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Course Criteria
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3.00 Credits
Staff. A survey of a number of actively-growing areas of mathematics, according to the interests of the students and the instructor. For example, the course might focus on famous unsolved problems, such as the Riemann Hypothesis. Explorations with computer packages for symbolic manipulation. Supervised Study. (C) Staff. Prerequisite(s): Permission of major adviser. Hours and credit to be arranged. Study under the direction of a faculty member. Intended for a limited number of mathematics majors. Geometry-Topology, Differential Geometry. (M) Staff. Prerequisite(s): Math 240/241. Point set topology: metric spaces and topological spaces, compactness, connectedness, continuity, extension theorems, separation axioms, quotient spaces, topologies on function spaces, Tychonoff theorem. Fundamental groups and covering spaces, and related topics. Geometry-Topology, Differential Geometry. (M) Staff. Prerequisite(s): Math 500 or with the permission of the instructor. Review of 2- and 3-dimensional vector calculus, differential geometry of curves and surfaces, Gauss-Bonnet theorem, elementary Riemannian geometry, knot theory, degree theory of maps, transversality.
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3.00 Credits
Staff. Prerequisite(s): Math 240. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513. An introduction to groups, rings, fields and other abstract algebraic systems, elementary Galois Theory, and linear algebra -- a more theoretical course than Math 370.
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3.00 Credits
Staff. Prerequisite(s): Math 502 or with the permission of the instructor. Students who have already received credit for either Math 370, 371, 502 or 503 cannot receive further credit for Math 312 or Math 313/513. Students can receive credit for at most one of Math 312 and Math 313/513. Continuation of Math 502. Graduate Proseminar in Mathematics. (A) Staff. This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material. Graduate Proseminar in Mathematics. (B) Staff. This course focuses on problems from Algebra (especially linear algebra and multilinear algebra) and Analysis (especially multivariable calculus through vector fields, multiple integrals and Stokes theorem). The material is presented through student solving of problems. In addition there will be a selection of advanced topics which will be accessible via this material.
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3.00 Credits
Staff. Prerequisite(s): Math 240/241. Math 200/201 also recommended. Construction of real numbers, the topology of the real line and the foundations of single variable calculus. Notions of convergence for sequences of functions. Basic approximation theorems for continuous functions and rigorous treatment of elementary transcendental functions. The course is intended to teach students how to read and construct rigorous formal proofs. A more theoretical course than Math 360.
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3.00 Credits
Staff. Prerequisite(s): Math 508 or with the permission of the instructor. Linear algebra is also helpful. Continuation of Math 508. The Arzela-Ascoli theorem. Introduction to the topology of metric spaces with an emphasis on higher dimensional Euclidean spaces. The contraction mapping principle. Inverse and implicit function theorems. Rigorous treatment of higher dimensional differential calculus. Introduction to Fourier analysis and asymptotic methods. Advanced Linear Algebra. Staff. Prerequisite(s): Math 114 or 115. Math 512 covers Linear Algebra at the advanced level with a theoretical approach. Students can receive credit for at most one of Math 312 and Math 512. Topics will include: Vector spaces, Basis and dimension, quotients; Linear maps and matrices; Determinants, Dual spaces and maps; Invariant subspaces, Cononical forms; Scalar products: Euclidean, unitary and symplectic spaces; Orthogonal and unitary operators; Tensor products and polylinear maps; Symmetric and skew-symmetric tensors and exterior algebra. (CIS 313, MATH313) Computational Linear Algebra. Staff. A number of important and interesting problems in a wide range of disciplines within computer science are solved by recourse to techniques from linear algebra. The goal of this course will be to introduce students to some of the most important and widely used algorithms in matrix computation and to illustrate how they are actually used in various settings. Motivating applications will include: the solution of systems of linear equations, applications matrix computations to modeling geometric transformations in graphics, applications of the Discrete Fourier Transform and related techniques in digital signal processing, the solution of linear least squares optimization problems and the analysis of systems of linear differential equations. The course will cover the theoretical underpinnings of these problems and the numerical algorithms that are used to perform important matrixcomputations such as Gaussian Elimination, LU Decomposition and Singular Value Decomposition.
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3.00 Credits
Staff. Prerequisite(s): Math 240, Stat 430. This course presents the basic mathematical tools to model financial markets and to make calculations about financial products, especially financial derivatives. Mathematical topics covered: stochastic processes, partial differential equations and their relationship. No background in finance is assumed.
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3.00 Credits
Staff. Corequisite(s): Math 508 or permission of the instructor. Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces.
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3.00 Credits
Staff. Corequisite(s): Math 508 or permission of the instructor. Informal introduction to such subjects as compact operators and Fredholm theory, Banach algebras, harmonic analysis, differential equations, nonlinear functional analysis, and Riemann surfaces.
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3.00 Credits
Staff. Prerequisite(s): Math 241. Introduction to calculus of variations. The topics will include the variation of a functional, the Euler-Lagrange equations, parametric forms, end points, canonical transformations, the principle of least action and conservation laws, the Hamilton-Jacobi equation, the second variation.
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3.00 Credits
Staff. The required background is (1) enough math background to understand proof techniques in real analysis (closed sets, uniform covergence, fourier series, etc.) and (2) some exposure to probability theory at an intuitive level (a course at the level of Ross's probability text or some exposure to probability in a statistics class). After a summary of the necessary results from measure theory, we will learn the probabist's lexicon (random variables, independence, etc.). We will then develop the necessary techniques (Borel Cantelli lemmas, estimates on sums of independent random variables and truncation techniques) to prove the classical laws of large numbers. Next come Fourier techniques and the Central Limit Theorem, followed by combinatorial techniques and the study of random walks. (STAT531) Stochastic Processes. (M) Staff. Topics in Analysis. (M) Staff. Prerequisite(s): Math 360/361 and Math 370; or Math 508/509 and Math 502. Topics may vary but typically will include an introduction to topological linear spaces and Banach spaces, and toHilbert space and the spectral theorem. More advanced topics may include Banach algebras, Fourier analysis, differential equations and nonlinear functional analysis.
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