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Course Criteria
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3.00 Credits
For students who wish to study calculus but do not know analytic geometry. Algebra review, graphs and functions, polynomial functions, rational functions, conic sections, systems of equations in two variables, exponential and logarithmic functions, trigonometric functions and trigonometric identities, applications of trigonometry, sequences, series, and limits. Prerequisites: Score of 550 on the mathematics portion of the SAT completed within the last year or the appropriate gade on the General Studies Mathematics Placement Examination. 3 points
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3.00 Credits
A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms. Prerequisites: Math V3007 3 points
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3.00 Credits
Fourier series and integrals, discrete analogues, inversion and Poisson summation formulae, convolution. Heisenberg uncertainty principle. Stress on the application of Fourier analysis to a wide range of disciplines. Prerequisites: three terms of calculus and linear algebra or four terms of calculus. General Education Requirement: Quantitative and Deductive Reasoning (QUA). 3 points
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3.00 Credits
The second term of this course may not be taken without the first. Prerequisite: Math V1102-Math V1202 and MATH V2010, or the equivalent. Groups, homomorphisms, rings, ideals, fields, polynomials, field extensions, Galois theory. General Education Requirement: Quantitative and Deductive Reasoning (QUA). 3 points
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3.00 Credits
Algebraic number fields, unique factorization of ideals in the ring of algebraic integers in the field into prime ideals. Dirichlet unit theorem, finiteness of the class number, ramification. If time permits, p-adic numbers and Dedekind zeta function. Prerequisites: MATH W4041-W4042 or the equivalent. General Education Requirement: Quantitative and Deductive Reasoning (QUA). 3 points
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3.00 Credits
Finite groups acting on finite sets and finite dimensional vector spaces. Group characters. Relations with subgroups and factor groups. Arithmetic properties of character values. Applications to the theory of finite groups: Frobenius groups, Hall subgroups and solvable groups. Characters of the symmetric groups. Spherical functions on finite groups. Prerequisites: Math V2010 and Math W4041 or the equivalent. 3 points
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3.00 Credits
Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem. Prerequisites: Mathematics W4041,W4042 and Mathematics V3007. 3 points
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3.00 Credits
Advanced topics in geometry and topology chosen by the instructor from the following list. Non-Euclidean geometry (e.g., hyperbolic, elliptic, projective), combinatorial topology, algebraic topology, knot theory, braid theory, Morse theory, dynamical systems, foliations, graph theory. Prerequisites: Math W4041 Not offered in 2009-2010. 3 points
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3.00 Credits
Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces. Prerequisites: MATH V1202, MATH V2010, and rudiments of group theory (e.g., MATH W4041). MATH V1208 or W4061 is recommended, but not required. General Education Requirement: Quantitative and Deductive Reasoning (QUA). 3 points
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3.00 Credits
The study of algebraic and geometric properties of knots in R^3, including but not limited to knot projections and Reidemeister's theorm, Seifert surfaces, braids, tangles, knot polynomials, fundamental group of knot complements. Depending on time and student interest, we will discuss more advanced topics like knot concordance, relationship to 3-manifold topology, other algebraic knot invariants. Prerequisites: Math V2010 or equivalent and Math W4041. Recommended: Math W4051 or equivalent. 3 points
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