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MATH V3050: Discrete Time Models In Finance
3.00 Credits
Columbia University in the City of New York
Elementary discrete time methods for pricing financial instruments, such as options. Notions of arbitrage, risk-neutral valuation, hedging, term-structure of interest rates.
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MATH V3050 - Discrete Time Models In Finance
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MATH V3386: Differential Geometry
3.00 Credits
Columbia University in the City of New York
Prerequisites: MATH V1202 or the equivalent. Local and global differential geometry of submanifolds of Euclidiean 3-space. Frenet formulas for curves. Various types of curvatures for curves and surfaces and their relations. The Gauss-Bonnet theorem.
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MATH V3386 - Differential Geometry
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MATH W4007: Analytic Number Theory
3.00 Credits
Columbia University in the City of New York
Prerequisites: Math V3007 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.
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MATH W4007 - Analytic Number Theory
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MATH W4032: Fourier Analysis
3.00 Credits
Columbia University in the City of New York
Prerequisites: three terms of calculus and linear algebra or four terms of calculus. 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.
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MATH W4032 - Fourier Analysis
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MATH W4041: Introduction To Modern Algebra
3.00 Credits
Columbia University in the City of New York
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.
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MATH W4041 - Introduction To Modern Algebra
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MATH W4043: Advanced Topics In Algebra: Algebraic Number Theory
3.00 Credits
Columbia University in the City of New York
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.
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MATH W4043 - Advanced Topics In Algebra: Algebraic Number Theory
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MATH W4044: Representations of Finite Groups
3.00 Credits
Columbia University in the City of New York
Prerequisites: Math V2010 and Math W4041 or the equivalent. 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.
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MATH W4044 - Representations of Finite Groups
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MATH W4045: Algebraic Curves
3.00 Credits
Columbia University in the City of New York
Prerequisites: Mathematics W4041,W4042 and Mathematics V3007. Plane curves, affine and projective varieties, singularities, normalization, Riemann surfaces, divisors, linear systems, Riemann-Roch theorem.
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MATH W4045 - Algebraic Curves
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MATH W4051: Topology
3.00 Credits
Columbia University in the City of New York
Metric spaces, continuity, compactness, quotient spaces. The fundamental group of topological space. Examples from knot theory and surfaces. Covering spaces.
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MATH W4051 - Topology
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MATH W4052: Introduction to Knot Theory
3.00 Credits
Columbia University in the City of New York
Prerequisites: Math V2010 or equivalent, Math W4041 and Math W4051. 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.
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MATH W4052 - Introduction to Knot Theory
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