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DepartmentofPureMathematics Graduate Courses
Winter 2009 Course #Course TitleInstructor MeetDays/Time Room PMath 640/440Analytic Number Theory &M. RubinsteinMWF 2:30 3:20MC 2035 Number Theory PMath 651/451Measure and IntegrationK.E. HareMWF 12:30 1:20MC 4041 AMath 431 PMath 664/464Algebraic CurvesR. MoraruMWF 9:30 10:20MC 4041 PMath 667/467Topology S.New MWF10:30  11:20MC 4041 PMath 711Topics in Mathematical Logic:R. MoosaTTH 10:30 11:50MC 5158A Model Theory PMath 744Topics in Number Theory:C.L. StewartTTh 9:00 10:30MC 5046 Padic analysis PMath 810Banach Algebras andK.R. DavidsonTTH 10:30 12:00MC 5046 C*Algebras PMath 833Topics in Harmonic Analysis:N. SpronkTTh 3:00 4:30MC 5045 Semisimple Lie Groups:SL(2,R)
 Students should discuss their course selection with their Supervisor, the Graduate Officer, or the course Professor.  You will require a ”Permission Number” in order to enroll in some courses through QUEST.  Please obtain your Permission Number from Shonn Martin in MC 5064
 A ’Drop/Add’ form is required if you wish to Audit a course.Please see Shonn in MC 5064.
Revised December 23, 2008
Please enroll in your courses by Friday, February 13th, 2009.
PMath 640Analytic Number TheoryM. Rubinstein (held with)PMath 440 Primitive roots, law of quadratic reciprocity, Gaussian sums, the Riemann zeta function, Dirichlet series, primes in arithmetical progressions.
RequiredNo textbook required Textbook: References 1. “AnIntroduction to the theory of numbers”, G.H. Hardly and E.M. Wright, Oxford University Press, (5th ed., 1979) 2. “Introductionto Analytic Number Theory”, T.M. Apostol. SpringerVerlag, 1976. 3. ’Lectureson Elementary Number Theory’ by H. Davenport’s ’Multiplicative Number Theory’, SpringerVerlag, 2000.
References:Theory of Algebraic Numbers”, 2nd Edition, H. Pollard and H.G.1. “The Diamond. CarusMathematical Monographs, No. 9, MAA 1975. 2. “AlgebraicNumber Theory”, 2nd ed.Serge Lang, Graduate Texts in Mathematics, Springer–Verlag, 1994. 3. “AlgebraicNumber Theory”, Ian Stewart and David Tall, John Wiley & Sons, 1979
PMath 651 (held with)PMath 451 AMath 431
Recommended Textbook:
Measure and Integration
Real and Complex Analysis, 3rd ed. Walter Rudin, ISBN: 0071002766, McGrawHill
K.E. Hare
General measures, measurability, Caratheodory extension theorem and construction of measures, p integration theory, convergence theorems,Lspaces, absolute continuity, RadonNikodym theorem, p product measures, Fubini’s theorem, signed measures, Riesz Representation theorems forLand C(X).
PMath 664Algebraic CurvesR. Moraru (held with)PMath 464 An introduction to the geometry of algebraic curves with applications to elliptic curves and computa tional algebraic geometry.Plane curves, affine varieties, the group law on the cubic, and applications.
Required Textbook:
Algebraic Curves Author: William Fulton This outofprint textbook available from Pixel Planet, MC 2018
PMath 667 (held with)PMath 467
S. New
RequiredA Basic Course in Algebraic Topology Textbook:Author: W.A. Massey Publisher: SpringerVerlalg Homotopy of spaces, the fundamental group, the classification of two dimensional manifolds, covering spaces, Euler characteristic, homology groups; applications to the fundamental theorem of algebra, the BorsukUlam theorem, and the ham sandwich theorem.
References 1. “Topology”,J.R. Munkres.PrenticeHall, 2000. 2. “AlgebraicTopology, by Allen Hatcher, published by Cambridge University Press.
PMath 711
Topics in Mathematical Logic Model Theory
Course Title:Topics in Mathematical Logic—Model Theory Course Number:PMATH 711 Time:T Th 10:3011:50 AM Location:MC 5158A Instructor:32453Rahim Moosa, MC 5049, ext.
R. Moosa
While this course is intended as a followup to the Model Theory and Set Theory course (PMath 433/633) that I taught in the Fall of 2008, it is open to all graduate students familiar with some basic first–order logic.
Prerequisites.Here are some topics I will expect people to have seen before taking the course: firstorder languages and structures, Tarski’s truth and satisfaction, theories and their models, the compactness theorem.
Useful background:definable sets, eleFamiliarity with the following additional topics will be useful: mentary equivalence and elementary substructures, quantifier elimination.There are PMath 433/633 lecture notes available that cover the above material.
Objectives.Here are topics I hope to cover:existentially closed models and model completions/companions, model theory of fields, types and saturation, Morley rank andωstability. hope to produce lecture notes along the way.None. However I
Reference.Model theory:an introduction by D. Marker, Springer, New York, 2002.
The first lecture will be held Tuesday, January 13, 2009
PMath 744
TopicsinNumberTheory: Padic analysis
C.L. Stewart
Local fields are fields which are complete with respect to a discrete valuation.After some preliminary work on valuations we intend to discuss the padic numbers and the padic analogue of the complex numbers. Afterthat we plan to turn to the basics of padic analysis together with some applications. A particularly significant application is to the study of height functions.It turns out that the most natural measure for the size or height, of an algebraic number involves its padic size for each prime p.
As Koblitz has remarked:“padic analysis can be of interest to students for several reasons.First of all, in many areas of mathematical research  such as number theory and representation theory padic techniques occupy an important place.More naively, for a student who has just learned calculus, the “brave new world” of nonArchimedean analysis provides an amusing perspective on the world of classical analysis.padic analysis, with a foot in classical analysis and a foot in algebra and number theory, provides a valuable point of view for a student interested in any of those areas.”
Required Textbook:
No textbook required
The first class will be held on
Tuesday, January 6, 2009
PMath 810
Banach Algebras and C*Algebras
K.R. Davidson Office:MC 5080
Background:PM 653 or equivalent background in functional analysis is required.
Course Description:
Banach algebras:functional calculus, Gelfand transform, Jacobson radical; C*algebras: ideals,states, GNS construction; von Neumann algebras:the density theorems; the spectral theorem for normal operators.
An Introduction to Operator Algebrasby Laurent Marcoux, posted online. C*Algebras by Exampleby K.R. Davidson, AMS, 1996.
The first lecture will be held Tuesday, January 6, 2009
PMath 833
Topics in Harmonic Analysis SemisimpleLieGroups:SL(2,R)
Nico Spronk
This is a course on harmonic analysis on locally compact, abelian groups.The course will cover the basic topics of Fourier analysis such as convolution, charac ters, dual group and the Fourier transform.Some of the theorems we will study are: Parseval’s/Plancherel theorem Pontryagin Duality theorem Inversion theorem Bochner’s theorem Other topics could include: Structure of lca groups Interpolation sets and almost periodic functions Introduction to harmonic analysis or compact nonabelian groups n The classical groups,Rand the torus, are important examples of lca groups and will be used to motivate the more abstract ideas we will investigate.
Background:Functional analysis, Lebesgue measure theory.Abstract measure theory would be helpful.
Textbook:Real Analysis, 2nd ed., Bruckner, Bruckner and Thomson paperback. Can be purchased from the Math Copy Centre References:“Fourier Analysis on Groups”, by Walter Rudin. “Abstract harmonic analysis”, by Hewitt and Ross. “An introduction to harmonic analysis”, by Yitzhak Katznelson. “Essays in commutative harmonic analysis”, by C.C. Graham and O.C. McGehee.
The first lecture will be held Tuesday, January 6, 2009