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# Mathematics

### Faculty List

**Professor and Chair of the Department **

J. Quastel, MSc, Ph D, FRSC

**Professors and Associate Chair (Research) **

S. Alexakis, BA, Ph D

**Professor and Associate Chair (Graduate) **

K. Rafi, B Sc, Ph D

**Professor and Associate Chair (Undergraduate) **

J. Repka, B Sc, Ph D (U)

**University Professors **

J.G. Arthur, MA, Ph D, FRSC, FRS

J. Friedlander, MA, Ph D, FRSC (UTSC)

I.M. Sigal, BA, Ph D, FRSC

**Professors **

D. Bar-Natan, B Sc, Ph D

E. Bierstone, MA, Ph D, FRSC

I. Binder, B Sc, M Sc, Ph D (UTM)

J. Bland, M Sc, Ph D

A. Braverman, B Sc, Ph D

A. Burchard, B Sc, Ph D

G. Elliott, B Sc, Ph D, FRSC

M. Goldstein, B Sc, Ph D (UTSC)

M. Gualtieri, B Sc, Ph D

V. Ivrii, MA, Ph D, Dr Math, FRSC

L. Jeffrey, AB, Ph D, FRSC (UTSC)

R. Jerrard, M Sc, Ph D (U), FRSC

J. Kamnitzer, B Sc, Ph D

V. Kapovitch, B Sc, Ph D

Y. Karshon, B Sc, Ph D (UTM)

K. Khanin, M Sc, Ph D (UTM)

B. Khesin, M Sc, Ph D

A. Khovanskii, M Sc, Ph D

H. Kim, B Sc, Ph D

S. Kudla, B A, MA, Ph D, FRSC

R. McCann, BSc, Ph D, FRSC

M. Marcolli, M Sc, Ph D

E. Meinrenken, B Sc, Ph D, FRSC

P. Milman, Dipl Maths, Ph D, FRSC

F. Murnaghan, M Sc, Ph D

K. Murty, B Sc, Ph D, FRSC

A. Nabutovsky, M Sc, Ph D

A. Nachman, B Sc, Ph D

D. Panchenko, B Sc, M Sc, Ph D

M. Pugh, BSc, Ph D

R. Rotman BA, Ph D

L. Seco, BA, Ph D (UTM)

C. Sulem, M Sc, Dr D’Etat, FRSC

S. Todorcevic, B Sc, Ph D, FRSC

B. Virag, BA, Ph D (UTSC)

W.A.R. Weiss, M Sc, Ph D (UTM)

M. Yampolsky, B Sc, Ph D (UTM)

**Associate Professors **

F. Herzig, BA, Ph D

J. Scherk, D Phil (UTSC)

J. Tsimerman, Ph D

**Associate Professors, Teaching Stream **

D. Burbulla, B Sc, B Ed, MA

**Assistant Professors **

S. Aretakis, MA, Ph D (UTSC)

C. Blois, B Sc, M Sc, Ph D

J. de Simoi, M Sc, Ph D (UTM)

V. Dimitrov, AB, M Sc, Ph D (Coxeter, CLTA)

M. Groechenig, B Sc, D Phil (UTM)

R. Haslhofer, B Sc, M Sc, Ph D (UTSC)

J. Lefebvre, B Sc, Ph D

Y. Liokumovich, B Sc, M Sc, Ph D (UTM)

F. Pusateri, BS, MS, Ph D

B. Rossman, BA, MA, Ph D

K. Serkh, Ph D

A. Shankar, B Sc, Ph D (UTM)

A. Stinchombe, BMath, Ph D

G. Tiozzo, MA, Ph D (UTSC)

I. Varma, Ph D

W. Yu, Ph D (UTSC)

H. Yuen, BA, Ph D

A. Zaman, B Sc, M Sc, Ph D

K. Zhang, B Sc, Ph D (UTM)

**Assistant Professors, Teaching Stream **

B. Galvao-Souza, Ph D

A. Gracia-Saz, Ph D

N. Hoell BA, MA, M Phil, Ph D - CLTA

S. Mayes-Tang, Bc, MS, Ph D

F. Parsch, B Sc, M Sc, Ph D

B. Rossman, BA, MA, Ph D

J. Siefken, HBS, MS, Ph D

**Lecturers **

S. Homayouni, B Sc, Ph D

N. Jung, BA, MSc, Ph D

E.A.P. LeBlanc, MA, Ph D

J. Tate, B Sc, B Ed

S. Uppal, M Sc

**Professors Emeriti **

M.A. Akcoglu, M Sc, Ph D, FRSC

E.J. Barbeau, MA Ph D (U)

T. Bloom, MA, Ph D, FRSC

M. D. Choi, MA, Ph D, FRSC

H.C. Davis, MA, Ph D (N)

E.W. Ellers, Dr Rer Nat

I.R. Graham, B Sc, Ph D (UTM)

S. Halperin, M Sc, Ph D, FRSC

V. Jurdjevic, MS, PhD

J.W. Lorimer, M Sc, Ph D (U)

E. Mendelsohn, M Sc, Ph D (UTSC)

K. Murasugi, MA, D Sc, FRSC

P. Rosenthal, MA, Ph D, LLB

P. Selick, B Sc, MA, Ph D (UTSC)

D.K. Sen, M Sc, Dr s Sc

F. D. Tall, AB, Ph D (UTM)

**Associate Professor Emeritus **

N.A. Derzko, B Sc, Ph D

S.M. Tanny, B Sc, Ph D (UTM)

**Associate Professors, Teaching Stream, Emiritus**

A. Lam, M Sc

A, Igelfeld, M Sc

**Senior Lecturer Emeritus **

P. Kergin, Ph D

F. Recio, MSc, Ph D

## Introduction

Mathematics is the study of shape, quantity, pattern and structure. It serves as a tool for our scientific understanding of the world. Knowledge of mathematics opens gateways to many different professions such as economics, finance, computing, engineering, and the natural sciences. Aside from practical considerations, mathematics can be a highly satisfying intellectual pursuit, with career opportunities in teaching and research.

The department counts many of Canada's leading research mathematicians among its faculty. Our mathematics programs are flexible, allowing students to select courses based on specialization and interest. Contents range from calculus and linear algebra in the non-specialist programs to more advanced topics such as real and complex analysis, ordinary and partial differential equations, differential geometry, topology, commutative algebra, graph theory, mathematical logic, number theory, and functional analysis.

The department offers eight specialist programs in addition to the major and minor programs.

In the **Mathematics, Applied Mathematics, Mathematics and Physics**, and **Mathematics and Philosophy **specialist programs, students acquire an in-depth knowledge and expertise in mathematical reasoning and the language of mathematics, with its emphasis on rigor and precision. These programs are designed for students wishing to pursue graduate studies; most of the graduates of these programs continue on to graduate school with some of them gaining admission to the world’s best graduate schools.

The **Mathematical Applications in Economics and Finance** specialist program is designed to prepare students for direct entry into the world of finance. It can also serve as a gateway to an MBA or a Master of Finance degree, possibly followed by an eventual doctorate.

The **Mathematics and its Applications** specialist programs offer three areas of concentration: teaching, physical science, and probability/statistics. These specialist programs are designed as `enhanced double majors.' The required courses for these concentrations are almost identical for the first two years, but they diverge in the upper years. Students in these programs can also continue on to graduate studies.

The **Major **and **Minor** programs are intended for students who want to combine mathematical skills with work in other subjects. These programs require less coursework than the specialist programs, but still require the completion of some upper year mathematics courses.

Students interested in becoming K-12 teachers should consider applying to the combined degree program --- a six-year program that leads to an Honours Bachelor of Science (HBSc) from the University of Toronto and a Master of Teaching (MT) from the Ontario Institute for Studies in Education (OISE). The HBSc part of this program involves completing a Math Major, a Minor in Education and Society (offered by Victoria College) and a Minor in an area that would lead to a second "teachable" subject. Please see the Victoria College website for more information.

The Professional Experience Year Co-op Program is available to eligible full-time Specialist and Major students after their second or third year of study. The PEY Co-op program is an optional 12-16 month work term providing industrial experience. It gives students an opportunity to apply their skills in the context of a paid internship.

The Department of Mathematics offers introductory courses for incoming students to foster the development of mathematics skills.

**PUMP Level 1 and PUMP Level 2 (Preparing for University Mathematics Program)**

**Both programs** are non-credit courses that equip students with the necessary background knowledge required to succeed in first year mathematics courses. The content for the courses may be viewed at http://www.math.toronto.edu/cms/undergraduate-program/current-students-ug/pump-courses-2/.

**PUMP Level 1** provides a quick math review during the months of July and August, for students who would like to take six weeks prior to the start of the first semester to practice pre-calculus math skills. During other terms, it is scheduled as a longer course, for students who have not taken the appropriate high school mathematics prerequisites for university calculus and linear algebra. This course is recommended for any student who wish to close any existing gap between high school math and University level math courses or anyone who wishes to review high school math before attempting University level math or other science courses.

**PUMP Level 2**, is an Introduction to Proofs course. The curriculum provides background knowledge that is a preparation for MAT137Y1, MAT157Y1, MAT240H1, MAT247H1, MAT237Y1, and other proof-oriented advanced courses. The course covers the reading and comprehension of mathematical statements, analyzing definitions and properties, formulation of arguments, and strategies for proofs. This course is recommended for any student who wish to add to their knowledge by joining the group of students who will commence their preparation for the more challenging concepts in the advance analytical programs, during the months of July and August.

Visit http://www.math.toronto.edu/cms/potential-students-ug/ for up-to-date information on the availability of PUMP Level 1 and PUMP Level 2.

If you have questions about the content of these courses, e-mail inquiries@math.utoronto.ca.

**Course Change Dates**

Some of the more advanced first- and second-year courses have "change dates" during the first few weeks of the academic year. The "change date" occurs after the general "add date" for courses and before the "drop date" for courses. For example, a student enrolled in MAT157Y1 can change their enrolment to MAT137Y1 or MAT135H1 at any time on or before the change date. For deadlines and further details, see http://www.math.toronto.edu/cms/change-dates.

**Contact Information**

Associate Chair for Undergraduate Studies

Enquiries and student counseling: Bahen Centre, Room 6291

Departmental Office: Bahen Centre, Room 6290 (416-978-3323)

Websites: http://www.math.toronto.edu/cms/potential-students-ug/

### Mathematics Programs

### Mathematics Specialist (Science Program) - ASSPE1165

**Completion Requirements:**

(12.5 FCE, including at least 3.0 FCE at the 400-level)

The Specialist Program in Mathematics is directed toward students who hope to pursue mathematical research as a career.

First Year:

MAT157Y1, MAT240H1, MAT247H1

Second Year:

MAT257Y1, MAT267H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department.

NOTE: Students may use the CR/NCR option with this H course and have it count toward the Mathematics Specialist program. Students in the VIC program may also use VIC172Y1.

2. MAT327H1

Third and Fourth Years:

1. MAT347Y1, MAT354H1, MAT357H1, MAT363H1/ MAT367H1 ( MAT363H1 can be taken in the second year, if desired)

2. 2.0 FCE of: MAT309H1, MAT351Y1, ANY 400-level APM/MAT

3. 3.0 FCE of APM/MAT at the 300+ level, including at least 2.0 FCE at the 400 level (these may include options above not already chosen)

4. MAT477H1

NOTE:

1. The Department recommends that PHY151H1 and PHY152H1 be taken in the First Year, and that CSC148H1 and STA257H1 be taken during the program. If you do not have a year-long course in programming from high school, the Department strongly recommends that you take CSC108H1 prior to CSC148H1.

2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

3. Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

### Applied Mathematics Specialist (Science Program) - ASSPE2053

**Completion Requirements:**

(13.0-13.5 FCE, including at least 1.5 FCE at the 400-level)

The Specialist Program in Applied Mathematics is directed toward students who hope to pursue applied mathematical research as a career.

First Year:

MAT157Y1, MAT240H1, MAT247H1, ( CSC108H1, CSC148H1)/ CSC150H1

Second Year:

MAT257Y1, MAT267H1, STA257H1, STA261H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department.

NOTE: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Third and Fourth Years:

1. MAT351Y1, MAT327H1, MAT347Y1, MAT354H1, MAT357H1, MAT363H1/ MAT367H1 ( MAT363H1 can be taken in the second year, if desired), STA347H1

2. At least 1.5 FCE chosen from: MAT332H1, MAT344H1, MAT454H1, MAT457H1, MAT458H1, MAT464H1, STA302H1, STA457H1, CSC336H1, CSC436H1, CSC446H1, CSC456H1

3. 1.0 FCE from: APM421H1, APM426H1, APM441H1, APM446H1, APM461H1, APM462H1, APM466H1

4. MAT477H1

NOTE:

1. The Department recommends that PHY151H1 and PHY152H1 be taken in the First Year, and that CSC148H1 and STA257H1 be taken during the program. If you do not have a year-long course in programming from high school, the Department strongly recommends that you take CSC108H1 prior to CSC148H1.

2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

3. Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

### Mathematics and Physics Specialist (Science Program) - ASSPE0397

**Completion Requirements:**

(14.5-15.5 FCE, including at least 1.0 FCE at the 400-level)

First Year:

MAT157Y1, MAT240H1, MAT247H1, PHY151H1, PHY152H1

Second Year:

MAT257Y1, MAT267H1, PHY224H1, PHY250H1, PHY252H1, PHY254H1, PHY256H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department.

NOTE: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

2. Note: PHY252H1 and PHY324H1 may be taken in the 2nd or 3rd year.

Third Year:

1. MAT351Y1, MAT334H1/ MAT354H1, MAT357H1

2. One of: MAT327H1, MAT347Y1, MAT363H1/ MAT367H1 ( MAT363H1 can be taken in the second year, if desired)

3. PHY324H1, PHY350H1, PHY354H1, PHY356H1

Fourth Year:

1. Two of: APM421H1, APM426H1, APM446H1, APM441H1

2. Two of: PHY450H1, PHY452H1, PHY454H1, PHY456H1, PHY460H1

3. One of: MAT477H1, PHY424H1, PHY478H1, PHY479Y1

NOTE:

1. Students who are intending to apply to graduate schools in mathematics would be well-advised to take MAT347Y1.

2. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

3. Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

### Mathematics and Philosophy Specialist (Science Program) - ASSPE1361

**Completion Requirements:**

Consult the Undergraduate Coordinators of the Departments of Mathematics and Philosophy.

(12.0 FCE including at least 1.0 FCE at the 400-level)

First Year:

MAT157Y1, MAT240H1, MAT247H1; PHL232H1 or PHL233H1

Higher Years:

1. MAT257Y1, MAT327H1, MAT347Y1, MAT354H1/ MAT357H1

2. PHL345H1, MAT309H1/ PHL348H1

3. Four of: PHL325H1, PHL331H1, PHL332H1, PHL346H1, PHL347H1, PHL349H1, PHL355H1, PHL451H1, PHL480H1

4. 1.0 FCE from PHL200Y1/ PHL205H1/ PHL206H1/ PHL210Y1

5. PHL265H1/ PHL275H1

6. 2.0 FCE of PHL/APM/MAT at the 300+ level, to a total of 12.0 FCE.

NOTE: Students with a CGPA of 3.5 and above may apply to have graduate level math courses count towards their 400-level course requirements.

### Mathematical Applications in Economics and Finance Specialist (Science Program) - ASSPE1700

**Completion Requirements:**

(12-12.5 FCE, including at least 1.5 FCE at the 400-level)

First Year:

ECO100Y/( ECO101H1, ECO102H1); MAT137Y1/ MAT157Y1, MAT223H1, MAT224H1

**(Please check the requirements for ECO206Y1 to ensure that you pass these first year courses with grades that allow registration in ECO206Y1)**

Second Year:

ECO206Y1; MAT237Y1, MAT244H1, MAT246H1 (waived for students taking MAT157Y1); STA257H1, STA261H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department. Note: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Third Year:

1. APM346H1; ECO358H1; ECO359H1; MAT337H1; STA302H1/ ECO375H1; STA347H1

2. One of: MAT332H1, MAT344H1, MAT334H1, MAT475H1

Fourth Year:

NOTE:

1. Students planning to take specific fourth year courses should ensure that they have the necessary third year prerequisites.

2. Please note that STA457H1 lists STA302H1 as one of the prerequisites so you are encouraged to plan ahead.

### Mathematics & Its Applications Specialist (Physical Science) (Science Program) - ASSPE1758

**Completion Requirements:**

(13.5-14.5 FCE, including at least 1.0 FCE at the 400 level)

Core Courses:

First Year:

( CSC108H1, CSC148H1)/ CSC150H1, MAT137Y1/ MAT157Y1, MAT223H1/ MAT240H1,

MAT224H1/ MAT247H1 (recommended, can also be taken in 2nd year)

Second Year:

MAT235Y1/ MAT237Y1/ MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/ MAT267H1, STA257H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department. Note: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Higher Years:

NOTE:

1. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

**Physical Sciences Concentration:**

2. PHY151H1, PHY152H1, AST221H1

3. Three of: AST222H1, PHY250H1, PHY252H1, PHY254H1, PHY256H1

4. APM346H1/ MAT351Y1

5. Three of: AST320H1, AST325H1, MAT337H1, MAT363H1/ MAT367H1, PHY350H1, PHY354H1, PHY356H1, PHY357H1, PHY358H1

6. Two of: APM421H1, APM426H1, APM441H1, APM446H1, PHY407H1, PHY408H1, PHY456H1

### Mathematics & Its Applications Specialist (Probability/Statistics) (Science Program) - ASSPE1890

**Completion Requirements:**

(11.5-13.0 FCE, including at least 1.0 FCE at the 400 level)

Core Courses:

First Year:

( CSC108H1, CSC148H1)/ CSC150H1; MAT137Y1/ MAT157Y1, MAT223H1/ MAT240H1, MAT224H1/ MAT247H1

Second Year:

MAT235Y1/ MAT237Y1/ MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/ MAT267H1; STA257H1

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department. Note: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Higher Years:

MAT301H1, MAT334H1

NOTE:

1. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

**Probability/Statistics Concentration:**

1. APM346H1/ MAT351Y1/ APM462H1; MAT337H1; STA261H1, STA302H1, STA347H1, STA352Y1/( STA452H1, STA453H1)

2. Additional 1.0 FCE at the 300+level from APM/MAT/STA

3. Two of: STA437H1, STA442H1, STA447H1, STA457H1

### Mathematics & Its Applications Specialist (Teaching) (Science Program) - ASSPE1580

**Completion Requirements:**

(11.5-12.0 FCE, including at least 1.0 FCE at the 400 level)

Core Courses:

First Year:

CSC108H1; MAT137Y1/ MAT157Y1, MAT223H1/ MAT240H1, MAT224H1/ MAT247H1 (recommended, can also be taken in 2nd year)

Second Year:

MAT235Y1/ MAT237Y1/ MAT257Y1, MAT246H1 (waived for students taking MAT157Y1), MAT244H1/ MAT267H1; STA257H1

NOTE:

1. MAT237Y1/ MAT257Y1 is a direct or indirect prerequisite for many courses in each of the areas of concentration except the Teaching Concentration. Students are advised to take MAT237Y1/ MAT257Y1 unless they have planned their program and course selection carefully and are certain that they will not need it.

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department. Note: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may also use VIC172Y1.

Higher Years:

MAT301H1, MAT334H1

NOTE:

1. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

**Teaching Concentration:**

For course selection, note that OISE requires students to have a second teachable subject.

1. MAT329Y1, HPS390H1/ MAT390H1, HPS391H1/ MAT391H1

2. Two of: MAT332H1/ MAT344H1, MAT335H1, MAT337H1, MAT363H1/ MAT367H1

3. Two of: MAT309H1, MAT315H1; STA302H1/ STA347H1

4. MAT401H1/ MAT402H1 and 0.5 FCE at the 400-level from MAT, APM, STA

### Mathematics Major (Science Program) - ASMAJ1165

**Completion Requirements:**

(7.5 full courses or their equivalent. These must include at least 2.5 full course equivalent (FCE) at the 300+ level. Of those 2.5 FCE,at least 0.5 FCE must be at the 400 level).

First Year:

( MAT135H1, MAT136H1)/ MAT137Y1/ MAT157Y1, MAT223H1/ MAT240H1, MAT224H1/ MAT247H1

Second Year:

MAT235Y1/ MAT237Y1/ MAT257Y1, MAT244H1, MAT246H1

NOTE:

1. MAT224H1 may be taken in first year

Second and Higher Years:

1. At least 0.5 FCE with a significant emphasis on ethics and social responsibility: ENV333H1/ ETH201H1/ ETH210H1/ ETH220H1/ HPS200H1/ JPH441H1/ PHL265H1/ PHL273H1/ PHL275H1/ PHL281H1 or another H course approved by the Department. Note: Students may use the CR/NCR option with this H course and have it count toward the program. Students in the VIC program may use VIC172Y1.

Higher Years:

1. MAT301H1, MAT309H1/ MAT315H1, MAT334H1

2. Additional 0.5 FCE at the 200+ level from: ACT240H1/ ACT230H1 APM236H1, MAT309H1/ MAT315H1/ MAT335H1/ MAT337H1, STA247H1/ STA257H1

3. Additional 0.5 FCE at the 300+level from: APM346H1, APM462H1, MAT309H1, MAT315H1, MAT332H1/ MAT344H1, MAT335H1, MAT337H1, MAT363H1, MAT475H1, HPS390H1, HPS391H1, PSL432H1

4. MAT401H1/ MAT402H1 or any other MAT/APM 400-level course

NOTES:

1. Students using MAT157Y1 towards the first year program requirements must replace the exclusion course MAT246H1 with a different H level MAT/APM course at the 200+ level.

2. In the major program, higher level courses within the same topic are acceptable substitutions. With a judicious choice of courses, usually including introductory computer science, students can fulfill the requirements for a double major in mathematics and one of several other disciplines.

3. Students planning to take specific fourth year courses should ensure that they have the necessary second and third year prerequisites.

4. Students interested in becoming K-12 teachers should consider applying to the combined degree program --- a six-year program that leads to an Honours Bachelor of Science (H B Sc) from the University of Toronto and a Master of Teaching (M T) from the Ontario Institute for Studies in Education (OISE). The HBSc part of this program involves completing a Math Major, a Minor in Education and Society (offered by Victoria College) and a Minor in an area that would lead to a second "teachable" subject. Please see the Victoria College website for more information.

### Mathematics Minor (Science Program) - ASMIN1165

**Completion Requirements:**

(4.0 FCE)

1. ( MAT135H1, MAT136H1)/ MAT137Y1/ MAT157Y1

2. MAT221H1(80%+)/ MAT223H1/ MAT240H1, MAT235Y1/ MAT237Y1/ MAT257Y1, MAT224H1/ MAT244H1/ MAT246H1/ APM236H1/ MAT247H1

Note: MAT221H1/ MAT223H1 should be taken in first year

3. Additional 1.0 FCE at the 300+ level from APM/MAT/ HPS390H1/ HPS391H1/ PSL432H1 [note that APM306Y1 will be counted as 0.5 FCE towards this requirement.].

NOTE:

1. In the minor program, higher level courses within the same topic are acceptable substitutions.

2. Students planning to take specific third and fourth year courses should ensure that they have the necessary first, second and third year prerequisites.

3. APM306Y1 will be counted for 0.5 credits of Society and its Institutions (BR3) and 0.5 credits of The Physical and Mathematical Universes (BR5).

### Joint Programs

- Economics and Mathematics, see Economics
- Statistics and Mathematics, see Statistics
- Combined Degree Program: STG, Honours Bachelor of Science, Major in Mathematics / Master of Teaching

### Combined Degree Program (CDP) in Science and Education: Mathematics (Major), Honours Bachelor of Science/Master of Teaching

The Combined Degree Program in Arts/Science and Education is designed for students interested in studying the intersections of teaching subjects and Education, coupled with professional teacher preparation. Students earn an Honours Bachelor’s degree from the Faculty of Arts and Science (St. George) and an accredited professional Master of Teaching (MT) degree from the Ontario Institute for Studies in Education (OISE). They will be recommended to the Ontario College of Teachers for an Ontario Teacher’s Certificate of Qualifications as elementary or secondary school teachers. The CDP permits the completion of both degrees in six years with 1.0 FCE that may be counted towards both the undergraduate and graduate degrees.

Program requirements:

1. Minor in Education and Society, Victoria College

2. Major in Mathematics (first teaching subject)

3. Minor in an area corresponding to the second teaching subject as determined by OISE (see http://pepper.oise.utoronto.ca/~jhewitt/mtresources/intermediate_senior_teaching_subject_prerequisites_2016-17.pdf)

See here for additional information on the CDP, including admission, path to completion and contact information.

## Mathematics Courses

### MAT133Y1 - Calculus and Linear Algebra for Commerce

**Hours:**72L

Mathematics of finance. Matrices and linear equations. Review of differential calculus; applications. Integration and fundamental theorem; applications. Introduction to partial differentiation; applications.

NOTE: please note Prerequisites listed below. Students without the proper prerequisites for MAT133Y1 may be deregistered from this course.

**Note that MAT133Y is not a valid prerequisite for a number of more advanced quantitative courses. Students who are considering a quantitative non-Commerce PoSt, such as a math minor or a stats minor, may want to consider MAT135H and MAT136H, MAT137Y, or MAT157Y instead of MAT133Y. Specifically, a student who took MAT133Y may need to subsequently take MAT135H and MAT136H as "extra" or take MAT137Y or MAT157Y in order to proceed in non-Commerce PoSts.**

**Prerequisite:**High school level calculus

**Exclusion:**MAT135H1, MAT136H1, MAT137Y1, MAT157Y1, MATA30H3, MATA31H3, MATA32H3, MATA33H3, MATA35H3, MATA36H3, MATA37H3, MAT133Y5, MAT134Y5, MAT135Y5, MAT137Y5, MAT138Y5, MAT186H, MAT187H, MAT194H, MAT195H, MAT196H & MAT197H

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT135H1 - Calculus 1(A)

**Hours:**36L/12T

Review of trigonometric functions, trigonometric identities and trigonometric limits. Functions, limits, continuity. Derivatives, rules of differentiation and implicit differentiation, related rates, higher derivatives, logarithms, exponentials. Trigonometric and inverse trigonometric functions, linear approximations. Mean value theorem, graphing, min-max problems, l’Hôpital’s rule; anti- derivatives. Examples from life science and physical science applications.

**Prerequisite:**High school level calculus

**Exclusion:**MAT133Y1, MAT136H1, MAT137Y1, MAT157Y1, MATA30H3, MATA31H3, MATA32H3, MATA33H3, MATA35H3, MATA36H3, MATA37H3, MAT133Y5, MAT134Y5, MAT135Y5, MAT137Y5, MAT138Y5, MAT186H, MAT187H, MAT194H, MAT195H, MAT196H & MAT197H

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT136H1 - Calculus 1(B)

**Hours:**36L/12T

Definite Integrals, Fundamental theorem of Calculus, Areas, Averages, Volumes. Techniques: Substitutions, integration by parts, partial fractions, improper integrals. Differential Equations: Solutions and applications. Sequences, Series, Taylor Series. Examples from life science and physical science applications.

**Prerequisite:**MAT135H1

**Exclusion:**MAT133Y1, MAT137Y1, MAT157Y1, MATA32H3, MATA33H3, MATA36H3, MATA37H3, MAT133Y5, MAT134Y5, MAT135Y5, MAT137Y5, MAT138Y5, MAT186H, MAT187H, MAT194H, MAT195H, MAT196H & MAT197H

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT137Y1 - Calculus

**Hours:**72L/24T

A conceptual approach for students with a serious interest in mathematics. Attention is given to computational aspects as well as theoretical foundations and problem solving techniques. Review of Trigonometry. Limits and continuity, mean value theorem, inverse function theorem, differentiation, integration, fundamental theorem of calculus, elementary transcendental functions, Taylor's theorem, sequence and series, power series. Applications.

**Prerequisite:**High school level calculus

**Exclusion:**MAT135H1, MAT136H1, MAT157Y1, MATA35H3, MATA36H3, MATA37H3, MAT135Y5, MAT137Y5, MAT138Y5, MAT187H, MAT194H, MAT195H, MAT196H & MAT197H

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT138H1 - Introduction to Proofs

**Hours:**36L/12T

The goal of this course is for students to become comfortable with abstraction, rigour, logic, and proofs. They will practice reading and understanding mathematical statements, analyzing definitions and properties, formulating conjectures and generalizations, providing and writing reasonable and precise arguments, writing and critiquing proofs. The instructor may use specific mathematical content, which could vary from year to year, to practice these skills. The course is aimed at students interested in the creative character of mathematics, particularly those planning to take any of our proof-oriented courses, and is an excellent preparation for MAT137Y1, MAT157Y1, or MAT240H1.

Note: students may take this course concurrently with MAT157Y1 or MAT137Y1, or prior to registering in MAT157Y1 or MAT137Y1. This course can also be used by students who have already taken MAT136H1 and wish to bridge the gap to MAT237Y1.

**Prerequisite:**High school level calculus

**Exclusion:**MAT157Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT157Y1 - Analysis I

**Hours:**72L/48T

A theoretical course in calculus; emphasizing proofs and techniques, as well as geometric and physical understanding. Trigonometric identities. Limits and continuity; least upper bounds, intermediate and extreme value theorems. Derivatives, mean value and inverse function theorems. Integrals; fundamental theorem; elementary transcendental functions. Techniques of integration. Taylor's theorem; sequences and series; uniform convergence and power series.

**Prerequisite:**High school level calculus

**Exclusion:**MAT137Y1, MATA37H3, MAT137Y5, MAT157Y5, MAT195H1, & MAT197H1

**Recommended Preparation:**Students should consider taking the Preparing for University Math Level II in order to prepare in advance for MAT157Y1. Students may also take MAT138H1 concurrently with MAT157Y1. Students will receive credit for both MAT157Y1 and MAT138H1 if MAT138H1 is taken before or along with MAT157Y1.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### JMB170Y1 - Biology, Models, and Mathematics

**Hours:**48L/24T

Applications of mathematics to biological problems in physiology, genetics, evolution, growth, population dynamics, cell biology, ecology, and behaviour. Mathematical topics include: power functions and regression; exponential and logistic functions; binomial theorem and probability; calculus, including derivatives, max/min, integration, areas, integration by parts, substitution; differential equations, including linear constant coefficient systems; dynamic programming; Markov processes; and chaos. This course is intended for students in Life Sciences.

**Corequisite:**BIO120H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT193H1 - Math and Magic

**Hours:**24S

In this course we will look at magic tricks! Not just any magic tricks, but ones that involve only Mathematics and maybe a flair for the presentation. Some magic tricks involve only elementary Mathematics, others involve very deep Mathematics. In the discussions, we will talk about the tricks and the Mathematics behind them. Restricted to first-year students. Not eligible for CR/NCR option.

**Prerequisite:**High school level algebra.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT198H1 - Cryptology: The Mathematics of Secrecy and Security

**Hours:**24S

How do we send our own confidential information through secure channels, and how can we break codes to uncover the secret information of our adversaries? The mathematical field of cryptology is dedicated to answering such questions. In this course we will study breakthroughs in cryptology, from secret messages in the ancient world and the Enigma cipher in World War II, to modern cryptosystems that facilitate online commerce. Along the way, you will develop a sophisticated understanding of how numbers interact and develop the ability to communicate messages secretly and mathematics clearly. Restricted to first-year students. Not eligible for CR/NCR option.

**Prerequisite:**High school level algebra.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT199H1 - Puzzles, Mind, and Math

**Hours:**24S

The course is offered in a seminar/workshop format. Each week students solve puzzles, and develop initial and essential intuitions for the fundamental ideas of each puzzle. Then in the lectures and through the reading materials, the mathematical foundations of each puzzle are studied. Students learn to apply mathematical techniques to strengthen and generalize the puzzles, and to enrich their initial problem solving intuitions.

Topics covered include mathematics of numbers, counting, base and modular arithmetic, geometry and geometric constructions, graph theory, games, decision theory, logic puzzles, and recursion. Restricted to first-year students. Not eligible for CR/NCR option.

**Prerequisite:**High school level algebra

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### JUM202H1 - Mathematics as an Interdisciplinary Pursuit

**Hours:**24L/12T

A study of the interaction of mathematics with other fields of inquiry: how mathematics influences, and is influenced by, the evolution of science and culture. Art, music, and literature, as well as the more traditionally related areas of the natural and social sciences may be considered. (Offered every three years)

JUM202H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.

**Exclusion:**JUM102H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### JUM203H1 - Mathematics as a Recreation

**Hours:**24L/12T

A study of games, puzzles and problems focusing on the deeper principles they illustrate. Concentration is on problems arising out of number theory and geometry, with emphasis on the process of mathematical reasoning. Technical requirements are kept to a minimum. A foundation is provided for a continuing lay interest in mathematics. (Offered every three years)

JUM203H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.

**Exclusion:**JUM103H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### JUM205H1 - Mathematical Personalities

**Hours:**24L/12T

An in-depth study of the life, times and work of several mathematicians who have been particularly influential. Examples may include Newton, Euler, Gauss, Kowalewski, Hilbert, Hardy, Ramanujan, Gödel, Erdös, Coxeter, Grothendieck. (Offered every three years)

JUM205H1 is particularly suited as a Science Distribution Requirement course for Humanities and Social Science students.

**Exclusion:**JUM105H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT221H1 - Applied Linear Algebra

**Hours:**36L/12T

An application-oriented approach to linear algebra, based on calculations in standard Euclidean space. Systems of linear equations, matrices, Gauss-Jordan elimination, subspaces, bases, orthogonal vectors and projections. Matrix inverses, kernel and range, rank-nullity theorem. Determinants, eigenvalues and eigenvectors, Cramer's rule, diagonalization. This course has strong emphasis on building computational skills in the area of algebra. Applications to curve fitting, economics, Markov chains and cryptography.

**Prerequisite:**High school level calculus

**Exclusion:**MAT223H1, MATA23H3, MAT223H5, MAT224H1, MAT240H1, MAT240H5, MAT247H1, MAT247H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT223H1 - Linear Algebra I

**Hours:**36L/12T

Systems of linear equations, matrix algebra, real vector spaces, subspaces, span, linear dependence and independence, bases, rank, inner products, orthogonality, orthogonal complements, Gram-Schmidt, linear transformations, determinants, Cramer's rule, eigenvalues, eigenvectors, eigenspaces, diagonalization.

**Prerequisite:**High school level calculus

**Exclusion:**MATA23H3, MAT223H5, MAT224H1, MAT240H1, MAT240H5, MAT247H1, MAT247H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT224H1 - Linear Algebra II

**Hours:**36L/12T

Fields, complex numbers, vector spaces over a field, linear transformations, matrix of a linear transfromation, kernel, range, dimension theorem, isomorphisms, change of basis, eigenvalues, eigenvectors, diagonalizability, real and complex inner products, spectral theorem, adjoint/self-adjoint/normal linear operators, triangular form, nilpotent mappings, Jordan canonical form.

**Prerequisite:**MAT221H1(80%)/ MAT223H1/ MAT240H1

**Exclusion:**MAT247H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT235Y1 - Calculus II

**Hours:**72L

Parametric equations and polar coordinates. Vectors, vector functions and space curves. Differential and integral calculus of functions of several variables. Line integrals and surface integrals and classic vector calculus theorems. Examples from life sciences and physical science applications.

**Prerequisite:**( MAT135H1/MATA30H3/MATA31H3, MAT136H1/MATA36H3/MATA37H3)/MAT135Y5/ MAT137Y1/MAT137Y5/ MAT157Y1/MAT157Y5

**Exclusion:**MAT237Y1, MAT257Y1, MATB41H3, MATB42H3, MAT232H5, MAT233H5, MAT236H5, MAT368H5, MAT291H & MAT294H

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM236H1 - Applications of Linear Programming

**Hours:**36L

Introduction to linear programming including a rapid review of linear algebra (row reduction, matrix inversion, linear independence), the simplex method with applications, the duality theorem, complementary slackness, the dual simplex method and the revised simplex method.

**Prerequisite:**MAT221H1/ MAT223H1/ MAT240H1 (Note: no waivers of prerequisites will be granted)

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT237Y1 - Multivariable Calculus

**Hours:**72L

Sequences and series. Uniform convergence. Convergence of integrals. Elements of topology in R^2 and R^3. Differential and integral calculus of vector valued functions of a vector variable, with emphasis on vectors in two and three dimensional euclidean space. Extremal problems, Lagrange multipliers, line and surface integrals, vector analysis, Stokes' theorem, Fourier series, calculus of variations.

**Prerequisite:**MAT137Y1/ MAT157Y1/( MAT135H1, MAT136H1(90%))/( MAT136H1(70%), MAT138H1(70%)), MAT223H1/ MAT240H1

**Exclusion:**MAT235Y1, MAT257Y1, MATB41H3, MATB42H3, MATB43H3 & MAT368H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT240H1 - Algebra I

**Hours:**36L/24T

A theoretical approach to: vector spaces over arbitrary fields, including C and Z_p. Subspaces, bases and dimension. Linear transformations, matrices, change of basis, similarity, determinants. Polynomials over a field (including unique factorization, resultants). Eigenvalues, eigenvectors, characteristic polynomial, diagonalization. Minimal polynomial, Cayley-Hamilton theorem.

**Prerequisite:**High school level calculus

**Corequisite:**MAT157Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT244H1 - Introduction to Ordinary Differential Equations

**Hours:**36L

First order ordinary differential equations: Direction fields, integrating factors, separable equations, homogeneous equations, exact equations, autonomous equations, modeling. Existence and uniqueness theorem. Higher order equations: Constant coefficient equations, reduction of order, Wronskian, method of undetermined coefficients, variation of parameters. Solutions by series and integrals. First order linear systems, fundamental matrices. Non-linear equations, phase plane, stability. Applications in life and physical sciences and economics.

**Prerequisite:**( MAT135H1/MATA35H3/MATA30H3/MATA31H3, MAT136H1/MATA36H3/MATA37H3)/MAT135Y5/ MAT137Y1/MAT137Y5/ MAT157Y1/MAT157Y5, MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5

**Corequisite:**MAT235Y1/ MAT237Y1/ MAT257Y1

**Exclusion:**MAT267H1, MAT212H5, MAT258Y5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT245H1 - Mathematical Methods in Data Science

**Hours:**36L/24P

An introduction to the mathematical methods behind scientific techniques developed for extracting information from large data sets. Elementary probability density functions, conditional expectation, inverse problems, regularization, dimension reduction, gradient methods, singular value decomposition and its applications, stability, diffusion maps. Examples from applications in data science and big data.

**Prerequisite:**MAT137Y1/ MAT157Y1, MAT223H1/ MAT240H1, MAT224H1/ MAT247H1

**Corequisite:**MAT237Y1/ MAT257Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT246H1 - Concepts in Abstract Mathematics

**Hours:**36L/12T

Designed to introduce students to mathematical proofs and abstract mathematical concepts. Topics may include modular arithmetic, sizes of infinite sets, and a proof that some angles cannot be trisected with straightedge and compass.

**Prerequisite:**MAT133Y1/( MAT135H1, MAT136H1)/ MAT137Y1, MAT223H1

**Exclusion:**MAT157Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT247H1 - Algebra II

**Hours:**36L

A theoretical approach to real and complex inner product spaces, isometries, orthogonal and unitary matrices and transformations. The adjoint. Hermitian and symmetric transformations. Spectral theorem for symmetric and normal transformations. Polar representation theorem. Primary decomposition theorem. Rational and Jordan canonical forms. Additional topics including dual spaces, quotient spaces, bilinear forms, quadratic surfaces, multilinear algebra.

**Prerequisite:**MAT240H1/MAT240H5

**Corequisite:**MAT157Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT257Y1 - Analysis II

**Hours:**72L/48T

Topology of R^n; compactness, functions and continuity, extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers. Integration; Fubini's theorem, partitions of unity, change of variables. Differential forms. Manifolds in R^n; integration on manifolds; Stokes' theorem for differential forms and classical versions.

**Prerequisite:**MAT157Y1/MAT157Y5, MAT247H1/MAT247H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT267H1 - Advanced Ordinary Differential Equations

**Hours:**36L/12T

A theoretical course on Ordinary Differential Equations. First-order equations: separable equations, exact equations, integrating factors. Variational problems, Euler-Lagrange equations. Linear equations and first-order systems. Fundamental matrices, Wronskians. Non-linear equations. Existence and uniqueness theorems. Method of power series. Elementary qualitative theory; stability, phase plane, stationary points. Oscillation theorem, Sturm comparison. Applications in mechanics, physics, chemistry, biology and economics.

**Prerequisite:**MAT157Y1/MAT157Y5, MAT247H1/MAT247H5

**Corequisite:**MAT257Y1

**Exclusion:**APM288H1, MAT244H1, MATB44H3, MAT242H5, MAT252H5, MAT234H1, MAT292H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT271H1 - Insights from Mathematics

**Hours:**36L/6T

This breadth course is accessible to students with limited mathematical background. Various mathematical techniques will be illustrated with examples from humanities and social science disciplines. Some of the topics will incorporate user friendly computer explorations to give participants the feel of the subject without requiring skill at calculations.

Note: This course cannot be used to satisfy requirements of program in the math department.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT282H1 - Topics in Mathematics

**Hours:**36L

A course in mathematics on a topic outside the current undergraduate offerings. For information on the specific topic to be studied and possible additional preqrequisites, go to http://www.math.toronto.edu/cms/current-students-ug/

**Prerequisite:**1.0 FCE in 100-level MAT courses. Possible additional topic-specific prerequisites.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT295H1 - Independent Reading in Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Similar workload to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT296H1 - Independent Reading in Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT297Y1 - Research Project in Mathematics

**Hours:**TBA

Independent research under the direction of a faculty member. Similar workload to a 72L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT299Y1 - Research Opportunity Program

**Hours:**TBA

Credit course for supervised participation in faculty research project. Details at http://www.artsci.utoronto.ca/current/course/rop. Not eligible for CR/NCR option.

**Distribution Requirements:**Science

### MAT301H1 - Groups and Symmetries

**Hours:**36L

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 theorem. Normal subgroups, quotient groups. Classification of finitely generated abelian groups. Emphasis on examples and calculations.

**Prerequisite:**MAT224H1/ MAT247H1, MAT235Y1/ MAT237Y1, MAT246H1/ CSC236H1/ CSC240H1. (These Prerequisites will be waived for students who have MAT257Y1.) For FASE students, MAT185H, MAT194H, MAT195H.

**Exclusion:**MAT347Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM306Y1 - Mathematics and Law

**Hours:**72L

This course examines the relationship between legal reasoning and mathematical logic; provides a mathematical perspective on the legal treatment of interest and actuarial present value; critiques ethical issues; analyzes how search engine techniques on massive databases transform legal research and considers the impact of statistical analysis and game theory on litigation strategies.

NOTE

This course counts as 0.5 FCE in BR3 and 0.5 FCE in BR5.

This course will only contribute 0.5FCE to the Math Minor program.

**Prerequisite:**( MAT135H1/ MAT136H1)/ MAT137Y1/ MAT157Y1, MAT221H1/ MAT223H1/ MAT240H1

**Exclusion:**JUM206Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5); Society and its Institutions (3)

### MAT309H1 - Introduction to Mathematical Logic

**Hours:**36L

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.

**Prerequisite:**MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5, MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5, MAT246H1/ CSC236H1/ CSC240H1 (These Prerequisites will be waived for students who have MAT257Y1)

**Exclusion:**CSC438H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT315H1 - Introduction to Number Theory

**Hours:**36L

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.

**Prerequisite:**( MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5, MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5, MAT246H1/ CSC236H1/ CSC240H1)/ MAT157Y1/MAT157Y5/ MAT247H1/MAT247H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT327H1 - Introduction to Topology

**Hours:**36L

Metric spaces, topological spaces and continuous mappings; separation, compactness, connectedness. Fundamental group and covering spaces. Brouwer fixed-point theorem. Students in the math specialist program wishing to take additional topology courses are advised to obtain permission to take MAT1300H, MAT1301H.

**Prerequisite:**MAT157Y1/MAT157Y5/( MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5, MAT246H1 and permission of the instructor)

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT329Y1 - Concepts in Elementary Mathematics

**Hours:**72L

This course is aimed at students intending to become elementary school teachers. Emphasis is placed on the formation and development of fundamental reasoning and learning skills required to understand and to teach mathematics at the elementary level. Topics may include: Problem Solving and Strategies, Sets and Elementary Logic, Numbers and Elements of Number Theory, Introductory Probability and Fundamentals of Geometry.

The course may include an optional practicum in school classrooms.

**Prerequisite:**MAT137Y1/ MAT138H1/ MAT223H1/ MAT246H1 and any 5.0 FCE with a CGPA of at least 2.5

**Exclusion:**MAT382H5

**Recommended Preparation:**Participation in the practicum requires the presentation of an Ontario Police Report that declares suitability to work with minors and other special groups.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT332H1 - Introduction to Graph Theory

**Hours:**36L

This course will explore the following topics: Graphs, subgraphs, isomorphism, trees, connectivity, Euler and Hamiltonian properties, matchings, vertex and edge colourings, planarity, network flows and strongly regular graphs. Participants will be encouraged to use these topics and execute applications to such problems as timetabling, tournament scheduling, experimental design and finite geometries.

**Prerequisite:**MAT224H1/MATB24H3/MAT224H5/ MAT247H1/MAT247H5

**Corequisite:**Recommended Corequisite: MAT301H1/ MAT347Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT334H1 - Complex Variables

**Hours:**36L

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.

**Prerequisite:**MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5, MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5/ MAT257Y1

**Exclusion:**MAT354H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT335H1 - Chaos, Fractals and Dynamics

**Hours:**36L

An elementary introduction to a modern and fast-developing area of mathematics. One-dimensional dynamics: iterations of quadratic polynomials. Dynamics of linear mappings, attractors. Bifurcation, Henon map, Mandelbrot and Julia sets. History and applications.

**Prerequisite:**MAT137Y1/(MATA30H3, MATA31H3, MATA37H3)/MAT137Y5/ MAT157Y1/MAT157Y5/ MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5, MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT336H1 - Elements of Analysis

**Hours:**36L/12T

This course provides the foundations of analysis and rigorous calculus for students who will take subsequent courses where these mathematical concepts are central of applications, but who have only taken courses with limited proofs. Topics include topology of Rn, implicit and inverse function theorems and rigorous integration theory.

**Prerequisite:**MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5, MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5; (for FASE students, MAT185H, MAT195H/ESC195H)

**Exclusion:**MAT257Y1, MAT337H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT337H1 - Introduction to Real Analysis

**Hours:**36L

Construction of Real Numbers. Metric spaces; compactness and connectedness. Sequences and series of functions, power series; modes of convergence. Interchange of limiting processes; differentiation of integrals. Function spaces; Weierstrass approximation; Fourier series. Contraction mappings; existence and uniqueness of solutions of ordinary differential equations. Countability; Cantor set; Hausdorff dimension.

**Prerequisite:**MAT224H1/MATB24H3/MAT224H5/ MAT247H1/MAT247H5, MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5, MAT246H1; NOTE: These Prerequisites will be waived for students who have MAT257Y1

**Exclusion:**MAT357H1 & MAT378H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT344H1 - Introduction to Combinatorics

**Hours:**36L

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.

**Prerequisite:**MAT223H1/MATA23H3/MAT223H5/ MAT240H1/MAT240H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM346H1 - Partial Differential Equations

**Hours:**36L

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.

**Prerequisite:**MAT235Y1/MAT235Y5/(MATB41H3, MATB42H3)/ MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5/ MAT257Y1, MAT244H1/MAT244H5/ MAT267H1

**Exclusion:**MAT351Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT347Y1 - Groups, Rings and Fields

**Hours:**72L/24T

Groups, subgroups, quotient groups, Sylow theorems, Jordan-Hölder theorem, finitely generated abelian groups, solvable groups. Rings, ideals, Chinese remainder theorem; Euclidean domains and principal ideal domains: unique factorization. Noetherian rings, Hilbert basis theorem. Finitely generated modules. Field extensions, algebraic closure, straight-edge and compass constructions. Galois theory, including insolvability of the quintic.

**Prerequisite:**MAT257Y1/(85% in MAT247H1/MAT247H5)

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM348H1 - Mathematical Modelling

**Previous Course Number:**MAT482

**Hours:**36L/22P

An overview of mathematical modelling. A variety of approaches for representing physical situations mathematically followed by analytical techniques and numerical simulations to gain insight. Questions from biology, economics, engineering, medicine, physics, physiology, and the social sciences formulated as problems in optimization, differential equations, and probability. Precise content varies with instructor.

**Prerequisite:**MAT244H1/ MAT267H1, MAT224H1/ MAT247H1, STA237H1/ STA247H1/ STA257H1/ MAT377H1

**Exclusion:**MAT482H1 (Topics in Mathematics: Topics in Mathematical Modelling), offered in Winter 2019

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT351Y1 - Partial Differential Equations

**Hours:**72L

This is a first course in Partial Differential Equations, intended for Mathematics students with interests in analysis, mathematical physics, geometry, and optimization. The examples to be discussed include first-order equations, harmonic functions, the diffusion equation, the wave equation, Schrodinger's equation, and eigenvalue problems. In addition to the classical representation formulas for the solutions of these equations, there are techniques that apply more broadly: the notion of well-posedness, the method of characteristics, energy methods, maximum and comparison principles, fundamental solutions, Green's functions, Duhamel's principle, Fourier series, the min-max characterization of eigenvalues, Bessel functions, spherical harmonics, and distributions. Nonlinear phenomena such as shock waves and solitary waves are also introduced.

**Prerequisite:**MAT257Y1/85% in MAT237Y1, MAT267H1

**Exclusion:**APM351Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT354H1 - Complex Analysis I

**Hours:**36L

Complex numbers, the complex plane and Riemann sphere, Möbius transformations, elementary functions and their mapping properties, conformal mapping, holomorphic functions, Cauchy's theorem and integral formula. Taylor and Laurent series, maximum modulus principle, Schwarz' lemma, residue theorem and residue calculus.

**Prerequisite:**MAT257Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT357H1 - Foundations of Real Analysis

**Hours:**36L

Function spaces; Arzela-Ascoli theorem, Weierstrass approximation theorem, Fourier series. Introduction to Banach and Hilbert spaces; contraction mapping principle, fundamental existence and uniqueness theorem for ordinary differential equations. Lebesgue integral; convergence theorems, comparison with Riemann integral, L^p spaces. Applications to probability.

**Prerequisite:**MAT257Y1/( MAT327H1 and permission of instructor)

**Exclusion:**MAT438H5

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT363H1 - Geometry of Curves and Surfaces

**Hours:**36L

Curves and surfaces in 3-spaces. Frenet formulas. Curvature and geodesics. Gauss map. Minimal surfaces. Gauss-Bonnet theorem for surfaces. Surfaces of constant curvature.

**Prerequisite:**MAT224H1/MATB24H3/MAT224H5/ MAT247H1/MAT247H5, MAT237Y1/(MATB41H3, MATB42H3, MATB43H3)/MAT237Y5/ MAT257Y1 ( MAT257Y1 can be taken concurrently). For FASE students, MAT185H, MAT194H, MAT195H, AER210H.

**Exclusion:**MAT367H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT367H1 - Differential Geometry

**Hours:**36L

Manifolds, partitions of unity, submersions and immersions, vector fields, vector bundles, tangent and cotangent bundles, foliations and Frobenius’ theorem, multillinear algebra, differential forms, Stokes’ theorem, Poincare-Hopf theorem

**Prerequisite:**MAT257Y1/( MAT224H1, MAT237Y1, MAT246H1,and permission of instructor)

**Recommended Preparation:**Multivariable calculus ( MAT257Y1), Linear algebra ( MAT240H1, MAT247H1)

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT377H1 - Mathematical Probability

**Hours:**36L/12T

This course introduces students to various topics in mathematical probability theory. Topics include basic concepts (such as probability, random variables, expectations, conditional probability) from a mathematical point of view, examples of distributions and stochastic processes and their properties, convergence results (such as the law of large numbers, central limit theorem, random series, etc.), various inequalities, and examples of applications of probabilistic ideas beyond statistics (for example, in geometry and computer science).

**Prerequisite:**MAT247H1/MAT247H5, MAT257Y1

**Exclusion:**STA347H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT382H1 - Topics in Mathematics

**Hours:**36L

A course in mathematics on a topic outside the current undergraduate offerings. For information on the specific topic to be studied and possible additional preqrequisites, go to http://www.math.toronto.edu/cms/current-students-ug/

**Prerequisite:**2.5 FCE in 100-level or 200-level APM or MAT courses. Possible additional topic-specific prerequisites.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT390H1 - History of Mathematics up to 1700

**Hours:**36L

A survey of ancient, medieval, and early modern mathematics with emphasis on historical issues. (Offered in alternate years)

**Prerequisite:**At least 1.0 FCE in APM/MAT at the 200 level.

**Exclusion:**HPS309H1, HPS310Y1, HPS390H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT391H1 - History of Mathematics after 1700

**Hours:**24L/12T

A survey of the development of mathematics from 1700 to the present with emphasis on technical development. (Offered in alternate years)

**Prerequisite:**At least 1.0 FCE in APM/MAT at the 200 level.

**Exclusion:**HPS309H1, HPS310H1, HPS391H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT395H1 - Independent Reading in Mathematics

**Hours:**TBA

Independent reading under the direction of a faculty member. Topic must be outside current undergraduate offerings. Similar workload to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM396H1 - Independent Reading in Applied Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Similar workload to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA 3.5 for APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT396H1 - Independent Reading in Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Similar workload to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT397Y1 - Research Project in Mathematics

**Hours:**TBA

Independent research under the direction of a faculty member. Workload similar to a 72L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT398H0 - Research Excursions

**Hours:**TBA

An instructor-supervised group project in an off-campus setting. Details at http://www.artsci.utoronto.ca/current/course/rep. Not eligible for CR/NCR option.

**Distribution Requirements:**Science

### MAT398Y0 - Research Excursions

**Hours:**TBA

An instructor-supervised group project in an off-campus setting. Details at http://www.artsci.utoronto.ca/current/course/rep. Not eligible for CR/NCR option.

**Distribution Requirements:**Science

### MAT399Y1 - Research Opportunity Program

**Hours:**TBA

Credit course for supervised participation in faculty research project. Details at http://www.artsci.utoronto.ca/current/course/rop. Not eligible for CR/NCR option.

### MAT401H1 - Polynomial Equations and Fields

**Hours:**36L

Commutative rings; quotient rings. Construction of the rationals. Polynomial algebra. Fields and Galois theory: Field extensions, adjunction of roots of a polynomial. Constructibility, trisection of angles, construction of regular polygons. Galois groups of polynomials, in particular cubics, quartics. Insolvability of quintics by radicals.

**Prerequisite:**MAT301H1

**Exclusion:**MAT347Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT402H1 - Classical Geometries

**Hours:**36L

Euclidean and non-euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces.

**Prerequisite:**MAT301H1/ MAT347Y1, MAT235Y1/ MAT237Y1/ MAT257Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT409H1 - Set Theory

**Hours:**36L

Set theory and its relations with other branches of mathematics. ZFC axioms. Ordinal and cardinal numbers. Reflection principle. Constructible sets and the continuum hypothesis. Introduction to independence proofs. Topics from large cardinals, infinitary combinatorics and descriptive set theory.

**Prerequisite:**MAT357H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT415H1 - Algebraic Number Theory

**Hours:**36L

A selection from the following: finite fields; global and local fields; valuation theory; ideals and divisors; differents and discriminants; ramification and inertia; class numbers and units; cyclotomic fields; diophantine equations.

**Prerequisite:**MAT347Y1 or permission of instructor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT417H1 - Analytic Number Theory

**Hours:**36L

A selection from the following: distribution of primes, especially in arithmetic progressions and short intervals; exponential sums; Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann zeta-function; sieve methods, large and small; diophantine approximation, modular forms.

**Prerequisite:**MAT334H1/ MAT354H1/permission of instructor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM421H1 - Mathematical Foundations of Quantum Mechanics

**Hours:**36L

Key concepts and mathematical structure of Quantum Mechanics, with applications to topics of current interest such as quantum information theory. The core part of the course covers the following topics: Schroedinger equation, quantum observables, spectrum and evolution, motion in electro-magnetic field, angular momentum and O(3) and SU(2) groups, spin and statistics, semi-classical asymptotics, perturbation theory. More advanced topics may include: adiabatic theory and geometrical phases, Hartree-Fock theory, Bose-Einstein condensation, the second quantization, density matrix and quantum statistics, open systems and Lindblad evolution, quantum entropy, quantum channels, quantum Shannon theorems.

**Prerequisite:**( MAT224H1, MAT337H1)/ MAT357H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT425H1 - Differential Topology

**Hours:**36L

Smooth manifolds, Sard's theorem and transversality. Morse theory. Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem. Vector fields and Euler characteristic. Hopf degree theorem. Additional topics may vary.

**Prerequisite:**MAT257Y1, MAT327H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM426H1 - General Relativity

**Hours:**36L

Einstein's theory of gravity. Special relativity and the geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature. Einstein's equations. Cosmological implications: big bang and inflationary universe. Schwarzschild stars: bending of light and perihelion precession of Mercury. Topics from black hole dynamics and gravitational waves. The Penrose singularity theorem.

**Prerequisite:**MAT363H1/ MAT367H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT436H1 - Introduction to Linear Operators

**Hours:**36L

The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced). The intention will be to discuss a number of the topics in Pedersen's textbook Analysis Now. Students will be encouraged to lecture on some of the material, and also to work through some of the exercises in the textbook (or in the suggested reference books).

**Prerequisite:**5.0 FCE from MAT, including MAT224H1/ MAT247H1 and MAT237Y1/ MAT257Y1.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT437H1 - K-Theory and C* Algebras

**Hours:**36L

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.

Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.

Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)

The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classification of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras---both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras---what Bratteli called AF algebras---is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

Students will be encouraged to prepare oral or written reports on various subjects related to the course, including basic theory and applications.

**Prerequisite:**5.0 FCE from MAT, including MAT224H1/ MAT247H1 and MAT237Y1/ MAT257Y1.

**Recommended Preparation:**Students are encouraged to execute basic research that answers the question, what is an abelian group?

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM441H1 - Asymptotic and Perturbation Methods

**Hours:**36L

Asymptotic series. Asymptotic methods for integrals: stationary phase and steepest descent. Regular perturbations for algebraic and differential equations. Singular perturbation methods for ordinary differential equations: W.K.B., strained co-ordinates, matched asymptotics, multiple scales. (Emphasizes techniques; problems drawn from physics and engineering)

**Prerequisite:**APM346H1/ MAT351Y1, MAT334H1/ MAT354H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT445H1 - Representation Theory

**Hours:**36L

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.

**Prerequisite:**MAT347Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM446H1 - Applied Nonlinear Equations

**Hours:**36L

Partial differential equations appearing in physics, material sciences, biology, geometry, and engineering. Nonlinear evolution equations. Existence and long-time behaviour of solutions. Existence of static, traveling wave, self-similar, topological and localized solutions. Stability. Formation of singularities and pattern formation. Fixed point theorems, spectral analysis, bifurcation theory. Equations considered in this course may include: Allen-Cahn equation (material science), Ginzburg-Landau equation (condensed matter physics), Cahn-Hilliard (material science, biology), nonlinear Schroedinger equation (quantum and plasma physics, water waves, etc). mean curvature flow (geometry, material sciences), Fisher-Kolmogorov-Petrovskii-Piskunov (combustion theory, biology), Keller-Segel equations (biology), and Chern-Simmons equations (particle and condensed matter physics).

**Prerequisite:**APM346H1/ MAT351Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT448H1 - Introduction to Commutative Algebra and Algebraic Geometry

**Hours:**36L

Basic notions of algebraic geometry, with emphasis on commutative algebra or geometry according to the interests of the instructor. Algebraic topics: localization, integral dependence and Hilbert's Nullstellensatz, valuation theory, power series rings and completion, dimension theory. Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, varieties over the complex numbers. This course will be offered in alternating years.

**Prerequisite:**MAT347Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT449H1 - Algebraic Curves

**Hours:**36L

Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout's theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities. This course will be offered in alternating years.

**Prerequisite:**MAT347Y1, MAT354H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT454H1 - Complex Analysis II

**Hours:**36L

Harmonic functions, Harnack's principle, Poisson's integral formula and Dirichlet's problem. Infinite products and the gamma function. Normal families and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem.

**Prerequisite:**MAT354H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT457H1 - Advanced Real Analysis I

**Hours:**36L

Lebesque measure and integration; convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem. Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p spaces, Hölder and Minkowski inequalities.

**Prerequisite:**MAT357H1

**Exclusion:**MAT457Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT458H1 - Advanced Real Analysis II

**Hours:**36L

Fourier series and transform, convergence results, Fourier inversion theorem, L^2 theory, estimates, convolutions. Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.

**Prerequisite:**MAT457H1

**Exclusion:**MAT457Y1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM461H1 - Combinatorial Methods

**Hours:**36L

A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.

**Prerequisite:**MAT224H1/ MAT247H1, MAT137Y1/ MAT157Y1, MAT301H1/ MAT347Y1

**Recommended Preparation:**MAT344H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM462H1 - Nonlinear Optimization

**Hours:**36L

An introduction to first and second order conditions for finite and infinite dimensional optimization problems with mention of available software. Topics include Lagrange multipliers, Kuhn-Tucker conditions, convexity and calculus variations. Basic numerical search methods and software packages which implement them will be discussed.

**Prerequisite:**MAT223H1, MAT224H1, MAT235Y1,

**Recommended Preparation:**MAT336H1/ MAT337H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT464H1 - Riemannian Geometry

**Hours:**36L

Riemannian metrics. Levi-Civita connection. Geodesics. Exponential map. Second fundamental form. Complete manifolds and Hopf-Rinow theorem. Curvature tensors. Ricci curvature and scalar curvature. Spaces of constant curvature.

**Prerequisite:**MAT367H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM466H1 - Mathematical Theory of Finance

**Hours:**36L

Introduction to the basic mathematical techniques in pricing theory and risk management: Stochastic calculus, single-period finance, financial derivatives (tree-approximation and Black-Scholes model for equity derivatives, American derivatives, numerical methods, lattice models for interest-rate derivatives), value at risk, credit risk, portfolio theory.

**Prerequisite:**APM346H1, STA347H1

**Corequisite:**STA457H1

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT475H1 - Problem Solving Seminar

**Hours:**TBA

This course addresses the question: How do you attack a problem the likes of which you have never seen before? Students will apply Polya's principles of mathematical problem solving, draw upon their previous mathematical knowledge, and explore the creative side of mathematics in solving a variety of interesting problems and explaining those solutions to others.

**Prerequisite:**MAT224H1/ MAT247H1, MAT235Y1/ MAT237Y1/ MAT257Y1, and at least 1.0 FCE at the 300+ level in APM/MAT

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT477H1 - Seminar in Mathematics

**Hours:**TBA

Seminar in an advanced topic. Content will generally vary from semester to semester. Student presentations are required.

**Prerequisite:**MAT347Y1, MAT354H1, MAT357H1; or permission of instructor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT478H1 - Seminar in Mathematics

**Hours:**TBA

Seminar in an advanced topic. Content will generally vary from semester to semester. Student presentations are required.

**Prerequisite:**MAT347Y1, MAT354H1, MAT357H1; or permission of instructor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT482H1 - Topics in Mathematics

**Hours:**36L

A course in mathematics on a topic outside the current undergraduate offerings. For information on the specific topic to be studied and possible additional preqrequisites, go to http://www.math.toronto.edu/cms/current-students-ug/

**Prerequisite:**6.0 FCE in 100-level, 200-level, and 300-level APM and MAT courses. Possible additional topic-specific prerequisites.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT483H1 - Topics in Mathematics

**Hours:**36L

**Prerequisite:**6.0 FCE in 100-level, 200-level, and 300-level APM and MAT courses. Possible additional topic-specific prerequisites.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT495H1 - Independent Reading in Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### APM496H1 - Independent Readings in Applied Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside current undergraduate offerings. Similar workload to a course that has 36 lecture hours. Not eligible for CR/NCR option.

**Prerequisite:**minimum GPA 3.5 for APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT496H1 - Independent Reading in Mathematics

**Hours:**TBA

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor.

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)

### MAT497Y1 - Research Project in Mathematics

**Hours:**TBA

Independent research under the direction of a faculty member. Not eligible for CR/NCR option. Similar workload to a 72L course.

**Prerequisite:**Minimum GPA of 3.5 in APM and MAT courses. Permission of the Associate Chair for Undergraduate Studies and of the prospective supervisor

**Distribution Requirements:**Science

**Breadth Requirements:**The Physical and Mathematical Universes (5)