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Course Criteria
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0.00 - 4.00 Credits
Riemannian geometry of surfaces. Surfaces in Euclidan space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Bonnet Theorem, uniformization of surfaces
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0.00 - 4.00 Credits
Basic facts about Fourier series, Fourier transforms, and applications to the classical partial differential equations will be covered. Also Fast Fourier Transforms, finite Fourier series, Dirichlet characters, and applications to properties of primes.
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0.00 - 4.00 Credits
Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The Riemann zeta function and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. Proof that every positive integer is the sum of four perfect squares. This course is the second semester of a four-semester sequence, but may be taken independently of the other semesters.
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0.00 - 4.00 Credits
The theory of Lebesgue integration in n-dimensional space. Differentiation theory. Hilbert space theory and applications to Fourier transforms, and partial differential equations. Introduction to fractals.
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0.00 - 4.00 Credits
Introduction to numerical methods with emphasis on algorithms, applications and computer implementation issues. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Direct and iterative methods for solving linear systems. Numerical solutions of differential equations, two-point boundary value problems. Topics in approximation theory. Lectures are supplemented with numerical examples using MATLAB.
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0.00 - 4.00 Credits
The course draws problems from the sciences and engineering for which mathematical models have been developed and analyzed in order to describe, understand and predict natural and man-made phenomena. Topics will range across the physical sciences and biology, including cognitive science and neurobiology. Model building strategies: analytical and computational methods; the manner in which applications motivate mathematical developments.
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0.00 - 4.00 Credits
(1) Wiener measure. (2) Stochastic differential equations. (3) Markov diffusion processes. (4) Linear theory of stationary processes. (5) Ergodicity, mixing, central limit theorem of processes, Gibbs random field. If time permits, the theory of products of random matrices and PDE with random coefficients will be discussed.
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0.00 - 4.00 Credits
Introduction to Algebraic Geometry; no previous knowledge of the topic is assumed. Familiarity with commutative algebra is helpful but will cover the necessary background
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0.00 - 4.00 Credits
Introductory, fast moving course in Analysis. In the first part of the course we will discuss some basic topics of general interest. In th second part, we will discuss some results in the theory of partial differential equations and will introduce topics of current research interest concerning geometric nonlinear PDE's.
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0.00 - 4.00 Credits
An introduction to the representation theory of finite groups, Lie groups and Lie algebras, and applications.
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