Mathematics
Mathematics: Introduction
Faculty Affiliation
Arts and Science
Degree Programs
Mathematics
MSc and PhD
Overview
The Department of Mathematics is a distinguished Faculty of more than 60 mathematicians, offering research opportunities in the areas of pure mathematics and applied mathematics. Faculty areas of research include, but are not limited to, real and complex analysis, ordinary and partial differential equations, harmonic analysis, nonlinear analysis, several complex variables, functional analysis, operator theory, C*algebras, ergodic theory, group theory, analytic and algebraic number theory, Lie groups and Lie algebras, automorphic forms, commutative algebra, algebraic geometry, singularity theory, differential geometry, symplectic geometry, classical synthetic geometry, algebraic topology, set theory, settheoretic topology, mathematical physics, fluid mechanics, probability, combinatorics, optimization, control theory, dynamical systems, computer algebra, cryptography, and mathematical finance.
Contact and Address
Web: www.math.toronto.edu/cms
Email: gradinfo@math.toronto.edu
Telephone: (416) 9787894
Fax: (416) 9784107
Department of Mathematics
University of Toronto
Room 6290, 40 St. George Street
Toronto, Ontario M5S 2E4
Canada
Mathematics: Graduate Faculty
Full Members
Members Emeriti
Associate Members
Mathematics: Mathematics MSc
Master of Science
Program Description
The MSc is a researchoriented program. Opportunities for graduate study and research are available in most of the main areas of pure and applied mathematics. There is a large selection of graduate courses and seminars, a diverse student body of domestic and international students, and yet classes are small and the ratio of graduate students to faculty is low.
Many recent graduates are engaged in university teaching, and a significant number hold administrative positions in universities or in the professional communities. Others are pursuing careers in industry (technological or financial) or in government.
The MSc program is offered:

for students with a complete undergraduate background in mathematics:

12 months fulltime

24 months parttime


for students who do not have a complete undergraduate background in mathematics. This option is not available on a parttime basis:

16 months fulltime

24 months fulltime

Provisional admission to the PhD program may be granted at the time of admission to the master's program.
MSc Program (12Month FullTime and 24Month PartTime)
Minimum Admission Requirements

Applicants are admitted under the General Regulations of the School of Graduate Studies. Applicants must also satisfy the Department of Mathematics' additional admission requirements stated below.

Evidence of an excellent academic background and mathematical ability.
Program Requirements

Students must complete the program in one of two ways:

3.0 approved fullcourse equivalents (FCEs) and a supervised research project (MAT 4000Y), or its equivalent, or

2.0 approved FCEs and an acceptable thesis. Two approved halfyear courses are considered the equivalent of a fullyear course.


With approval, two prerequisite undergraduate half courses can be substituted for 0.5 graduate FCE.

Students may, with approval, take courses outside the department as part of a coherent program.

Students who undertake the MSc parttime must, at a minimum, satisfy the requirements of the 12month program.

Students who plan to continue to the PhD program may select 2.0 FCEs in core courses from the approved list in the PhD program requirements section. Students who obtain a grade of A– or higher in each of the corresponding core courses may count coursework towards the PhD comprehensive examination requirement in the particular subject areas.
Program Length
3 sessions fulltime (typical registration sequence: F/W/S);
6 sessions parttime
Time Limit
3 years fulltime;
6 years parttime
MSc Program (16Month FullTime)
Minimum Admission Requirements

Applicants are admitted under the General Regulations of the School of Graduate Studies. Applicants must also satisfy the Department of Mathematics' additional admission requirements stated below.

Evidence of an excellent academic background and mathematical ability.

Students who do not have a complete undergraduate background in mathematics may be accepted into the 16month program. This possibility may interest students who have some background in a subject in which mathematics is applied and/or who are interested in industrial applications of mathematics.
Program Requirements

Students must complete the program fulltime in one of two ways:

3.0 approved fullcourse equivalents (FCEs) and a supervised research project (MAT 4000Y), or its equivalent, or

2.0 approved FCEs and an acceptable thesis. Two approved halfyear courses are considered the equivalent of a fullyear course.


Students must also complete an approved selection of prerequisites and other courses: an additional 2.0 FCEs in Year 2, 3, or 4 undergraduate courses in any of the following subjects: algebra, analysis, partial differential equations, probability, and topology.

With approval, two prerequisite undergraduate half courses can be substituted for 0.5 graduate FCE.

Students may, with approval, take courses outside the department as part of a coherent program.

Students who plan to continue to the PhD program may select 2.0 FCEs in core courses from the approved list in the PhD program requirements section. Students who obtain a grade of A– or higher in each of the corresponding core courses may count coursework towards the PhD comprehensive examination requirement in the particular subject areas.
Program Length
4 sessions fulltime (typical registration sequence: F/W/S/F)
Time Limit
3 years fulltime
MSc Program (24Month FullTime)
Minimum Admission Requirements

Applicants are admitted under the General Regulations of the School of Graduate Studies. Applicants must also satisfy the Department of Mathematics' additional admission requirements stated below.

Evidence of an excellent academic background and mathematical ability.

Students who do not have a complete undergraduate background in mathematics may be accepted into the 24month program. This possibility may interest students who have some background in a subject in which mathematics is applied and/or who are interested in industrial applications of mathematics.
Program Requirements

Students must complete the program fulltime in one of two ways:

3.0 approved fullcourse equivalents (FCEs) and a supervised research project (MAT 4000Y), or its equivalent, or

2.0 approved FCEs and an acceptable thesis. Two approved halfyear courses are considered the equivalent of a fullyear course.


Students must also complete an approved selection of prerequisites and other courses: an additional 3.0 FCEs in Year 2, 3, or 4 undergraduate courses in any of the following subjects: algebra, analysis, partial differential equations, probability, and topology.

With approval, two prerequisite undergraduate half courses can be substituted for 0.5 graduate FCE.

Students may, with approval, take courses outside the department as part of a coherent program.

Students who plan to continue to the PhD program may select 2.0 FCEs in core courses from the approved list in the PhD program requirements section. Students who obtain a grade of A– or higher in each of the corresponding core courses may count coursework towards the PhD comprehensive examination requirement in the particular subject areas.
Program Length
6 sessions fulltime (typical registration sequence: F/W/S/F/W/S)
Time Limit
3 years fulltime
Mathematics: Mathematics PhD
Doctor of Philosophy
Program Description
The PhD is a researchoriented program consisting of coursework, comprehensive examinations, and a thesis embodying the results of original research. Opportunities for graduate study and research are available in most of the main areas of pure and applied mathematics.
Applicants may enter the PhD program via one of two routes: 1) following completion of an appropriate MA or 2) direct entry following completion of a bachelor’s degree.
PhD Program
Minimum Admission Requirements

Applicants are admitted under the General Regulations of the School of Graduate Studies. Applicants must also satisfy the Department of Mathematics' additional admission requirements stated below.

A master's degree from a recognized university. Students must satisfy the department of their ability to do independent research at an advanced level. They must show evidence of an excellent academic background and mathematical ability.
Program Requirements

Coursework. Students must successfully complete at least 3.0 fullcourse equivalents (FCEs). Out of the following 12 core courses, students must complete 6 courses.
Core Courses
MAT 1000H

Real Analysis I

MAT 1001H

Real Analysis II

MAT 1002H

Complex Analysis

MAT 1060H

Partial Differential Equations I

MAT 1061H

Partial Differential Equations II

MAT 1100H

Algebra I

MAT 1101H

Algebra II

MAT 1300H

Topology I

MAT 1301H

Topology II

MAT 1600H

Mathematical Probability I

MAT 1601H

Mathematical Probability II

MAT 1850H

Linear Algebra and Optimization


Comprehensive examinations.

Students must pass comprehensive examinations in basic mathematics before beginning an area of research. These examinations are scheduled at the start of the Fall session (usually September) and should be taken no later than the start of the third session.

Students who obtain a grade of A– or higher in each of the corresponding core courses for the general areas of mathematics will be exempted from the comprehensive examination requirement in the specific area of study.


Students must pass a qualifying oral examination or give a seminar presentation in their particular area of study before embarking on serious thesis research.

The main requirement of the degree is an acceptable thesis embodying original research of a standard that warrants publication in the research literature.
Program Length
4 years
Time Limit
6 years
PhD Program (DirectEntry)
Minimum Admission Requirements

Applicants are admitted under the General Regulations of the School of Graduate Studies. Applicants must also satisfy the Department of Mathematics' additional admission requirements stated below.

Exceptionally strong BSc students with a grade point average (GPA) of 3.7 or higher may apply for direct admission to the PhD program. Students must satisfy the department of their ability to do independent research at an advanced level. They must show evidence of an excellent academic background and mathematical ability.
Program Requirements
 Coursework. Students must complete at least 4.0 fullcourse equivalents (FCEs). Out of the following 12 core courses, students must complete 6 courses (3.0 FCEs). Students must also complete 1.0 elective FCE.
Core Courses
MAT 1000H  Real Analysis I 
MAT 1001H  Real Analysis II 
MAT 1002H  Complex Analysis 
MAT 1060H  Partial Differential Equations I 
MAT 1061H  Partial Differential Equations II 
MAT 1100H  Algebra I 
MAT 1101H  Algebra II 
MAT 1300H  Topology I 
MAT 1301H  Topology II 
MAT 1600H  Mathematical Probability I 
MAT 1601H  Mathematical Probability II 
MAT 1850H  Linear Algebra and Optimization 

Students must complete MAT 4000Y^{+} Supervised Research Project (1.0 FCE) or its equivalent.

Comprehensive examinations.

Students must pass comprehensive examinations in basic mathematics before beginning an area of research. These examinations are scheduled at the start of the Fall session (usually September) and should be taken no later than the start of the third session.

Students who obtain a grade of A– or higher in each of the corresponding core courses for the general areas of mathematics will be exempted from the comprehensive examination requirement in the specific area of study.


Students must pass a qualifying oral examination or give a seminar presentation in their particular area of study before embarking on serious thesis research.
The main requirement of the degree is an acceptable thesis embodying original research of a standard that warrants publication in the research literature.
Program Length
5 years
Time Limit
7 years
Mathematics: Mathematics MSc, PhD Courses
Each year the department offers a selection of courses chosen from the following list, with the possibility of further additions. The courses MAT 1000H, 1001H, 1100H, 1101H, 1300H, 1301H, 1600H, and 1601H will be offered each year; the complete list of courses is available from the department. In addition, it may be possible for a student to arrange to take one of the listed courses as an individual reading course. Students should consult the office of the Coordinator at the beginning of the academic year.
PhD students are expected to attend and contribute to seminars in the research areas.
MAT 1000H

Real Analysis I

MAT 1001H

Real Analysis II

MAT 1002H

Complex Analysis

MAT 1003H

Theory of Several Complex Variables

MAT 1004H

Theory of Approximation

MAT 1005H

Fourier Analysis

MAT 1006H

Topics in Real Analysis

MAT 1007H

Topics in Complex Variables

MAT 1008H

Functions of a Complex Variable

MAT 1010H

Functional Analysis

MAT 1011H

Introduction to Linear Operators

MAT 1012H

Real Analysis II

MAT 1013H

Theory of Several Complex Variables II

MAT 1015H

Topics in Operator Theory

MAT 1016Y

Topics in Operator Algebras

MAT 1017H

Introduction to Ktheory for Operator Algebras

MAT 1034H

Topics in Harmonic Analysis

MAT 1037H

Von Neumann Algebras

MAT 1044H

Potential Theory

MAT 1045H

Topics in Ergodic Theory

MAT 1051H

Introduction to Ordinary Differential Equations

MAT 1052H

Topics in Ordinary Differential Equations

MAT 1060H

Partial Differential Equations I

MAT 1061H

Partial Differential Equations II

MAT 1062H

Topics in Partial Differential Equations I

MAT 1100H

Algebra I

MAT 1101H

Algebra II

MAT 1103H

Topics in Algebra I

MAT 1104H

Topics in Algebra II

MAT 1105H

Topics in Representation Theory

MAT 1109H

Classical Groups

MAT 1110H

Algebraic Groups

MAT 1120H

Lie Groups and Lie Algebras I

MAT 1122H

Lie Groups and Representations I

MAT 1126H

Lie Groups and Fluid Dynamics

MAT 1128H

Topics in Probability

MAT 1155H

Commutative Algebra

MAT 1190H

Algebraic Geometry

MAT 1191H

Topics in Algebraic Geometry

MAT 1192H

Advanced Topics in Algebraic Geometry

MAT 1196H

Representation Theory

MAT 1197H

Automorphic Forms and Representation Theory I

MAT 1198H

Automorphic Forms and Representation Theory II

MAT 1199H

Automorphic Forms

MAT 1200H

Algebraic Number Theory

MAT 1202H

Analytic Number Theory

MAT 1203H

Computational Aspects of Number Theory

MAT 1210H

Topics in Number Theory

MAT 1300H

Differential Topology

MAT 1301H

Algebraic Topology

MAT 1302H

Combinatorial Methods

MAT 1303H

Combinatorial Designs

MAT 1304H

Topics in Combinatorics

MAT 1305H

Topics in Geometric Topology

MAT 1309H

Geometrical Inequalities

MAT 1312H

Topics in Geometry

MAT 1313Y

Seminar in Geometry

MAT 1314H

Introduction to Noncommutative Geometry

MAT 1318H

Seminar in Geometry and Topology

MAT 1340H

Differential Topology

MAT 1341H

Topics in Differential Geometry

MAT 1342H

Introduction to Differential Geometry

MAT 1343H

Riemannian Manifolds

MAT 1344H

Symplectic Geometry

MAT 1346H

Homotopy Theory

MAT 1347H

Topics in Symplectic Geometry and Topology

MAT 1351H

Topics in Homotopy Theory

MAT 1355H

Singularity Theory

MAT 1359H

Moduli Spaces of Flat Connections

MAT 1360H

Complex Manifolds

MAT 1392H

Algebra Seminar

MAT 1399H

Advanced Point Set Topology

MAT 1403H

Model Theory

MAT 1404H

Introduction to Model Theory and Set Theory

MAT 1430H

Set Theory

MAT 1435H

Topics in Set Theory

MAT 1436H

Large Cardinals, Structure Theory of Ideals, and Applications (prerequisites: MAT 309H or MAT 409H)

MAT 1449H

Seminar in Foundations

MAT 1450H

Topics in Foundations

MAT 1498H  Communicating Mathematics to a General Audience (Credit/No Credit) 
MAT 1499H

Teaching Large Mathematics Courses (Credit/No Credit)

Applied Mathematics
MAT 1500Y

Applied Analysis

MAT 1501H

Applied Analysis I

MAT 1502H

Topics in Geometric Analysis

MAT 1507H

Asymptotic and Perturbation Methods

MAT 1508H

Techniques of Applied Mathematics

MAT 1509H  Mathematical and Computational Linguistics 
MAT 1520H

Wave Propagation

MAT 1525H  Topics in Inverse Problems and Image Analysis 
MAT 1525Y

Inverse Problems of XRay and Radar Imaging

MAT 1600H

Mathematical Probability I

MAT 1601H

Mathematical Probability II

MAT 1638H

Fluid Mechanics

MAT 1639Y

Topics in Fluid Mechanics

MAT 1700H

General Relativity

MAT 1710H

Group Theory and Quantum Mechanics

MAT 1722H

C* Algebras and Quantum Mechanics

MAT 1723H

Foundations of Quantum Mechanics

MAT 1724H

Functional Analysis in Quantum Mechanics

MAT 1725Y

Scattering Theory

MAT 1739H

Topics in Mathematical Physics

MAT 1750H

Computational Mathematics

MAT 1751H  Quantum Computing, Foundations to Frontier 
MAT 1760H

Computer Algebra

MAT 1761H

Algorithms in Algebraic Geometry

MAT 1800H  Methods of Applied Mathematics I 
MAT 1801H  Methods of Applied Mathematics II 
MAT 1840H

Control Theory

MAT 1841H

Mathematics of Massive Data Analysis: Fundamentals and Applications

MAT 1845H

Dynamical Systems

MAT 1847H

Holomorphic Dynamics

MAT 1850H  Linear Algebra and Optimization 
MAT 1855H

Mathematical Problems in Economics

MAT 1856H

Mathematical Finance

MAT 1880H

Case Studies in Applied Mathematics

Individual Reading Courses
MAT 1900Y

Readings in Pure Mathematics

MAT 1901H

Readings in Pure Mathematics

MAT 1902H

Readings in Pure Mathematics

MAT 1951H

Readings in Applied Mathematics

MAT 2001H

Readings in Theoretical Mathematics I

MAT 2002H

Readings in Theoretical Mathematics II

Seminars
MAT 3001H

Seminar in Pure Mathematics (Credit/No Credit)

MAT 3002H

Seminar in Applied Mathematics (Credit/No Credit)

Research Project
MAT 4000Y^{+}

Supervised Research Project

^{+} Extended course. For academic reasons, coursework is extended into session following academic session in which course is offered.