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  • MATH 6560: Differential Geometry II

    3.00 Credits
    Tulane University of Louisiana

    Staff. Differentiable manifolds. Vector fields and flows. Tangent bundles. Frobenius theorem. Tensor fields. Differential forms, Lie derivatives. Integration and deRham’s theorem. Riemannian metrics, connections, curvature, parallel translation, geodesics, and submanifolds, including surfaces. First and second variation formulas, Jacobi fields, Lie groups. The Maurer-Cartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.
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  • MATH 6610: Algebra I

    3.00 Credits
    Tulane University of Louisiana

    Staff. Pre-requisite: MATH 3090 and 3110. Vector spaces: matrices, eigenvalues, Jordan canonical form. Elementary number theory: primes, congruences, function, linear Diophantine equations, Pythagorean triples. Group theory: cosets, normal subgroups, homomorphisms, permutation groups, theorems of Lagrange, Cayley, Jordan-Hölder, Sylow. Finite abelian groups, free groups, presentations. Ring theory: prime and maximal ideals, fields of quotients, matrix and Noetherian rings. Fields: algebraic and transcendental extensions, survey of Galois theory. Modules and algebras: exact sequences, projective and injective and free modules, hom and tensor products, group algebras, finite dimensional algebras. Categories: axioms, subobjects, kernels, limits and colimits, functors and adjoint functors.
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  • MATH 6620: Algebra II

    3.00 Credits
    Tulane University of Louisiana

    Staff. Pre-requisite: MATH 3090 and 3110. Vector spaces: matrices, eigenvalues, Jordan canonical form. Elementary number theory: primes, congruences, function, linear Diophantine equations, Pythagorean triples. Group theory: cosets, normal subgroups, homomorphisms, permutation groups, theorems of Lagrange, Cayley, Jordan-Hölder Sylow. Finite abelian groups, free groups, presentations. Ring theory: prime and maximal ideals, fields of quotients, matrix and Noetherian rings. Fields: algebraic and transcendental extensions, survey of Galois theory. Modules and algebras: exact sequences, projective and injective and free modules, hom and tensor products, group algebras, finite dimensional algebras. Categories: axioms, subobjects, kernels, limits and colimits, functors and adjoint functors.
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  • MATH 6650: Differential Equations I

    3.00 Credits
    Tulane University of Louisiana

    Staff. ODE: existence and uniqueness, stability and linearized stability, phase plane analysis, bifurcation and chaos. PDE: heat, wave, and Laplace equations, functional analytic (Sobolev space) and geometric (characteristic) methods. Maximum principle. Introduction to nonlinear PDE’s.
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  • MATH 6660: Differential Equations II

    3.00 Credits
    Tulane University of Louisiana

    Staff. ODE: existence and uniqueness, stability and linearized stability, phase plane analysis, bifurcation and chaos. PDE: heat, wave, and Laplace equations, functional analytic (Sobolev space) and geometric (characteristic) methods. Maximum principle. Introduction to nonlinear PDE’s.
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  • MATH 6710: Analysis I

    3.00 Credits
    Tulane University of Louisiana

    Staff. Pre-requisite: MATH 3050, 3090, and 4060. Lebesgue measure on R. Measurable functions (including Lusin’s and Egoroff’s theorems). The Lebesgue integral. Monotone and dominated convergence theorems. Radon-Nikodym Theorem. Differentiation: bounded variation, absolute continuity, and the fundamental theorem of calculus. Measure spaces and the general Lebesgue integral (including summation and topics in Rn such as the Lebesgue differentiation theorem). Lp spaces and Banach spaces. Hahn-Banach, open mapping, and uniform boundedness theorems. Hilbert space. Representation of linear functionals. Completeness and compactness. Compact operators, integral equations, applications to differential equations, self-adjoint operators, unbounded operators.
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  • MATH 6720: Analysis II

    3.00 Credits
    Tulane University of Louisiana

    Staff. Pre-requisite: MATH 3050, 3090, and 4060. Lebesgue measure on R. Measurable functions (including Lusin’s and Egoroff’s theorems). The Lebesgue integral. Monotone and dominated convergence theorems. Radon-Nikodym Theorem. Differentiation: bounded variation, absolute continuity, and the fundamental theorem of calculus. Measure spaces and the general Lebesgue integral (including summation and topics in Rn such as the Lebesgue differentiation theorem). Lp spaces and Banach spaces. Hahn-Banach, open mapping, and uniform boundedness theorems. Hilbert space. Representation of linear functionals. Completeness and compactness. Compact operators, integral equations, applications to differential equations, self-adjoint operators, unbounded operators.
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  • MATH 6750: Computation I, II

    3.00 Credits
    Tulane University of Louisiana

    Staff. Floating point arithmetic (limitations and pitfalls). Numerical linear algebra, solving linear systems by direct and iterative methods, eigenvalue problems, singular value decompositions, numerical integration, interpolation. Iterative solution of nonlinear equations. Unconstrained optimization. Solution of ODE, both initial and boundary value problems. Numerical PDE. Introduction to fluid dynamics and other areas of application.
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  • MATH 6760: Computation I, II

    3.00 Credits
    Tulane University of Louisiana

    Staff. Floating point arithmetic (limitations and pitfalls). Numerical linear algebra, solving linear systems by direct and iterative methods, eigenvalue problems, singular value decompositions, numerical integration, interpolation. Iterative solution of nonlinear equations. Unconstrained optimization. Solution of ODE, both initial and boundary value problems. Numerical PDE. Introduction to fluid dynamics and other areas of application.
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  • MATH 6810: Applied Mathematics I

    3.00 Credits
    Tulane University of Louisiana

    Staff. Pre-requisite: MATH 3050, 3090, 3470, and 4060. Formulating mathematical models. Introduction to differential equations and integral equations. Fourier series and transforms, Laplace transforms. Generating functions. Dimensional analysis and scaling. Regular and singular perturbations. Asymptotic expansions. Boundary layers. The calculus of variations and optimization theory. Similarity solutions. Difference equations. Stability and bifurcation. Introduction to probability and statistics, and applications. Note: Mathematics 651, 652, 655, 656, 661, 662, 665, 666, 671, 672, 675, 676, 681, 682 are particularly recommended for students planning to do graduate work in mathematics.
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