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, set-theoretic 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) 978-7894
Fax: (416) 978-4107

Department of Mathematics
University of Toronto
Room 6290, 40 St. George Street
Toronto, Ontario M5S 2E4
Canada

Mathematics: Graduate Faculty

Full Members

Alexakis, Spyridon - BA, PhD
Arthur, James - BSc, MSc, PhD
Bar-Natan, Dror - BSc, PhD
Bierstone, Edward - BSc, MA, PhD
Binder, Ilia - PhD
Bland, John - BSc, MSc, PhD
Braverman, Alexander - BSc, PhD
Burchard, Almut - MS, PhD (Associate Chair - Graduate)
Colliander, James - BA, MS, PhD
Elliott, George - BSc, MSc, PhD
Friedlander, John - BSc, BS, MA, PhD
Goldstein, Michael - BA, MMath, ScD, PhD
Graham, Ian - BSc, ScD
Gualtieri, Marco - BSc, DPhil
Herzig, Florian - BA, PhD
Ivrii, Victor - MA, PhD, DSc
Jeffrey, Lisa - BA, MA, PhD
Jerrard, Robert - AB, PhD
Kamnitzer, Joel - BMath, PhD
Kapovitch, Vitali - BS, PhD
Karshon, Yael - PhD
Khanin, Konstantin - PhD
Khesin, Boris - MS, PhD
Khovanski, Askold - MS, PhD, DSc
Kim, Henry - BSc, PhD
Kudla, Stephen - BA, MA, PhD
Marcolli, Matilde - MS, PhD
McCann, Robert - BS, PhD
Meinrenken, Eckhard - PhD
Milman, Pierre - MA, PhD
Murnaghan, Fiona - BSc, MSc, PhD
Murty, Vijayakumar - BSc, PhD
Nabutovsky, Alexander - MSc, PhD
Nachman, Adrian - BSc, MA, PhD
Panchenko, Dmitriy - MSc, PhD
Pugh, Mary - BA, MS, PhD
Quastel, Jeremy - BSc, MS, PhD (Chair and Graduate Chair)
Rafi, Kasra - BSc, PhD
Repka, Joe - BSc, PhD
Rosenthal, Jeffrey - BSc, AM, PhD, FRSC
Rotman, Regina - BA, PhD
Scherk, John - BSc, MSc, DPhil
Seco, Luis - PhD
Selick, Paul - BSc, MSc, PhD
Sigal, Israel Michael - BA, PhD
Sulem, Catherine - MMath, PhD
Tanny, Stephen - BSc, PhD
Todorcevic, Stevo - PhD
Tsimerman, Jacov - BSc, PhD
Virag, Balint - BA, MA, PhD
Weiss, William - BSc, MSc, PhD
Yampolsky, Michael - DPhil

Members Emeriti

Akcoglu, Mustafa - MSc, PhD
Andrews, David - BSc, MSc, PhD
Bloom, Thomas - BSc, MA, PhD
Davis, H. Chandler - BS, MA, PhD
Ellers, Erich - DrRerNat, DrRerNat
Fraser, Donald AS - BA, MA, PhD, FRSC
Halperin, J Stephen - BSc, MSc, PhD, FRSC
Haque, Wahidul - MA, MS, PhD
Jurdjevic, Velimir - BS, MS, PhD
Kupka, Ivan - BSc, PhD, PhD
McCool, James - BSc, PhD
Murasugi, Kunio - BSc, DSc
Sen, Dipak - MSc, DSc
Sharpe, Richard - BSc, MA, PhD
Smith, Stuart - BSc, PhD
Tall, Franklin - AB, PhD

Associate Members

Desjardins, Julie - BA, MA, PhD
Dimitrov, Vesselin - MS, PhD
Farah, Ilijas - PhD
Groechenig, Michael - BSc, PhD
Haslhofer, Robert - BSc, MSc, PhD
Lefebvre, Jeremie - BSc, PhD
Leuschke, Graham - BA, MA, PhD
Liokumovich, Yevgeny - BSc, MSc, PhD
Pusateri, Fabio Giuseppe - BS, MS, PhD
Ramdorai, Sujatha - BSc, MSc, PhD
Serkh, Kirill - BS, MS, PhD
Stinchcombe, Adam - BMath, PhD
Varma, Ila - BS, MSc, PhD
Yu, Yun William - BA, MPH, MRes, PhD
Yuen, Henry - PhD

Mathematics: Mathematics MSc

Master of Science​

Program Description

The MSc is a research-oriented 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 full-time

    • 24 months part-time

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

    • 16 months full-time

    • 24 months full-time

Provisional admission to the PhD program may be granted at the time of admission to the master's program.

 

MSc Program (12-Month Full-Time and 24-Month Part-Time)

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 full-course equivalents (FCEs) and a supervised research project (MAT 4000Y), or its equivalent, or

    • 2.0 approved FCEs and an acceptable thesis. Two approved half-year courses are considered the equivalent of a full-year 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 part-time must, at a minimum, satisfy the requirements of the 12-month 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 full-time (typical registration sequence: F/W/S);
6 sessions part-time

Time Limit

3 years full-time;
6 years part-time

 

MSc Program (16-Month Full-Time)

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 16-month 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 full-time in one of two ways:

    • 3.0 approved full-course equivalents (FCEs) and a supervised research project (MAT 4000Y), or its equivalent, or

    • 2.0 approved FCEs and an acceptable thesis. Two approved half-year courses are considered the equivalent of a full-year 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 full-time (typical registration sequence: F/W/S/F)

Time Limit

3 years full-time

 

MSc Program (24-Month Full-Time)

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 24-month 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 full-time in one of two ways:

    • 3.0 approved full-course equivalents (FCEs) and a supervised research project (MAT 4000Y), or its equivalent, or

    • 2.0 approved FCEs and an acceptable thesis. Two approved half-year courses are considered the equivalent of a full-year 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 full-time (typical registration sequence: F/W/S/F/W/S)

Time Limit

3 years full-time

Mathematics: Mathematics PhD

Doctor of Philosophy​

Program Description

The PhD is a research-oriented 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 full-course 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 (Direct-Entry)

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 full-course 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 K-theory 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 X-Ray 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.