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Course Criteria
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3.00 Credits
Foundations of the modern theory of probability with applications. Probability spaces, random variables, independence, conditioning, expectation, generating functions, and Markov chains.
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3.00 Credits
Advanced concepts in modern probability. Convergence theorems and laws of large numbers. Stationary processes and ergodic theorems. Martingales. Diffusion processes and stochastic integration.
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3.00 Credits
Methods of analysis for hyperbolic, elliptic, and parabolic equations, including characteristic manifolds, distributions, Green's functions, maximum principles and Fourier analysis.
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3.00 Credits
Tools for PDEs and special topics: spherical means, method of descent, subharmonic functions, Hamilton- Jacobi equations, Riemann invariants, conservation laws for linear and non-linear waves.
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3.00 Credits
Naive set theory, topological spaces, product spaces, subspaces, continuous functions, connectedness, compactness, countability, separation axioms, metrization, complete metric spaces, function spaces, and Baire spaces.
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3.00 Credits
Fundamental group, retractions and fixed points, homotopy types, separation theorems, classification of surfaces, Seifert-van Kampen Theorem, classification of covering spaces, and applications to group theory.
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3.00 Credits
Basic definitions and theorems on affine, projective, and quasi-projective varieties.
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3.00 Credits
Local properties of quasi-projective varieties. Divisors and differential forms.
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3.00 Credits
A rigorous treatment of the theory of differential geometry.
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3.00 Credits
Special classes of matrices, canonical forms, matrix and vector norms, localization of eigenvalues, matrix functions, applications.
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