University of Toronto at Scarborough 2002/2003 Calendar
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(B.Sc.)
Discipline Representative:
Until June 30, 2002
C. Albanese (416-287-7261)
July 1, 2002 to June 30, 2003 - T.B.A.
Our Mathematics began in the ancient Mesopotamian civilizations. The Babylonians already knew much of the mathematics taught traditionally in our schools. Their algebra and geometry was phrased in terms of crops and fields and money. Since the Renaissance, much of mathematics has come from problems in physics and astronomy; for example, calculus arose from problems in mechanics. In turn mathematics has provided the theoretical framework and tools in the Physical Sciences. In the 19th century some parts of mathematics appeared to develop away from their origins in the physical world. To the great surprise of many scientists and mathematicians, some of the "pure" mathematics has turned out to be essential in many aspects of 20th century science. Differential geometry provides the language for general relativity and cosmology, and Hilbert space theory and group representations are the tools for quantum mechanics. Similarly, graph theory, combinatorics and number theory play a major role in computer science.
The Specialist Programs in Mathematics and in Mathematics and Its Applications, and the Major Program in Mathematical Sciences, are eligible for inclusion in the Co-operative Program in Physical Sciences and the Early Teacher Project in Physical Sciences. Please refer to the Physical Sciences section (page 162) and to the Co-operative Program (page 65) sections of this Calendar for further information.
Please refer to the Physical
Sciences Scarborough preamble on page 162 for a list of the Programs
offered. Descriptions of these Programs will be found on subsequent
pages of this section.
The Specialist Program in Mathematics is designed to give students a thorough grounding in the main areas of Mathematics, together with an understanding of the close relationship between Mathematics and other Sciences. It is aimed at students who may be interested in teaching, law, government or industry, or who may decide to pursue a career in research.
[This Program is comparable to the Specialist Program in Mathematics and Applications on the St. George Campus.]
First Year:
CSCA58H Introduction to Computer Science
MATA23H Linear Algebra I
MATA26Y Calculus
PHYA10H Dynamics of Classical Systems
PHYA21H Principles of Modern Physics
First or Second Year:
CSCB70H Fundamental Data Structures and Techniques
Second Year:
MATB24H Linear Algebra II
MATC01H Groups and Symmetry
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
MATB43H Introduction to Analysis
STAB22H Statistics
Second or Third Year:
MATC02H Fields and Groups
MATC25H Classical Plane Geometries and their Transformations
or
MATC15H Introduction to Number Theory
Third Year:
MATB44H Ordinary Differential Equations
Third or Fourth Year:
MATC46H Differential Equations
MATC34H Complex Variables
STAB47H Introduction to Probability Theory and Mathematical Statistics
two of:
MATB61H Linear Programming and Optimization
MATC35H Chaos, Fractals and Dynamics
MATC38H Introduction to Real Analysis
MATC61H Introduction to Mathematical Finance
MATC65H Complex Variables II
one of:
MATC44H Introduction to Combinatorics
MATC32H Graph Theory and Algorithms for Its Applications
CSCC50H Numerical Algebra and Optimization
CSCC51H Numerical Approximation, Integration and Ordinary Differential Equations
1.0 F.C.E. from MAT at B-, C-, or D-level.
Fourth Year:
PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society
NOTE: PSCD01H is a required course for ETP students
Recommended course: PHYB21H
The Specialist Program in Mathematics has been withdrawn. Students currently registered in it will be allowed to complete it. Please consult with the Supervisor of Studies.
Interested students should consider the Statistics Stream of the Specialist Program in Mathematics and Its Applications (below).
CORE FOR ALL PROGRAM STREAMS:
First Year:
[CSCA58H Introduction to Computer Science
or
PSCB57H Introduction to Scientific Computing (if
PSCB57H is selected it should be taken in second year)]
MATA23H Linear Algebra I
MATA26Y Calculus
First or Second Year:
STAB22H Statistics
Second Year:
MATB24H Linear Algebra II
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
MATB43H Introduction to Analysis
MATB44H Ordinary Differential Equations
Second or Third Year:
STAB47H Introduction to Probability Theory and Mathematical Statistics**
Third Year:
MATC01H Groups and Symmetry
Third or Fourth Year:
MATC34H Complex Variables
* PSCB57H is required for the Computational Physical Sciences stream
** STAB47H must be taken in second year for the Statistics stream
AREAS OF CONCENTRATION:
Teaching Stream:
MATC02H Fields and Groups
MATC15H Introduction to Number Theory
MATC25H Classical Plane Geometries and their Transformations
[MATC32H Graph Theory and Algorithms for its Applications
or
MATC44H Introduction to Combinatorics]
Three of:
MATB61H Linear Programming and Optimization
MATC46H Differential Equations
MATC35H Chaos, Fractals and Dynamics
MATC38H Introduction to Real Analysis
MATC63H Differential Geometry I
MATC65H Complex Variables II
Four C- or D-level CSC, MAT or STA half-courses
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
NOTE: Students following this stream are encouraged to apply for acceptance into the Early Teacher Project. ETP students are required to take PSCD01H.
Statistics Stream:
MATB61H Linear Programming and Optimization
MATC02H Fields and Groups
MATC25H Classical Plane Geometries and their Transformations
MATC46H Differential Equations
MATC61H Introduction to Mathematical Finance
Two of:
MATC35H Chaos, Fractals and Dynamics
MATC38H Introduction to Real Analysis
MATC65H Complex Analysis II
2.0 F.C.E.'s from C-level STA courses and 300- and
400-level STA courses on the St. George campus.
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
Computational Physical Sciences Stream:
ASTA03Y Introduction to Astronomy
PHYA10H Dynamics of Classical Systems
PHYA21H Principles of Modern Physics
MATB61H Linear Programming and Optimization
MATC35H Chaos, Fractals and Dynamics
MATC44H Introduction to Combinatorics
MATC46H Differential Equations
CSCC50H Numerical Algebra and Optimization
CSCC51H Numerical Approximation, Integration and Ordinary Differential Equations
Three of:
PHYC20H Vibrations and Waves
PHYB21H Electricity and Magnetism
PHYB24H Introduction to Quantum Physics
ASTB21H Solar System and Stellar Astrophysics
ASTC22H Galactic and Extragalactic Astrophysics
One of:
CSCC54H Computer-Based Simulation Models
CSCD18H Computer Graphics
MATC61H Introduction to Mathematical Finance
MATC65H Complex Variables II
MATD01H Readings in Mathematics
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
Computer Science Stream:
See Joint Mathematics stream in Computer Science
Specialist Program
Design Your Own Stream:
12 half-courses chosen with the approval of the program
supervisor for Mathematics and Its Applications
[PSCD02H Current Questions in Mathematics and Science
or
PSCD03H Computers in Contemporary Society]
b) 0.5 F.C.E. from:
MATC25H (B30H), MATB61H, MATC02H (B32H), MATC15H (B70H), MATC30H, MATC44H (C31H), MATC32H, MATC61H, MATC63H (C54H)
c) 1.0 F.C.E. from:
CSCB28H, CSCB38H, CSCB58H, CSCB70H, CSCC24H, CSCC50H, CSCC51H, CSCC54H, CSCC64H, CSCC78H, CSCC85H, CSC300/PSCD03, CSC318, CSC340, CSC D-level, CSC 400-level, MATC44H (C31H), MATC32H, MATD01H, MATD02H, STAB47H, STAC42H, STAC62H, STAC67H, STA300-level, STA400-level
b) 1.0 F.C.E. from:
MATB43H, MATB44H, MATC35H, MATC38H, MATC46H, MATC34H,
C65H, MATC25H, MATC91H, MATC15H, C30H, C32H, C43H, MATC44H, MATC32H,
MATC63H (C54H), STAC42H, STAC52H, STAC57H, STAC62H, STAC67H, any
300- and 400-level STA course on St. George campus. CSCB28H, CSCB38H,
CSCB58H, CSCC24H, CSCC50H, CSCC51H, CSCC54H, CSCC64H, CSCC78H,
CSCC85H, CSC300/PSCD03, CSC318, CSC340, MATC44H CSCC32H, any D-level
or 400-level CSC course.
Matrices, linear systems, elementary matrices and
the inverse of a matrix. Vector spaces over R, subspaces, basis
and dimension. Real inner product spaces, geometry in Rn,
lines and hyperplanes. Linear transformation, kernel, range, matrix
representation, isomorphisms. The determinant, Cramer's rule,
the adjoint matrix. Eigenvalues, eigenvectors, similarity, diagonalization.
Projections, Gram-Schmidt process, orthogonal transformations
and orthogonal diagonalization, isometries, quadratic forms, conics,
quadric surfaces.
Two one-hour lectures per week and one two-hour tutorial
per week.
First term of MATA26Y. Students in academic difficulty
in MATA26Y may withdraw from MATA26Y and enrol in MATA24S in the
Spring term.
These students are informed of this option by the
end of the Fall term. Classes begin in the first week of the Spring
term; late enrolment is not permitted. MATA24S together with MATA25F
is equivalent for program and prerequisite purposes to MATA26Y.
Students not enrolled in MATA26Y in the Fall term will not be
allowed to enrol in MATA24S, with the following exception. A student
who has successfully completed PHYA10F and wishes to enrol in
PHYA21S, but is not enrolled in MATA26Y, may enrol in MATA24S
with the consent of the instructor.
Two one-hour lectures per week and a one hour tutorial
per week.
Second term content of MATA26Y; the final examination
includes topics covered in MATA24H. Offered in the Summer Session
only; students who have not previously enrolled in MATA24H will
NOT be allowed to enrol in MATA25H. MATA24H together with MATA25F
is equivalent for program and MAT prerequisite purposes to MATA26Y.
Two one-hour lectures per week and a one hour tutorial
per week.
This course includes: limits and continuity, derivatives,
related rates, extremum problems, graph sketching, Newton's method,
indefinite and definite integrals, numerical integration, Taylor
approximation and differential equations.
MATA26Y introduces the basic techniques of calculus
with a strong emphasis on methods of approximation. The course
will develop these ideas by the investigation of specific examples.
MATA26Y is a demanding course which will equip the student for
most sciences and for further work in mathematics.
Two one-hour lectures per week and a one hour tutorial
per week.
Areas to be covered include: R^n as a model space
in economics, contour maps; exponential functions and logarithms,
with applications to finance; review of differential calculus
in one variable, marginal analysis; vectors, differential calculus
of several variables and optimization, with applications to economics.
Two one-hour lectures per week and a one hour tutorial
per week.
NOTE:
This course is intended for students in Management Programs. It
does not satisfy the requirements for any Program in the Physical
or Life Sciences, and it may not be used as a prerequisite for
any further Mathematics course.
An introductory course in Mathematics built around
applications in Life Science and Environmental Science. Discrete
probability theory, applications to genetics, exponential and
logarithm functions, plotting data; basic calculus: limits and
derivatives, maxima and minima, integration, growth rates, diffusion
rates; differential equations, population dynamics; vectors and
matrices in two and three dimensions; basic statistics: hypothesis
testing, distributional analysis.
Two one-hour lectures per week and a one hour tutorial
per week.
NOTE: This
course will not satisfy the Mathematics requirements for most
Programs in Physical Sciences, nor will it normally serve as a
pre-requisite for further courses in Mathematics. Students who
are not sure which MAT A-level course they should choose are encouraged
to consult with the supervisor(s) of Programs in their area(s)
of interest.
Fields. Vector spaces over a field. Linear transformations,
dual spaces. Diagonalizability, direct sums. Invariant subspaces,
Cayley-Hamilton theorem. Complex inner product, orthogonality,
the adjoint of a linear operator, the projection matrix and the
method of least squares. Normal, self-adjoint and unitary operators.
Spectral theorem. Conditioning and Rayleigh quotient. Jordan canonical
form.
Two one-hour lectures per week and a two hour tutorial
per week.
A study of Vector algebra in Rn, lines and planes
in R3, complex numbers, matrices, determinants and linear equations,
functions of several variables, partial derivatives, gradient,
tangent plane, Jacobian matrix and chain rule, Taylor series,
extremal problems, extremal problems with constraints and Lagrange
multipliers, multiple integrals, spherical and cylindrical coordinates,
law of transformation of variables.
Two one-hour lectures per week and a one hour tutorial
per week.
Fourier series. Vector fields in Rn, Divergence and
curl, curves, parametric representation of curves, path and line
integrals, surfaces, parametric representations of surfaces, surface
integrals. Green's, Gauss', and Stokes' theorems will also be
covered. An introduction to differential forms, total derivative.
Two one-hour lectures per week and a one hour tutorial
per week.
This course is designed for students whose interest
in mathematics has been stimulated by their experience in first
year mathematics courses, and who wish to acquire the analytic
techniques which are essential for more advanced work. There will
be a fundamental emphasis on rigorous analytic proofs. Students
will study the least upper bound principle for R, limits in R
and R2, continuous functions in one and two variables, space filling
curves and nowhere differentiable functions, existence of extrema
on closed and bounded sets, mean value theorems and the fundamental
theorems of the calculus, the Riemann integral.
Two one-hour lectures per week and a one hour tutorial
per week.
(Formerly MATC51H)
Ordinary differential equations of the first and
second order, existence and uniqueness; solutions by series and
integrals; linear systems of first order; non-linear equations;
difference equations.
Two one-hour lectures per week and a one hour tutorial
per week.
Co-requisite: MATB41H & MATB24H
Linear programming, simplex algorithm, duality theory,
interior point method; quadratic and convex optimization, stochastic
programming; applications to portfolio optimization and operations
research.
Two one-hour lectures per week and a one hour tutorial
per week.
Co-requisite: MATB42H
Congruences and fields. Permutations and permutation
groups. Linear groups. Abstract groups, homomorphisms, subgroups.
Symmetry groups of regular polygons and Platonic solids, wallpaper
groups. Group actions, class formula. Cosets, Lagrange's theorem.
Normal subgroups, quotient groups. Emphasis on examples and calulations.
Two one-hour lectures per week and a one hour tutorial
per week.
(Formerly MATB32H)
Introduction to abstract group theory: Sylow theorems,
groups of small order. Simple groups. Classification of finite
abelian groups. Fields and Galois theory: Polynomials over a field.
Field extensions, adjunction of roots of a polynomial. Constructibility,
trisection of angles, construction of regular polygons. Galois
groups of polynomials, in particular cubics, quartics. Solvable
groups. Insolvability of quintics by radicals.
Two one-hour lectures per week and a one hour tutorial
per week.
(Formerly MATB70)
Elementary topics in number theory; arithmetic functions;
polynomials over the residue classes modulo m, characters on the
residue classes modulo m; quadratic reciprocity law, representation
of numbers as sums of squares.
Two one-hour lectures per week and a one hour tutorial
per week.
An introduction to geometry with a selection of topics
from the following: symmetry and symmetry groups, finite geometries
and applications, non-Euclidean geometry.
Two one-hour lectures per week and a one hour tutorial
per week.
Co-requisite: MATC01H
Graphs, subgraphs, isomorphism, trees, connectivity,
Euler and Hamiltonian properties, matchings, vertex and edge colourings,
planarity, network flows and strongly regular graphs. A selection
of applications to such problems as timetabling, personnel assignment,
tank form scheduling, traveling salesmen, tournament scheduling,
experimental design and finite geometries. Explicit algorithms
and their computational complexity will be discussed whenever
possible.
Two one-hour lectures per week and a one hour tutorial
per week.
(Formerly MATC60H)
Theory of functions of one complex variable, analytic
and meromorphic functions. Cauchy's theorem, residue calculus,
conformal mappings, introduction to analytic continuation and
harmonic functions.
Two one-hour lectures per week and a one hour tutorial
per week.
(Formerly MATC56)
Sturm-Liouville problems, Green's functions, special
functions (Bessel, Legendre), partial differential equations of
second order, separation of variables, integral equations, Fourier
transform, stationary phase method.
Two one-hour lectures per week and a one hour tutorial
per week.
Co-requisite: MATB42
Brownian motions, Fokker-Planck equation, stopping
times, reflection principle, Girsanov theorem; Stochastic calculus,
Ito's lemma, martingales; stochastic optimization, Black-Scholes
equation. The course provides an introduction to methods of interest
in financial mathematics.
Two one-hour lectures per week and a one hour tutorial
per week.
Recommended: STAC62H
Curves and surfaces in Euclidean 3-space. Serret-Frenet
frames and the associated equations, the first and second fundamental
forms and their integrability conditions, intrinsic geometry and
parallelism, the Gauss-Bonnet theorem.
Two one-hour lectures per week and a one hour tutorial
per week.
Applications of complex analysis to geometry, physics
and number theory. Fractional linear transformations and the Lorentz
group. Solution to the Dirichlet problem by conformal mapping
and the Poisson kernel. The Riemann mapping theorem. The prime
number theorem.
Two one-hour lectures per week and a one hour tutorial
per week.
Independent study under the direction of a faculty
member.
Independent study under the direction of a faculty
member.
University of Toronto at Scarborough 2002/2003 Calendar
(416-287-7274)
CSCA58H Introduction to Computer Science
MATA23H Linear Algebra I
MATB24H Linear Algebra II
MATA26Y Calculus
CSCB70H Fundamental Data Structures and Techniques
MATB41H Techniques of the Calculus of Several Variables I
MATB42H Techniques of the Calculus of Several Variables II
STAB22H Statistics
STAB47H Introduction to Probability Theory and Mathematical Statistics
a) 2.0 F.C.E. from:
STAC42H, STAC52H, STAC57H, STAC62H, STAC67H, any
300- and 400-level STA courses on St. George campus.
MATA23H3 Linear Algebra I
Exclusion: (MATA04), MAT223
Prerequisite: OAC Calculus & OAC Algebra and
Geometry
MATA24H3 Calculus A
Exclusion: MATA26Y, (MATA28Y), MATA27H; MATA29Y,
MAT130, 133, 134, 135, 137, 139, 149
Prerequisites: Enrolment in MATA26Y, and withdrawal
from MATA26Y after midterm, with a mark of at least 20% in the
midterm.
MATA25H3 Calculus B
Exclusion: MATA26Y, (MATA28Y), MATA27H MATA29Y, MAT123,
124, 125, 126, 133, 135, 137
Prerequisites: MATA24S successfully completed.
MATA26Y3 Calculus
Exclusion: (MATA28Y), MATA27H, MATA29Y, MAT123, 124,
125, 126, 133, 135, 137,
Prerequisite: [OAC Calculus] & [one of OAC Algebra
and Geometry or OAC Finite Mathematics]
MATA27H3 Introduction to Optimization
Exclusions: MATA26Y (MATA28Y), MATA29Y, MATA24H,
MATA25H, MAT123, 124, 125, 126, 133, 135, 137
Prerequisite: OAC Calculus
MATA29Y3 Introduction to Mathematical Modeling
Exclusion: MATA26Y, MATA28Y, MATA27H, MATA24H, MATA25H,
MATB41H, MAT123, 124, 125, 126, 133, 135, 137; JMB170
Prerequisites: OAC Calculus and [OAC Algebra and
Geometry or OAC Finite Mathematics]
MATB24H3 Linear Algebra II
Exclusion: MATA04Y, MAT224
Prerequisite: MATA23H, MAT223
MATB41H3 Techniques of the Calculus of Several Variables I
Exclusion: MATA28Y, MATA29Y, MAT230, 234, 235, 237,
239, 257
Prerequisite: MATA23H, MATA26Y
MATB42H3 Techniques of the Calculus of Several Variables II
Exclusion: MAT230, 234, 235, 237, 239, 257
Prerequisite: MATB41H
MATB43H3 Introduction to Analysis
Exclusion: (MATA27Y); MAT246
Prerequisite: MATA26Y & [MATA23H & MATB24H
(MATA04Y)]
Corequisite: MATB42H
MATB44H3 Differential Equations I
Exclusion: (MATC51), MAT244, 267
Prerequisite: MATA26Y & MATA23H
MATB61H3 Linear Programming and Optimization
Exclusion: APM261H
Prerequisite: MATA23H
MATC01H3 Groups and Symmetry
Exclusion: (MATC91H, MATB31), MAT301, (MAT347)
Prerequisite: MATB24H
MATC02H3 Fields and Groups
Exclusion: (MATB32H), MAT302 (MAT347)
Prerequisite: MATC01H
MATC15H3 Introduction to Number Theory
Exclusion: (MATB70), MAT315
Prerequisite: MATA26Y & MATB24H, (MATA04Y)
MATC25H3 Classical Plane Geometries and their Transformations
Exclusion: (MATB30), MAT365 (325)
Prerequisite: MATA23H
MATC32H3 Graph Theory and Algorithms for its Applications
Prerequisite: [MATB24H or CSCB38H] & at least
one other B-level course in Mathematics or Computer Science
MATC34H3 Complex Variables
Exclusion: (MATC60H), MAT334
Prerequisite: MATB42
MATC46H3 Differential Equations II
Exclusion: (MATC56), MAT267
Prerequisite: MATB44H & MATB24H
MATC61H3 Introduction to Mathematical Finance
Prerequisite: MATB42H, STAB47H
Corequisites: MATB61H & MATC46H
MATC63H3 Differential Geometry
Exclusion: MATC54, MAT363
Prerequisite: MATB43
MATC65H3 Complex Variables II
Exclusion: MAT354
Prerequisite: MATC34H
MATD01H3 Readings in Mathematics
Prerequisite: A GPA of 2.5 or more and consent of
the instructor.
MATD02H3 Readings in Mathematics II
Prerequisite: A GPA of 2.5 or more and consent of
the instructor.
COURSES NOT OFFERED 2002/2003
MATB30H3 Geometry I
Exclusion: (MATB25Y)
Prerequisite: MATA04Y
Co-requisite: MATB31H (MATB44H)
MATC30H3 Geometry II
Exclusion: (MATB25Y)
Prerequisite: MATC25H
MATC31H3 Combinatorics
Prerequisites: MATB44H or (CSCB38) & at least
one other B-level course in mathematics or computer science
MATC35H3 Chaos, Fractals
and Dynamics
Exclusion: MAT335
Prerequisites: MATB43H
MATC38H3 Introduction
to Real Analysis
Exclusion: MAT337 (MAT338)
Prerequisite: MATB42H & MATB43H
MATC44H3 Introduction
to Combinatorics
Exclusion: (MATC31), MAT344
Prerequisite: MATB24H
MATC53Y3 Real Analysis
Exclusion: (MATC50H), (MATC55H), MAT338, 350
Prerequisite: MATB32H (MATB49H) & MATB41H &
MATB42H & MATB43H
Full Listing of Courses Not Offered
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