An in-depth treatment of multivariable calculus. Extends the material covered in Mathematics 2210. Chain rule, inverse and implicit function theorems, Riemann integration in Euclidean n-space, Gauss-Green-Stokes theorems, applications.

Abstract vector spaces, quotient spaces, linear transformations, dual spaces, determinants. Solvable groups. Field extensions, Galois theory, solvability of equations by radicals.

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.

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).

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.

The complex number system, complex integration and differentiation, conformal mapping, Cauchy’s theorem, calculus of residues.

An introduction to topology. Elementary point set topology: topological spaces, compactness, connectedness, continuity, homeomorphisms, product and quotient spaces. Classification of surfaces and other geometric applications.

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.

This course is a general introduction to Concurrency, i.e., the Mathematical modeling of systems made up of several processes interacting with each other. The process algebra CSP (Communicating Sequential Processes) will be studied, both on the syntactic and semantic level. The denotational, operational, and algebraic models used to reason about the language will be presented, and examples will be used throughout to illustrate the theory.

This course covers a variety of advances in topics in mathematics and exposes students to recent developments not available in other parts of the mathematics curriculum. It meets ini conjunction with graduate level courses MATH 7710-7790 Topics covered will vary from semester to semester. Recent topics offered include Knot Theory and 3-Manifolds, Algebraic Combinatorics, Cardiac Modeling, Number Theory. Students may receive credit for MATH 490 more than once, when the topics covered are distinct. Each section will have the specific topic listed as a subtitle and will have specific prerequisites to the 300 level or above.

No more than four hours of 4910-4920 may be counted toward satisfying the major requirements.

Thesis may serve to satisfy part of the departmental honors requirements.

Mathematics Department, 424 Gibson Hall, New Orleans, LA 70118 504-865-5727 math@math.tulane.edu