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Course Criteria
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3.00 Credits
Staff. Corequisite(s): Math 600/601. Complex numbers, analytic functions, Cauchy's theorem and consequences, isolated singularities, analytic continuation, open mapping theorem, infinite series and products, harmonic and subharmonic functions, maximum principle, fractorial linear transformations, geometric and local properties of analytic functions, Weierstrauss Theorem, normal families, residues, conformal mapping, Riemann mapping theorem, branch points, second order linear O.D.E.'s. Algebraic Topology, Part I. (A) Staff. Prerequisite(s): Math 600/601 or with the permission of the instructor. Homotopy groups, Hurewicz theorem, Whitehead theorem, spectral sequences. Classification of vector bundles and fiber bundles. Characteristic classes and obstruction theory. Algebraic Topology, Part I. (B) Staff. Prerequisite(s): Math 618 or with the permission of the instructor. Rational homotopy theory, cobordism, K-theory, Morse theory and the h-corbodism theorem. Surgery theory.
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3.00 Credits
Staff. Corequisite(s): Math 600/601. Lebesgue measure and integral, Borel measures, convergence theorems. Banach spaces, Hahn-Banach Theorem, Lpspaces, Riesz-Fischer theorem, Stone-Weierstrass theorem, Radon-Nikodym theorem. Applications to Fourier series and integrals, Plancherel Theorem, Distributions, convolutions and mollifiers. Partitions of unity. Applications to P.D.E.'s
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3.00 Credits
Staff. Corequisite(s): Math 502 or permission of the instructor. Informal introduction to such subjects as homological algebra, number theory, and algebraic geometry.
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3.00 Credits
Staff. Corequisite(s): Math 500 or permission of the instructor. Informal introduction to such subjects as homology and homotopy theory, classical differential geometry, dynamical systems, and knot theory.
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3.00 Credits
Staff. Introduction to concepts and methods of calculus for students with little or no previous calculus experience. Polynomial and elementary transcendental functions and their applications, derivatives, extremum problems, curve- sketching, approximations; integrals and the fundamental theorem of calculus.
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3.00 Credits
Staff. Brief review of High School calculus, applications of integrals, transcendental functions, methods of integration, infinite series, Taylor's theorem. Use of symbolic manipulation and graphics software in calculus.
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3.00 Credits
Staff. Prerequisite(s): Math 104. Functions of several variables, vector-valued functions, partial derivatives and applications, double and triple integrals, conic sections, polar coordinates, vectors and analytic geometry, first and second order ordinary differential equations. Applications to physical sciences. Use of symbolic manipulation and graphics software in calculus.
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3.00 Credits
Staff. Prerequisite(s): Math 104. Functions of several variables, partial derivatives, multiple integrals, differential equations; introduction to linear algebra and matrices with applications to linear programming and Markov processes. Elements of probability and statistics. Applications to social and biological sciences. Use of symbolic manipulation and graphics software in calculus.
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3.00 Credits
Staff. This course allows Penn students to teach a series of hands-on activities to students in math classes at University City High School. The semester starts with an introduction to successful approaches for teaching math in urban high schools. The rest of the semester will be devoted to a series of weekly hands-on activities designed to teach fundamental aspects of geometry. The first class meeting of each week, Penn faculty teach Penn students the relevant mathematical background and techniques for a hands-on activity. During the second session of each week, Penn students will teach the hands-on activity to a small group of UCHS students. The Penn students will also have an opportunity to develop their own activity and to implement it with the UCHS students.
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3.00 Credits
Natural Science & Mathematics Sector. Class of 2010 and beyond. Staff. May also be counted toward the General Requirement in Natural Science & Mathematics. Topics from among the following: logic, sets, calculus, probability, history and philosophy of mathematics, game theory, geometry, and their relevance to contemporary science and society.
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