Mathematics
Faculty
List
Associate Chair: L.C. Jeffrey (416-287-7265) 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 and Major Programs in Mathematics are eligible for
inclusion in the Co-operative Program in Physical Sciences and in
the Concurrent Teacher Education Program (CTEP). Please refer to the
Physical Sciences section, the Co-operative Programs section and the
Concurrent Teacher Education section of this Calendar for
further information. The Supervisor of Studies for the Co-operative
programs is S. Chrysostomou (chrysostomou@utsc.utoronto.ca). Service Learning and Outreach (Previously Known as Science
Engagement) Mathematics ProgramsSPECIALIST PROGRAM IN MATHEMATICS (SCIENCE) Supervisor of Studies: E. Moore (416-287-7267) Email: emoore@utsc.utoronto.ca Program Objectives This program provides the student with a sound foundation in the main areas of mathematics, and some exposure to computer programming and statistics. It comprises four streams: Comprehensive, Statistics, Teaching, and Design-Your-Own, each serving a more specific goal. The Comprehensive Stream provides a broad and deep knowledge of mathematics at the undergraduate level. It is the recommended program for students who plan to pursue graduate study in mathematics, but it is also suitable for other career paths. The Statistics Stream provides greater exposure to statistics, and the areas of mathematics most closely associated with it. This stream prepares students for careers in industry, or for graduate study in certain mathematically-oriented subjects, including statistics and financial mathematics. The Teaching Stream is intended for students with a serious interest in mathematics but whose career objectives lie in mathematics education at the elementary or secondary level. The Design-Your-Own Stream allows students to tailor their studies in mathematics to specific interests, with guidance from (and approval of) the program supervisor. Program Requirements The Program requirements consist of a core 14 courses (7 credits), common to all four streams, and additional requirements that depend on the stream, for a total of 25-27 courses (12.5-13.5 credits). The structure of the programs allows for easy switching between streams until relatively late. Consequently, these programs should not be viewed as rigidly separated channel's feeding students to different career paths, but as a flexible structure that provides guidance to students in their course selection based on their broad (but possible fluid) interests. Core (7 credits) 1. Writing Requirement (0.5 credit)(*) 2. A-level courses (2 credits) 3. B-level courses (3.5 credits) 4. C-level courses (1 credit) A. Comprehensive Stream 5. Elementary courses in closely related disciplines (1.5
credits): (***) 6. Additional courses in analysis and algebra (1.5
credits): 7. Courses in key areas of mathematics (1.5 credits):
8. Mathematics of computation (0.5 credit): 9. Electives (1.5 credits): B. Statistics Stream 5. Algebra and Analysis (1.5 credits): 6. Regression Analysis (0.5 credit): 7. Discrete mathematics and geometry (0.5 credit): 8. Upper-level MAT electives (1 credit): 9. Upper-level STA electives (2 credits): C. Teaching Stream 5. Algebra, analysis, and geometry (2 credits): 6. Discrete mathematics (0.5 credit): 7. MAT electives (1.5 credits): 8. MAT/STA/CSC electives (1.5 credits): D. Design-Your-Own-Stream 5. Electives (5.5 credits): Supervisor of Studies:N. Cheredeko (416-287-7226) Email: n.cheredeko@utoronto.ca Program Objectives 1. Foundational courses (5 credits) (*) MATA31H3 is required for MATA37H3 2. Further analysis courses (1 credit) 3. Further algebra geometry, and discrete mathematics courses
(1 credit) 4. Electives (1 credit) Recommended Writing Course: Students are urged to
take a course from the following list of courses by the end of their
second year. See the Statistics section of this Calendar for program requirements. Mathematics CoursesMATA02H3 The Magic of Numbers A selection from the following topics: the number sense (neuroscience
of numbers); numerical notation in different cultures; what
is a number; Zeno’s paradox; divisibility, the fascination of
prime numbers; prime numbers and encryption; perspective in art and
geometry; Kepler and platonic solids; golden mean, Fibonacci sequence;
elementary probability. Systems of linear equations, matrices, Gaussian elimination; basis,
dimension; dot products; geometry to Rn; linear transformations;
determinants, Cramer's rule; eigenvalues and eigenvectors, diagonalization. An introduction to the basic techniques of Calculus. Elementary functions:
rational, trigonometric, root, exponential and logarithmic functions
and their graphs. Basic calculus: limits, continuity, derivatives,
derivatives of higher order, analysis of graphs, use of derivatives;
integrals and their applications, techniques of integration. A theoretical course in calculus emphasizing proofs and techniques,
as well as the intuition behind them. Axioms and basic properties
of real numbers. Functions, including transcendentals. Limits and
continuity. Least upper bounds, extreme and intermediate value theorems.
Derivatives and applications. Integrals and the fundamental theorem
of calculus. This is a calculus course with most examples and applications of
an economic nature. Topics to be covered: linear programming (geometric);
introduction to financial mathematics; continuous functions including
exponential and logarithmic functions with applications to finance;
differential calculus of one variable; marginal analysis; optimization
of single variable functions; techniques of integration. This course will introduce the students to multivariable calculus
and linear algebra. Topics will include: matrix algebra; multi-variable
functions; contour maps; partial and total differentiation; optimization
of multi-variable functions; optimization of constrained multi-variable
functions; Lagrange multipliers. A calculus course emphasizing examples and applications in the biological
and environmental sciences. Discrete probability; basic statistics:
hypothesis testing, distribution analysis. Basic calculus: extrema,
growth rates, diffusion rates; differential equations; population
dynamics; vectors and matrices in 2 and 3 dimensions; genetics applications. This course is intended to prepare students for the physical sciences.
Topics to be covered include: Newton's method, approximation of functions
by Taylor polynomials, numerical methods of integration, complex numbers,
sequences, series, Taylor series, differential equations. A continuation of MATA31H3,
emphasizing proofs and techniques, as well as the intuition behind
them. Transcendental functions revisited. Techniques and applications
of integration. Taylor polynomials and remainder term. Sequences
and series. Uniform convergence and power series. Fields, vector spaces over a field, linear transformations; inner
product spaces, coordinatization and change of basis; diagonalizability,
orthogonal transformations, invariant subspaces, Cayley-Hamilton theorem;
hermitian inner product, normal, self-adjoint and unitary operations.
Some applications such as the method of least squares and introduction
to coding theory. 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.
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. Generalities of sets and functions, countability. Topology
and analysis on the real line: sequences, compactness, completeness,
continuity, uniform continuity. Topics from topology and analysis
in metric and Euclidean spaces. Sequences and series of functions,
uniform convergence. 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. Linear programming, simplex algorithm, duality theory, interior point
method; quadratic and convex optimization, stochastic programming;
applications to portfolio optimization and operations research. 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 calculations. Predicate calculus. Relationship between truth and provability; Gödel's
completeness theorem. First order arithmetic as an example of a first-order
system. Gödel's incompleteness theorem; outline of its proof.
Introduction to recursive functions. 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. The main problems of coding theory and cryptography are defined.
Classic linear and non-linear codes. Error correcting and decoding
properties. Cryptanalysis of classical ciphers from substitution to
DES and various public key systems [e.g. RSA] and discrete logarithm
based systems. Needed mathematical results from number theory, finite
fields, and complexity theory are stated. Fundamentals of set theory, topological spaces and continuous functions,
connectedness, compactness, countability, separatability, metric spaces
and normed spaces, function spaces, completeness, homotopy. Graphs, subgraphs, isomorphism, trees, connectivity, Euler and Hamiltonian
properties, matchings, vertex and edge colourings, planarity, network
flows and strongly regular graphs; applications to such problems as
timetabling, personnel assignment, tank form scheduling, traveling
salesmen, tournament scheduling, experimental design and finite geometries. 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. Topics covered include: metric spaces, dynamics on the real line,
fixed points, periodic points, attractors, repellers, Sharkovski's
theorem parametrized families of functions and bifurcations, period
doubling, dynamics of the logistic map, symbolic dynamics, chaos,
topological equivalence of the logistic map and the shift map, Newton's
method; dynamics on the complex line, iterations of rational functions,
Julia sets, Mandelbrot set. Topics in measure theory: the Lebesgue integral, Riemann-Stieltjes
integral, Lp spaces, Hilbert and Banach spaces, Fourier series. Basic counting principles, generating functions, permutations with
restrictions. Fundamentals of graph theory with algorithms; applications
(including network flows). Combinatorial structures including block
designs and finite geometries. 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. Mathematical analysis of problems associated with biology, including
models of population growth, cell biology, molecular evolution, infectious
diseases, and other biological and medical disciplines. A review of
mathematical topics: linear algebra (matrices, eigenvalues and eigenvectors),
properties of ordinary differential equations and difference equations. 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. The course discusses the Mathematics curriculum (K-12) from the following
aspects: the strands of the curriculum and their place in the world
of Mathematics, the nature of proofs, the applications of Mathematics,
and its connection to other subjects. Mathematical problems which have arisen repeatedly in different cultures,
e.g. solution of quadratic equations, Pythagorean theorem; transmission
of mathematics between civilizations; high points of ancient mathematics,
e.g. study of incommensurability in Greece, Pell's equation in
India. 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, constructibility;
Galois groups of polynomials, in particular cubics; insolvability
of quintics by radicals. An introduction to geometry with a selection of topics from the following:
symmetry and symmetry groups, finite geometries and applications,
non-Euclidean geometry. A variety of topics from geometry, analysis, combinatorics, number
theory and algebra, to be chosen by the instructor. 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. Monte Carlo Method (mean time between failures, servicing requests),
Data Manipulation (z-transform, filters, Bode Plots), Discrete Fourier
Transform (real time processing , FFT, image processing), Regression
(best fit to discrete data, Hilbert Space, Gram's theorem), Frequency-Domain
Methods, Numerical Models for PDE, Galerkin's methods, Cubic Splines. Independent study under direction of a faculty member. |
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