applied to a wide variety of problems in geometry and other sciences. Towards the end of the course, the student develops tools to tackle a great many types of integrals of varying difficulty.

In order to take this course, a student should have completed Pre-Calculus Mathematics 40S. As this course is a prerequisite for many courses, both within and outside the major, students are encouraged to take MATH-1101(6) in their first year of studies.

This course covers calculus of a function of one variable. Topics covered include limits, continuity, differentiation and integration of elementary functions (algebraic, exponential, logarithmic, trigonometric and inverse trigonometric), indeterminate forms and l’Hospital’s Rule, and improper integrals. Applications covered include maximization and minimization problems, related rates, curve sketching, area, volume, and arc length. A grade of at least C in this course is required to take MATH-2105(3). The material covered in this course is equivalent to the combined material from MATH-1103 (3) Introduction to Calculus I and MATH-1104 (3) Introduction to Calculus II.

PREREQUISITES: Pre-Calculus Mathematics 40S or permission of the instructor or Department Chair.

NOTE: A grade of at least C in this course is required to take MATH-2101(6).

RESTRICTIONS: A student may not receive credit for MATH-1101 (6), and any of the following: MATH-1102 (3), MATH-1103 (3) or MATH-1104 (3).

**MATH-1103 (Le3,La1.5) Introduction to Calculus I*** *

This** ** course covers differential calculus of a function of one variable. Topics include: limits, continuity, and the differentiation of algebraic, trigonometric, inverse trigonometric, exponential and logarithmic functions.** **Applications include curve sketching, optimization problems, and related rate problems.

The combined material from MATH-1103(3) Introduction to Calculus I and MATH-1104(3) Introduction to Calculus II is equivalent to MATH-1101 (6) Introduction to Calculus.

PREREQUISITE: Pre-Calculus Mathematics 40S or permission of the instructor or Department Chair.

RESTRICTIONS: A student may not receive credit for more than one of MATH-1101 (6), MATH-1102 (3), or MATH-1103 (3).

NOTE: If you have not obtained a minimum grade of 60% in 40S Pre-calculus it is recommended you take MATH-0042 Mathematics Access Course II.

**MATH -1104 (Le3,La1.5) Introduction to Calculus II**

This course covers single variable integral calculus. Topics include: definite and indefinite integrals, the natural logarithm defined as an integral, L’Hôpital’s rule, techniques of integration, and improper integrals. Applications covered include areas between curves, volumes of solids of revolution, and arc length of a curve. The combined material from MATH-1103(3) Introduction to Calculus I and MATH-1104(3) Introduction to Calculus II is equivalent to MATH-1101 (6) Introduction to Calculus.

PREREQUISITE: Minimum grade of C in MATH-1103(3) or permission of instructor or Department Chair.

RESTRICTIONS**:** A student may not receive credit for more than one of MATH-1101 (6), MATH-1102 (3), or MATH-1104(3).

**NOTE: MATH-1103 (3) AND MATH-1104 (3) together are equivalent to MATH-1101 (6)**

**MATH-1102(3) Basic Calculus (Terminal)**

Calculus is not just for students of mathematics and the natural sciences; it also has applications in areas such as business and economics. Many courses in the Department of Economics have 1102 as a prerequisite or required course for majors.

In this course, you will learn how to differentiate and integrate algebraic, logarithmic and exponential functions. Limits, continuity, maximizing and minimizing problems, curve sketching, and the fundamental theorem of calculus are also covered.

This course covers a portion of the calculus of a function of one variable. Topics covered include: limits, continuity, differentiation and integration of elementary functions (algebraic, exponential, and logarithmic). Applications covered include: maximizing and minimizing problems, curve sketching, and area.

PREREQUISITES: Pre-Calculus Mathematics 40S or Applied Mathematics 40S.

RESTRICTIONS: A student may not receive credit for MATH-1102 (3) and any of the following: MATH-1101 (6), MATH-1103 (3) OR MATH-1104 (3).

NOTE: Students intending to take senior courses in Mathematics, Statistics, Physics, Chemistry, or similar scientific and technical areas should not take MATH-1102 (which is a terminal course). Such students should take MATH-1101 (6) or both MATH-1103 (3) AND MATH-1104 (3).

**MATH-1201(3) Linear Algebra I**

Linear Algebra is one of the cornerstones of modern mathematics It is an essential tool in other fields of mathematics and a major tool in the natural sciences, social sciences, engineering, and management science.

In many applications it is necessary to represent an entity by means of several numbers e.g. a person in medical study might be represented by their height (in cms), weight (in kgs), age, and blood pressure e.g. [182, 82, 54, 140, 80]. Such a string of numbers is called a vector. Certain types of relations between vectors can be represented by matrices. A matrix is simply a rectangular array of numbers e.g.

[ 2 3 1 0 ] [ 0 3 -1 4 ] [ 5 6 9 -3 ]

In the Linear Algebra course the basic algebra and arithmetic of vectors and matrices are developed.

This is an introduction to fundamental results concerning systems of linear equations, matrices, determinants, properties of complex numbers, vector geometry, and vector space properties of n-dimensional Euclidean space.

PREREQUISITE: Pre-Calculus Mathematics 40S or Applied Mathematics 40S.

RESTRICTIONS: A student may not receive credit for both MATH-1201(3) and the former MATH-2201(6).

**MATH-1401(3) Discrete Mathematics**

This course serves as the gateway to modern mathematics. Simultaneously, the course contributes to three goals. First, students are exposed to the notation as well as several important theorems and techniques of mathematics and statistics. Topics include: logic; set theory and manipulation of sets; basic number theory and related algorithms; functions and their properties; relations; and counting techniques, including the binomial theorem.

More important perhaps is the role this course plays in introducing the student to the mathematical discipline. Emphasis is placed on the development of logical arguments and rigorous irrefutable proofs of mathematical truths. Beginning with very simple, rigid proofs, each student cultivates his or her unique style of devising proofs.

Finally, this course also exposes students to aspects of mathematics which have found important applications in computer science and related areas. Logic is the basis of most computer programming, functions are a paradigm for program modules, and relations provide the basis for the theory of data structures. This course is very similar to introductory courses found in many computer science departments and it is a required course for those students pursuing a 4-year BSc in Applied Computer Science.

All Mathematics and Statistics majors are encouraged to take Discrete Mathematics in their first year of study.

This course includes the study of permutations and combinations, the binomial theorem, basic set theory and logic, functions, relations, partial orders, and mathematical induction. This course is primarily intended for students planning to major in Mathematics.

PREREQUISITE: Pre-Calculus Mathematics 40S or Applied Mathematics 40S.

# Second Year Courses

*MATH-2105(3) Intermediate Calculus I*

MATH-2106(3) Intermediate Calculus II

*MATH-2105(3) Intermediate Calculus I*

MATH-2106(3) Intermediate Calculus II

MATH-2106(3) Intermediate Calculus II

These courses expand significantly on the body of material introduced in our Introduction to Calculus. On one hand, we delve a bit deeper into the theory of Calculus, exploring the mystery of the limit and studying limits of sequences and series. On the other hand, the theories of differentiation and integration are extended far beyond the real number line.

While much attention is paid to the calculus of several variables (e.g., partial derivatives, multiple integrals), we also undertake a study of alternative coordinate systems such as polar, spherical and cylindrical coordinates. These topics involve an intriguing interplay of geometry, calculus and algebra.

Applications of calculus are as numerous and as varied as the disciplines of Pure, Applied and Social Science. The techniques covered in this course provide the student with added flexibility to more accurately model and analyze real-world problems.

* MATH-2105(3) Intermediate Calculus I *

*MATH-2105(3) Intermediate Calculus I*

Topics covered are selected from the following list: Formal definition of a limit, limit theorems, sequences, infinite series (convergence tests, power series, Taylor's theorem), parametric equations, polar coordinates. Students who have already completed MATH-1201(3) or the former MATH-2201(6) should inform registration staff that they have standing in one of the corequisite courses.

PREREQUISITE: A grade of at least C in MATH-1101(6) or in both MATH-1103 (3) AND MATH-1104 (3).

COREQUISITES: MATH-1201(3) or the former MATH-2201(6).

NOTE: Mathematics majors are strongly advised to take MATH-2105(3) in their second year of studies.

RESTRICTIONS: A student may not receive credit for both MATH-2105(3) and the former MATH-2101(6).

*MATH-2106(3) Intermediate Calculus I I*

*MATH-2106(3) Intermediate Calculus I I*

This course covers topics in multivariable and vector calculus.

PREREQUISITE: MATH-2105(3) and MATH-1201(3) or the former MATH-2201(6).

RESTRICTIONS: A student may not receive credit for both MATH-2106(3) and the former MATH-2101(6).

Note: Mathematics majors are strongly advised to take MATH-2106(3) in their second year of studies.

**MATH-2102(3) Differential Equations I**

**MATH-2103(3) Differential Equations II**

A differential equation is simply an equation involving derivatives

e.g. x² d²y/dx² + 3x dy/dx + (4x³ +5x)y=0.

To solve a differential equation means to find functions which satisfy the equation.

The study of differential equations has been a major area of applied mathematics for several centuries since many phenomena in the physical world can be modeled by differential equations. Fields such as physics, chemistry, biology, engineering, and economics make extensive use of differential equations.

The following is a simple example of a model used in the study of electrical circuits. In a certain circuit (a series circuit containing a resistor, an inductor, a capacitor, and an alternating voltage source) the current, I, as a function of time t, satisfies the equation

*d*^{2}*I*/*dt*^{2} + *6* *dI*/*dt* + *25* *I* = *60* cos *5t* with *I* = *7* and *dI*/*dt* = *-23* when *t*= *0*.

The solution of this model is *I* = *2* sin *5t* + *7* *e*^{-3t} sin *4t*.

The differential equation courses use techniques from many other areas of mathematics. The Differential Equations I course uses differentiation and integration techniques from first year calculus, polynomial algebra, and complex numbers. The Differential Equations II course uses improper integrals and power series from intermediate calculus, and linear algebra techniques from second year linear algebra.

*MATH-2102(3) Differential Equations I *

This is an introduction to differential equations including the following topics: solution of first order equations, reduction techniques, and solution of nth order linear differential equations. Most applications considered will be from Physics. Students who have already completed MATH-1201(3) or the former MATH-2201(6) should inform registration staff that they have standing in one of the corequisite courses

PREREQUISITE: MATH-1101(6) or both MATH-1103 (3) AND MATH-1104 (3).

COREQUISITES: MATH-1201(3) or the former MATH-2201(6).

** MATH-2103(3) Differential Equations II **

This course covers further concepts and methods of solving differential equations. The contents include: Laplace transform method, power and Frobenius series solutions, matrix methods for systems of linear differential equations, and an introduction to partial differential equations including the method of separation of variables.

PREREQUISITE: MATH-2102(3).

COREQUISITES: MATH-2106(3) or the former MATH-2101(6), MATH-2203(3) (formerly MATH-2221(3) or the former MATH-2201(6)). Students who have already completed MATH-1201(3) or the former MATH-2201(6) should inform registration staff that they have standing in one of the corequisite courses.

RESTRICTIONS: A student may not receive credit for both MATH-2103(3) and the former MATH-2112(3).

**MATH-2202(3) Applied Algebra**

Why is it that small scratches on a compact disc cause no appreciable loss of sound quality? You may be surprised to learn that this is due in part to the fact that the way in which music or data is stored on the CD exploits some sophisticated techniques from algebra!

This course serves as an introduction to modern applications of algebra to such diverse fields as communications, computer science and statistics. After learning some basics about different algebraic structures and methods, the student will learn about such applications as cryptography, error-correcting codes, data compression, design of experiments, computer graphics and fast arithmetic.

Linear Algebra I (MATH-1201(3)) and Discrete Mathematics (MATH-1401(3)) are prerequisites for this course. It is recommended that Linear Algebra II (MATH-2203(3)) be taken concurrently with this course.

This course will introduce students to modern applications of algebraic structures. It begins with a study of the fundamental properties of finite fields and their relationship with geometry. The course continues by studying such applications as error-correcting codes, cryptography, design of experiments and fast arithmetic.

PREREQUISITES: MATH-1201(3) and MATH-1401(3) or permission of instructor.

RESTRICTIONS: A student may not receive credit for both MATH-2202(3) and the former MATH-2301(3).

**MATH-2203(3) Linear Algebra II**

This course is potentially the most interesting and valuable undergraduate mathematics course you will study. This course affords an excellent opportunity for students to develop a capability for handling abstract concepts. Having been introduced to finite dimensional vector spaces in 1201(3), students will study vector spaces in more depth in this course. Topics covered include linear transformations, bases for vector spaces, eigenvalues and eigenvectors, diagonalization, and inner products.

This course covers finite dimensional vector spaces; linear transformations and matrices; change of bases; eigenvalues and eigenvectors and diagonalization.

PREREQUISITES: MATH-1201(3) and MATH-1101(6) or both MATH-1103 (3) AND MATH-1104 (3)..

RESTRICTIONS: A student may not receive credit for both this course and the former MATH-2201(6) or the former MATH-2221(3).

**MATH-2501(3) Introduction to Number Theory**

This course begins an in-depth study of the integers. Beginning with the elementary concepts of divisibility and factorization, a rich theory is developed with some surprises and a fine line between solved and unsolved problems. Topics may include unique factorization, solution of congruences, multiplicative functions, quadratic congruences, and dipohantine equations. The method of rigorous proof will be emphasized as students are taken through some elegant proofs of powerful theorems such as the Möbius Inversion Formula and the Quadratic Reciprocity Theorem. Meanwhile, students are given a wide variety of interesting problems--of varying difficulty-- to solve throughout the term. Efficient techniques for number-theoretic computations are also developed and students in this course are expected to become skilful at such computations.

In order to take this course, a student must have satisfactorily completed MATH-1401 Discrete Mathematics and must have at least 3 additional credit hours in Mathematics at the 1000 level or higher.

This course explores properties of integers, linear Diophantine equations, Fermat's Theorem, congruences, and quadratic residues.

PREREQUISITES: MATH-1401(3) and an additional 3 credit hour or 6 credit hour course in Mathematics at the 1000 level or above.

**MATH-2701(3) Linear Optimization**

How do you solve huge scheduling problems such as timetabling flights for an airline like Air Canada or Canadian? What's the best way to play poker? In this course we study a tool applicable to both these problems: linear programming.

Real life linear programming takes hours on large computers, but we introduce the basic methods, and outline applications. Some of the course is cookbook, but we also have occasional generous doses of theory.

Topics for this course are selected from the Simplex algorithm, game theory, sensitivity analysis, duality theory, and efficient implementations of Simplex.

PREREQUISITES: MATH-2203(3) (formerly MATH-2221(3)), the former MATH-2201(6) or permission of instructor.

RESTRICTIONS: A student may not receive credit for this course and the former MATH-4702(3) or MATH-4702(6).

**MATH-2901(3) History of Calculus **

This course gives an overview of the main ideas of Calculus, together with their historical development. It will investigate issues such as the definition of Calculus; how and when it developed; what problems inspired its creation; and how it changed the way mathematicians and others think about mathematical knowledge. Readings of original sources in English translation may range from Babylonian mathematical tablets, through Euclid and the Greeks, past Galileo, Leibniz and Newton, to Cauchy, Riemann and Robinson.

Prerequisites: Pre-Calculus Mathematics 40S, Applied Mathematics 40S, MATH-0042 or the former MATH-0040

Education students in the Early, Early/Middle and/or Middle Years Streams cannot use this course to satisfy the math distribution requirement unless they have also received credit for an additional Mathematics course at the 1000 level or higher, excluding MATH-2902(3). For all other students, this course can be used to fulfill either the Science Requirement or the Humanities Requirement.

**MATH-2902(3) Mathematics Prior to 1640 **

This course examines selected topics in mathematics and traces their development in the Old World prior to 1640. It focuses on the theory and methodologies of algebra, number theory, trigonometry, and combinatorics. As well, the course reviews geometry and number systems. Students are expected to solve problems and prove certain theorems.

Pre-requisites: Pre-Calculus Mathematics 40S, Applied Mathematics 40S, MATH-0042 or the former MATH-0040

NOTE: Education students in the Early, Early/Middle and/or Middle Years Streams cannot use this course to satisfy the math distribution requirement unless they have also received credit for an additional Mathematics course at the 1000 level or higher, excluding MATH-2902(3) or MATH-2901(3). For all other students, this course can be used to fulfil either the Science Requirement or the Humanities Requirement. Only ONE of MATH-2901 or MATH-2902 may be used towards degree credit.

**MATH-2903(3) Mathematics for Early/Middle Years Teachers**

This course is for students planning to become elementary or middle years teachers who wish to gain a more thorough understanding of the mathematics underpinning many of the topics taught in the K-8 curriculum.

Topics include discrete mathematics (logic, sets, proof techniques), number theory (numeration systems, Euclidean algorithm, prime factorization), Euclidean geometry (Euclid's axioms, congruence, Pythagorean Theorem) and combinatorics (counting and probability, Pascal's triangle).

Pre-requisites: A grade of 65% or higher in Applied Mathematics 40S, or a grade of 65% or higher in MATH-0041, or Pre-Calculus Mathematics 40S or MATH-0042 or the former MATH-0040.

Restrictions (ineligible students): This course may not be used towards the major or minor requirement for a degree in Mathematics.Students who are not registered in the Faculty of Education may not register in this course.

**MATH-2904(3) Math for Early/Middle Years Teachers II (Le3,La2)**

This course is for prospective elementary and middle years teachers who wish to gain a thorough understanding of the mathematics underpinning many of the topics in the K-8 curriculum. Topics include probability and statistics (measures of centre and variation, permutations and combinations, probability rules, expected value), measurement (metric system and US measurement system, perimeter, area, volume, Pythagorean theorem), and Euclidean geometry (angles, polygons and 3-dimensional shapes, Euclid’s axioms, congruence, Euclidean constructions, coordinate geometry, transformations of the plane).

**Prerequisites: **MATH-2903 OR 3 credit hours of mathematics at the 1000 level or above, excluding MATH-2901 and MATH-2902.

**Restrictions: **Students who are not registered in the Faculty of Education may not take this course. This course may not be used towards the major or minor requirement in Mathematics.

# Third and Fourth Year Courses

#### MATH-3101(6) Advanced Calculus and Analysis

Most likely, the proofs that you have seen of the Mean Value Theorem and the Fundamental Theorem of Calculus (the two most important theorems from your Introduction to Calculus class) were crucially dependent upon the Extreme Value Theorem: any continuous real-valued function defined on a closed bounded interval of the real line attains absolute maximum and minimum values. It is, however, extremely unlikely that you have seen a proof of this important theorem.

In Advanced Calculus and Analysis we provide an extensive study of the real number system, sequences, series, continuous functions, and integration of real and complex-valued functions. In the process, we will fill many of the theoretical gaps left open in your introductory and intermediate calculus courses. For example, we will prove very general versions of both the Extreme and Intermediate Value Theorems from Introduction to Calculus. The real line and the complex plane are examples of metric spaces, and with only a little more work we will often obtain our theorems in much greater generality by working in the abstract setting of an arbitrary metric space.

Some of the specific topics that we will study in this class are countability and cardinality, topological properties such as compactness and connectivity of the real line and other metric spaces, completeness of metric spaces, sequences and subsequences, continuity and uniform continuity of functions, Riemann-Stieltjes integration, uniform convergence of sequences/series of functions, Fourier series, and, as time permits, various topics in vector calculus such as Green’s and Stokes’ theorems.

This course is very theoretical in nature and is designed to bring students beyond the classical study of calculus and toward the modern discipline of mathematical analysis.

This course studies construction of the reals, uniform convergence and sup-norms, Stone-Weierstrass theorem, theory of integration, Fourier analysis, line and surface integrals, Green's, Gauss' and Stoke's theorems.

PREREQUISITE: MATH-1401(3), MATH-2106(3) (or the former MATH-2101(6)), and MATH-2203(3)(or the former MATH-2201(6) or MATH-2221(3)).

#### MATH-3103 (3) Methods: Advanced Calculus

This course covers methods and applications of advanced calculus. Topics are chosen from: differentiation and integration of vector valued functions; arc length and speed; curvature and general motion in 3-dimensional space; vector fields, line integrals and surface integrals; the fundamental theorems of vector analysis (Green's Theorem, Stokes' Theorem, and the Divergence Theorem); Fourier series and other topics in harmonic analysis.

PREREQUISITE: MATH-2106(3) or the former MATH-2101(6)

**MATH-3203(3) Linear Algebra III**

This course introduces students to inner product spaces; properties of Hermitian and normal matrices; unitary matrices; factorization theorems; Schur’s Theorem; the Spectral

Theorem; the Cayley-Hamilton Theorem; and quadratic forms. At the discretion of the instructor, other advanced topics in linear algebra may be covered.

PREREQUISITES: MATH-1401(3) and MATH-2203(3) or the former MATH-2201(6)

**MATH-3202(3) Group Theory**

A group is a set of elements with a binary operation that satisfies certain properties (closure, associativity, identity property, inverse property). A familiar group is the set of integers with the binary operation of addition. Since the set of symmetries of an object form a group (under function composition), group theory, often called the “official language of symmetry”, is an important tool in scientific theory. Chemists use group theory to understand molecular structure; physicists to study forces and particles.

In this course, the abstract properties of groups are studied, ending with the celebrated Fundamental Theorem of Finite Abelian Groups.

This course studies symmetry groups of regular polygons and Platonic solids, permutations and permutation groups; abstract groups, Cosets, Homomorphisms, Subgroups, Normal subgroups and quotient groups; isomorphism theorems, Sylow theorems, classification of finitely generated Abelian groups; group actions and counting with Burnside's lemma.

PREREQUISITES: MATH-1401(3) and MATH-2203(3) or the former MATH-2221(3) or the former MATH-2201(6)

RESTRICTIONS: A student may not receive credit for this course and MATH-4201(6) or the former MATH-3201(6).

MATH-4202(3) Rings and Fields

*MATH-4202(3) Rings and Fields*

Simply put, a ring is a generalization of the set of integers. Somewhat more formally, a ring is a set together with two binary operations (usually called addition and multiplication) that satisfy certain properties. The integers form a ring under ordinary addition and multiplication. Elements in a ring need not have multiplicative inverses and a ring need not have a multiplicative identity.

A field is a ring in which multiplication is commutative and every nonzero element has a multiplicative inverse. The integers, for instance, are not a field since they do not satisfy the latter property. Some familiar fields are the rational numbers, the complex numbers, and the real numbers.

In this course, the abstract theory of rings and fields is covered in depth. Topics include integral domains, division rings, polynomial rings, PID’s, UFD’s and classification of finite fields.

**MATH-4402(3) Networks and Their Applications**

A large communication network consists of many locations, some of which have direct connections to others. What is the most efficient way to send information between two given locations in the network?

We wish to design a network connecting various information centers. We would like to do so in such a way that, even if a few links in the network go down or are disabled, it will still be possible to send information from any center to any other center. How can this be done?

A number of workshops are to be set up at a conference. Certain workshops cannot be given at the same time as others because of various kinds of conflicts. How many different time periods are needed to arrange all the workshops?

A large project consists of many jobs to be carried out. Some jobs must be completed before others can be started. How can we design a schedule to carry out all of the jobs of the project? What is the best way to schedule the jobs?

In a transportation network connecting various cities by air links, what is the maximum number of people that can be transported from one city to another taking the capacity constraints of the planes into consideration?

In a communications network (such as the internet of the telecommunications network) having many nodes one often wishes to connect a smaller subcollection of nodes by using as few links as possible so that as much communication as possible can still take place in the remaining network. How can this be done?

These problems are typical problems in the area of graph theory and combinatorial optimization. The concept of a graph is an extremely simple and basic one in mathematics: a graph or network consists of a collection of objects, together with some relationship of interest which some objects have to others. It has direct and important applications to problems in many areas of interest in industry, business and science, including the examples above in communication networks, transportation networks and job scheduling.

The three courses MATH-3401(3), MATH-4401(3) and MATH-4402(3) provide a comprehensive introduction to the basic concepts, methods, and applications of graphs. The subject matter is an appealing balance of theory, algorithms, and applications.

Any student wishing to take MATH-3401(3) must already have taken at least one full course credit in Mathematics at the first year level. The course MATH-4401(3) requires MATH-3401(3) as a prerequisite; it develops more advanced topics, and focuses on many optimization problems, building directly on the material studied in MATH-3401(3). The course MATH-4402(3) requires MATH-1401(3) and MATH-3401(3) as prerequisites; it introduces the student to a variety of network problems with emphasis on their algorithmic aspects.

**MATH-3401(3) GRAPH THEORY **(Le3)

This course includes the following topics: graph isomorphism, shortest path problem, Euler tours, trees, graph colourings and bipartite matchings.

**MATH-4401(3) ADVANCED GRAPH THEORY AND COMBINATORIAL OPTIMIZATION**

Topics covered in this course will be chosen from trees, connectivity, graph colourings, optimal matchings, packings and coverings, planar graphs, extremal graph theory, Ramsey theory, ordered sets.

**MATH-3402(3) COMBINATORICS**

Most of us have been counting since we were about three years old, so what can we possibly have to learn about counting? Plenty, as it turns out.

Many deep results in mathematics are obtained by counting objects in sophisticated ways. There are also many real world applications in which counting is essential. For example, computer scientists analyze the efficiency of certain computer algorithms by enumerating the possible data configurations which might be generated. In genetics, researchers are studying DNA sequences looking for patterns which occur more frequently than one would expect.

This course provides an introduction to enumerative combinatorics (counting configurations with certain well-defined properties). The student, already familiar with counting simple types of configurations such as permutations and combinations from Discrete Math, will learn some very powerful techniques for counting more complicated types of objects.

This course includes the following topics: generating functions and recurrence relations, the principle of inclusion and exclusion, symmetric groups, finite fields, and combinatorial designs.

PREREQUISITES: MATH-1401(3) plus one of MATH-1101(6), MATH-1102(3), MATH-1103 (3) or MATH-1104 (3).

RESTRICTIONS: A student may not receive credit for this course and the former MATH-2022(3).

**MATH-3403(3) Mathematical Logic**

Can you imagine a real number bigger than 0, but smaller than 1/n for each natural number n? Such a creature is called an infinitesimal.

You might think such a thing doesn't make sense . . . but it can.

In this course we prove some wild and philosophically interesting things about Mathematics and Proof, but it takes a lot of careful work. Nothing is `hard', but everything is `theoretical'.

This course introduces the student to mathematical logic. Course topics include propositional calculus, first order logic, completeness, compactness, decidability, Gödel's Incompleteness theorem and models.

PREREQUISITES: MATH-1401(3) plus one of MATH-3401(3) (formerly MATH-2011(3)), MATH-3402(3) (formerly MATH-2022(3)), MATH-2203(3) (formerly MATH-2221(3)), the former MATH-2201(6), or MATH-2501(3).

RESTRICTIONS: A student may not receive credit for this course and the former MATH-3401(3).

**MATH-3701(3) Numerical Methods**

Methods for finding roots of functions, numerical differentiation and numerical integration, interpolation and polynomial appropriation, solving systems of equations, and solving the initial value problem in ordinary differential equations are studied in this course. Computer programming is used to implement the relevant numerical algorithms,

PREREQUISITES: MATH-2102(3) and MATH-2106(3) or the former MATH-2101(6) and either MATH-1201(3) or MATH-2203(3) for the former MATH-2201(6).

RESTRICTIONS: A student may not receive credit for this course and the former MATH-3701(6).

**MATH-4101(3) Complex Analysis **

This course studies construction of the complex numbers from the reals, Cauchy's theorems, Laurent Series, evaluating line integrals by means of residues, Cauchy-Riemann equations, conformal mapping, harmonic functions, Riemann sphere, Riemann surfaces, analytic continuation and monodromy theorem.

**MATH-4403(3) Set Theory**

How many points are there in a line? Do all infinite sets have the same number of elements? Can we measure and compare the sizes of infinite sets?

These questions and the investigations related to answering them, undertaken by G. Cantor and others in the late 1800's, mark the beginning of the current period of "modern" mathematics. The concept of a set is by now pervasive in all areas of mathematics. All basic mathematical structures in Algebra, Geometry, Topology and Analysis are defined directly in terms of the set concept, and a basic knowledge of the properties and theory of sets is essential in studying these disciplines. Set Theory is now widely regarded as being the foundation of Mathematics.

The purpose of this course is two-fold: Firstly, to develop the basic concepts and theorems of Set Theory which are essential to the study of all areas of modern mathematics, and to learn how and why Set Theory has such a fundamental role in Mathematics; and secondly, to study the subject of Set Theory in its own right, as a very interesting and valuable area of Mathematics.

In addition to studying the concept of cardinality, or size, of infinite sets as suggested at the beginning of this description, this course will also examine the extent to which ideas like mathematical induction and recursive definitions can be used in contexts involving infinite sets. Given an infinite set we may ask "How many elements does it have?" We may also inquire "In what ways may its elements be arranged or ordered?"

Among the orderings most important in Mathematics are partial orderings, total orderings and well-orderings. These concepts play a prominent role in this course. In addition to their importance throughout Mathematics it will be seen how these concepts can be used to define and develop, from first principles, the concept of number, and to be able to answer questions such as :

What is a number ? How can we define the real number system? How do we know that the basic number systems of Mathematics like the integers and the reals satisfy their familiar properties?

Another aspect of the subject of Set Theory which will be discussed in this course is its axiomatic side. Why is it important to develop the concept of set in an axiomatic way? Using which axiomatic system? These questions are very much involved with and related to mathematical logic, in which the concepts of definability, proof, and provability are prominent.

Set Theory is a fundamental and essential part of any modern mathematical education. The material in this course is essential for further advanced study in all areas of Mathematics.

This course introduces concepts of set theory essential to modern mathematics. Topics include axioms for sets, infinite sets, cardinality, ordinal and cardinal numbers, and ordered sets.

PREREQUISITES: MATH-1401(3) plus one of MATH-3401(3) (formerly MATH-2011(3)), MATH-3402(3) (formerly MATH-2022(3)), MATH-2203(3) (formerly MATH-2221(3)), the former MATH-2201(6), or MATH-2501(3).

RESTRICTIONS: A student may not receive credit for this course and the former MATH-3402(3), the former MATH-3601(3), or the former MATH-2402(6).

**MATH-4601(3) Introduction to Topology and Analysis**

In this course the basic concepts of topology are studied in the setting of metric spaces: open sets, convergence, continuity, connectedness, compactness, and completeness. The emphasis will be on Euclidean spaces, normed linear spaces, and function spaces, where the concepts of topology apply to mathematical analysis. Connections with and applications to analysis will be emphasized throughout, such as the monotone convergence theorem, the intermediate value property, the Heine-Borel property, Baire category, uniform boundedness, and the fixed-point property for complete metric spaces.

**MATH-4602(3) Real Analysis**

The Riemann integral that you have extensively studied in your calculus courses suffers from serious drawbacks in terms of the interchange of limit and the integral. In Advanced Calculus you will learn (or have already learned) that to guarantee the validity of such an exchange, the limit convergence must be uniform over the interval on which you are integrating. At the beginning of the 20th century, Henri Lebesgue developed a new theory of integration with the advantages that

(a) there are more Lebesgue integrable functions than there are Riemann integrable functions, and, more importantly,

(b) the theorems relating the interchange of integral and limit are valid under much less stringent conditions than those required for Riemann integration.

The need for such an interchange is often highly desirable in practice, so the Lebesgue integral is often much more convenient than the Riemann integral.

In this class we will define abstract measure spaces, prove the Carathéodory and Hahn extension theorems, define the Lebesgue measure and study its special properties, study measurable functions, integration taken with respect to arbitrary and Lebesgue measure, prove the Monotone and Lebesgue Dominated Convergence Theorems, and we will study the classical Banach spaces Lp(X) (including the Hölder and Minkowski inequalities, the Riesz-Fischer Theorem, and Lusin’s Theorem). We will also discuss the Radon-Nikodým theorem, the Jordan Decomposition theorem, the Riesz representation theorem for Lp(X), and, time permitting, product measures and the theorems of Fubini and Tonelli.

This class is strongly recommended to all students intending to pursue graduate degrees in mathematics or statistics.

**MATH-4603(3) Topology**

Concepts such as intervals in ordered sets, pointwise convergence of sequences of functions, and directed collections of sets, all indicate the need for generalized notions of neighbourhood and convergence in a more general setting than that of the real line, or even a metric space, as was considered in Advanced Calculus and Analysis. Point-set topology provides the context for developing such general notions through the study of topological spaces. The subject of topology is filled with fascinating examples of spaces with interesting properties and its theory is deep and broadly developed.

Topology, though very interesting in its own rite, is also fundamental to the study of a wide variety of mathematical disciplines such as algebra, lattice theory, logic, classical analysis, and functional analysis. As such, applications of topology are as widespread as the subjects to which it applies itself, and topology is widely regarded as an essential part any undergraduate student’s training in modern mathematics.

In this class, the basic properties of topological spaces are studied. These include separation properties, covering properties (such as compactness), countability properties (such as first and second countability / separability), product spaces and the Tychonoff theorem, the Tietze extension theorem, local compactness and one-point compactifications, and sequential properties (such as the Baire category theorem and sequential compactness). Time permitting, topics such as net and filter convergence, the Stone-Cech compactification, and topological groups may be covered in class or assigned as individual/group projects.

This class is strongly recommended to all students intending to pursue graduate degrees in mathematics.

This course is a study of topological spaces and their applications. Topics to be studied will include separation axioms, covering properties, product spaces, quotient spaces, filters, nets, convergence, compactness and connectedness.

Corequisite: MATH-3101(6)

read the Course Calendar full course descriptions here