## Mathematics

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**Description:** Review of numbers, inequalities, functions, analytic geometry; limits, continuity; derivatives and applications, Taylor polynomials; log, exp, and inverse trig functions. Integration, fundamental theorem of calculus substitution, trapezoidal and Simpson's rules. Prerequisites: Pure Mathematics 30 or Mathematics 30-1 or equivalent, and Mathematics 31. Notes: (1) Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

**Description:** Area between curves, techniques of integration. Applications of integration to planar areas and lengths, volumes and masses. First order ordinary differential equations: separable, linear, direction fields, Euler's method, applications. Infinite series, power series, Taylor expansions with remainder terms. Polar coordinates. Rectangular, spherical and cylindrical coordinates in 3-dimensional space. Parametric curves in the plane and space: graphing, arc length, curvature; normal binormal, tangent plane in 3-dimensional space. Volumes and surface areas of rotation. Prerequisite: MATH 100. Notes: (1) Credit can be obtained in at most one of MATH 101, 115, 118, 146, or SCI 100. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

**Description:** Vectors and matrices, solution of linear equations, equations of lines and planes, determinants, matrix algebra, orthogonality and applications (Gram-Schmidt), eigenvalues and eigenvectors and applications, complex numbers. Prerequisite or corequisite MATH 100. Notes: (1) Credit can be obtained in at most one of MATH 102, 120, 125, or 127. (2) Students in all sections of this course will write a common final examination. (3) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

**Description:** Review of analytic geometry. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. Applications of the derivative. Integration. Fundamental Theorem of Calculus. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Students who have taken Mathematics 31 are advised to take MATH 114. See Note (3) at the beginning of the Mathematic course listings. Credit can be obtained in at most one of MATH 100, 113, 114, 117, 144, or SCI 100.

**Description:** Review of analytic geometry. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. Applications of the derivative. Integration. Fundamental Theorem of Calculus. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100.

**Description:** Inverse trigonometric functions. Techniques of integration. Improper integrals. Applications of the definite integral. Introduction to differential equations. Prerequisite: MATH 100, 113, 114, 117, 134, or 144, or equivalent. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 146, or SCI 100.

**Description:** Functions, continuity, and the derivative. Applications of the derivative. Extended limits and L'Hospital's rule. Prerequisites: Pure Mathematics 30 or Mathematics 30-1 or equivalent, and Mathematics 31 or equivalent, or consent of Department. Notes: (1) This course is designed for students with at least 80 percent in Pure Mathematics 30 or Mathematics 30-1 and Mathematics 31. (2) Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100. (3) Engineering students will receive a weight of 4.0 units for this course.

**Description:** Integration and the Fundamental Theorem. Techniques and applications of integration. Derivatives and integrals of the exponential, and trigonometric functions. Introduction to infinite series. Introduction to partial derivatives. Prerequisite: MATH 117 or its equivalent. Students with MATH 100, 113, 114, 134, or 144 will be admitted with consent of Department. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 146, or SCI 100. Engineering students will receive a weight of 4.0 units for this course.

**Description:** Systems of linear equations. Vectors in n-space, vector equations of lines and planes. Matrix algebra, inverses and invertibility. Introduction to linear transformations. Subspaces of n-space. Determinants. Introduction to eigenvalues and eigenvectors. The dot product and orthogonality. Applications in a variety of fields, numerical methods. Prerequisites: Mathematics 30-1. Credit can be obtained in at most one of MATH 102, 120, 125 or 127.

**Description:** Systems of linear equations; vectors in Euclidean n-space; span and linear independence in Euclidean n-space; dot and cross product; orthogonality; lines and planes; matrix arithmetic; determinants; introduction to eigenvectors and eigenvalues; introduction to linear transformations; complex numbers; vector space axioms; subspaces and quotients. Prerequisite: Mathematics 30-1. Note: Credit can be obtained in at most one of Math 102, 120,125, or 127.

**Description:** The derivative as a rate of change. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The definite integral as a summation. Integration. The Fundamental Theorem of Calculus. Applications of the derivative and the definite integral in the context of the life sciences. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Note: Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100.

**Description:** Tangents, velocity, and derivatives. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. The derivative as a rate of change. Area, displacement, and the definite integral. Integration. Fundamental Theorem of Calculus. Applications of the derivative and the definite integral in the context of the physical sciences. Prerequisite: Pure Mathematics 30 or Mathematics 30-1 or equivalent. Credit can be obtained in at most one of MATH 100, 113, 114, 117, 134, 144, or SCI 100.

**Description:** Inverse trigonometric functions. Techniques of integration. Improper integrals. Volumes, arc length, surface area, and work. Introduction to differential equations. Applications in the context of the physical sciences. Prerequisite: MATH 100 or 113 or 114 or 117 or 134 or 144 or equivalent. Note: Credit can be obtained in at most one of MATH 101, 115, 118, 146, or SCI 100.

**Description:** Elementary Number Theory, Numeration Systems, Number Systems and Elementary Probability Theory. Math Fair. Prerequisite: Mathematics 30-1 or 30-2, or consent of Department. Notes: (1) This course is restricted to Elementary Education students. (2) This course cannot be used for credit towards a Science degree.

**Description:** First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) This course may not be taken for credit if credit has already been obtained in any of MATH 205, 334, or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

**Description:** First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to Engineering students and Science students in the following programs: Specialization Physics, Specialization Geophysics, Specialization Computing Science, or Specialization Geography (Meteorology). (2) This course may not be taken for credit if credit has already been obtained in any of MATH 205, 334, or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

**Description:** First-order equations; second-order linear equations: reduction of order, variation of parameters; Laplace transform; linear systems; power series; solution by series; separation of variables for PDEs. Prerequisite or corequisite: MATH 209 or 214. Notes: (1) Open only to Engineering students and Science students in the following programs: Specialization Physics, Specialization Geophysics, Specialization Computing Science, or Specialization Geography (Meteorology). (2) This course may not be taken for credit if credit has already been obtained in any of MATH 205, 334, or 336. (3) Students in all sections of this course will write a common final examination. Non-Engineering students who take this course will receive *3.0.

**Description:** Partial differentiation, derivatives of integrals. Multiple integration using rectangular, cylindrical, and spherical coordinates. Vector Field Theory. Prerequisite: MATH 101. Prerequisite or corequisite: MATH 102. Notes: (1) Students in all sections of this course will write a common final examination. (2) Restricted to Engineering students. Non-Engineering students who take this course will receive *3.0.

**Description:** Infinite Series. Plane curves and polar coordinates. Three dimensional analytic geometry. Partial derivatives. This course may not be taken for credit if credit has already been obtained in MATH 209 or MATH 217. Prerequisite: MATH 101, 115, 118, 146, or SCI 100 or equivalent.

**Description:** First order and second order linear differential equations with constant coefficients. Curves, tangent vectors, arc length, integration in two and three dimensions, polar cylindrical and spherical coordinates, line and surface integrals. Green's divergence and Stokes' theorems. Note: This course may not be taken for credit if credit has already been obtained in MATH 209 or 317. Prerequisite: MATH 214 or equivalent.

**Description:** Axiomatic development of the real number system. Topology of Rn. Sequences, limits and continuity. Multi-variable calculus: differentiation and integration, including integration in spherical and polar coordinates. The differential and the chain rule. Taylor's Formula, maxima and minima. Introduction to vector field theory. Prerequisites: MATH 118 (or MATH 101, 115, 146, or SCI 100 with consent of Department) and any linear algebra course. Engineering students will receive a weight of 4.0 units for this course.

**Description:** A problem-solving approach to discrete mathematics, covering secret codes, public-key codes, error-correcting codes, enumeration, recurrence relations, induction, graph theory, graph algorithms and parallel algorithms. Prerequisite: Any 100-level mathematics course or SCI 100, MATH 120, 125 or 127 recommended.

**Description:** Vector spaces. Inner product spaces. Examples of n-space and the space of continuous functions. Gram-Schmidt process, QR factorization of a matrix and least squares. Linear transformations, change of basis, similarity and diagonalization. Orthogonal diagonalization, quadratic forms. Applications in a variety of fields, numerical methods. Prerequisites: A 100-level linear algebra course, and one of Mathematics 31 or a 100-level calculus course. Note: Credit can be obtained in at most one of MATH 225 or 227.

**Description:** Review of vector space axioms, subspaces and quotients; span; linear independence; Gram-Schmidt process; projections; methods of least squares; linear transformations and their matrix representations with respect to arbitrary bases; change of basis; eigenvectors and eigenvalues; triangularization and diagonalization; canonical forms (Schur, Jordan, spectral theorem). Prerequisite: MATH 127 or equivalent. Students with MATH 102 or 125 will be admitted with consent of Department. Note: Credit can be obtained in at most one of MATH 225 or 227.

**Description:** Integers. Mathematical induction. Equivalence relations. Commutative rings, including the integers mod n, complex numbers and polynomials. The Chinese remainder theorem. Fields and integral domains. Euclidean domains, principal ideal domains and unique factorization. Quotient rings and homomorphisms. Construction of finite fields. Applications such as public domain encryption, Latin squares and designs, polynomial error detecting codes, and/or addition and multiplication of large integers. Prerequisite: Any linear algebra course. Credit for MATH 228 cannot be obtained if credit for MATH 326 has been already obtained.

**Description:** Basic Euclidean geometry, congruence, parallelism, area, and similarity. Sound axiomatic development with emphasis on problem solving. Constructions and loci, inequalities, maxima and minima, circles, isometries, and additional topics. Prerequisites: Any 100-level Mathematics course or SCI 100.

**Description:** Transformation geometry, isometry and homothety, applications in Euclidean geometry; the algebra of transformations, the Classification Theorem, frieze patterns and wall-paper groups. Prerequisite: MATH 241

**Description:** Accumulation and amount functions, effective, nominal, simple, and compound rates, force of interest and discount, simple and general annuities certain, variable annuities and perpetuities, amortization schedules and sinking funds, bonds and other securities, applications, installment loans, depreciation, depletion, capitalized cost. Prerequisite: MATH 101, 115, 118 , 146, or SCI 100 or equivalent. Corequisite: MATH 209 or 214.

**Description:** Reasoning and problem solving in the context of logic, algebra, geometry, and combinatorics. Prerequisite: MATH 160, or consent of Department. Notes: (1) This course is restricted to Elementary Education students. (2) This course cannot be used for credit towards a Science degree.

**Description:** Derivation of the classical partial differential equations of applied mathematics, solutions using separation of variables. Fourier expansions and their applications to boundary value problems. Introduction to Fourier Transforms. Emphasis on building an appropriate mathematical model from a physical problem, solving the mathematical problem, and carefully interpreting the mathematical results in the context of the original physical problem. Prerequisites: MATH 201 and 209 or equivalents. Notes: (1) Open only to students in Engineering, Specialization Physics, and Specialization Geophysics. (2) This course may not be taken for credit if credit has already been obtained in MATH 337.

**Description:** Complex numbers, analytic functions, Cauchy-Riemann equation, Cauchy Theorem, power series and Laurent expansions, residues, inverse Laplace transform. Complex inner product spaces, orthogonal expansions, Gram-Schmidt orthogonalization completeness, Fourier expansions applied to signals, Parseval's relation and Bessel's inequality. Prerequisite: MATH 209. Notes: (1) Restricted to Engineering students. (2) This course may not be taken for credit if credit has already been obtained in MATH 311 or 411.

**Description:** Complex numbers. Complex series. Functions of a complex variable. Cauchy's theorem and contour integration. Residue Theorem and its applications. Prerequisite or corequisite: MATH 209 or 215.

**Description:** Construction of real numbers, Heine-Borel and related theorems, differentiation and Riemann integral of functions, topological concepts in metric spaces, sequences, continuous maps, contraction maps, and applications. Prerequisite: MATH 209 or 215 or equivalent. Note: This course may not be taken for credit if credit has already been obtained in MATH 217.

**Description:** Implicit function theorem. Transformations of multiple integrals. Line integrals, theorems of Green, Gauss and Stokes. Sequences and series of functions. Uniform convergence. Prerequisite: MATH 217.

**Description:** Graphs, paths and cycles, trees, planarity and duality, coloring problems, digraphs, matching problems, matroid theory. Prerequisite: MATH 120 or 125 or equivalent and any 200-level MATH course. MATH 222 recommended.

**Description:** Divisibility, prime numbers, congruences, quadratic residues, quadratic reciprocity, arithmetic functions and diophantine equations; sums of squares. Prerequisites: MATH 228 (or 128 or 223).

**Description:** Hermitian and unitary matrices, spectral theorem. Jordan canonical form. Cayley-Hamilton Theorem. Bilinear forms, positive-definiteness, Sylvester's Law of inertia, geometric lattices. Numerical methods. Application to discrete system evolution, matrix exponentials and differential equations. Prerequisites: MATH 225. Note: May not be taken for credit if credit has already been obtained in MATH 227.

**Description:** Rings, fields, polynomials, algebras. Homomorphisms, ideals, quotients. Ring extensions, field extensions, construction of finite fields. Integral domains: Euclidean, principal ideal, unique factorization. Chain conditions. Introduction to modules. Modules over a principal ideal domain, finitely generated abelian groups, matrix canonical forms. Prerequisite MATH 227, or both MATH 225 and 228.

**Description:** Groups, subgroups, homomorphisms. Symmetry groups. Matrix groups. Permutations, symmetric group, Cayley's Theorem. Group actions. Cosets and Lagrange's Theorem. Normal subgroups, quotient groups, isomorphism theorems. Direct and semidirect products. Finite Abelian groups. Prerequisite: MATH 227 or 228. This course may not be taken for credit if credit has already been obtained in MATH 229.

**Description:** First order equations, linear equations of higher order. Power series solution. Laplace transform methods. Introduction to special functions. Introduction to linear systems. Prerequisites: MATH 125 or 127 and one of MATH 209, 214 or 217. Note: This course may not be taken for credit if credit has already been obtained in MATH 201.

**Description:** Boundary value problems of classical Math Physics, orthogonal expansions, classical special functions. Advanced transform techniques. Note: This course may not be taken for credit if credit has already been obtained in either MATH 300 or equivalent. Prerequisites: MATH 215, and MATH 334 or 336.

**Description:** Combinatorial geometry and topology, convex sets, sets with constant width, Helly-type problems, extremal problems. Prerequisite: MATH 120 or 125 or equivalent, MATH 222 or MATH 241.

**Description:** Projective geometry, Poncelet-Steiner constructions, inverse geometry, Mohr-Mascheroni constructions, Principle of Duality, conic sections. Prerequisite: MATH 241.

**Description:** Frenet-Seret theory of curves in the plane and in 3-space, examples; local theory of surfaces in 3-space: first and second fundamental forms, Gauss map and Gauss curvature, geodesics and parallel transport, theorema egregium, mean curvature and minimal surfaces. Prerequisites: One of MATH 102, 125 or 127 and one of MATH 209, 215 or 217.

**Description:** Simple Market Model: one-step binomial model, basic notions and assumptions. Risk-Free Assets: simple interest, zero-coupon bonds, money market account. Risky Assets: dynamic of stock prices, binomial tree model, trinomial tree model. Discrete time market model: stock and money market model, extended models. Portfolio management: risk, two securities, capital asset pricing model. Prerequisite: MATH 253 and STAT 265 or consent of the Department.

**Description:** Forward and futures contracts: forward and futures prices, hedging with futures. Options: put-call parity, bounds on option prices, time value of options. Option pricing: European and American options in the binomial tree model, Black-Scholes formula. Financial engineering: hedging option positions, hedging business risk. Variable interest rates: maturity-independent yields, general term structure. Stochastic interest rates: arbitrage pricing of bonds, interest rate derivative securities. Prerequisite: MATH 356 or consent of the Department.

**Description:** Model development, computation, and analysis for problems in the life sciences. Models include differential equations, difference equations and stochastic formulations. Model evaluation and prediction. Applications are chosen from epidemiology, ecology, population biology, physiology and medicine. Prerequisites: MATH 102, 120, 125, or 127; MATH 209, 214, or 217. No previous computing experience is needed.

**Description:** This course is designed to develop the students' problem-solving abilities along heuristic lines and to illustrate the processes of Applied Mathematics. Students will be encouraged to recognize and formulate problems in mathematical terms, solve the resulting mathematical problems and interpret the solution in real world terms. Typical problems considered include nonlinear programming, optimization problems, diffusion models. Prerequisites: MATH 102, 120, 125, or 127; MATH 209, 214, or 217.

**Description:** Introduction to optimization. Problem formulation. Linear programming. The simplex method and its variants (revised Simplex method, dual simplex method). Extreme points of polyhedral sets. Theory of linear inequalities (Farkas Lemma). Complementary slackness and duality. Post-optimality analysis. Interior point methods. Applications (elementary games, transportation problems, networks, etc.). Prerequisites: MATH 102, 120, 125, or 127; MATH 209, 214, or 217.

**Description:** Approximation of functions by Taylor series, Newton's formulae, Lagrange and Hermite interpolation. Splines. Orthogonal polynomials and least-squares approximation of functions. Direct and iterative methods for solving linear systems. Methods for solving non-linear equations and systems of non-linear equations. Introduction to computer programming. Prerequisites: MATH 102, 120, 125, or 127; MATH 209, 214, or 217. Note: Credit can be obtained for at most one of MATH 280, 381, CMPUT 340. Note: Extra classes may be held for students lacking a background in one of the major programming languages such as Fortran, C, C++ or Matlab.

**Description:** Required by all students who have just completed a Science Internship Program. Must be completed during the first academic term following return to full-time studies. Note: A grade of F to A+ will be determined by the student's job performance as evaluated by the employer, by the student's performance in the completion of an internship practicum report, and by the student's ability to learn from the experience of the Internship as demonstrated in an oral presentation. Prerequisites: WKEXP 955 and 956.

**Description:** Principles of Monte Carlo methods. Essentials of stochastics. Introduction to financial derivatives pricing. Generating random variables. Simulating stochastic differential equations. Application to financial derivatives pricing and interest rate models. Variance reduction techniques. Prerequisite: STAT 471, or E E 387 and consent of the Department.

**Description:** Complex number system. Analytic functions. Cauchy's Integral theorem and formula. Applications including the maximum modulus principle, Taylor expansion and Laurent expansion. Harmonic functions. The residue theorem with applications; calculus of residues, argument principle, and Rouche's theorem. Basics of analytic continuation. Additional topics at the instructor's discretion such as: Normal families, The Riemann mapping Theorem, Picard's Theorem. Note: This course is primarily for Honors students in Mathematics or Physics. Offered in alternate years. It may be offered in intervening years if demand is sufficient. Prerequisite: MATH 314 or 317.

**Description:** Differentiation of maps in Rn, implicit function and mapping theorems, sequences of functions, Riemann-Stielties integration, additional topics at the discretion of the instructor. Prerequisite: MATH 314. Note: This course may not be taken for credit if credit has already been obtained in MATH 317.

**Description:** Review of probability tools for discrete financial analysis; Conditional probabilities/expectations. Filtrations, adapted and predictable processes. Martingales, submartingales and supermartingales in discrete-time. Doob decomposition for supermartingales. Predictable representation. Discrete- time financial modes: Arbitrage, complete and incomplete markets. Self-financing property, value and gain processes. Valuation of contingent claims. Binomial model: Model specifications, Perfect hedging. Utility functions and consumption/ investment problems. European and American options in discrete time. Futures and forward contracts in discrete time. Transition to the continuous-time framework. Corequisite: STAT 471 or consent of the Department.

**Description:** Brief review of set operations and countable sets. Measure theory, integration theory, Lebesgue measure and integrals on R^n, product measure, Tonelli-Fubini theorem. Functions of bounded variation, absolutely continuous functions. Prerequisite: MATH 317 or 414 or equivalent.

**Description:** Classical Banach spaces. Hahn-Banach, open mapping and closed graphs theorems. Hilbert spaces, orthonormal bases. Elements of spectral theory, spectra of compact operators, spectral theorem for compact self-adjoint operators. Prerequisite: MATH 417. Corequisite: MATH 447.

**Description:** Permutations and combinations, Binomial Theorem, Principle of Inclusion-Exclusion, recurrence relations, generating functions, orthogonal Latin squares, balanced incomplete block designs, Steiner triple systems, perfect difference sets, Boolean algebra and Finite State Machines. Prerequisites: Either (1) MATH 326 or (2) one of MATH 222 or 228 and a 300-level MATH course, MATH 322 recommended.

**Description:** Elements of group theory, cosets, Lagrange's theorem, binary group codes, polynomials, finite field theory, error correcting codes. Prerequisites: MATH 228 (or 223, or 128), any 300-level MATH course.

**Description:** Field extensions. Groups of automorphisms of fields. Galois theory. Finite fields and applications. Solvable groups, the insolvability of the quintic equation. Ruler and compass construction. Prerequisite: MATH 326 (or MATH 228 by consent of the instructor) and MATH 328. Note: (1) This course cannot be taken for credit if credit has already been obtained in MATH 427 or 329.

**Description:** Introduction to commutative algebra, algebraic geometry, and homological algebra. Additional topics at the discretion of the instructor. Prerequisite: MATH 326 or consent of Department.

**Description:** Group actions, Sylow Theory, solvable and nilpotent groups, Galois Theory. Prerequisite: MATH 328 or consent of the Department.

**Description:** Elementary existence and uniqueness theorems. Systems of equations, stability, perturbation theory. Introduction to numerical methods. Introduction to phase plane analysis. Prerequisite: MATH 334 or 336.

**Description:** Partial differential equations as physical models. Introduction to basic generalized functions. Theory of linear and quasi-linear first-order equations: general solution, initial value problem, generalized solutions and propagation of singularities, characteristic surfaces, shock formation. Theory of fully nonlinear first order equations: complete solution and the initial value problem. Hamilton-Jacobi equation and its applications. Second order linear equations in n dimensions: classification, canonical form, characteristic surfaces and shock formation, initial and boundary value problem. Prerequisite: MATH 337.

**Description:** Set Theory, metric spaces and general topology. Compactness, connectedness. Urysohn's Lemma and Tietze's Theorem. Baire Category Theorem. The Tychonoff Theorem. Homotopy and covering spaces. Primarily intended for third and fourth year students with a good background in Mathematics. Prerequisite: MATH 217 or 314. MATH 347 is recommended as a companion course. Offered in alternate years. It may be offered in intervening years if demand is sufficient.

**Description:** Riemannian geometry of n-space, metric tensors, various curvature concepts and their relationships, covariant differentiation, geodesics, parallel transport. Additional topics at the discretion of the instructor. Prerequisites: MATH 348, or MATH 217 and one of MATH 225 or 227. Offered in alternate years. It may be offered in intervening years if demand is sufficient.

**Description:** This course is designed to give credit to mature and able students for reading in areas not covered by courses, under the supervision of a staff member. A student, or group of students, wishing to use this course should find a staff member willing to supervise the proposed reading program. A detailed description of the material to be covered should be submitted to the Chair of the Department Honors Committee. (This should include a description of testing methods to be used.) The program will require the approval of both the Honors Committee, and the Chair of the Department. The students' mastery of the material of the course will be tested by a written or oral examination. This course may be taken in Fall or Winter and may be taken any number of times, subject always to the approval mentioned above. Prerequisite: Any 300-level MATH course.

**Description:** This course provides students in Specialization and Honors programs an opportunity to pursue research in mathematics under the direction of a member of the Department. Course requirements include at least one oral presentation and a written final report. Students interested in taking this course should contact the course coordinator two months in advance. Credit for this course may be obtained more than once. Prerequisites: a 300-level MATH course and consent of the course coordinator.

**Description:** Discrete-time stochastic analysis: Stochastic basis, filtration, stochastic sequences. Absolute continuity of probability measures and conditional expectations. Martingale-like and predictable stochastic sequences. Doob's decomposition. Stopping times and related properties. Uniformly integrable stochastic sequences. Transition from discrete-time to continuous-time stochastic analysis. Introduction to stochastic integration with respect to Brownian motion. Prerequisites: STAT 471 or consent of the Department.

**Description:** A review and some extensions of single variable complex analysis. Complex linearity and holomorphicity in several variables, Hartog's theorem, Weierstrass preparation theorem, Riemann extension theorem, Weierstrass division theorem, analytic Nullstellensatz, implicit and inverse function theorems, complex manifolds and analytic subvarieties, meromorphic maps. Prerequisite: MATH 411.

**Description:** Principles of Monte Carlo methods. Essentials of stochastics. Introduction to financial derivatives pricing. Generating random variables. Simulating stochastic differential equations. Application to financial derivatives pricing and interest rate models. Variance reduction techniques. Prerequisite: STAT 471 or FIN 654 or ECON 598 or consent of the Department. Note: This course may not be taken for credit if credit has already been obtained in MATH 408.

**Description:** Continuous semimartingales and quadratic variation. Stochastic integrals for continuous semimartingales. Ito's formula. Change of probability measure (Girsanov transformation). Martingale representation theorem for Brownian filtrations. Stochastic differential equations, diffusions. Introduction to discontinuous semimartingales with emphasis on Poisson processes. Prerequisites: MATH 505 or consent of the Department.

**Description:** Valuations and their extensions, ramifications; integral dependence, algebraic number fields, ideals and divisors, class number. Prerequisite: MATH 427.

**Description:** Review of probability tools for discrete financial analysis; Conditional probabilities/expectations. Filtrations, adapted and predictable processes. Martingales, submartingales and supermartingales in discrete-time. Doob decomposition for supermartingales. Predictable representation. Discrete-time financial modes: Arbitrage, complete and incomplete markets. Self-financing property, value and gain processes. Valuation of contingent claims. Binomial model: Model specifications, Perfect hedging. Utility functions and consumption/investment problems. European and American options in discrete time. Futures and forward contracts in discrete time. Transition to the continuous-time framework. Prerequisite: STAT 471 or consent of the Department. Note: This course may not be taken for credit if credit has already been obtained in MATH 415.

**Description:** Classical Banach spaces. Hahn-Banach, open mapping and closed graphs theorems. Hilbert spaces, orthonormal bases. Elements of spectral theory, spectra of compact operators, spectral theorem for compact self-adjoint operators. Prerequisite: MATH 417. Corequisite: MATH 447.

**Description:** Locally convex spaces, weak topologies and duality in Banach spaces, weak compactness in Banach spaces, structure of classical Banach spaces, local structures, infinite-dimensional geometry of Banach spaces and applications. Prerequisite: MATH 516. Corequisite: MATH 447 or consent of Department.

**Description:** Banach algebras and spectral theory, compact and Fredholm operators, the spectral theorem for bounded normal operators, operator algebras, representations of C+-algebras, elementary von Neumann algebra theory, and other topics. Prerequisite: MATH 516. Corequisite: MATH 447 or consent of Department.

**Description:** Financial markets in continuous-time: Arbitrage, completeness, self-financing strategies. Black Scholes model. Option pricing and hedging: European, American and exotic options. Consumption-investment problem: Utility maximization, optimal portfolio and optimal consumption. Prerequisite: MATH 515. Corequisite: MATH 510 or consent of the Department.

**Description:** Finite dimensional manifolds/submanifolds; tangent bundle, differential, inverse, and implicit function theorems, partitions of unity; imbeddings, immersions, submersions; vector fields and associated flows; Lie derivative, Lie bracket; tensor analysis, differential forms, orientation, integration, Stokes' theorem; basics of smooth bundle theory, Riemannian metrics; notion of a Lie group with basic examples, smooth Lie group actions, principal bundles. Prerequisite: MATH 446 or 448.

**Description:** Existence theorems, uniqueness theorems; linear systems (basic theory); stability (basic theory); nonlinear systems (local theory); nonlinear systems (global theory); bifurcations. Prerequisite: MATH 334 or equivalent.

**Description:** Asymptotics; boundary value problems; Poincare-Bendixson theory. Additional material will be chosen from among the following topics at the option of the instructor: separation; dichotomies; comparison and oscillation theory; bifurcation theory; nonautonomous systems; dynamical systems; functional differential equations; contingent equations; differential equations in Banach spaces. Prerequisite: MATH 524 or equivalent.

**Description:** Notions; Elliptic PDE's; Parabolic PDE's; Hyperbolic PDE's; Nonlinear Integrable PDE's. Prerequisite: MATH 436 or equivalent; corequisite: MATH 516.

**Description:** Particular background from point set topology (pasting and quotienting constructions); homotopy relation between maps and spaces; fundamental group; Seifert VanKampen theorem; covering spaces. Additional topics at the discretion of the instructor. Prerequisites: MATH 227, 317 and 447 or consent of Department.

**Description:** Direct and iterative methods for solving linear systems, iterative methods for nonlinear systems, polynomial and spline interpolations, least square approximation, numerical differentiation and integration, initial value problems for ODE's (one-step, multistep methods, stiff ODE's). Prerequisite: 400-level MATH course. Students are required to have knowledge of advanced Calculus and introductory knowledge in Analysis and Linear Algebra and some computer programming. Note 1: Restricted to graduate students only. Note 2: May not be taken for credit if credit has already been obtained in MATH 381, 481 or 486 or equivalent.

**Description:** Finite difference and finite element methods for boundary-value problems of elliptic equations. Numerical algorithms for large systems of linear algebraic equations: direct, classical relaxation, multigrid and preconditioned conjugate gradient methods. Algorithms for vector/parallel computers and the domain decomposition method. Prerequisites: MATH 337, 436 or equivalent and some computer programming.

**Description:** Asymptotic analysis of integrals: Laplace, stationary phase, and steepest descent methods. Regular and singular perturbations: trained coordinates, multiple scales, asymptotic matching, renormalization techniques, WKB theory, Hamiltonian perturbation theory, center manifolds and stability. Singularities in differential equations. Applications to algebraic, ordinary and partial differential equations. Prerequisite: MATH 438 or equivalent.

**Description:** Review, theory and extension of Fourier series for square integrable functions; orthonormal systems, Bessel's inequality, completeness, Parseval's identity, Riesz-Fischer Theorem. Extension to Fourier series for functions in other Lebesgue classes; Fejer means, conjugate series, Dirichlet, Fejer and Poisson kernels. Norm convergence; remarks on pointwise convergence. Fourier transforms and series in several dimensions; inverse transform, Plancherel formula, Poisson Formula, maximal functions, Riesz-Thorin Theorem and applications. Elementary distribution theory; D, D', S, S' and some elementary results, Fourier transforms of tempered distributions. Examination of some earlier results with tempered distributions instead of functions and getting familiar with basic concepts. Prerequisite: MATH 418.

**Description:** Abstract measures. Integration. Lp spaces. Radon-Nikodym theorem. Hahn and Lebesgue decomposition theorems. Product measures. Fubini's theorem. Prerequisite: MATH 418.

**Description:** Fundamentals including continuum hypothesis surface tension, classical thermodynamics, and transport phenomena. Introduction to Cartesian tensors. Kinematics of flow including Lagrangian and Eulerian descriptions, streamline, path line, streak line, vorticity and circulation. Derivation of the conservation laws for mass, momentum, and energy and a detailed description of the Boussinesq approximation. Conservation laws in a rotating frame. Vortex lines and tubes, role of viscosity in vortices, Kelvin's circulation theorem, the vorticity equation in nonrotating and rotating frames. Irrotational flow including its relevance, velocity potential, sources and sinks, and flow past various shapes. Gravity waves in deep and shallow water with and without surface tension in both the linear and nonlinear contexts. Dynamic similarity and Buckingham's Pi Theorem. Prerequisites: One of MATH 311, 411 and MATH 436 or consent of Instructor.

**Description:** Mathematical modeling in the biological and medical sciences. Students will learn how to apply mathematical methods and theory to a variety of different biological problems. Topics will be taken from: (i) continuous and discrete dynamical systems describing interacting and structured populations, resource management, biological control, reaction kinetics, biological oscillators and switches, the dynamics of infectious diseases and genetics and (ii) models of spatial processes in biology including random walks, pattern formation in morphogenesis and ecology, applications of traveling waves to population dynamics, epidemiology, chemical reactions, and models for neural patterns. Prerequisites: MATH 524 and a 400 or 500 level course on Partial Differential Equations or consent of Instructor.

**Description:** Group actions, Sylow Theory, solvable and nilpotent groups, Galois Theory. Additional topics at the discretion of the instructor. Prerequisite: MATH 328 or consent of the Department.

**Description:** Introduction to commutative algebra, algebraic geometry, and homological algebra. Additional topics at the discretion of the instructor. Prerequisite: MATH 326 or consent of the Department.

**Description:** Students registered in this course are supervised by individual staff members in areas of interest of the staff members. Students will be allowed to take this course only in exceptional circumstances and with the permission of the Chairman of the Department. This course shall not be counted against the minimum course requirement for graduate students.

**Description:** Credit for this course can be obtained twice.

*No description available for this course.*

*No description available for this course.*

*No description available for this course.*

**Description:** Topics among: Incomplete markets; Models with imperfection (markets with transaction costs, constraints or defaults); Risk measures; Interplay between finance and insurance; Mathematical models for the term structure of interest rates. Prerequisites: MATH 520 or consent of the Department.

**Description:** Prerequisite: MATH 519.

**Description:** Credit for this course may be obtained more than once.

**Description:** Credit for this course may be obtained more than once.

**Description:** Credit for this course may be obtained more than once.

**Description:** Credit for this course may be obtained more than once.

**Description:** Credit for this course may be obtained more than once.

*No description available for this course.*

**Description:** Credit for this seminar course can be obtained more than once.

**Description:** Credit for this course may be obtained more than once.

*No description available for this course.*

*No description available for this course.*

*No description available for this course.*

*No description available for this course.*

*No description available for this course.*

*No description available for this course.*

**Description:** Open only to students taking the MSc non-thesis option in mathematics.

**Description:** Open only to students taking the MSc non-thesis option in mathematics.