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MATH W4053: Introduction to Algebraic Topology
3.00 Credits
Columbia University in the City of New York
Prerequisites: MATH V21010, MATH W4041, MATH W4051 The study of topological spaces from algebraic properties, including the essentials of homology and the fundamental group. The Brouwer fixed point theorem. The homology of surfaces. Covering spaces.
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MATH W4053 - Introduction to Algebraic Topology
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MATH W4061: Introduction To Modern Analysis
3.00 Credits
Columbia University in the City of New York
Real numbers, metric spaces, elements of general topology. Continuous and differential functions. Implicit functions. Integration; change of variables. Function spaces.
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MATH W4061 - Introduction To Modern Analysis
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MATH W4065: Honors Complex Variables
3.00 Credits
Columbia University in the City of New York
Prerequisites: MATH V1207 and Math V1208 or MATH W4061. A theoretical introduction to analytic functions. Holomorphic functions, harmonic functions, power series, Cauchy-Riemann equations, Cauchy's integral formula, poles, Laurent series, residue theorem. Other topics as time permits: elliptic functions, the gamma and zeta function, the Riemann mapping theorem, Riemann surfaces, Nevanlinna theory.
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MATH W4065 - Honors Complex Variables
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MATH W4071: Introduction To the Mathematics of Finance
3.00 Credits
Columbia University in the City of New York
The mathematics of finance, principally the problem of pricing of derivative securities, developed using only calculus and basic probability. Topics include mathematical models for financial instruments, Brownian motion, normal and lognormal distributions, the BlackûScholes formula, and binomial models.
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MATH W4071 - Introduction To the Mathematics of Finance
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MATH W4081: Introduction To Differentiable Manifolds
3.00 Credits
Columbia University in the City of New York
The implicit function theorem. Concept of a differentiable manifold. Tangent space and tangent bundle, vector fields, differentiable forms. Stoke's theorem, tensors. Introduction to Lie groups.
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MATH W4081 - Introduction To Differentiable Manifolds
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MATH W4155: Probability Theory
3.00 Credits
Columbia University in the City of New York
Prerequisites: MATH W4061 or MATH V3007. A rigorous introduction to the concepts and methods of mathematical probability starting with basic notions and making use of combinatorial and analytic techniques. Generating functions. Convergence in probability and in distribution. Discrete probability spaces, recurrence and transience of random walks. Infinite models, proof of the law of large numbers and the central limit theorem. Markov chains.
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MATH W4155 - Probability Theory
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MATH W4390: General Relativity
3.00 Credits
Columbia University in the City of New York
Prerequisites: MATH W4081 This course will focus on mathematical aspects of general relativity. It will start with an introduction to the basic notions of Lorentzian geometry and then proceed with a treatment of the Einstein equations. The following topics will be covered: null structure equations, trapped surfaces, Penrose singularity theorem, black holes, Schwarzschild and Kerr spacetimes. No previous training in relativity is required.
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MATH W4390 - General Relativity
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MATH W4391: Quantum Mechanics: An Introduction for Mathematicans and Physicists
3.00 Credits
Columbia University in the City of New York
Prerequisites: MATH V1202 or the equivalent and MATH V2010. This course will focus on quantum mechanics, paying attention to both the underlying mathematical structures as well as their physical motivations and consequences. It is meant for undergraduates with no previous formal training in quantum theory. The measurement problem and issues of non-locality will be stressed.
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MATH W4391 - Quantum Mechanics: An Introduction for Mathematicans and Physicists
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MDES G4000: Theory and Methods in Middle East and Asian Studies
3.00 Credits
Columbia University in the City of New York
Prerequisites: the instructor's permission. Explores recent studies on the Middle East with explicitly stated theoretical orientations that may be grouped under three broad catagories of nationalism, discipline, and power and resistance. Methodologies as diverse as comparative method, post-structuralism, narrative, and ethnography are not investigated in the abstract but in the context of rich empirical case studies.
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MDES G4000 - Theory and Methods in Middle East and Asian Studies
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MDES G4247: Islamicate Culture in its Islamic and Jewish Forms
3.00 Credits
Columbia University in the City of New York
The historian Marshall Hodgson invented the term "Islamicate" to refer to cultural phenomena which do not pertain to the Islamic religion but which have been historically associated with places in which Muslims live. Thus a synagogue built in Egypt might exhibit Islamicate architecture but would have no formal association with Islam itself. In this course we will read some of the great works written by Muslims and Jews in the medieval Islamic world. We will examine what features of these works made them appealing across religious boundaries. We will explore what makes a work Islamicate and in what ways these features were considered by these authors to be separate from Islam itself. Thus, for example, we will investigate how the works of the Jewish philosopher Maimonides can be Islamicate, but not Islamic and how this made it possible for them to be read and enjoyed by Muslim audiences. All texts will be provided in English translation.
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MDES G4247 - Islamicate Culture in its Islamic and Jewish Forms
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