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.mathematics.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

Aretakis, Stefanos - MA, 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

Bremer Jr, James - BSc, BSc, PhD

Burchard, Almut - MS, PhD

Collins, Tristan - PhD

De Simoi, Jacopo - PhD

Desjardins, Julie - BA, MA, PhD

Elliott, George - BSc, MSc, PhD

Friedlander, John - BSc, BS, MA, PhD

Graham, Ian - BSc, ScD

Gualtieri, Marco - BSc, DPhil

Haslhofer, Robert - BSc, MSc, PhD

Herzig, Florian - BA, PhD

Ingram, Patrick - PhD

Ivrii, Victor - MA, PhD, DSc

Jeffrey, Lisa - BA, MA, PhD

Jerrard, Robert - AB, PhD (Chair and Graduate Chair)

Kamnitzer, Joel - BMath, PhD

Kapovitch, Vitali - BS, PhD

Karshon, Yael - PhD

Khanin, Konstantin - PhD

Khesin, Boris - MS, PhD

Khovanski, Askold - MS, DSc, PhD

Kim, Henry - BSc, PhD

Kopparty, Swastik - BS, MS, PhD

Kudla, Stephen - BA, MA, PhD

Litt, Daniel - PhD

Marcolli, Matilde - MS, PhD

McCann, Robert - BS, PhD

Meinrenken, Eckhard - PhD

Milman, Pierre - MA, PhD

Murphy, Emmy - BS, 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

Rafi, Kasra - BSc, PhD

Repka, Joe - BSc, PhD

Rosenthal, Jeffrey - BSc, AM, PhD, FRSC

Rotman, Regina - BA, PhD

Rozenblyum, Nick - PhD

Saraf, Shubhangi - BS, MS, PhD

Scherk, John - BSc, MSc, DPhil

Seco, Luis - PhD

Serkh, Kirill - BS, MS, PhD

Shankar, Arul - BSc, PhD

Sigal, Israel Michael - BA, PhD

Stinchcombe, Adam - BMath, PhD

Sulem, Catherine - MMath, PhD

Tanny, Stephen - BSc, PhD

Tiozzo, Giulio - BA, MA, PhD

Todorcevic, Stevo - PhD

Tsimerman, Jacov - BSc, PhD

Uriarte-Tuero, Ignacio - BS, MSc, PhD (Associate Chair - Graduate)

Virag, Balint - BA, MA, PhD

Weiss, William - BSc, MSc, PhD

Yampolsky, Michael - DPhil

Yu, Yun William - BA, MPH, MRes, PhD

Zhang, Ke - BS, PhD

Members Emeriti

Akcoglu, Mustafa - MSc, PhD

Andrews, David - BSc, MSc, PhD

Ellers, Erich - DrRerNat, DrRerNat

Halperin, J. Stephen - BSc, MSc, PhD, FRSC

Jurdjevic, Velimir - BS, MS, PhD

Kupka, Ivan - BSc, PhD, PhD

McCool, James - BSc, PhD

Murasugi, Kunio - BSc, DSc

Murnaghan, Fiona - BSc, MSc, PhD

Selick, Paul - BSc, MSc, PhD

Sen, Dipak - MSc, DSc

Sharpe, Richard - BSc, MA, PhD

Smith, Stuart - BSc, PhD

Tall, Franklin - AB, PhD

Associate Members

Dauvergne, Duncan - BSc, MSc, PhD

Landon, Benjamin Christopher - BSc, MSc, PhD

Liokumovich, Yevgeny - BSc, MSc, PhD

Olano Espinosa, Sebastian - PhD

Pan, Wenyu - PhD

Shlapentokh-Rothman, Yakov - BS, PhD

Spink, Hunter - PhD

Varma, Ila - BS, MSc, PhD

Mathematics: Mathematics MSc

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.

Completion Requirements

Students must complete the program in one of two ways:
- 3.0 approved full-course equivalents (FCEs) and a supervised research project (MAT4000Y), 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.

Mode of Delivery: In person
Program Length: 3 sessions full-time (typical registration sequence: FWS); 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.

Completion 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 (MAT4000Y), 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.

Mode of Delivery: In person
Program Length: 4 sessions full-time (typical registration sequence: FWS-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.

Completion Requirements

Students must complete the program full-time as follows:
- 3.0 approved full-course equivalents (FCEs) and a supervised research project (MAT4000Y), 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.

Mode of Delivery: In person
Program Length: 6 sessions full-time (typical registration sequence: FWS-FWS)
Time Limit: 3 years full-time

Mathematics: Mathematics PhD

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.

Completion 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:
- MAT1000H Real Analysis I.
- MAT1001H Real Analysis II.
- MAT1002H Complex Analysis.
- MAT1060H Partial Differential Equations I.
- MAT1061H Partial Differential Equations II.
- MAT1100H Algebra I.
- MAT1101H Algebra II.
- MAT1300H Topology I.
- MAT1301H Topology II.
- MAT1600H Mathematical Probability I.
- MAT1601H Mathematical Probability II.
- MAT1850H Linear Algebra and Optimization.

Comprehensive examination.
- Students must pass a comprehensive examination in basic mathematics before beginning an area of research. This examination is scheduled at the start of the Fall session (usually September) and should be taken no later than the start of the third session.
- Students have the option to write the final exam of any core course to obtain core credit. This requires approval of the Graduate Office.
- 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.

Mode of Delivery: In person
Program Length: 4 years full-time (typical registration sequence: Continuous)
Time Limit: 6 years full-time

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.

Completion 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):
  - MAT1000H Real Analysis I.
  - MAT1001H Real Analysis II.
  - MAT1002H Complex Analysis.
  - MAT1060H Partial Differential Equations I.
  - MAT1061H Partial Differential Equations II.
  - MAT1100H Algebra I.
  - MAT1101H Algebra II.
  - MAT1300H Topology I.
  - MAT1301H Topology II.
  - MAT1600H Mathematical Probability I.
  - MAT1601H Mathematical Probability II.
  - MAT1850H Linear Algebra and Optimization.
- Students must also complete 1.0 elective FCE.

Students must complete MAT4000Y Supervised Research Project or its equivalent.
Comprehensive examination.
- Students must pass a comprehensive examination in basic mathematics before beginning an area of research. This examination is scheduled at the start of the Fall session (usually September) and should be taken no later than the start of the third session.
- Students have the option to write the final exam of any core course to obtain core credit. This requires approval of the Graduate Office.
- 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.

Mode of Delivery: In person
Program Length: 5 years full-time (typical registration sequence: Continuous)
Time Limit: 7 years full-time

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 MAT1000H, 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.

Course Code	Course Title
MAT1000H	Real Analysis I
MAT1001H	Real Analysis II
MAT1002H	Complex Analysis
MAT1004H	Theory of Approximation
MAT1005H	Fourier Analysis
MAT1006H	Topics in Real Analysis
MAT1007H	Topics in Complex Variables
MAT1008H	Functions of a Complex Variable
MAT1010H	Functional Analysis
MAT1011H	Introduction to Linear Operators
MAT1012H	Real Analysis II
MAT1013H	Theory of Several Complex Variables II
MAT1015H	Topics in Operator Theory
MAT1016Y	Topics in Operator Algebras
MAT1017H	Introduction to K-theory for Operator Algebras
MAT1034H	Topics in Harmonic Analysis
MAT1037H	Von Neumann Algebras
MAT1045H	Topics in Ergodic Theory
MAT1051H	Introduction to Ordinary Differential Equations
MAT1060H	Partial Differential Equations I
MAT1061H	Partial Differential Equations II
MAT1062H	Topics in Partial Differential Equations I
MAT1064H	Elliptic Boundary Value Problems on Nonsmooth Domains
MAT1100H	Algebra I
MAT1101H	Algebra II
MAT1103H	Topics in Algebra I
MAT1104H	Topics in Algebra II
MAT1105H	Topics in Representation Theory
MAT1109H	Classical Groups
MAT1110H	Algebraic Groups
MAT1120H	Lie Groups and Lie Algebras I
MAT1126H	Lie Groups and Fluid Dynamics
MAT1128H	Topics in Probability
MAT1155H	Commutative Algebra
MAT1190H	Algebraic Geometry
MAT1191H	Topics in Algebraic Geometry
MAT1192H	Advanced Topics in Algebraic Geometry
MAT1194H	Algebraic Curves
MAT1196H	Representation Theory
MAT1197H	Automorphic Forms and Representation Theory I
MAT1198H	Automorphic Forms and Representation Theory II
MAT1199H	Automorphic Forms
MAT1200H	Algebraic Number Theory
MAT1202H	Analytic Number Theory
MAT1203H	Computational Aspects of Number Theory
MAT1210H	Topics in Number Theory
MAT1300H	Differential Topology
MAT1301H	Algebraic Topology
MAT1302H	Combinatorial Methods
MAT1303H	Combinatorial Designs
MAT1304H	Topics in Combinatorics
MAT1305H	Topics in Geometric Topology
MAT1306H	The Discrete Mathematics Toolkit
MAT1309H	Geometrical Inequalities
MAT1312H	Topics in Geometry
MAT1313Y	Seminar in Geometry
MAT1314H	Introduction to Noncommutative Geometry
MAT1318H	Seminar in Geometry and Topology
MAT1340H	Differential Topology
MAT1341H	Topics in Differential Geometry
MAT1342H	Introduction to Differential Geometry
MAT1343H	Riemannian Manifolds
MAT1344H	Symplectic Geometry
MAT1346H	Homotopy Theory
MAT1347H	Topics in Symplectic Geometry and Topology
MAT1351H	Topics in Homotopy Theory
MAT1355H	Singularity Theory
MAT1359H	Moduli Spaces of Flat Connections
MAT1360H	Complex Manifolds
MAT1392H	Algebra Seminar
MAT1399H	Advanced Point Set Topology
MAT1403H	Model Theory
MAT1404H	Introduction to Model Theory and Set Theory
MAT1430H	Set Theory
MAT1435H	Topics in Set Theory
MAT1449H	Seminar in Foundations
MAT1497H	Professional Development
MAT1498H	Communicating Mathematics to a General Audience
MAT1499H	Teaching Large Mathematics Courses

Applied Mathematics

Course Code	Course Title
MAT1500H	Topics in Graph Theory
MAT1500Y	Applied Analysis
MAT1502H	Topics in Geometric Analysis
MAT1507H	Asymptotic and Perturbation Methods
MAT1508H	Techniques of Applied Mathematics
MAT1509H	Mathematical and Computational Linguistics
MAT1520H	Wave Propagation
MAT1525H	Topics in Inverse Problems and Image Analysis
MAT1525Y	Inverse Problems of X-Ray and Radar Imaging
MAT1600H	Mathematical Probability I
MAT1601H	Mathematical Probability II
MAT1639Y	Topics in Fluid Mechanics
MAT1700H	General Relativity
MAT1710H	Group Theory and Quantum Mechanics
MAT1722H	C* Algebras and Quantum Mechanics
MAT1723H	Foundations of Quantum Mechanics
MAT1724H	Functional Analysis in Quantum Mechanics
MAT1725Y	Scattering Theory
MAT1739H	Topics in Mathematical Physics
MAT1750H	Computational Mathematics
MAT1751H	Quantum Computing, Foundations to Frontier
MAT1760H	Computer Algebra
MAT1761H	Algorithms in Algebraic Geometry
MAT1800H	Methods of Applied Mathematics I
MAT1801H	Methods of Applied Mathematics II
MAT1839H	Integral Equation Methods for the Numerical Solution of PDEs
MAT1840H	Control Theory
MAT1841H	Mathematics of Massive Data Analysis: Fundamentals and Applications
MAT1844H	Nonlinear Dynamical Systems
MAT1845H	Dynamical Systems
MAT1847H	Holomorphic Dynamics
MAT1850H	Linear Algebra and Optimization
MAT1855H	Mathematical Problems in Economics
MAT1856H	Mathematical Finance
MAT1880H	Case Studies in Applied Mathematics

Individual Reading Courses

Course Code	Course Title
MAT1900Y	Readings in Pure Mathematics
MAT1901H	Readings in Pure Mathematics
MAT1902H	Readings in Pure Mathematics
MAT1950Y	Readings in Applied Mathematics
MAT1951H	Readings in Applied Mathematics

Seminars

Course Code	Course Title
MAT3001H	Seminar in Pure Mathematics
MAT3002H	Seminar in Applied Mathematics

Research Project

Course Code	Course Title
MAT4000Y	Supervised Research Project