### Lectures on the theory of maxima and minima of functions of several variables. (Weierstrass theory.)

Prerequisite: MTH Description: This is a comprehensive and rigorous course in the study of real valued functions of one real variable. Topics include sequences of numbers, limits and the Cauchy criterion, continuous functions, differentiation, inverse function theorem, Riemann integration, sequences and series, uniform convergence. This course is a prerequisite for most advanced courses in analysis.

Note: The MTH prerequisite for this course is strictly enforced. Students who have not completed MTH , but who have had an equivalent course, need to obtain a waiver from the director of graduate studies. Prerequisite: MTH Description: This is a rigorous course in the study of analysis in dimensions greater than one. Three basic theorems: the inverse function theorem, the implicit function theorem, and the change of variables theorem in multiple integrals are among the subjects studied in detail.

Topics in this course include continuously differentiable functions, the chain rule, inverse and implicit function theorems, Riemann integration, partitions of unity, change of variables theorem. Sufficient conditions, Hamilton-Jacobi Theory. Basic Existence theorems. Measure, outer measure, measurable sets, including Lebesgue measure.

Measure theoretic modeling and the Borel-Cantelli lemmas. Measurable functions. The Lebesgue integral. Convergence theorems. The relation between the Riemann integral and the Lebesgue integral.

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Substitution ciphers. Hill ciphers. Congruences and modular exponentiation. Digital Encryption Standard. Public key and RSA cryptosystems.

## Lectures on the Theory of Maxima and Minima of Functions of Several Variables. by Harris Hancock

Pseudoprimes and primality testing. Pollard rho method. Basic finite field theory. Discrete log. Digital signatures.

## MAT Mathematics

Newton-Cotes quadrature formulas, Gaussian quadrature and orthogonal polynomials. Romberg quadrature, difference equations, numerical solution of ordinary differential equations, predictor-corrector methods, Runge-Kutta methods. Note: cross-listed with Computer Science Prerequisite: MTH , MTH or concurrent registration Description: Solution of nonlinear equations and simultaneous linear equations, linear least-square approximations.

Chebyshev polynomials, minimax approximations, calculation of eigenvalues and eigenvectors. Function spaces definitions, applications: Fourier series, orthogonal polynomials, finite elements. Integral equations classification, solution methods, domain, range, adjoint, Fredholm alternative, spectral theory. Singular perturbation theory multiple scales analysis, singular perturbation theory for algebraic equations and boundary layer problems, WKB approximation, homogenization theory.

Prerequisite: Differential equations, linear algebra, or consent from instructor Description: Emphasis is on the application of mathematical techniques to help unravel underlying mechanisms involved in various biological processes. Topics will be chosen from a broad range, among the possibilities being reaction kinetic, biological oscillations, population ecology, developmental biology, neurobiology, epidemiology, physiological fluid dynamics, sensory biology, etc.

Prerequisite: MTH or equivalent Description: Mathematical formulation and analysis of models for phenomena in the natural sciences. Includes derivation of relevant differential equations from conservation laws and constitutive relations. Potential topics include diffusion, stationary solutions, traveling waves, linear stability analysis, scaling and dimensional analysis, perturbation methods, variational and phase space methods, kinematics and laws of motion for continuous media.

Examples from areas might include, but are not confined to, biology, fluid dynamics, elasticity, chemistry, astrophysics, geophysics.

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Prerequisite: MTH and consent of instructor Description: Existence and uniqueness of solutions, continuation of solutions, dependence on initial conditions and parameters; linear systems of equations with constant and variable coefficients; autonomous systems, phase space and stability. Prerequisite: Consent of instructor Description: A rigorous study of the wave, heat, and potential equations in two dimensions, focusing on fundamental concepts, methods and properties of solutions.

General properties of second order linear equations in two dimensions, classification, characteristics, well-posed problems and approximation. Solution of the three types of equations by the method of separation of variables and Fourier series. Poisson representation formulas. Nonhomogeneous problems and Green's function.

Formulation and properties of the Tricomi problem. Discussion of a simplied problem in fluid dynamics.

Mathematical models for the valuation of derivative products will be derived and analyzed. Prerequisite: MTH or MTH Description: Describes the mathematical development of both the theoretical and the computational techniques used to analyze financial instruments. Specific topics include utility functions; forwards, futures, and swaps; and modeling of derivatives and rigorous mathematical analysis of the models, both theoretically and computationally. Develops, as needed, the required ideas from partial differential equations and numerical analysis.

Prerequisite: Consent of instructor Description: Introduction to von Neumann's theory of games with applications to optimal strategies, decision theory, and linear programming.

Treats problems, advanced techniques and recent developments in algebra. Note: Can be taken more than once for credit. Treats problems, advanced techniques and recent developments in analysis. Prerequisite: Consent of instructor Description: A topics course.

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## Calculus of variations

Treats problems, advanced techniques and recent developments in applied mathematics. Treats problems, advanced techniques and recent developments in combinatorial analysis. Treats problems, advanced techniques and recent developments in geometry. Treats problems, advanced techniques and recent developments in logic and set theory. Treats problems, advanced techniques and recent developments in number theory.

Treats problems, advanced techniques and recent developments in computational mathematics. Treats problems, advanced techniques and recent developments in topology. Prerequisite: Admission only by consent of the Department Chairman Description: Teaching assignments within the Department will be delegated to all registrants, whose work will be supervised by a member of the department staff.

May be taken more than once for credit, hour-allowance of which will depend upon type and amount of instructional duties. Prerequisite: MTH or equivalent Description: Propositional and predicate logic; consistency and completeness results from each. First order theories, particularly arithmetic. Godel's incompleteness theorem.

Prerequisite: MTH or equivalent Description: Development of Godel-Bernays axioms for set theory, ordinal numbers, ordinal arithmetic, cardinal numbers, cardinal arithmetic, constructible sets, large cardinal axioms. Recent consistency and independence results Godel, Cohen. Prerequisites: MTH or equivalent Description: Basic aspects of monoid theory, group theory, ring theory including algebras , module theory, field theory, and category theory. The following is a representative list of topics which may be covered.

Of course, individual instructors may modify this list. MODULES: exact sequences, projective and injective modules, tensor products, exterior and symmetric algebras over a module, finitely generated modules, torsion, modules over a PID, Jordan and rational canonical forms for matrices, Cayley-Hamilton theorem;. FIELDS: transcendental extensions, separable and inseparable extensions, cyclotomic extensions, Kummer extensions, algebraic closure, finite fields, Galois theory;.

Cauchy theorem, Cauchy integral formula, power series, Laurent series, calculus of residues, analytic continuation, monodromy theorem. Riemann surfaces, theorems of Liouville, Weierstrass and Mittag-Leffler. Riemann mapping theorem. Picard theorems, approximation by rational functions and polynomials. Prerequisite: MTH - MTH or the equivalent Description: General topology: topological spaces, continuous maps, connected spaces, compact spaces.

Homotopy theory: homotopy classes of maps, fundamental groups, Van Kampen's theorem, covering spaces, classification of covering spaces. Elementary manifold theory: tangent vectors, derivative of maps, transversality, Sard's theorem, differential forms, exterior derivative, de Rham cohomology. Singular homology theory: chain complex, relative homology, long exact sequence, excision, Mayer-Vietoris exact sequence. Prerequisite: MTH MTH , MTH MTH or the equivalent Description: Classical number theory, binomial coefficients, combinational problems, prime factorization, arithmetic functions, congruences, residue systems, linear congruences, congruences of higher degree, primitive roots, indices, quadratic reciprocity.

Analytic number theory, primes, elementary estimates on sums of primes and functions of primes, estimates for sums of arithmetic functions. Selberg's theorem, prime number theorem. Prerequisites: and or or or Combinatorics Fall: 9 units A major part of the course concentrates on algebraic methods, which are relevant in the study of error correcting codes, and other areas. Topics covered in depth include permutations and combinations, generating functions, recurrence relations, the principle of inclusion and exclusion, and the Fibonacci sequence and the harmonic series.