Mathematics

Mathematics: Introduction

​​​​​​​​​​​​​​​​​​​​​​​Faculty Affiliation

Arts and Science

Degree Programs

Mathematics

​​MSc
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.utoronto.ca
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
Buchweitz, Ragnar-Olaf - ScD, DrHab
Burchard, Almut - 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
Khovanskii, Askold - MS, PhD, DSc
Kim, Henry - BSc, PhD
Kudla, Stephen - BA, MA, 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 (Graduate Chair)
Rafi, Kasra - BSc, PhD
Repka, Joseph - BSc, PhD
Rosenthal, Jeffrey - BSc, AM, PhD, FRSC
Rotman, Regina - BA, PhD (Associate Chair - Graduate)
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
Barbeau, Edward - BA, MA, PhD
Choi, Man-Duen - BSc, MSc, PhD
Davis, H Chandler - BS, MA, PhD
Ellers, Erich - DrRerNat, DrRerNat
Fraser, Donald AS - BA, MA, PhD, FRSC
Greiner, Peter - BSc, MA, PhD
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

Associate Members

Aretakis, Stefanos - MA, PhD
Cunningham, Clifton - MSc, PhD
De Simoi, Jacopo - PhD
Faifman, Dmitry - BSc, MSc, PhD
Farah, Ilijas - PhD
Fortier Bourque, Maxime - BSc, MSc, MSc, PhD
Haslhofer, Robert - BSc, MSc, PhD
Jaimungal, Sebastian - BSc, MSc, PhD
Lefebvre, Jeremie - BSc, PhD
Rossman, Benjamin - BA, MA, PhD
Shankar, Arul - BSc, PhD
Shi, Xianwen - PhD
Tiozzo, Giulio - BA, MA, 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.

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 and 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 a 16-month or 24-month program which includes an approved selection of prerequisite and other courses in addition to the program requirements below. 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.

  • 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.

  • This option is completed full-time; it is not available on a part-time basis.

Program Length

4 sessions full-time over 16 months (typical registration sequence: F/W/S/F);
6 sessions full-time over 24 months (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.

 

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.

  • Normally, a master's degree from a recognized university. However, exceptionally strong BSc students may apply for direct admission to the PhD program. In all cases, 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 3.0 full-course equivalents (FCEs) (six half courses).

  • Students must pass a comprehensive examination in basic mathematics before beginning an area of research. This examination should be taken as soon as possible, and not later than the beginning of the third session of PhD study. The usual examination covers the three general areas of analysis, algebra, and topology, at the level of Year 1 graduate courses offered by the department in these subjects. Students planning to specialize in applied mathematics must take the analysis and/or algebra portion of the comprehensive examination, but may substitute from several areas of applied mathematics for the remaining portions.

  • 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 full-time; 5 years direct-entry

Time Limit

6 years full-time; 7 years direct-entry

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 will be available from the department in May. 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 An​alysis 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 1044​H
​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 1063H
​Topics in Partial Differential Equations II
​MAT 1075H
​Differential Analysis
​MAT 1100H
​Algebra I
​MAT 1101H
​Algebra II
​MAT 1102H
​Topics in the Theory of Groups
​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 1124H
​Topics in Matrix Theory
​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 1194H
​​Algebraic Curves
​MAT 1195H
​Elliptic Curves and Cryptography
​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
​Topology I
​MAT 1301H
​Topology II
​MAT 1302H
​Combinatorial Theory
​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 1350H
​Topics in Algebraic Topology I
​MAT 1351H
​Topics in Homotopy Theory
​MAT 1352H
​Topics in Algebraic Topology II
​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 1448H
​Topics in Set Theoretic Topology
​MAT 1449H
​Seminar in Foundations
​MAT 1450H
​Topics in Foundations
​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 1520H
​Wave Propagation
​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 1711H
​Topics in 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 1760H
​Computer Algebra
​MAT 1761H
​Algorithms in Algebraic Geometry
​MAT 1840H
​Control Theory
​MAT 1841H
​Mathematics of Massive Data Analysis: Fundamentals and Applications
​MAT 1845H
​Dynamical Systems
​MAT 1847H
​Holomorphic Dynamics
​MAT 1855H
​Mathematical Economics
​MAT 1856H
​Mathematical Finance
​MAT 1880H
​Case S​tudies 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

MSc Project

MAT 4000Y​+
Supervised Research Project

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