Graduate Students outside of the Mathematics Department normally enroll in 6000-level courses.
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Graduate course descriptions
1000 level| 2000 level | 3000 level | 4000 level | Graduate courses
Math
6020/7020
Mathematical Statistics (3)
Prerequisites: Math 2210, 3070, 3080. Thorough review of key
distributions for probability and statistics, including the
multivariate calculus needed to develop them. Full derivativation of
sampling distribution. Classical principles of inference including best
tests and estimations. Methods of finding tests and estimators.
Introduction to Bayesian estimators.
Math 6030/7030
Stochastic Processes (3)
Prerequisite: Math 3070/6070, 3080/6080 Markov processes, Poisson
processes, queueing models, introduction to Brownian motion.
Math
6040/7260 Linear Models (3)
Prerequisite: Math 3070/6070, 3080/6080. Corequisite: Math 3090 or
approval of instructor. Review of linear algebra pertinent to
least squares regression. Review of multivariate normal, chi-square,
t, F distributions. Classical theory of linear regression and related
inference. Regression diagnostics. Extensive practice in data analysis.
Math 6050/3050 Real Analysis I (3)
Prerequisite: Math 2210. Introduction to analysis. Real numbers, limits, continuity, uniform continuity, sequences and series, compactness, convergence, Riemann integration. An in-depth treatment of the concepts underlying calculus.
Math 6090/3090 Linear Algebra (4)
Prerequisite: Math 2210. An introduction to linear algebra emphasizing matrices and their applications. Gaussian elimination, determinants, vector spaces and linear transformations, orthogonality and projections, eigenvector problems, diagonalizability, Spectral Theorem, quadratic forms, applications. MATLAB is used as a computational tool.
Math 6110/3110
Abstract Algebra (3)
Prerequisite: Math 2210 An introduction to abstract
algebra. Elementary number theory and congruences. Basic
group theory: groups, subgroups, normality, quotient groups,
permutation groups. Ring theory: polynomial rings, unique
factorization domains, elementary ideal theory. Introduction to
field theory.
Math 6210/4210 Differential Geometry (3)
Prerequisites: Math 6050 and 6090. Theory of plane and space curves including arc length, curvature, torsion, Frenet equations, surfaces in three-dimensional space. First and second fundamental forms, Gaussian and mean curvature, differentiable mappings of surfaces, curves on a surface, special surfaces.
Math 6240/4240 Ordinary
Differential Equations (3)
Prerequisites: Math 3090/6090. Review of linear algebra,
first-order equations (models, existence, uniqueness, Euler method,
phase line, stability of equilibria), higher-order linear equations,
Laplace transforms and applications, power series of solutions, linear
first-order, systems (autonomous systems, phase plane), application of
matrix normal forms, linearization and stability of nonlinear systems,
bifurcation, Hopf bifurcation, limit cycles, Poincare-Bendixson
theorem, partial differential equations (symmetric boundary-value
problems on an interval, eigenvalue problems, eigenfunction expansion,
initial-value problems in 1D).
Math 6250/4250 Mathematical Foundations of Computer Security (3)
Prerequisites: Calculus, Math 2170 and Math 3110 or the permission of the instructor.
This course studies the mathematics underlying computer security, including both public key and symmetric key cryptography, crypto-protocols and information flow. The course includes a study of the RSA encryption scheme, stream and clock ciphers, digital signatures and authentication. It also considers semantic security and an analysis of secure information flow.
Math 6280/3280 Information Theory (3,3)
Prerequisites: Math 3050 or 3090and familiarity with discrete probability. This introduction to information theory will address fundamental concepts, such as information, entropy, relative entropy, and mutual information. In addition to giving precise definitions of these concepts, the course will include a probabilistic approach based on equipartitions. Many of the applications of information will be discussed, including Shannon's basic theorems on channel capacity and related coding theorems. In addition to channels and channel capacity, the course will discuss applications of information theory to mathematics, statistics ,and computer science.
Math 6310/3310 Scientific Computing (3)
Prerequisites: Math 2210, 2240, and Computer Science 1010 or equivalent. Errors. Curve fitting and function approximation, least squares approximation, orthogonal polynomials, trigonometric polynomial approximation. Direct methods for linear equations. Iterative methods for nonlinear equations and systems of nonlinear equations. Interpolation by polynomials and piecewise polynomials. Numerical integration. Single-step and multi-step methods for initial-value problems for ordinary differential equations, variable step size. Current algorithms and software.
Math
6350
Optimization (3)
Prerequisite: Math 3090 or equivalent. Constrained and
unconstrained non-linear optimization; Linear programming,
combinatorial optimization as time allows. Emphasis is on realistic
problems whose solution requires computers, using Maple or Mathematica.
Math
6470/4470
Analytical Methods of Applied Mathematics (3)
Prerequisites: Math 2210 and 2240. Derivations of
transport, heat/reaction-diffusion, wave. Poisson's equations;
well-posedness; characteristics for first order PDE's; D'Alembert
formula and conservation of energy for wave equations; propagation of
waves; Fourier transforms; heat kernel, smoothing effect; maximum
principles; Fourier series and Sturm-Liouville eigenexpansions; method
of separation of variables; frequencies of wave equations, stable and
unstable modes, long-time behavior of heat equations; delta function;
fundamental solution of Laplace equation, Newton potential; Green's
function and Poisson formula; Dirichlet Principle.
Math 6550/7510, 6560/7520 Differential Geometry I, II (3, 3)
Differential 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
Math
7010/6510,
7020/6520 Topology I and II (3, 3)
Prerequisites: Math 3050 and 4060. Point set topology.
Connectedness, product and quotient spaces, separation properties,
metric spaces. Classification of compact connected surfaces. Homotopy.
Fundamental group and covering spaces. Singular and simplicial
homology. Eilenberg-Steenrod axioms. Computational techniques,
including long exact sequences. Mayer-Vietoris sequences, excision, and
cellular chain complexes. Introduction to singular cohomology.
Math
7110/6610,
7120 /6620 Algebra I and II (3, 3)
Prerequisites: 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.
Math
7210/6710,
7220/6720 Analysis I and II (3, 3)
Prerequisites: 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
such as the Lebesgue differentiation theorem).
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.
Math
7240
Mathematical Statistics (3)
Prerequisites: Math 6070, 6080and 7210 or permission of the instructor. Consists of Math 6020 and additional meetings and readings to cover advanced limit theorems and foundations of mathematical statistics.
Math 7260/6040 Linear Models (3)
Prerequisite: Math 6070, 6080. Corequisite: Math 6090 or approval of instructor. Review of linear algebra pertinent to least squares regression. Review of multivariate normal, chi-square, t, F distributions. Classical theory of linear regression and related inference. Regression diagnostics. Extensive practice in data analysis.
Math
7310, 7320
Applied Mathematics I, II (3, 3)
This is a first year graduate course in Applied Mathematics. A solid
working knowledge of linear algebra and advanced calculus is the
necessary background for this class. The topics covered
include a mix of analytical and numerical methods that are used to
understand models described by differential equations. We will
emphasize applications from science and engineering, as they are the
driving force behind each of the topics addressed.
Math 7350 Scientific Computing I (3)
Prerequisites: MATH 3310 or MATH 7310-7320. Introduction to numerical analysis: well-posedness and condition number, stability and convergence of numerical methods, a priori and a-posteriori analysis, sources of error in computational models, machine representation of numbers. Linear operators on normed spaces. Root finding for nonlinear equations. Polynomial interpolation. Numerical integration. Orthogonal polynomials in approximation theory. Numerical solution of ordinary differential equations. Detailed syllabus
Math 7360
Data Analysis (3)
Prerequisites: Math 6070, 6080 and 6040/7260 (or equivalent
background in mathematical statistics and linear models).This course covers the
statistical analysis of datasets using the R
software package. The R
environment, an Open Source system based on the S language, is
one of the most versatile and powerful tools available for statistical
data analysis, and is widely used in both academic and industrial
research. Key topics include graphical methods, generalized
linear models, clustering, classification, time series analysis, and
spatial statistics. No prior knowledge of R
is required.
Math 7510/6550, 7520/6560 Differential Geometry I, II (3, 3)
Differentiable manifolds. Vector fields and flows. Tangent bundles. Frobenius theorem. Tensor fields. Differentiable 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
Math 7530, 7540 Partial Differential
Equations I and II (3, 3)
Prerequisites: MATH 3050, 4060, 4470/6470/7310, 7210 and 7220
or by instructor’s approval. Classical weak and strong
maximum principles for 2nd order elliptic and parabolic equations, Hopf
boundary point lemma, and their applications. Sobolev spaces, weak
derivatives, approximation, density theorem, Sobolev
inequalities, Kondrachov compact imbedding. 
theory for second order
elliptic equations, existence via Lax-Milgram Theorem, Fredholm
alternative, a brief introduction to estimates, Harnack inequality,
eigenexpansion. theory for second order parabolic and hyperbolic
equations, existence via Galerkin method, uniqueness and regularity via
energy method. Semigroup theory applied to second order parabolic and
hyperbolic equations. A brief introduction to elliptic and parabolic
regularity theory, the and Schauder estimates. Nonlinear
elliptic equations, variational methods, method of upper and lower
solutions, fixed point method, bifurcation method. Nonlinear parabolic
equations, global existence, stability of steady states, traveling wave
solutions. Conservation laws, Rankine-Hugoniot jump condition,
uniqueness issue, entropy condition, Riemann problem for Burger's
equation, p-systems.
Math 7550, 7560 Probability and Statistics I and II (3)
Various types of convergence, independent incraments, stable laws, central limit problem. Central limit theorems, x^2 distribution, contingency tables. Sampling distributions for normal populations (t, x^2, F). Estimation of parameters: minimum variance, maximum likelihood, sufficiency, nonparametric estimation. Hypothesis testing: Neyman-Pearson lemmas, general linear models, analysis of variances and covariance, regression. Introdution to time series, sampling design, and Bayesian theory.
Math 7570 Scientific Computing II (3)
Prerequisites: MATH 7350. Flotaing point arithmetic (limitations and pitfalls). Numerical linear algebra, solving linear systems by direct and iterative models, eigenvalue problems, singular value decompositions, numerical integration, interpolation, iterative solution of nonlinear equations, unconstrained optimization.
Math 7580 Scientific Computing III (3)
Prerequisites: MATH 7350 and 7570. Solution of ODE, both initial and boundary value problems. Numerical PDE. Introduction to fluid dynamics and other areas of application. Detailed syllabus
Math
7710 - 7790
Special Topics (3)
Prerequisites: defined by the instructor. Courses on
special topics list of subject titles include: Algebra, Analysis,
Applied Math, Computation, Differential Equations, Geometry, Probability
and Statistics, Theoretical Computer Science and Topology offered every
year. Each course is designed
to cover advanced material not included in one of the
regular courses listed above.
| Mathematics
Department Tulane University 6823 St. Charles Ave New Orleans, LA 70118 phone: (504) 865-5727 fax: (504) 865-5063 |
Last Updated:
April 6, 2010
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