2003-2004 GRADUATE STUDIES INMATHEMATICS HANDBOOK

TABLE OF CONTENTS

1 Introduction................................................................................................................................................................ 12 Department of Mathematics ...................................................................................................................................... 13 The Graduate Program .............................................................................................................................................. 54 Graduate Courses....................................................................................................................................................... 85 Research Activities...............................................................................................................................................….206 Admission Requirements and Application Procedures .......................................................................................... 207 Fees and Financial Assistance ................................................................................................................................ 218 Other Information ................................................................................................................................................... 22Appendix A Comprehensive Examination Syllabi for Pure Mathematics Ph.D. Studies .......................................................... 26Appendix B

Oral Examination Syllabi for Applied Mathematics Ph.D. Studies ....................................................................... 28Appendix C Ph.D. Degrees Conferred from 1993-2003 ....................................................................................….................... 30Appendix D The Fields Institute for Research in Mathematical Sciences .................................................................................. 33

1. INTRODUCTION

The purpose of this handbook is to provide information about the graduate program of the Department of Mathematics,University of Toronto. It includes detailed information about the department, its faculty members and students, a listingof courses offered in 2003-2004, a summary of research activities, admissions requirements, application procedures,fees and financial assistance, and information about similar matters of concern to graduate students and prospectivegraduate students in mathematics.

This handbook is intended to complement the calendar of the university’s School of Graduate Studies, where full detailson fees and general graduate studies regulations may be found.

For further information, or for application forms, please contact:

The Graduate OfficeDepartment of Mathematics

University of Toronto100 St. George Street, Room 4067

Toronto, Ontario, Canada M5S 3G3Telephone: (416) 978-7894

Fax: (416) 978-4107Email: ida@math.toronto.edu

Website: http://www.math.toronto.edu/graduate

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2. DEPARTMENT OF MATHEMATICS

Mathematics has been taught at the University of Toronto since 1827. Since the first Canadian Ph.D. degree inmathematics was conferred to Samuel Beatty (under the supervision of John Charles Fields) in 1915, more than 350Ph.D. degrees and 900 Master’s degrees have been awarded in this University. Many of our recent graduates areengaged in university teaching, and a significant number hold administrative positions in universities or in theprofessional communities. Others are pursuing careers in industry (technological or financial), or in government.

The Department of Mathematics, University of Toronto is a distinguished faculty of more than sixty mathematicians.We have a large selection of graduate courses and seminars, and a diverse student body of domestic and internationalstudents, yet classes are small and the ratio of graduate students to faculty is low. We are in a unique position to takemaximum advantage of the presence of the Fields Institute, which features special programs in pure and appliedmathematics. Currently the department has 86 graduate students, of whom 25 are enrolled in the Master’s program, and61 in the Ph.D. program.

Opportunities for graduate study and research are available in most of the main fields of pure and applied mathematics.These fields include real and complex analysis, ordinary and partial differential equations, harmonic analysis, nonlinearanalysis, 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, algebraicgeometry, singularity theory, differential geometry, symplectic geometry, classical synthetic geometry, algebraictopology, set theory, set theoretic topology, mathematical physics, fluid mechanics, probability (in cooperation with theDepartment of Statistics), combinatorics, optimization, control theory, dynamical systems, computer algebra,cryptography, and mathematical finance.

We offer a research-oriented Ph.D., and Master’s program. Very strong students may be admitted directly to the Ph.D.program with a Bachelor’s degree; otherwise; it is normal to do a 1-year Master’s degree first. (Provisional admissionto the Ph.D. program may be granted at the time of admission to the Master’s program.) The Master’s program may beextended to 16 months or 24 months for students who do not have a complete undergraduate preparation, or forindustrial students engaged in a project.

There is a separate Master’s of Mathematical Finance Program not directly under the Department’s jurisdiction, butwith which some of our faculty members are associated.

During their studies here, graduate students are encouraged to participate in the life of the close community of U of Tmathematics. Almost all of them do some work in connection with undergraduate teaching, either as tutorial leaders,markers, or, especially in later years of their program, instructors. There is a Mathematics Graduate StudentAssociation which organizes social and academic events and makes students feel welcome.

GRADUATE FACULTY MEMBERS

AKCOGLU, M.A. (Professor Emeritus) Ph.D. 1963 (Brown)• Ergodic theory, functional analysis, harmonic analysis ALBANESE, C. (Associate Professor) Ph.D. 1988 (ETH, Zürich)• Statistical mechanics, dynamical systems, mathematical financeALMGREN, R. (Associate Professor) Ph.D. 1989 (Princeton)• Scientific computing, numerical methods, partial differential equationsARTHUR, J. (University Professor) B.Sc. 1966 (Toronto), M.Sc. 1967 (Toronto), Ph.D. 1970 (Yale)• Representations of Lie groups, automorphic forms BARBEAU, E. (Professor) B.Sc. 1960 (Toronto), M.A. 1961 (Toronto), Ph.D. 1964 (Newcastle-upon-Tyne)• Functional analysis, optimization under constraint, history of analysis, number theory BAR-NATAN, D. (Associate Professor) Ph.D. 1991 (Princeton)• Theory of quantum invariants of knots, links and three manifolds BIERSTONE, E. (Professor) B.Sc. 1969 (Toronto), Ph.D. 1973 (Brandeis)• Singularity theory, analytic geometry, differential analysisBINDER, I. (Assistant Professor) Ph.D. 1997 (Caltech)

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• Harmonic and complex analysis, conformal dynamics BLAND, J. (Professor) Ph.D. 1982 (UCLA)• Several complex variables, differential geometry BLOOM, T. (Professor) Ph.D. 1965 (Princeton)• Several complex variables BUCHWEITZ, R.-O. (Professor) Ph.D. (Dr.rer.nat.) 1976 (Hannover), Doctorat d’Etat 1981 (Paris VII)• Commutative algebra, algebraic geometry, singularities. CHOI, M.-D. (Professor) M.Sc. 1970 (Toronto), Ph.D. 1973 (Toronto)• Operator theory, operator algebras, matrix theoryCOLLIANDER, James (Assistant Professor) Ph.D. 1997 (Illinois, Urbana-Champaign)• Partial differential equations, harmonic analysisCONSANI, C. (Associate Professor) Ph.D. 1996 (Chicago)• Arithmetic algebraic geometryDAVIS, C. (Professor Emeritus) Ph.D. 1950 (Harvard)• Operators on Hilbert spaces, matrix theory and applications (including numerical analysis) DEL JUNCO, A. (Professor) B.Sc. 1970 (Toronto), M.Sc. 1971 (Toronto), Ph.D. 1974 (Toronto)• Ergodic theory, functional analysis DERZKO, N. (Associate Professor) B.Sc. 1970 (Toronto), Ph.D. 1965 (Caltech)• Functional analysis, structure of differential operators, optimization and control theory with applications to

economics ELLERS, E. (Professor Emeritus) Dr.rer.nat. 1959 (Hamburg)• Classical groups ELLIOTT, G. A. (Canada Research Chair and Professor) Ph.D. 1969 (Toronto)• Operator algebras, K-theory, non-commutative geometry and topology FRIEDLANDER, J. (University Professor) B.Sc. 1965 (Toronto), Ph.D. 1972 (Penn State)• Analytic number theoryGOLDSTEIN, M. (Professor) Ph.D. 1977 (Tashkent), Doctorat d’Etat 1987 (Vilnius)• Spectral theory of Schroedinger operators and localizationGRAHAM, I. (Professor) B.Sc. 1970 (Toronto), Ph.D. 1973 (Princeton)• Several complex variables, one complex variable GREINER, P.C. (Professor) Ph.D. 1964 (Yale)• Partial differential equations HALPERIN, S. (Professor Emeritus) B.Sc. 1965 (Toronto), M.Sc. 1966 (Toronto), Ph.D. 1970 (Cornell)• Homotopy theory and loop space homology HAQUE, W. (Professor Emeritus) Ph.D. 1964 (Stanford)• Mathematical economicsHORI, K. (Assistant Professor) Ph.D. 1994 (Tokyo)• Gauge field theory, string theory IVRII, V. (Professor) Ph.D. 1973 (Novosibirsk)• Partial differential equations JEFFREY, L. (Professor) Ph.D. 1992 (Oxford)• Symplectic geometry, geometric applications of quantum field theory JERRARD, Robert (Associate Professor) Ph.D. 1994 (Berkeley)• Nonlinear partial differential equations, Ginzburg-Landau theory JURDJEVIC, V. (Professor) Ph.D. 1969 (Case Western)• Systems of ordinary differential equations, control theory, global analysisKAPRANOV, M. (Professor and Mossman Chair) Ph.D. 1988 (Steklov Institute, Moscow)• Algebra, algebraic geometry and category theoryKARSHON, Y. (Associate Professor) Ph.D. 1993 (Harvard)• Equivariant symplectic geometryKHESIN, B. (Professor) Ph.D. 1989 (Moscow State)• Poisson geometry, integrable systems, topological hydrodynamics KHOVANSKII, A. (Professor) Ph.D. 1973, Doctorat d’Etat 1987 (Steklov Institute, Moscow)

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• Algebra, geometry, theory of singularitiesKIM, Henry (Professor) Ph.D. 1992 (Chicago)• Automorphic L-functions, Langlands’ programLEHMAN, A. (Professor Emeritus) Ph.D. 1954 (Florida)• Combinatorial theory, networks, lattice theory LORIMER, J.W. (Professor) Ph.D. 1971 (McMaster)• Rings and geometries, topological Klingenberg planes, topological chain rings

LU, Steven (Adjunct Professor) Ph.D. 1990 (Harvard)• Algebraic geometry, several complex variablesLYUBICH, M. (Canada Research Chair and Professor) Ph.D. 1984 (Tashkent State University)• Dynamical systems, especially holomorphic and low dimensional dynamicsMASCHLER, Gideon (Assistant Professor) Ph.D. 1997 (SUNY, Stony Brook)• Complex geometry and four-manifoldsMASSON, D. (Professor Emeritus) Ph.D. 1963 (London)• Special functions, Jacobi matrices, orthogonal polynomials, difference equations, continued fractions and q-series McCANN, R. (Associate Professor) Ph.D. 1994 (Princeton)• Mathematical physics, mathematical economics, inequalities, optimization, partial differential equations McCOOL, J. (Professor Emeritus) Ph.D. 1966 (Glasgow)• Infinite group theory MEINRENKEN, E. (Associate Professor) Ph.D. 1994 (Universität Freiburg)• Symplectic geometry MENDELSOHN, E. (Professor) Ph.D. 1968 (McGill)• Block designs, combinatorial structuresMILMAN, P. (Professor) Ph.D. 1975 (Tel Aviv)• Singularity theory, analytic geometry, differential analysis MURASUGI, K. (Professor Emeritus) D.Sc. 1960 (Tokyo)• Knot theory MURNAGHAN, F. (Professor) Ph.D. 1987 (Chicago)• Harmonic analysis and representations of p-adic groups MURTY, V.K. (Professor) Ph.D. 1982 (Harvard)• Number theoryNABUTOVSKY, A. (Associate Professor) Ph.D. 1992 (Weizmann Institute of Science)• Geometry and logicNACHMAN, A. (Professor) Ph.D. 1980 (Princeton)• Inverse problems, partial differential equations, medical imaging QUASTEL, J. (Professor) Ph.D. 1990 (Courant Institute)• Probability, stochastic processes, partial differential equationsPUGH, M. (Associate Professor) Ph.D. 1993 (Chicago)• Scientific computing, nonlinear PDEs, fluid dynamics, computational neuroscienceRANGER, K.B. (Professor Emeritus) Ph.D. 1959 (London)• Fluid mechanics, applied analysis with particular reference to boundary value problems, biofluid dynamics REPKA, J. (Professor) B.Sc. 1971 (Toronto), Ph.D. 1975 (Yale)• Group representations, automorphic forms ROONEY, P.G. (Professor Emeritus) Ph.D. 1952 (Caltech)• Integral operators, functional analysis ROSENTHAL, P. (Professor) Ph.D. 1967 (Michigan)• Operators on Hilbert spaces ROTMAN, R. (Assistant Professor) Ph.D. 1998 (SUNY, Stony Brook)• Riemannian geometry SCHERK, J. (Associate Professor) D.Phil. 1978 (Oxford)• Algebraic geometry

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SECO, L. (Professor) Ph.D. 1989 (Princeton)• Harmonic analysis, mathematical physics, mathematical finance SELICK, P. (Professor) B.Sc. 1972 (Toronto), M.Sc. 1973 (Toronto), Ph.D. 1977 (Princeton)• Algebraic topologySEMENOV, Y. (Adjunct Professor) Ph.D. 1973 (Kiev), Doctorat d’Etat 1982 (Steklov Inst., Moscow)• Partial differential equations, qualitative methods, spectral theorySEN, D.K. (Professor Emeritus) Dr.es.Sc. 1958 (Paris)• Relativity and gravitation, mathematical physics SHARPE, R. (Professor Emeritus) B.Sc. 1965 (Toronto), M.Sc. 1966 (Toronto), Ph.D. 1970 (Yale)• Differential geometry, topology of manifolds SHERK, F.A. (Professor Emeritus) Ph.D. 1957 (Toronto)• Finite and discrete geometry SIGAL, I.M. (University Professor, Norman Stuart Robertson Chair in Applied Math) Ph.D. 1975 (Tel Aviv)• Mathematical physics SMITH, S.H. (Professor Emeritus) Ph.D. 1963 (London)• Fluid mechanics, particularly boundary layer theory SULEM, C. (Professor) Doctorat d’Etat 1983 (Paris-Nord)• Partial differential equations, nonlinear analysis, numerical computations in fluid dynamics TALL, F.D. (Professor) Ph.D. 1969 (Wisconsin)• Set theory and its applications, set-theoretic topology TANNY, S.M. (Associate Professor) Ph.D. 1973 (M.I.T.)• Combinatorics, mathematical modeling in the social sciences TODORCEVIC, S. (Adjunct Professor) Ph.D. 1979 (Belgrade)• Set theory and combinatoricsVIRAG, B. (Junior Canada Research Chair and Assistant Professor) Ph.D. 2000 (Berkeley)• ProbabilityWEISS, W. (Professor) M.Sc. 1972 (Toronto), Ph.D. 1975 (Toronto)• Set theory, set-theoretic topologyYAMPOLSKY, M. (Associate Professor) Ph.D. 1997 (SUNY, Stony Brook)• Holomorphic and low-dimensional dynamical systems

3. THE GRADUATE PROGRAM

The Department of Mathematics offers graduate programs leading to Master of Science (M.Sc.) and Doctor ofPhilosophy (Ph.D.) degrees in pure and applied mathematics.

The M.Sc. Program

The M.Sc. program may be done on either a full- or part-time basis. Full-time students normally complete theprogram in one full year of study; part-time students may take up to three years to complete the program. Thedegree requirements are as follows:

1a. Completion of 6 half-courses (or the equivalent combination of half- and full-year courses), to be selectedfrom those described in the Graduate Courses section of this handbook. The normal course load for full-time graduate students is 3 courses in the fall term and 3 in the spring term. Part-time students may take upto 2 courses per term, although only one is recommended. Students who do not have a strongundergraduate background in the core areas (real and complex analysis, algebra, and topology) and who areinterested in proceeding to the Ph.D. program are advised to take some of the core courses.

1b. Completion of the Supervised Research Project (MAT 4000Y). This project is intended to give the student

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the experience of independent study in some area of advanced mathematics, under the supervision of afaculty member. The supervisor and the student, with the approval of the graduate coordinator decide thetopic and program of study. The project is normally undertaken during the summer session, after the othercourse requirements have been completed, and has a workload roughly equivalent to that of a full-yearcourse.

2. M.Sc. Thesis Option (less common than option 1). Students who take this option will be required to takeand pass two approved full-year courses and submit an acceptable thesis.

The Ph.D. Program

The Ph.D. program normally takes three or four years of full-time study beyond the Master’s level to complete.A Master’s degree is normally a prerequisite; however, exceptionally strong B.Sc. students may apply for directadmission to the Ph.D. program. The program requirements are as follows:

1. Coursework: Completion of at least 6 half-courses (or the equivalent combination of half-year and full-yearcourses), to be selected from those described in the Graduate Courses section of this handbook. Normally,6 half-courses are taken in the first year of study (3 half-courses in the fall term and 3 in the spring term).It is strongly recommended that the student take some additional courses in other years.

2. Comprehensive Examination: The student is required to pass a comprehensive examination in basic

mathematics before beginning an area of specialization. The purpose of the examination is to ensure thatthe student has mastered enough of the basic material in the various areas of mathematics to qualify as aprospective member of the profession. The examinations in the three general areas (analysis, algebra andtopology) last three hours each, and take place during a one-week period in early September and early May.The student is responsible for learning the basic mathematics in these areas at a level covered by the fourcore courses. Exemptions from individual exams will be given if the student has obtained a grade of A orbetter in the corresponding core course(s), provided that a written exam has formed a major component ofthe grading process in the course. Syllabi for the pure mathematics comprehensive exams appear inAppendix A. Copies of mock examination questions and/or past written examination papers are accessibleto all candidates.

A student planning to specialize in applied mathematics must take a total of three exams, including at leastone of algebra and analysis, but may make substitutions for the remaining pure mathematics exam(s) fromsix areas of applied mathematics. The six possible areas of applied mathematics are quantum theory,relativity, fluid mechanics, combinatorics, control theory and optimization, and mathematical finance.Syllabi for the comprehensive examinations in these areas can be found in Appendix B. The departmentreserves the right to determine which pure mathematics exams must be taken in combination with particularareas of applied mathematics. All exams are to be taken within 12 months of entering the Ph.D. program unless the ExaminationCommittee grants permission for a deferral in writing.

3. Specialist Oral Examination: The student is required to pass an oral examination in the area ofspecialization. The examination committee consists of the prospective thesis supervisor and two otherdepartment members. At the option of the supervisor (in consultation with the Graduate Coordinator), thespecialist oral may consist either of an examination on a specified syllabus or a seminar presentation (orpresentations) on an advanced topic. In either case the examination committee must be satisfied that thestudent has demonstrated the knowledge expected of a specialist in the area. The examination should betaken before the end of the second year of Ph.D. studies.

4. Thesis: The main requirement of the degree is an acceptable thesis. This will embody original research of a

standard that warrants publication in the research literature. It must be written under the supervision of one

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or more members of the department. The student presents the thesis results in three stages.

(i) Thesis Content Seminar. This is the first opportunity for a student, after obtaining thesis results, topresent them before interested department members. The seminar consists of an informal presentationof the thesis results. The presentation frequently takes place within the context of one of the regulardepartmental research seminars.

(ii) Departmental Oral Examination. The student gives a 20-minute summary of the thesis and must defendit before a departmental examination committee. Copies of the thesis should be available two weeksbefore the departmental oral examination. The committee may approve the thesis without reservations,or approve the thesis on condition that revision be made, or require the student to take anotherdepartmental oral examination.

(iii) Final Oral Examination. Upon successful completion of the departmental oral, the student proceeds tothe final oral examination conducted by the School of Graduate Studies. The thesis is sent to anexternal reader who submits a report two weeks prior to the examination; this report is circulated tomembers of the examination committee and to the student. The examination committee consists of fourto six faculty members; it is optional for the external reader to attend the examination. The proceduresare similar to those of the departmental examination.

5. Language: Although the Department does not require any formal examination, it is strongly recommendedthat the student acquire an adequate reading knowledge of at least one of French, German or Russian. Insome areas of mathematical research, important results often appear in foreign language journals: studentsare expected to become familiar with the literature in their area in whatever language it is written.

6. Residence: The School of Graduate Studies no longer has a formal residence requirement. However,students are expected to become extensively involved in departmental activities.

Administration of the Graduate Program

A central administration authority called the School of Graduate Studies establishes the basic policies andprocedures governing all graduate study at the University of Toronto. Detailed information about the School isobtained in its calendar, distributed to all graduate students during registration week, and in the admissionspackage that accompanies the application.

The Department of Mathematics has its own graduate administrative body—the graduate committee—composedof 6-8 faculty members appointed by the chair of the department, and five graduate students elected by theMathematics Graduate Students Association. One of the faculty members is the graduate coordinator, who isresponsible for the day-to-day operation of the program. The graduate committee meets frequently throughoutthe year to consider matters such as admissions, scholarships, course offerings, and departmental policiespertaining to graduate students. Student members are not permitted to attend meetings at which the agendaconcerns confidential matters relating to other students. Information regarding appeals of academic decisions isgiven in the Grading Procedures section of the Calendar of the School of Graduate Studies. Students may alsoconsult the Graduate Coordinator (or the student member of the departmental Graduate Appeals Committee)regarding information about such appeals.

In accordance with School of Graduate Studies regulations, a supervisory committee will be established for eachPh.D. student who has chosen a research area and a supervisor. This committee consists of three facultymembers including the supervisor. It is responsible (in consultation with the Graduate Coordinator) formonitoring the student’s progress. A pre-supervisory committee may be established at an earlier stage at eitherthe Department’s or the student’s option. Information about general graduate supervision is available in the SGSdocument Graduate Supervision, Guidelines for Students, Faculty, and Administrators.

General Outline of the 2003-2004 Academic

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Year

Registration August 5 – September 12, 2003Fall Term Monday, September 8 – Friday,

December 5, 2003Spring Term Monday, January 5 – Friday,

April 8, 2004Reading Week February 16 – 20, 2004

Official Holidays (University Closed):

Labour Day Monday, September 1, 2003Thanksgiving Day Monday, October 13, 2003Christmas/New Year Monday, December 22, 2003

– Friday, January 2, 2004Good Friday Friday, April 9, 2004Victoria Day Monday, May 24, 2004Canada Day Thursday, July 1, 2004Civic Holiday Monday, August 2, 2004

4. GRADUATE COURSES

The following is a list and description of the courses offered to graduate students in the 2003-2004 academicyear. Some of these courses are cross-listed with senior (400-level) undergraduate courses; that is, the coursesare open to both graduate and senior undergraduate students. Others are intended for graduate students. Four ofthe courses are core courses. These are the basic beginning graduate courses. They are designed to help thestudent broaden and strengthen his/her general background in mathematics prior to specializing towards athesis. A student with a strong background in the area of any of the core courses should not take that particularcourse. Some students will take all four core courses before the Ph.D. Comprehensive Examination; most willtake some but not all of them. In addition, graduate students may take several intermediate (300-level)undergraduate courses (listed in the Faculty of Arts and Science Calendar) if their background is felt to be weakin some area; no graduate course credit is given for these courses.

There are three other means by which graduate students may obtain course credit, apart from completing theformal courses listed on the following pages. In each of these cases, prior approval of the graduate coordinator isrequired.

1. Students may take a suitable graduate course offered by another department (e.g. statistics or computerscience). Normally at least two-thirds of the course requirements for each degree should be in theMathematics Department.

2. It is sometimes possible to obtain course credit for appropriately extensive participation in a researchseminar (see Research Activities section).

3. It is also possible to obtain a course credit by working on an individual reading course under the supervisionof one of the faculty members, provided the material covered is not available in one of the formal courses orresearch seminars. (Note: this is distinct from the MAT 4000H Supervised Research Project required of allM.Sc. students.)

Most courses meet for three hours each week, either in three one-hour sessions or two longer sessions. For somecourses, particularly those cross-listed with undergraduate courses, the times and locations of classes will be setin advance of the start of term. For other courses, the times and locations of classes will be established atorganizational meetings during the first week of term, so that a time convenient for all participants may bearranged. During registration week, students should consult the math department website or the graduatebulletin board outside the mathematics office for class and organization meeting times and locations.

CORE COURSES

MAT 1000Y (MAT 457Y)REAL ANALYSISV. Jurdjevic1. Lebesgue integration, measure theory, convergence theorems, the Riesz representation theorem, Fubini’s

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theorem, complex measures.

2. L p -spaces, density of continuous functions, Hilbert space, weak and strong topologies, integral operators.3. Inequalities.4. Bounded linear operators and functionals. Hahn-Banach theorem, open-mapping theorem, closed graph

theorem, uniform boundedness principle.5. Schwartz space, introduction to distributions, Fourier transforms on the circle and the line (Schwartz space

and L 2 ).

6. Spectral theorem for bounded normal operators.Textbooks:Royden, H.L.: Real Analysis, Macmillan, 1988.Kolmogorov, A.N. and Fomin, S.V.: Introductory Real Analysis, 1975.References:Rudin, W.: Real and Complex Analysis, 1987.Taylor, A.: Introduction to Functional Analysis, Wiley, 1954.Yosida, K.: Functional Analysis, Springer, 1965.Folland, G.B.: Real Analysis: Modern Techniques and their Applications, 1999.

MAT 1001S (MAT 454S)COMPLEX ANALYSIST. Bloom1. Review of elementary properties of holomorphic functions. Cauchy’s integral formula, Taylor and Laurent

series, residue calculus.2. Harmonic functions. Poisson’s integral formula and Dirichlet’s problem.3. Conformal mapping, Riemann mapping theorem.4. Elliptic functions and the modular function.5. Analytic continuation, monodromy theorem, little Picard theorem.References:Ahlfors, L.: Complex Analysis, 3rd Edition, McGraw-Hill, New York, 1966.Rudin, W.: Real and Complex Analysis, 2nd Edition, McGraw-Hill, New York, 1974.

MAT 1100YALGEBRAJ. Repka1. Linear Algebra. Students will be expected to have a good grounding in linear algebra, vector spaces, dual

spaces, direct sum, linear transformations and matrices, determinants, eigenvectors, minimal polynomials,Jordan canonical form, Cayley-Hamilton theorem, symmetric, alternating and Hermitian forms, polardecomposition.

2. Group Theory. Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct andsemidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups,nilpotent and solvable groups, generators and relations.

3. Ring Theory. Rings, ideals, rings of fractions and localization, factorization theory, Noetherian rings,Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraicvarieties.

4. Modules. Modules and algebras over a ring, tensor products, modules over a principal ideal domain,applications to linear algebra, structure of semisimple algebras, application to representation theory of finitegroups.

5. Fields. Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem ofGalois theory, solution of equations by radicals.

Textbooks:Alperin and Bell: Groups and Representations.Dummit and Foote: Abstract Algebra, 2nd EditionOther References:

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Jacobson: Basic Algebra, Vols. I and II.Lang: Algebra.Artin, M.: Algebra.

MAT 1300Y (MAT 427S)TOPOLOGYP. Selick1. Point set topology: Topological spaces and continuous functions, connectedness and compactness,

countability and separation axioms.2. Homotopy: Fundamental group, Van Kampen theorem, Brouwer's theorem for the 2-disk. Homotopy of

spaces and maps, higher homotopy groups. Covering theory, universal coverings.3. Homology: Simplicial and singular homology, homotopy invariance, exact sequences, Mayer-Vietoris,

excision, Brouwer's theorem for the n-disk, CW-complexes, Euler characteristic.4. Cohomology: Cohomology groups, cup products, cohomology with coefficients.5. Topological manifolds: Orientation, fundamental class, de Rham cohomology for smooth manifolds.

References:Greenberg and Hatcher: Lectures on Algebraic TopologyMunkres: TopologyAdditional References:Munkres: Elements of Algebraic Topology, Perseus Books, 1993Selick: Introduction to Homotopy Theory, Fields Institute Monographs, 1997Massey: Basic Course in Algebraic Topology, Springer, 1997Hatcher: Algebraic Topology, Cambridge University Press, 2001Bredon: Topology and Geometry, Springer, 1993Fulton: Algebraic Topology: A First Course, Springer, 1994Bott, Tu: Differential Forms in Algebraic Topology, Springer, 1997

2003-2004 GRADUATE COURSES

MAT 1003HTHEORY OF SEVERAL COMPLEX VARIABLESIan GrahamThis course will introduce the basic theory of several complex variables. In the later part of the course we willdiscuss some recent results in holomorphic mappings. The topics covered will include the following:

Versions of the Cauchy integral formula in several variables. The inhomogeneous Cauchy-Riemann equations(the p -operator). Domains of holomorphy, pseudoconvexity. Theorems of H. Cartan about holomorphic

mappings; questions of biholomorphic equivalence. Special classes of holomorphic and biholomorphicmappings of the

unit ball in ÷ n .

References:S. Krantz, Function Theory of Several Complex Variables, Reprint of the 1992 Edition, AMS ChelseaPublishing, 2001.I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York2003.

MAT 1015STOPICS IN OPERATOR THEORYM.-D. Choi

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This course is concerned with the major developments in operator theory, including topics on norms, numericalranges, dilations, almost commutativity, and spectral variations. The lectures will be self-contained.

Graduate students taking this course for credit will be required to present a written report on a topic pertinent totheir mathematical backgrounds.

MAT 1037SINTRODUCTION TO THE THEORY OF VON NEUMANN ALGEBRASG. ElliottThis course will discuss the idea of a von Neumann algebra—in which a commutative von Neumann algebra isjust the algebra of bounded functions on a measure space, whereas a non-commutative von Neumann algebraincludes the algebra of all bounded operators on a Hilbert space, but also any subalgebra of this, assumed to beclosed under the adjoint operation, and under convergence in what is referred to as the strong operator topology.Examples of such subalgebras arise in areas as diverse as group theory, ergodic theory, geometry, and quantumfield theory, but a very large class of examples, the so-called amenable ones, can be constructed by a very simpleprocedure involving just finite-dimensional ones—namely, just construct an increasing sequence of finite-dimensional von Neumann algebras, on a fixed Hilbert space, and take the closure of the union in the strongoperator topology. These algebras, incidentally, can now be classified, by well-known work of Connes andothers, but the details of this classification—aside from the statement itself, which is elementary—would beconsidered as beyond the scope of the course. Nevertheless, the existence of this theorem provides a simplesyllabus for the course—namely, what are the basic techniques for studying von Neumann algebras, which,when combined together into a whole that is greater than the sum of its parts, provide the proof of theclassification theorem? In particular, suitable items for the syllabus would be how the roots of algebraic K-theorymay be found already in the work of Murray and von Neumann, and how a routine pursuit of this von Neumannalgebra K-theory led Jones to new invariants for knots. (These two items are rather elementary.)

References:Alain Connes: Noncommutative Geometry, Academic Press, San Diego, 1994.Jacques Dixmier: Les algèbres d'opérateurs dans l'espace Hilbertien. (Algèbres de von Neumann), secondedition, Gauthier-Villars, Paris, 1969.David E. Evans and Yasuyuki Kawahigashi: Quantum Symmetries on Operator Algebras, Clarendon Press,Oxford, 1998.Peter A. Fillmore: A User's Guide to Operator Algebras, John Wiley & Sons, Inc., Toronto, 1996.

MAT 1051F (MAT 468F)ORDINARY DIFFERENTIAL EQUATIONSM. GoldsteinSturm-Liouville problem and oscillation theorems for second-order linear equations. Qualitative theory; integralinvariants, limit cycles. Dynamical systems; invariant measures; bifurcations, chaos. Elements of the calculus ofvariations. Hamiltonian systems. Analytic theory; singular points and series solution. Laplace transform.

MAT 1060YPARTIAL DIFFERENTIAL EQUATIONS (offered at the Fields Institute)W. CraigThis is a one year course that is intended to be a graduate level introduction to the theory of partial differentialequations (PDEs). The course material will start with an overview of the basic properties of the wave equation,Laplace's equation and the heat equation; introducing Fourier transform techniques, distributions, Green'sfunctions, and some of the basic notions of the theory of PDE. We will proceed to cover the general theory ofPDE, including first order theory, the Cauchy Kowalevskaya theorem and generalizations, the Malgrange -Ehrenpreis theorem, and subsequent counterexamples to existence. On a more general level, we will take up thetheory of elliptic equations and their regularity, symmetric hyperbolic systems and energy estimates, and

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parabolic systems. We will then move to the study of more advanced techniques, such as the development ofBrownian motion and Wiener measure, pseudodifferential and Fourier integral operators, and methods ofnonlinear functional analysis. Throughout this course, an attempt will be made to connect the theory to relevantexamples of current research interest in mathematics and its physical applications.

MAT 1062FPDE'S AND POTENTIAL THEORYP. Greiner

I shall discuss the ∂ -Neumann problem, of importance in the study of several complex variables, whichprovides the first natural nonelliptic boundary value problem for the Laplace-Beltrami operator. Such questionslead to second order subelliptic operators, whose fundamental solutions, heat kernels and wave kernels, may begiven in terms of invariants of the induced subRiemannian geometry. We shall pay particular attention to theconnection between classical Hamiltonian mechanics and a new "complex Hamiltonian formalism". Timepermitting I shall discuss path integrals on Heisenberg groups and applications.

MAT 1104FK-THEORY AND C*-ALGEBRASG. ElliottThe course will be an introductory one, but one of the most important applications of C*-algebra K-theory maybe described, somewhat poetically perhaps, as follows. The circle is of course a basic notion in mathematics. The circle, taken together with finite Cartesianproducts of copies of itself—the finite-dimensional tori—provides a microcosm for topology. The phenomenathus represented may be summed up in a single space, the product of infinitely many circles. This space, theinfinite-dimensional torus, has rather deep properties which can be expressed only in the language of non-commutative operator algebras. For instance, the commutative C*-algebra of continuous functions on this space(the study of which is equivalent to the study of the space) has the property that its tensor product with the C*-algebra of compact operators (on a separable infinite-dimensional Hilbert space) is isomorphic to the algebra oftwo-by-two matrices over a different algebra. In fact, this is only the first of a number of surprising properties ofC*-algebras directly related to the infinite-dimensional torus, discovered only recently. These properties, oncethe language of C*-algebras is admitted, are surprisingly simple to state, and even to verify. (One of them is thepresence of both finite and infinite projections in a certain simple C*-algebra.) Each of the properties envisagednot only provides insight into the infinite-dimensional torus—and by extension into the circle—but alsoprovides the answer to an outstanding question in the theory of C*-algebras.

References:Florin-Petre Boca: Rotation C*-Algebras and Almost Mathieu Operators, The Theta Foundation, Bucharest,2001.Alain Connes: Noncommutative Geometry, Academic Press, San Diego, 1994.Kenneth R. Davidson: C*-Algebras by Example, Fields Institute Monographs 6, American MathematicalSociety, Providence, 1996.David E. Evans and Yasuyuki Kawahigashi: Quantum Symmetries on Operator Algebras, Clarendon Press,Oxford, 1998.Huaxin Lin: An Introduction to the Classification of Amenable C*-Algebras, World Scientific Publishing Co.Pte. Ltd., Singapore, 2001.Mikael Rørdam: Classification of Nuclear, Simple C*-Algebras, article in Volume 126 of Encyclopaedia ofMathematical Sciences, Springer, Berlin, 2001 (Volume VII of subseries, Operator Algebras and Non-Commutative Geometry, editors J. Cuntz and V.F.R. Jones), 146 pages.Mikael Rørdam et al.: An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000.

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MAT 1121SGEOMETRY OF INFINITE-DIMENSIONAL LIE GROUPSB. KhesinI. Main notions on Lie groups, Lie algebras, adjoint and coadjoint representations.

1. A Lie group and Lie algebra.2. The adjoint representation.3. Group adjoint orbits.4. The coadjoint representation and orbits.5. Central extensions.

II. Geometry of infinite-dimensional Lie groups and their orbits.1. Affine Kac--Moody Lie algebras and groups.2. The Virasoro algebra and group.3. The Lie--Poisson (or Euler) equations for Lie groups.4. Groups of diffeomorphisms.5. The double loop (or elliptic) Lie groups and Lie algebras.6. Calogero--Moser systems.7. Groups of (pseudo)differential operators.

III. Around holomorphic bundles.1. Definition and integrability of holomorphic bundles.2. Hitchin systems.

IV. Poisson structures on moduli spaces.1. Moduli spaces of flat connections.2. The Poincaré residue and Cauchy--Stokes formula.3. Moduli spaces of holomorphic bundles.

V. Around the Chern--Simons functional.1. A reminder on the Lagrangian formalism.2. Main example: the Chern--Simons functional.3. The holomorphic Chern--Simons functional.4. The CS functional and linking number.

VI. Polar homology.1. The table of the real--complex correspondence.2. Polar homology of a complex projective manifold.3. Axiomatic definition and various generalizations.4. A reminder on the push-forward of holomorphic differential forms.5. Polar homology of a quasi-projective manifold.

References:A. Pressley and G. Segal: Loop Groups, Clarendon Press, Oxford (1986)M. Atiyah: Collected Works, 5th Volume, Gauge TheoryS. Kobayashi: Differential Geometry of Complex Vector Bundles, Iwanami Shoten and Princeton Univ. Press(1987)

MAT 1190FREAL ALGEBRAIC GEOMETRY AND FEWNOMIALSA.G. KhovanskiiFirst results in real algebraic geometry are due to Newton and Descartes. Beautiful Sturm theorem allows to findexplicitly the number of real roots of a given real polynomial. Famous Tarsky theorem is a multidimensionalvariant of Sturm theorem. A lot of modern results in real algebraic geometry were found recently in connectionwith the 16th Hilbert problem.

The Fewnomials theory was discovered in the early 1980's. This far reaching generalization of real algebraic

geometry applies apart from polynomials to functions like exp x, log x, x α and many others. Using this theory

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one can explicitly estimate from above the complexity of the intersections of level sets of such functions.This theory has applications in numerous branches of pure and applied mathematics, for example in thetheory of abelian integrals, in real algebraic geometry, in dynamical systems, in logic and in computer science.

I will present a new proof of Sturm and Tarskii theorems and an introduction to the modern algebraic geometryand the theory of Fewnomials. Basic undergraduate preparation in mathematics suffices. No special preliminaryknowledge is required.

MAT 1191SELLIPTIC CURVESK. ConsaniThe aim of this course is to cover basic properties of the geometry and arithmetic of elliptic curves.The following are the main topics that will be covered in the course. Weierstrass equation, the group law on anelliptic curve, differentials on the curve, the Weil pairing, elliptic curves over finite and local fields. If timepermits: the Hasse-Weil L-function of an elliptic curve and few notions of elliptic modular forms. Some basicknowledge of algebraic geometry is assumed such as: definition and main properties of Zariski topology, thenotion of affine and projective variety and in particular that of an algebraic curve and of divisor on the curve.

This course may be useful to graduate students wanting to work in algebraic geometry or number theory as theywould get a motivational, example-oriented introduction on the subject of elliptic curves. In addition, advancedundergraduates would take advantage from this course and could start afterwards independent study projects,senior-theses and seminar work.

References: J. Silverman: The Arithmetic of Elliptic Curves, Springer-Verlag.N. Koblitz: Introduction to Elliptic Curves and Modular Forms, Springer VerlagR. Hartshorne: Algebraic Geometry, Springer-Verlag.W. Fulton: Algebraic Curves, Addison-Wesley.

MAT 1194S (MAT 449S)ALGEBRAIC CURVESE. BierstoneProjective 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.

References:E. Brieskorn and H. Knorrer, Plane Algebraic Curves, Birkhauser, 1986.W. Fulton, Algebraic Curves, Benjamin, 1969. (It may be reprinted by another publisher.)R.J. Walker, Algebraic Curves, Springer, 1978. (Perhaps also reprinted.)

MAT 1197FREPRESENTATIONS OF REDUCTIVE p-ADIC GROUPSF. MurnaghanThe course will cover many of the basic results in representation theory of reductive p-adic groups. To avoidtechnical complications, many of the results will be stated and proved for representations of p-adic generallinear groups. No prior knowledge of structure of reductive groups or their representations is assumed. Coursenotes will be provided. Topics will include: valuations and local fields, topological groups and smoothrepresentations, Haar measure and characters, induced representations and Jacquet modules, supercuspidalrepresentations, Hecke algebras and K-types, discrete series representations, matrix coefficients of admissiblerepresentations, principal series representations, intertwining maps, Satake's isomorphism, spherical

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representations.

References:1. Course notes2. P. Cartier, Representations of p-adic groups: a survey, in Proc. Symp. Pure Math. (1979) volume 33, part 1,

pp.111-155.3. J.W.S. Cassels, Local Fields, London Mathematical Society Student Texts, volume 3, Cambridge University

Press , 1986.4. H. Jacquet, Representations des groupes lineaires p-adiques, Theory of Group Representations and

Harmonic Analysis (C.I.M.E. II Ciclo, Montecatini terme, 1970) Edizioni Cremonese, Roma, 1971, pp.119-220.

5. Harish-Chandra, Harmonic analysis on reductive p-adic groups, Lecture Notes in Mathematics, volume162, Springer 1970.

MAT 1200F (MAT 415F)ALGEBRAIC NUMBER THEORYH. KimIntroduce techniques of modern number theory via p-adic numbers, adeles, ideles and harmonic analysis onlocally compact groups. The main goal will be Tate's proof of the analytic continuation and the functionalequation of Hecke's L-functions and their applications such as class number formulas, Dirichlet's theorem onarithmetic progressions, Tchebotarev's density theorem and quadratic reciprocity law, etc. We will follow thefirst half of Analytic Number Theory by Larry Goldstein.

References:S. Lang: Algebraic Number TheoryE. Hecke: Lectures on the Theory of Algebraic Numbers

MAT 1303F (APM 461F/CSC 2413F)COMBINATORIAL METHODSE. MendelsohnAdvanced topics in combinatorial theory chosen from a variety of areas, including enumeration, combinatorialdesigns, combinatorial identities, generating functions, graphs, finite geometries, coding theory, andcombinatorial cryptography.

MAT 1343FDIFFERENTIABLE MANIFOLDSL. Jeffrey1. Smooth manifolds; smooth maps between manifolds; submanifolds; immersions, embeddings, the inverse

and implicit function theorems; the rank theorem2. Smooth manifolds3. Tangent spaces; submanifolds and smooth maps between manifolds4. The tangent and cotangent bundle; vector bundles5. Differential forms, exterior derivative6. Vector fields; flows on manifolds; Lie derivative7. Integration on manifolds; Stokes' theorem8. Poincaré lemma, de Rham cohomology, Mayer-Vietoris sequence in de Rham cohomology9. Lie groups

If time permits, an introductory treatment of a selection of the following additional topics will be given:

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10. Riemannian metrics11. Connections and curvature

Suggested reading:L. Conlon, Differentiable manifolds: a first course (Birkhäuser, 1993)V. Guillemin, A. Pollack, Differential topology (Prentice-Hall, 1974

MAT 1350FKNOT THEORYD. Bar-NatanSynopsis. At first sight the natural numbers are dull. You can add and multiply and everybody knows how. Butnumber theory is anything but dull - it is rich and deep and relates to almost everything in mathematics.Likewise knots seem dull until you start playing with them and realize that their theory too is rich and deep andrelates to almost everything in mathematics - even to number theory! Our goal in this class will be to taste just abit of that as follows:

Part I. Introduction to knots and links: Reidemeister moves, colorings, linking numbers, the fundamental group,Seifert surfaces, knot factorization, the Kauffman bracket and the Jones polynomial, briefly on the Alexander,HOMFLY and Kauffman polynomials, alternating links.

Part II. Introduction to finite type (Vassiliev) invariants: Basics, the classical polynomials, the bialgebrastructure, framed and unframed knots, the relation with Lie algebras, universal finite type invariants, theKontsevich integral, briefly on Chern-Simons theory, configuration spaces and parenthesized tangles.

For part I the primary reference will be Lickorish's book "An Introduction to Knot Theory". For part II theprimary references will be research articles that will be distributed in class.

MAT 1352SMORSE THEORYE. MeinrenkenTopics: Gradient flows, homotopy type and critical values, Morse inequalities, Jacobi fields, conjugate points,Morse-Bott functions, applications to loop spaces, Lie groups, symmetric spaces.

References:Milnor: Morse TheoryMatsumoto: Introduction to Morse TheoryBott: Marston Morse and his Mathematical WorksBott: Lectures on Morse Theory, Old and NewHarvey-Lawson: Morse Theory and Stokes' Theorem

MAT 1404F (MAT 409F)SET THEORYF. TallWe will introduce the basic principles of axiomatic set theory, leading to the undecidability of the continuumhypothesis. We will also explore those aspects of infinitary combinatorics most useful in applications to otherbranches of mathematics.

Textbooks:W. Just and M. Weese, Discovering Modern Set Theory, I and II, AMS.K. Kunen, Set Theory, Elsevier

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MAT 1507SASYMPTOTIC METHODS FOR PDE'S (offered at the Fields Institute)V. Buslaev1. WKB asymptotics for ODE's.2. Oscillating solutions of stationary wave-type equations: formal asymptotic solutions, eikonal equation, wave

fronts, rays, asymptotical properties of formal solutions.3. Oscillating solutions of non-stationary wave-type equations. Oscillating solutions of Schroedinger-type

equation (semiclassical approximation). Uniform asymptotic representations.4. Generalized solutions of PDE's. Singular solutions of wave-type equations. Propagation of singularities.References:Wasow, Wolfgang: Linear turning point theory. Applied Mathematical Sciences 54, Springer-Verlag, 1985.ix+246 pp.Fedoryuk, Mikhail V.: Asymptotic analysis. Linear ordinary differential equations. Springer-Verlag,1993.viii+363 pp.Babich, V.M.. Buldyrev, V.S.: Short-wavelength diffraction theory. Asymptotic methods. Springer Series onWave Phenomena 4, Springer-Verlag, 1991. xi+445 pp.

MAT 1508S (APM 446S)APPLIED NONLINEAR EQUATIONSR. McCannAn introduction to nonlinear partial differential equations as they arise in physics, geometry, and optimization.A key theme will be the development of techniques for studying non-smooth solutions to these equations, inwhich the nature of the non-smoothness or its absence is often the phenomenon of interest. The course willbegin with a survey at the level of Evans' textbook, followed by an excursion into the mathematics of fluids.

References:L.C. Evans: Partial Differential Equations, Providence AMS 1998

GSM/19 ISBN 0-8218-0772-2 $75 ($60 AMS members)A.J. Majda and A.L. Bertozzi: Vorticity and Incompressible Flow,

Cambridge Univ. Press 2002. ISBN 0521639484 $40

MAT 1520FWAVE PROPAGATION (offered at the Fields Institute)C. SulemDerivation of canonical equations of mathematical physics. Small amplitude dispersive waves: Basic concepts,the nonlinear Schrödinger (NLS) equation as an envelope equation. Structural properties of the NLS equation:Lagrangian and Hamiltonian structure, Noether theorem, invariances and conservation laws The initial valueproblem: Existence theory, finite-time blowup, Stability/instability of solitary waves; long-time dynamicsAnalysis of the blow-up: self-similarity; modulation analysis; rate of blow-up.

References:W. Strauss : Nonlinear Wave equations, CBMS, Vol. 73, American Mathematical Society, 1989.C.Sulem and P.-L. Sulem: The Nonlinear Schrödinger equation: Self-focusing and Wave collapse,Appl. Math. Sciences, vol 139, 1999, Springer.

MAT 1525SINVERSE PROBLEMS (offered at the Fields Institute)A. NachmanThis course will expand on some of the lectures of the Symposium on Inverse Problems (taking place October 1-

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4 at the Fields Institute) and will provide the necessary background material. We will choose topics whichcombine methods from differential geometry and from partial differential equations, such as inverse scatteringon hyperbolic manifolds and recovering riemannian metrics from boundary data.

MAT 1700S (APM 426S)GENERAL RELATIVITYG. Maschler

Special relativity. The geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature.Einstein's equation and the Hilbert action. Exact solutions: Schwarzschild metrics and black holes, Robertson-Walker metrics and Cosmology. Applications: bending of light, perihelion precession of Mercury, Hubble's law,the big bang. Additionally, as time permits: gravitational waves, black hole dynamics, spinors and twistors,Penrose diagrams, topology of spacetime singularities.

Textbook:R. Wald: General Relativity

Additional References:O'Neill: Semi-Riemannian GeometryHawking and Ellis: The Large Scale Structure of the UniverseJohn Baez: General Relativity on-line tutorial - http://math.ucr.edu/home/baez/gr/gr.htmlPenrose & Rindler: Spinors and Space-time Vol. 1,2.

MAT 1723F (APM 421F)MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICSR. JerrardThe Hilbert space language in quantum mechanics: states as vectors, observables as operators. The role of self-adjointness. Interpretation of the observation process: eigenvalues. Statistical interpretation of quantummechanics. Simultaneous observation and commutativity; the Heisenberg indeterminacy relations.Representations of the canonical commutation relations. Classical mechanics as limiting case of quantum:Poisson brackets vs. commutators. Schroedinger equation in abstract form and in the coordinate representation.Elements of scattering theory.

Textbook:F. Riesz and B. Sz.-Nagy: Functional Analysis, Dover.

MAT 1739FSUPERSYMMETRIC QUANTUM FIELD THEORIESK. HoriThis course will give an accessible introduction to supersymmetric quantum field theories in low dimensions.Supersymmetry is relevant for many recent developments in various fields in mathematics, including Gromov-Witten invariants, Donaldson invariants, Floer homology, various fixed point theorems, etc. No specialbackground is assumed.

1. Gaussian integrals and Feynman diagrams, fermionic integrals.2. Supersymmetric quantum mechanics and Morse theory.3. Non-linear sigma models, Landau-Ginzburg models, Linear sigma models.4. Renormalization group flows, supersymmetric non-renormalization theorems.5. Chiral rings and topological field theory.6. Mirror symmetry.

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Textbook: K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil and E. Zaslow, ``Mirrorsymmetry'' (AMS/CMI, 2003).References:P. Deligne, P. Etingof, D.S. Freed, L.C. Jeffrey, D. Kazhdan, J.W. Morgan, D.R. Morrison, E. Witten, editors,Quantum Fields and Strings: A Course for Mathematicians, vols. 1 & 2 (AMS, 1999).E. Witten, ``Supersymmetry And Morse Theory'', J. Diff. Geom. 17 (1982) 661

MAT 1839FOPTIMAL TRANSPORTATION AND NONLINEAR DYNAMICS (offered at the Fields Institute)R. McCannThe optimal transportation problem of Monge and Kantorovich is now understood to be a mathematicalcrossroads, where problems from economics, fluid mechanics, and physics meet geometry and nonlinear PDE.This course gives a survey of these unexpected developments, using variational methods and duality to addressfree boundary problems, nonlinear elliptic equations (Monge-Ampere), regularity, geometric inequalities withsharp constants, metric and Riemannian geometry of probability measures, nonlinear diffusion, fluid mixing,and atmospheric flows.

Reference:C. Villani: Topics in Optimal Transportation, Providence: AMS 2003. GSM/58 ISBN 0-8218-3312-X $59 ($47 AMS members)

MAT 1856S (APM 466S)MATHEMATICAL THEORY OF FINANCEL. SecoIntroduction 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.STA 2111FGRADUATE PROBABILITY IJ. RosenthalA rigorous introduction to probability theory: Probability spaces, random variables, independence, characteristicfunctions, Markov chains, limit theorems.

STA 2211SGRADUATE PROBABILITY IIJ. RosenthalContinuation of Graduate Probability I, with emphasis on stochastic processes: Poisson processes and Brownianmotion, Markov processes, Martingale techniques, weak convergence, stochastic differential equations.

STA 3077SRANDOM GRAPHSB. ViragThis course is an introduction to the theory of random graphs with a special emphasis on trees and planargraphs. Connectedness properties of a random graph, the size of the largest component. The connectionbetween random walks, branching processes and random trees. The canonical continuum random tree.Techniques for counting planar graphs: Feynman matrix integrals and labelled tree representations. Growth,distance and spectrum. The connection to circle packings and conformal maps. Conjectures and open questions.

There are no special prerequisites for this course, although some familiarity with basic probability and analysis

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is helpful.

COURSES FOR GRADUATE STUDENTS FROM OTHER DEPARTMENTS

(Math graduate students cannot take the following courses for graduate credit.)

MAT 2000YREADINGS IN THEORETICAL MATHEMATICSMAT 2001HREADINGS IN THEORETICAL MATHEMATICS IMATH 2002HREADINGS IN THEORETICAL MATHEMATICS II(These courses are used as reading courses for engineering and science students in need of instruction in specialtopics in theoretical mathematics. These course numbers can also be used as dual numbers for some third andfourth year undergraduate mathematics courses if the instructor agrees to adapt the courses to the special needsof graduate students. A listing of such courses is available in the 2003-2004 Faculty of Arts and ScienceCalendar. Students taking these courses should get an enrolment form from the graduate studies office of theMathematics Department. Permission from the instructor is required.)

COURSE IN TEACHING TECHNIQUES

The following course is offered to help train students to become effective tutorial leaders and eventuallylecturers. It is not for degree credit and is not to be offered every year.

MAT 1499HTEACHING LARGE MATHEMATICS CLASSESJ. RepkaThe goals of the course include techniques for teaching large classes, sensitivity to possible problems, anddeveloping an ability to criticize one’s own teaching and correct problems. Assignments will include such things as preparing sample classes, tests, assignments, course outlines,designs for new courses, instructions for teaching assistants, identifying and dealing with various types ofproblems, dealing with administrative requirements, etc. The course will also include teaching a few classes in a large course under the supervision of the instructor.A video camera will be available to enable students to tape their teaching for later (private) assessment.

5. RESEARCH ACTIVITIES

The Department of Mathematics offers numerous research activities, in which graduate students are encouragedto participate. Research seminars are organized informally at the beginning of each year by one or more facultymembers, and are offered to faculty and graduate students on a weekly basis throughout the year. The level andspecific content of these seminars varies from year to year, depending upon current faculty and student interest,and upon the availability and interests of invited guest lecturers. The following weekly research seminars havebeen offered in the past few years:

Research Seminars

Algebraic GeometryAlgebraic TopologyApplied MathematicsAutomorphic FormsCombinatorics

CryptographyDynamicsErgodic TheoryGanitaGeometry

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Geometry and TopologyGraduate Student SeminarHomotopy TheoryIndustrial MathematicsIntegrable Hamiltonian Systems SeminarJoint Math/PhysicsLie GroupsMathematical FinanceNumber TheoryOperator Theory

Partial Differential EquationsProbabilityRepresentation TheoryScientific ComputingSet TheorySeveral Complex VariablesString TheorySymplectic GeometrySingularitiesToric Varieties, Convex Geometry & Combinatorics

Note: It is possible for graduate students to obtain course credit for sufficiently extensive participation in researchseminars (see Graduate Courses section).

In addition to the weekly seminars, there are numerous special seminars throughout the year, a series of colloquia,and an active program of visiting lecturers. Graduate students are also welcome to attend lectures and seminarsoffered by other departments.

6. ADMISSION REQUIREMENTS AND APPLICATION PROCEDURES

All applications for admission to the graduate program in mathematics are reviewed both by the graduatecommittee of the Department of Mathematics, and by the School of Graduate Studies, which administers graduateapplications for the entire university.

The School of Graduate Studies establishes the basic admissions requirements and application procedures for allstudents wishing to pursue graduate study at the University of Toronto. They are described in detail in theadmissions package that accompanies the application forms, and in the School of Graduate Studies calendar.Please read them carefully and note the deadlines! In addition, the Department of Mathematics requires threeletters of reference. The letters must be from three people familiar with your mathematical work, giving theirassessment of your potential for graduate study and research in mathematics.

Students who do not have a degree from a university where the language of instruction is English must satisfy theUniversity of Toronto’s English language proficiency requirement prior to registration. There are three ways tosatisfy this requirement: (1) a score of at least 213 on the computer-based Test of English as a Foreign Language(TOEFL) and a score of at least 4.0 on the Test of Written English (TWE); (2) a score of at least 85 on theMichigan English Language Assessment Batter (MELAB); (3) a score of at least 7.0 on the International EnglishLanguage Testing Service (IELTS). Students are encouraged to satisfy this requirement by February (forSeptember admission), so that first scores are available at the time applications are considered.

7. FEES AND FINANCIAL ASSISTANCE

Fees

Information concerning fees is in the admissions package that accompanies the application forms; more detailedinformation and a schedule of fees for 2003-04 will be available in August 2003 from the School of GraduateStudies. Listed below are the fees for the 2002-03 academic session:

Domestic students: Full-time $5,890Visa students: Full-time $10,428

Financial Assistance

Below is a list of those types of financial assistance most commonly awarded to mathematics graduate students in2002-03 This information should also be applicable for students who wish to apply for the 2003-04 academic year;the deadlines for applications will be altered slightly in accordance with the 2003-04 calendar. Some awards areavailable from external funding agencies; others come from within the University.Less common scholarships, offered by smaller or foreign funding agencies, are also available; information aboutthese is generally distributed to the graduate offices of local universities.

Natural Sciences and Engineering Research Council (NSERC) Post-Graduate ScholarshipsValue: approx. $17,300-$19,100 for a twelve month periodEligibility: Canadian citizens, permanent residents; first class academic standing; full-time attendance (restrictedalmost exclusively to Canadian universities)Application: apply through the university you are currently attending; application available at www.nserc.caDeadline: around mid October. Contact your university for departmental deadline

Ontario Graduate Scholarships (OGS)Value: approx. $5,000 per term for two or three termsEligibility: no citizenship restrictions; first class academic standing; full-time attendance at an Ontario universityApplication: apply through the university you are currently attending or contact the OGS office in Thunder Bay,Ontario directly at 1-800-465-3957 (http://osap.gov.on.ca/NOT_SECURE/ogs.htm).Deadline: around the end of October

University of Toronto FellowshipsValue: minimum $1,000

Eligibility: no citizenship restrictions; at least an A − average; full-time attendance at the University of Toronto(or, in the case of the Master’s program and for students with a disability, part-time registration together with aletter from the Director of Special Services to Persons with a Disability confirming that part-time study is de factofull-time study for the student). See below for policy.Application: graduate school applicants will be considered automaticallyDeadline: February 1

Connaught ScholarshipValue: $15,000 plus full fees for first yearEligibility: no citizenship restrictions; full-time attendance at the University of Toronto; awarded only toexceptionally outstanding scholarsApplication: graduate school applicants will be considered automaticallyDeadline: February 1

Research AssistantshipsValue: a limited amount of funds is available for academically worthy studentsEligibility: no citizenship restrictions; full-time attendance; high academic standingApplication: graduate school applicants will be considered automaticallyDeadline: February 1

Teaching AssistantshipsValue: approx. $27-$30 per hour; number of hours per week will not exceed a maximum average of 8Eligibility: all full-time students who are accepted by the Mathematics Department (subject to satisfactoryperformance); may be held in conjunction with other awardsApplication: forms available in May from the Graduate Office, Department of MathematicsDeadline: June 10

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Department of Mathematics Policy on Financial Support of Graduate Students

Ph.D. Students: At the time of admission to the Ph.D. program, students will normally be guaranteed support for aperiod of four years (five years in the case of students admitted directly from a Bachelor’s program), except thatstudents who complete their degree requirements earlier will not be supported past the end of the academic year inwhich they finish. This guarantee will be made up of a mix of fellowships (including external awards such asNSERC), teaching assistantships and other sources of funding, at the discretion of the Department; and is subjectto satisfactory academic progress, the maintenance of good standing, and in the case of teaching assistantships,satisfactory performance in that role, as judged by the Department. Absent this, support may be reduced,suspended, or discontinued.

In exceptional circumstances some funding may be provided to students in a subsequent year, but the Departmentexpects that students will normally have completed their degree requirements within the four year period.

M.Sc. Students: Students who are granted provisional admission to the Ph.D. program at the time of admissionwill receive financial support, normally for one year only.

All full-time students in the first or second year of a Master’s program are eligible for teaching assistant work(subject to satisfactory performance).

8. OTHER INFORMATION

The Department of Mathematics is located in the heart of the University of Toronto, which in turn is located in theheart of downtown Toronto. Students therefore have access to a wide range of facilities and services. A list of theseappear below.

Facilities and Services

Library FacilitiesThe University of Toronto Library System is the 4th largest academic research library in North America. It hasover 4 million volumes. The Gerstein Science Information Centre has a comprehensive collection of mathematicalbooks. The Mathematics & Statistics Library, which is in the same building as the Mathematics Department, is thelocation for all mathematical journals on campus. It also houses a selective collection of over 16,000 titles ofmathematical books.

The resources of the Library System are electronically linked by UTLink. It includes the catalogues of all librarieson campus, electronic abstracts and indexes, and other electronic resources such as electronic journals and internet.

The Mathematics & Statistics Library has computers with Netscape which give access to MathSciNet, UTLink andother electronic resources. It has study spaces for reading journals/books and studying. The Library has aphotocopier for copying library materials. The Mathematics Library gives each graduate student a photocopyingallowance.

Computer FacilitiesEvery graduate student may obtain an account on the computer system. The Department of Mathematics has atotal of three main UNIX servers: an IBM RS6000 machine running AIX and two AMD Athlon machines runningLinux. The fastest of these, for floating point computations, is a dual processer IBM 44P-270 with 1GB of RAM.The Athlons, running at 950MHz and 1200MHz, with 512MB and 768MB of RAM respectively, have excellentinteger performance. These servers can support multiple users through several X-terminals, and ascii-terminals.There are a few PC's and a Macintosh G3. Programming facilities include C, Fortran, Maple, Mathematica andTeX (LaTeX, AMS-TeX, AMS-LaTeX).

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HousingThe university operates five graduate student residences-apartment complexes on or near the campus, ranging fromunfurnished family apartments to the more conventional bed-and-board residences. In addition, the UniversityHousing Service provides a listing of privately owned rooms, apartments and houses available for students to rent(see Contacts below). More detailed information about these facilities may be found in the admissions packagewhich accompanies the application forms.

Students should keep in mind that accommodation could be expensive and limited, particularly in downtownToronto. It is therefore advisable to make inquiries well in advance and to arrive in Toronto a few days prior to thestart of term. Students can expect to pay anywhere between $300 to $800 per month on accommodation and from$200 to $300 per month on food, travel and household necessities.

Health ServicesThe University of Toronto Health Service offers medical services and referrals to private physicians for Universityof Toronto students. Most of these services are free of charge if you are covered under Ontario Health Coverage(OHIP), or the University Health Insurance Plan (UHIP) for visa students. OHIP application forms and informationare available from the University Health Services (see Contacts below).UHIP coverage for visa students is compulsory and is arranged during registration at the International StudentCentre.

Disabled StudentsServices and facilities for disabled students are available at the University of Toronto. Further information may beobtained from the Coordinator of Services for Disabled Persons (see Contacts section below).

Foreign StudentsThe International Student Centre offers many services to foreign students, including an orientation program in lateAugust – early September, individual counselling whenever appropriate, and an English language program. Inaddition, the International Student Centre contacts all foreign students once they have been accepted into thegraduate program, to provide information and advice concerning immigration procedures (visa and studentauthorization forms), employment restrictions and authorization while in Canada, and other relevant matters (seeContacts below).

Athletics & RecreationA wide range of athletic facilities are available within the university, including an arena and stadium, playingfields, swimming pools, squash, tennis, badminton, volleyball and basketball courts, running tracks, archery andgolf ranges, fencing salons, exercise and wrestling rooms, dance studios, saunas, lockers and a sports store.Instruction courses, exercise classes and fitness testing are regularly offered, and there is an extensive intramuralprogram with several levels of competition in more than 30 sports.

Other recreational activities and facilities are also available within the university, such as theatre, music, pubs,dances, art exhibitions, a wide range of clubs, debates lectures and seminars, reading rooms, cafeterias and chapels.

University of Toronto students also enjoy easy access (walking distance or only a few minutes by subway) tosymphony concerts, theatres, ballet, operas, movies, restaurants and shopping.

Graduate Student AssociationsEvery graduate student at the University of Toronto is automatically a member of the Graduate Student Union(GSU). Graduate students in the Department of Mathematics are also members of the Mathematical GraduateStudents Association (MGSA). Between them, these associations sponsor many events every year, includingparties, pubs, dances, outings and more serious endeavours such as seminars and lectures.

Contacts and Sources of Information

Graduate OfficeDepartment of Mathematics

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University of Toronto100 St. George St., Room 4067Toronto, Ontario M5S 3G3(416) 978-7894(416) 978-4107 (Fax)ida@math.toronto.eduhttp://www.math.toronto.edu/graduate• all matters relating to graduate studies in mathematics at the University of Toronto• location of registration in September/January

Mathematics LibraryUniversity of Toronto100 St. George Street, Room 622Toronto, Ontario M5S 3G3(416) 978-8624(416) 978-4107 (Fax)mathlib@math.toronto.edu

School of Graduate StudiesUniversity of Toronto63 St. George StreetToronto, OntarioCanada M5S 2Z9(416) 978-5369(416) 978-4367 (Fax)graduate.information@utoronto.cahttp://www.sgs.utoronto.ca• general information concerning graduate studies at the University of Toronto

Fees DepartmentOffice of the ComptrollerUniversity of Toronto215 Huron Street, 3rd FloorToronto, Ontario M5S 1A1(416) 978-2142(416) 978-2610 (Fax)fees@finance.utoronto.cahttp://www.utoronto.ca/fees.htm• enquiries concerning fees• payment of fees

Special Services to Persons with a DisabilityUniversity of Toronto214 College Street, 1st FloorToronto, Ontario M5T 2Z9(416) 978-8060(416) 978-8246http://www.library.utoronto.ca/www/equity/ssd.htm• facilitates the inclusion of students with hidden or obvious disabilities and health conditions into university life

University Housing ServiceUniversity of Toronto214 College Street, 1st FloorToronto, Ontario M5T 2Z9416-978-8045

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416-978-1616 (Fax)housing.services@utoronto.cahttp://eir.library.utoronto.ca/StudentHousing/• information concerning university residences and rental accommodations

University Health ServiceUniversity of Toronto214 College Street, 2nd FloorToronto, Ontario M5T 2Z9416-978-8030416-978-2089 (Fax)http://www.utoronto.ca/health/• medical assistance for University of Toronto students• application forms for Ontario Health Coverage

International Student CentreUniversity of Toronto33 St. George StreetToronto, Ontario M5S 2E3(416) 978-2564(416) 978-4090 (Fax)isc.information@utoronto.cahttp://www.library.utoronto.ca/www/isc/• information and assistance for international students, including UHIP registration

The Athletic CentreUniversity of Toronto55 Harbord StreetToronto, Ontario M5S 2W6(416) 978-3437416-978-6978 (Fax)http://www.utoronto.ca/physical/fac_serv/index.html• athletics and recreation information

Hart HouseUniversity of Toronto7 Hart House CircleToronto, Ontario M5S 3H3416-978-4411colin.furness@utoronto.cahttp://www.utoronto.ca/harthouse• athletics and recreation informationGraduate Students’ UnionUniversity of Toronto16 Bancroft AvenueToronto, Ontario M5S 1C1416-978-2391 or -6233416-971-2362 (Fax)gsunion@chass.utoronto.cahttp://www.utoronto.ca/gsunion/

Sexual Harassment OfficeUniversity of Toronto40 Sussex Avenue416-978-3908

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http://www.library.utoronto.ca/equity/sexual.html• Students are covered by the Sexual Harassment Policy while on university premises or carrying on a

university-related activity. Complaints and requests for information are confidential.

Human Resources Development Canada (HRDC)811 Danforth Avenue, 1st Floor, or25 St. Clair Avenue East, 1st Floor1-800-206-7218• To obtain a Social Insurance Number (in person only). Office hours: Monday-Friday, 08:30-16:00• Applications available from Graduate Office of the Department of Mathematics or HRDC office. Supporting

documentation must be original, e.g. student authorization• Takes an average of 4 weeks to process

APPENDIX A: COMPREHENSIVE EXAMINATION SYLLABI FOR PURE MATHEMATICSPH.D. STUDIES

Algebra1. Linear algebra. Students will be expected to have a good grounding in linear algebra, vector spaces, dual

spaces, direct sum, linear transformations and matrices, determinants, eigenvectors, minimal polynomials,Jordan canonical form, Cayley-Hamilton theorem, symmetric, alternating and Hermitian forms, polardecompositon.

2. Group Theory. Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct andsemidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups,nilpotent and solvable groups, generators and relations.

3. Ring Theory. Rings, ideals, rings of fractions and localization, factorization theory, Noetherian rings, Hilbertbasis theorem, invariant theory, Hilbert Nullstellensatz, primary decompositon, affine algebraic varieties.

4. Modules. Modules and algebras over a ring, tensor products, modules over a principal ideal domain,applications to linear algebra, structure of semisimple algebras, application to representation theory of finitegroups.

5. Fields. Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem ofGalois theory, solution of equations by radicals.

No reference is provided for the linear algebra material.

References for the other material:Dummit & Foote: Abstract Algebra, Chapters 1-14 (pp. 17-568).Alperin & Bell: Groups and Representations, Chapter 2 (pp. 39-62), 5, 6 (pp. 107-178).

Complex Analysis1. Review of elementary properties of holomorphic functions. Cauchy’s integral formula, Taylor and Laurent

series, residue calculus.2. Harmonic functions. Poisson integral formula and Dirichlet’s problem.3. Conformal mapping, Riemann mapping theorem.4. Analytic continuation, monodromy theorem, little Picard theorem.

L. Ahlfors, Complex Analysis, Third Edition, McGraw-Hill, Chapters 1-4, 5.1, 5.5, 6.1, 6.2, 6.3.W. Rudin, Real and Complex Analysis, Second Edition, Chapter 16 (except 16.4-16.7).

Note: The material in Ahlfors can largely be replaced by Chapters 10, 11, 12.1-12.6, and 14 of Rudin. But Ahlforsis the official syllabus for this material. The second edition of Ahlfors can be used if it is noted that Section 5.5 in

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the third edition is Section 5.4 in the second edition.)

Real AnalysisReferences:H.L. Royden, Real Analysis, Third Edition, Prentice Hall, 1988.Gerald B. Folland, Real Analysis, Modern Techniques and Their Applications, John Wiley & Sons, 1984.Yitzhak Katznelson, An Introduction to Harmonic Analysis, Dover, 1976.

1. Background: Royden, Chapters 1 and 2; Folland (Prologue).2. Basic Measure Theory: Royden, Chapters 3 and 4, for the classical case on the real line (which contains all the

basic ideas and essential difficulties), then Chapter 11, Sections 1-4, for the general abstract case; Folland,Chapters 1 and 2.

3. Differentiation: Royden, Chapter 5, for the classical case, then Chapter 11, Sections 5 and 6 for the general

case; Folland, Chapter 3 (For differentiation on ú n one can restrict the attention to the one dimensional case,which contains all the basic ideas and essential difficulties.)

4. Basic Functional Analysis: Royden, Chapter 10, Sections 1,2,3,4,8; Folland, Chapter 5, Sections 1,2,3,5.

5. L p -Spaces: Royden, Chapter 6 for the classical case, and Chapter 11, Section 7 for the general case, Chapter13, Section 5 for the Riesz Representation Theorem; Folland, Chapter 6, Sections 1 and 2, Chapter 7, Section1.

6. Harmonic Analysis: Katznelson, Chapter 1, Chapter 2, Sections 1 and 2, and Chapter 6, Sections 1 to 4;Folland, Chapter 8, Sections 1,2,3,4,5, and 8. One can restrict the attention to the one dimensional case, asdone in Katznelson.

Topology1. Set Theory: Zorn’s lemma, axiom of choice. Point set topology: metric spaces, topological space, basis,

subbasis, continuous maps, homeomorphism, subspace, quotient space, Tychonov theorem, properties ofspaces (normal, Hausdorff, completely regular, paracompact), Urysohn lemma, Tietze’s extension theorem,parallelotopes, Stone-Cech compactification.

Dugundji, Topology, Allyn Bacon Inc., Boston, 1966, Chapters 1-7, 11-12. Kelley, General Topology, Graduate Texts in Mathematics 27, Springer-Verlag, 1955, Chapters 1-5, 7, andAppendix.

2. Algebraic topology: homotopic maps, fundamental group, Van Kampen theorem, higher homotopy groups.Covering spaces, covering transformations, map lifting, classification of covering spaces. Singular homologytheory, axioms for homology. CW complexes, cellular homology, Euler characteristic. Cohomology, universalcoefficient theorem, cup product. Manifolds, orientation, Poincaré duality. Applications (Brouwer fixed pointtheorem, Jordan-Brouwer separation theorem, invariance of domain, etc.)

Greenberg and Harper, Lectures on Algebraic Topology, pp. 1-229, Second Edition, Addison-Wesley, 1981.Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics 127, Springer-Verlag, 1991,pp. 117-146.

APPENDIX B: ORAL EXAMINATION SYLLABI FOR APPLIED MATH PH.D. STUDIES

A student planning to specialize in applied mathematics must take either the algebra or the analysis portion of thegeneral written examination, but may make substitutions from five areas of applied mathematics for the remainingportions (topology and/or at most one of algebra and analysis). The six possible areas of applied mathematics are:quantum theory, relativity, fluid mechanics, combinatorics, control theory and optimization, and mathematicalfinance. The following are the syllabi for the oral examinations:

Basics of Quantum Field Theory

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Nonrelativistic quantum mechanics: Quantum observables are self-adjoint operators. The spectral theorem forself-adjoint operators in terms of spectral measures, and the physical interpretation of the quantum formalism.Complete sets of observables. The essential self-adjointness of the Schrödinger operator: the Kato and Rellichtheorems. Eigenvalue problems and the energy spectrum: the harmonic oscillator; the hydrogen atom. TheSchrödinger, Heisenberg and interaction pictures. Regular and singular perturbation theory. Canonicalcommutation relations and von Neumann’s theorem. *Statistical operators and the trace class. *The Hilbert-Schmidt class as Liouville space and von Neumann’s equation.

Group theory and quantum mechanics: Representations of space and time translations. Stone’s theorem for one-parameter. Abelian groups of unitary operators. SU(2) and the concept of spin. The Heisenberg-Weyl group andcanonical commutation relations. Ray representations of the Galilei group. Relativistic invariance andrepresentations of the Poincaré group. The Lie algebras of the Galilei and Poincaré groups, and of their mainkinematical subgroups. *Systems of imprimitivity and particle localization.

Quantum scattering theory: Time-dependent scattering theory, wave operators and their intertwining properties.The S-operator and scattering cross-sections. Stations. The Coulomb potential and scattering for long-rangeinteractions. Eigenfunction expansions and Green functions. The T-matrix, the scattering amplitude and the Bornseries. *Asymptotic completeness. *Channel wave operators and asymptotic states.

Relativistic quantum field theory: The Klein-Gordon and Dirac equations. Fock space for spin 0 and spin ½particles. Creation and annihilation operators. Quantum fields as operator-valued distributions. The Gupta-Bleulerformalism for photons and gauge freedom.

References:Intermediate:J.M. Jauch: Foundations of Quantum Mechanics, 1968.W. Miller, Jr.: Symmetry Groups and their Applications, 1972.E. Prugovecki: Quantum Mechanics in Hilbert Space, 2nd Edition, 1981.S.S. Schweber: An Introduction to Quantum Field Theory, 1961.Advanced:A.O. Barut & R. Raczka: Theory of Group Representations and Applications, 1978.N.N. Bogulubov, A.A. Logunov & I.T. Todorov: Introduction to Axiomatic Quantum Field Theory, 1975.T. Kato: Perturbation Theory for Linear Operators, 2nd Edition, 1976.M. Reed & B. Simon: Methods of Modern Mathematical Physics, vols. 1-4, 1972-78.

Fluid Mechanics

It is expected that a student has a basic knowledge of real and complex analysis including ordinary differentialequations. The extra mathematics required includes:

1. Partial differential equations: Laplace’s equation, properties of harmonic functions, potential theory, heatequation, wave equation. Solutions through series and transform techniques. Bessel functions and Legendrefunctions. Distributions. (e.g. Duff and Naylor)

2. Asymptotic and perturbation techniques: Asumptotic series solutions of ordinary differential equations,asymptotic expansion of integrals. Singular perturbation problems, boundary layer methods, WKB theory,multiple time-scale analysis. (e.g. Bender and Orszag)

Basic physical properties of fluids. Derivation of the Navier-Stokes equations for a viscous compressible fluid;vorticity; energy balance. Simple exact solutions of the Navier-Stokes equations. Slow viscous motions; Stokesflows; Oseen flow. Irrotational flow; sources and sinks; complex variable methods. Boundary layer approximation.Blasius flow; separation; jets and wakes. Rotating flows; geostrophic behaviour. Free surface flows; wavepropagation. Simple unsteady boundary layer flows; Stokes layers. Shock waves in a tube; supersonic flow.

General Relativity and Classical Mechanics

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Space-times as Lorentz manifolds. Differential geometry (curvature, etc.) and local and global properties ofLorentz manifolds.

Field equations of general relativity, stationary and static space-times. Exact solutions. Schwarzschild, Kruskal,Kerr solutions. Cosmological models: Robertson-Walker and Friedman models and their properties.

Cauchy problem for the field equations. Classification of space-times.

Symplectic geometry, symplectic structure of cotangent bundles, Poisson brackets.

Hamiltonian equations, canonical transformations, Legendre transformations, Lagrangian systems, Hamilton-Jacobi theory.

References:Hawking & Ellis: Large Scale Structure of Space-Time, Chapters 2,3,4,5.O’Neill: Semi-Riemannian Geometry with Applications to Relativity.Abraham & Marsden: Foundations of Mechanics, Chapters 3,4,5.

Combinatorics

Enumeration: Ordinary and exponential generating functions. Difference equations and recursions. Partition andpermutations. Polya counting, Fruchts Theorem, systems of distinct representatives.

Graph Theory: Trees, connectivity, bipartite graphs, minimal spanning trees, Eulerian and Hamiltonian graphs,travelling salesman and chinese postman problems, matchings, chromatic number, perfect graphs.

Design Theory: Definitions, examples, finite fields, finite affine and projective spaces, Fisher’s inequality,symmetric designs, statement of Wilson’s Theorem and Wilson’s Fundamental Construction.

Coding Theory: Linear codes, sphere packing, Hamming and Plotkin bounds, perfect codes, polynomial overfinite fields.

Algorithms and Complexity: Algorithms for listing permutations, combinations, subsets. Analysis of algorithms,basic concepts such as NP, and #P, and NPC.

References:P.J. Cameron: Combinatorics: Topics, Techniques Algorithms, Cambridge Univ. Press, ISBN 0521457610.

Control Theory and Optimization

Control Theory: Qualitative properties of thereachable sets, Lie bracket and Lie determined systems, linear theory, stability and feedback.

Reference:V. Jurdjevic: Geometric Control Theory, Cambridge University Press, Chapters 1,2 and 3.Optimal Control: Linear-quadratic problems, symplectic form, Lagrangians, the Riccati equation, the MaximumPrinciple and its relation to the calculus of variations.

Reference:V. Jurdjevic: Geometric Control Theory, Cambridge University Press, Chapters 7, 8 ,11.

Linear Programming: Convex analysis, simplex algorithm, duality, computational complexity and Karmarkar’sAlgorithm.

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Reference:Bazaraa, Jarvis & Sherali: Linear Programming and Network Flows, Wiley, 1990, Chapters 2,3,4,6,8.

Nonlinear Programming: Unconstrained and constrained nonlinear problems. Introduction to computationalmethods.

Reference:Luenberger: Linear and Nonlinear Programming, Addison-Wesley, 1984, Chapters 6,7,10.

Mathematical Finance

Stochastic calculus: Martingales, Ito’s lemma, Girsanov’s theorem, stochastic differential equations, stoppingtimes.

Reference:Baxter & Rennie, Financial Calculus.

Finance: Equity derivatives, interest rate derivatives, market risk, credit risk, portfolio theory, numerical methods.

References:D. Duffie: Dynamic Asset Pricing Theory.P. Wilmott et al.: Mathematics of Financial Derivatives.RiskMetrics and CreditMetrics documents.

APPENDIX C: PH.D. DEGREES CONFERRED FROM 1993-2003

1993DANZIGER, Peter (Combinatorics) Uniformly Resolved DesignsGAO, Jianmin (Applied Differential Equations) Global Existence for Semilinear Wave Equations on OpenRobertson-Walker Space-TimesLEVY, Jason (Lie Algebras) A Modified Trace FormulaLINEK, Vaclav (Combinatorics) Colourings of 3-(v,4,1) Designs, Packings and CoveringsMANJI, Imtiaz (Algebraic Geometry) Hochschild (Co)Homology of Complete IntersectionsMCINTOSH, David (Ergodic Theory) Ergodic Averages for Weight Functions Moved by Non-Linear

Transformations on ú n

MORENZ, Phillip (Operator Theory) The Structure of C*-Convex SetsRODNEY, Peter (Combinatorics) Balance in Tournament DesignsROUTLIFFE, Jim (Control Theory) Sufficient Conditions for Optimality for Control SystemsSZEPTYCKI, Paul (Set Theoretic Topology) Some Consistency and Independence Results on CountablyMetacompact Spaces

1994BODNAR, Andrea (Fluid Mechanics) Low Reynolds Number Particle-Fluid InteractionsFERRANDO, Sebastian (Ergodic Theory) Moving Convergence for Superadditive Processes and HilbertTransformFRY, Robb (Functional Analysis) Approximation on Banach Spaces

HA, Minh Dzung (Ergodic Theory) Operators with Gaussian Distributions, L 2 -Entropy and a.e. ConvergenceHA, Xianwei (Representation Theory) Invariant Measure on Sums of Symmetric Matrices and its Singularities andZero PointsºABA, Izabella (Mathematical Physics) N-Particle Scattering in a Constant Magnetic FieldLISI, Carlo (Operator Theory) Perturbation by Rank-Two ProjectionsMA, Kenneth (Fluid Mechanics) Low Reynolds Number Flow in the Presence of a Corrugated Boundary

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PRANOTO, Iwan (Control Theory) Distributed Parameter System: Controllability and its Related ProblemsSTEVENS, Ken (Operator Algebras) The Classification of Certain Non-Simple Approximate Interval Algebras

1995GUZMAN-GOMEZ, Marisela (PDE) Regularity Properties of the Davey-Stewartson System for Gravity-CapillaryWavesLI, Liangqing (Operator Algebras) Classification of Simple C*-Algebras: Inductive Limit of Matrix Algebras over1-Dimensional SpaceLOUKANIDIS, Dimitrios (Algebraic Groups) Bounded Generation of Certain Chevalley GroupsMONROY-PEREZ, Felipe (Control Theory) Non-Euclidean Dubins’ Problem: A Control Theoretic ApproachPYKE, Randall (PDE) Time Periodic Solutions of Nonlinear Wave EquationsSTANLEY, Catherine (Algebra) The Decomposition of Automorphisms of Modules over Rings

TIE, Jingzhi (PDE) Analysis on the Heisenberg Group and Applications to the ∂ -Neumann Problem

1996DEPAEPE, Karl (Lie Algebras) Primitive Effective Pairs of Lie AlgebrasJUNQUEIRA, Lucia (Set Theory) Preservation of Topological Properties by Forcing and by ElementarySubmodelsSCHANZ, Ulrich (PDE) On the Evolution of Gravity-Capillary Waves in Three Dimension

1997AKBARY-MAJDABADNO, Amir (Number Theory) Non-Vanishing of Modular L-functions with Large LevelAUSTIN, Peter (Group Theory) Products of Involutions in the Chevalley Groups of Type F 4 (K)

CLOAD, Bruce (Operator Theory) Commutants of Composition OperatorsCUNNINGHAM, Clifton (Automorphic Forms) Characters of Depth-Zero Supercuspidal Representations ofSp 4 (F): From Perverse Sheaves to Shalika Germs

DOOLITTLE, Edward (PDE) A Parametrix for Stable Step Two Hypoelliptic Partial Differential OperatorsFARAH, Ilijas (Set Theory) Analytic Ideals and their QuotientsHOMAYOUNI-BOROOJENI, Soheil (Set Theoretic Topology) Partition Calculus for Topological SpacesSTANLEY, Donald (Algebraic Topology) Closed Model Categories and Monoidal Categories

SUN, Heng (Automorphic Forms) The Residual Spectrum of GL N( ) : The Borel Case

THERIAULT, Stephen (Algebraic Topology) A Reconstruction of Anick’s Fibration S 2 1n− → T 2 1n− (p r ) →

ΩS 2 1n+

1998BALLANTINE, Cristina (Automorphic Forms) Hypergraphs and Automorphic FormsCENTORE, Paul (Differential Geometry) A Mean-Value Laplacian for Finsler SpacesCHEN, Qun (Algebraic Geometry) Hilbert-Kunz Multiplicity of Plane Curves and a Conjecture of K. PardueGRUNBERG ALMEIDA PRADO, Renata (Set-theoretic Topology) Applications of Reflection to TopologyHILL, Peter (Knot Theory) On Double-Torus KnotsLI, Mingchu (Combinatorics) Hamiltonian Properties of Claw-Free GraphsMEZO, Paul (Automorphic Forms) A Global Comparison for General Linear Groups and their MetaplecticCoveringsSTEVENS, Brett (Combinatorics) Transversal Covers and PackingsSTEVENS, Irina (C*-Algebras) Hereditary Subalgebras of AI AlgebrasTRAVES, William (Algebraic Geometry) Differential Operators and Nakai’s Conjecture

1999DEAN, Andrew (C*-Algebras) A Continuous Field of Projectionless C*-Algebras

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FRIIS, Peter (C*-Algebras) Normal Elements with Finite Spectrum in C*-Algebras of Real Rank ZeroGUSTAFSON, Stephen (Mathematical Physics) Some Mathematical Problems in the Ginzburg-Landau Theory ofSuperconductivityMAHMOUDIAN, Kambiz (Number Theory) A Non-Abelian Analogue of the Least Character NonresidueMAHVIDI, Ali (Operator Theory) Invariant Subspaces of Composition OperatorsSCHIPPERS, Eric (Complex Variables) The Calculus of Conformal Metrics and Univalence Criteria forHolomorphic FunctionsYANG, Q. (Lie Algebras) Some Graded Lie Algebra Structures Associated with Lie Algebras and Lie Algebroids

2000CALIN, Ovidiu (Differential Geometry) The Missing Direction and Differential Geometry on HeisenbergManifoldsDERANGO, Alessandro (C*-Algebras) On C*-Algebras Associated with Homeomorphisms of the Unit CircleHIRSCHORN, James (Set Theory) Cohen and Random RealsMADORE, Blair (Ergodic Theory) Rank One Group Actions with Simple Mixing Z SubactionsMARTINEZ-AVENDAÑO, Rubén (Operator Theory) Hankel Operators and GeneralizationsMERKLI, Marco (Mathematical Physics) Positive Commutator Method in Non-Equilibrium Statistical MechanicsMIGHTON, John (Knot Theory) Topics in Ramsey Theory of Sets of Real NumbersMOORE, Justin (Set Theory ) Topics in Ramsey Theory of Sets of Real NumbersRAZAK, Shaloub (C*-Algebras) Classification of Simple Stably Projectionless C*-AlgebrasSCOTT, Jonathan (Algebraic Topology) Algebraic Structure in Loop Space HomologyZHAN, Yi (PDE) Viscosity Solution Theory of Nonlinear Degenerate

2001COLEMAN, James (Nonlinear PDE's) Blowup Phenomena for the Vector Nonlinear Schrödinger EquationIZADI, Farz-Ali (Differential Geometry) Rectification of Circles, Spheres, and Classical GeometriesKERR, David (C*-Algebras) Pressure for Automorphisms of Exact C*-Algebras and a Non-CommutativeVariational PrincipleOLIWA, Chris (Mathematical Physics) Some Mathematical Problems in Inhomogeneous CosmologyPIVATO, Marcus (Mathematical Finance) Analytical Methods for Multivariate Stable Probability DistributionsPOON, Edward (Operator Theory) Frames of Orthogonal ProjectionsSAUNDERS, David (Mathematical Finance) Mathematical Problems in the Theory of Incomplete MarketsSOLTYS-KULINICZ, Michael (Complexity) The Complexity of Derivations of Matrix IdentitiesVASILIJEVIC, Branislav (Mathematical Physics) Mathematical Theory of Tunneling at Positive TemperaturesYUEN, Waikong (Probability) Application of Geometric Bounds to Convergence Rates of Markov Chains and

Markov Processes on úú n

2002HERNANDEZ-PEREZ, Nicholas (Math. Finance) Applications of Descriptive Measures in Risk ManagementKAVEH, Kiumars (Algebraic Geometry) Morse Theory and Euler Characteristic of Sections of SphericalVarietiesMOHAMMADALIKANI, Ramin (Symplectic Geometry) Cohomology Ring of Symplectic ReductionsSOPROUNOV, Ivan (Algebraic Geometry) Parshin's Symbols and Residues, and Newton PolyhedraSOPROUNOVA, Eugenia (Algebraic Geometry) Zeros of Systems of Exponential Sums and TrigonometricPolynomialsTOMS, Andrew (Operator Algebras) On Strongly Performated K0 Groups of Simple C*-AlgebrasVUKSANOVIC, Vojkan (Set Theory) Canonical Equivalence RelationsZIMMERMAN, Jason (Control Theory) The Rolling Stone Problem

2003JONG, Peter (Ergodic Theory) On the Isomorphism Problem of p-EndomorphismsPEREIRA, Rajesh (Operator Theory) Trace Vectors in Matrix AnalysisSTAUBACH, Wolfgang (PDE) Path Integrals, Microlocal Analysis and the Fundamental Solution for HörmanderLaplacians

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TSANG, Kin Wai (Operator Algebras) A Classification of Certain Simple Stably Projectionless C*-Algebras

APPENDIX D: THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

The Fields Institute for Research in Mathematical Sciences was created in November 1991 with major fundingfrom the Province of Ontario, the Natural Sciences and Engineering Research Council of Canada, and McMasterUniversity, the University of Toronto, and the University of Waterloo. In September 1996 it moved from itstemporary location in Waterloo to its permanent site, a new building located at 222 College Street in Toronto, nextto the University of Toronto Bookstore. In addition to the three principal sponsoring universities about twentyuniversities across Canada are affiliated with it.

The mandate of the Fields Institute specifically includes the training of graduate students and this function is givena higher profile than at other similar mathematics research institutes. All major programs run at the institutioncontain graduate courses which students at any university affiliated with the institute may take for credit and theorganizers of major programs are expected to set aside some money to make it possible for graduate students toparticipate in their program.

In 2003-2004 the Fields Institute will have a thematic program in the area of Partial Differential Equations(August 2003 – June 2004). For further information on these programs please visit their website:http://www.fields.utoronto.ca/programs/scientific/