This page is a listing of graduate course descriptions. Course descriptions are subject to change.

Prerequisite: MTH 141-MTH 142 or equivalent

Description: A first course in probability. Introduces the basic
concepts of probability theory and addresses many concrete
problems. A list of basic concepts includes axioms of probability,
conditional probability, independence, random variables (continuous
and discrete), distribution functions, expectation, variance, joint
distribution functions, limit theorems.

Description: Topics include: review of probability, conditional probability, Bayes' Theorem; random variables and distributions; expectation and properties; covariance, correlation, and conditional expectation; special distributions; Central Limit Theorem and applications; estimations, including Bayes; estimators, maximum likelihood estimators, and their properties. Includes use of sufficient statistics to 'improve' estimators, distribution of estimators, unbiasedness, hypothesis testing, linear statistical models, and statistical inference from the Bayesian point of view.

Prerequisite: MTH
313 and consent of instructor

Description: Informal and formal development of propositional
calculus, predicate calculus and predicate calculus with equality.
Completeness theorem and some consequences. Additional reading on
selected topics.

Prerequisite: MTH 513 or consent of instructor

Description: Godel's incompleteness theorem, decidability and
recursiveness. Consistency problems. Additional reading on selected
topics.

Prerequisite: MTH
306 or equivalent

Description: Survey of functions of several variables,
differentiation, composite and implicit functions, maxima and
minima, differentiation under the integral sign, line integrals,
Green's theorem. Vector field theory: gradient, divergence and
curl, divergence theorem. Stokes' theorem, applications. Review of
general theory of sequences and series. Additional reading on
selected topics.

Prerequisite: MTH 141-MTH 142 or equivalent

Description: A first course in probability. Introduces the basic
concepts of probability theory and addresses many concrete
problems. A list of basic concepts includes axioms of probability,
conditional probability, independence, random variables (continuous
and discrete), distribution functions, expectation, variance, joint
distribution functions, limit theorems.

Prerequisite: MTH 420 or consent of instructor

Description: Definitions and elementary properties of groups and
fields, vector space, linear space, linear dependence, dimension,
vector space homomorphisms, kernel and cokernel of a vector space
homomorphism. Application to linear equations, duality. Rings and
ideals. Quotient rings. Integral domains, field of fractions.
Polynomial rings. Principal ideal rings, unique factorization,
lemma of Gauss. Eisenstein criterion of irreducible polynomials.
(Example of irreducible polynomials.) Extension of commutative
fields, finite multiplicative subgroup of a field is cyclic
characteristic of a field. Roots of unity. Applications to
elementary number theory. (Wilson's theorem, Fermat's theorem,
etc.) Additional reading on selected topics.

Prerequisite: MTH 309, with MTH 311 recommended; or consent
of instructor

Description: Topics in advanced linear algebra.

Prerequisite: Consent of instructor

Description: Partial differential equations of physics, separation
of variables and superposition of solutions; orthonormal sets.
Fourier series. Fourier transforms; application to boundary value
problems. Additional reading on selected topics.

Prerequisite: MTH
432 or consent of instructor

Description: The notion of analyticity. Calculus over the complex
numbers. Cauchy's theorems, residues, singularities, conformal
mapping. Weierstrass convergence theorem, analytic continuation.
Additional reading on selected topics.

Prerequisite: MTH 431 or equivalent and consent of
instructor

Description: Elementary set theory, functions and relations,
partially ordered sets. Zorn's Lemma, abstract topological spaces,
semi-metric and metric spaces, bases and subbases, convergence,
filters and nets, separation axioms, continuity and homeomorphisms,
connectedness, separability, compactness. Additional reading on
selected topics

Prerequisite: MTH
419 and consent of instructor

Description: The Euclidean Algorithm and unique factorization,
arithmetical functions, congruences, reduced residue systems,
primitive rotos, magic squares, certain diophantine equations.
Additional reading on selected topics.

Prerequisite: MTH
420 or consent of instructor

Description: Irrational numbers, continued fractions from a
geometric viewpoint, best rational approximations to real numbers,
the Fermat-Pell equation, quadratic fields and integers,
applications to diophantine equations. Additional reading on
selected topics.

Prerequisite: MTH
311

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 311 prerequisite for this course is strictly enforced. Students who have not completed MTH 311, but who have had an equivalent course, need to obtain a waiver from the director of graduate studies.

Prerequisite: MTH 431

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.

Prerequisite: MTH
432 or equivalent

Description: Necessary conditions in the calculus of variations.
Sufficient conditions, Hamilton-Jacobi Theory. Basic Existence
theorems

Prerequisite: MTH
432 or consent of instructor

Description: Bernoulli sequences, measure zero, Strong Law of Large
Numbers for Bernoulli sequences. 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. Fubini’s Theorem.

Prerequisite: MTH 419 or MTH 429 or consent of
instructor

Description: Cryptosystem definitions and basic types of attack.
Substitution ciphers. Hill ciphers. Congruences and modular
exponentiation. Digital Encryption Standard. Public key and RSA
cryptosystems. Pseudoprimes and primality testing. Pollard rho
method. Basic finite field theory. Discrete log. Digital
signatures.

Prerequisite: MTH 145, MTH 241 and MTH 306

Description: Lagrangian interpolation. 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. Additional reading on selected topics. **Note:
cross-listed with Computer Science 537**

Prerequisite: MTH 241, MTH 537 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. Additional reading on selected topics. **Note:
cross-listed with Computer Science 53.**

Prerequisite: MTH306 , MTH309 , MTH 418 or
consent of instructor

Description: Vector spaces and linear systems (linear vector
spaces, basis vectors, spectral theory, adjoint matrices,
eigenvalue problem, Fredholm alternative theorem, least squares
solutions, singular value decomposition). Function spaces
(definitions, applications: Fourier series, orthogonal polynomials,
finite elements). Integral equations (classification, solution
methods, domain, range, adjoint, Fredholm alternative, spectral
theory). Differential equations and Green's functions (delta
functions, Green’s functions , distribution theory, weak
solutions, construction of Green’s functions, spectral theory
of differential operators, adjoint, Fredholm alternative, Sturm
Liouville boundary value problems and solution by eigenfunction
expansions). Calculus of variations (Euler-Lagrange equations,
Hamilton’s principle, minimization of functions and relation
to eigenvalues of Sturm-Liouville operators).

Prerequisite: MTH306, MTH309, MTH 418 or consent of
instructor

Description: Transform theory for linear operators (Fourier
transforms, Laplace transforms, Hankel transforms). Partial
differential equations (theory of distributions, fundamental
solutions to Laplace, wave and heat equations, construction of
Green’s functions using method of images, partial transforms,
complete transforms, eigenfunction expansions).
Asymptotic evaluation of integrals (integration by
parts, Laplace’s method, Watson’s lemma, method of
steepest descents, method of stationary phase). Regular
perturbation theory (applications, method of strained
coordinates, eigenvalues of nonlinear boundary-value problems,
stationary and Hopf bifurcations). 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
306 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.

Prerequisite: MTH 543

Prerequisite: MTH
431 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. Additional reading
on selected topics.

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. Additional reading on selected topics.

Prerequisite: MTH 241, MTH 309, MTH 306

Description: This course will introduce the mathematical theory and
computation of modern financial products used in the banking and
corporate world. Mathematical models for the valuation of
derivative products will be derived and analyzed.

Prerequisite: MTH 458 or MTH 558

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. Additional reading on selected topics.

Prerequisite: MTH 420 and consent of instructor

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

Prerequisite: MTH
432 and consent of instructor

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

Prerequisite: Consent of instructor

Description: A topics course. Treats problems, advanced techniques
and recent developments in applied mathematics.

Prerequisite: Consent of instructor

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

Prerequisite: MTH 309, MTH 419 or equivalent and consent of
instructor

Description: A topics course. Treats problems, advanced techniques
and recent developments in geometry. Note: Can be taken more than
once for credit.

Prerequisite: Consent of instructor

Description: A topics course. Treats problems, advanced techniques
and recent developments in logic and set theory. Note: Can be taken
more than once for credit.

Prerequisite: Consent of instructor

Description: A topics course. Treats problems, advanced techniques
and recent developments in number theory. Note: Can be taken more
than once for credit.

Prerequisite: Consent of instructor

Description: A topics course. Treats problems, advanced techniques
and recent developments in computational mathematics.Note: Can be
taken more than once for credit.

Prerequisite: Consent of instructor

Description: A topics course. Treats problems, advanced techniques
and recent developments in topology. Note: Can be taken more than
once for credit.

Description: Effective Fall 2006, MTH 570 is the number for the topics course previously known as MTH 595

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
313 or equivalent

Description: Propositional and predicate logic; consistency and
completeness results from each. First order theories, particularly
arithmetic. Godel's incompleteness theorem.

Prerequisite: MTH
313 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 419-420 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.)

GROUPS: homomorphism theorems, symmetric groups, linear groups, Sylow theorems, solvable groups, group actions;

RINGS: prime and maximal ideals, the radical, UFDs, PIDs, Noetherian and Artinian rings, the Hilbert Basis theorem, localization, I-adic topologies and completions, algebraic varieties;

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;

ALGEBRAS: Morita equivalence, semi-simple rings, Wedderburn-Artin theorem, group algebras, Maschke's theorem, representation theory of groups and algebras;

CATEGORY THEORY: categories, functors, natural transformations, representable functors, adjoint functors, universal properties, limits, colimits, Yoneda's lemma.

Prerequisite: MTH 431-MTH 432, or the equivalent

Description: Functions (analytic, entire, meromorphic, etc.) of one
complex variables, conformal mappings, singularities, complex
integration. 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
431-MTH
432 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. Homology of manifold: cohomology, cup and cap products,
Poincaré duality, Lefschetz fixed point theorem.

Prerequisite: MTH 419-MTH 420, MTH 431-MTH 432 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.

Prerequisite: MTH 431-MTH 432 or the equivalent

Description: Metric spaces, Baire category argument,
Stone-Weierstrass theorem, Daniell integral, theory of measure,
measurable functions. Lusin's theorem, Egoroff's theorem. Lebesgue
integral, Fatou's lemma, convergence in measure, mean convergence,
almost uniform convergence. Dominated Convergence Theorem. Riesz
representation theorem, absolute continuity. Radon-Nikodym theorem,
bounded variation, Lebesgue's differentiation theorem, F.T.C. for
Lebesgue integral, density, approximate continuity. Radon-Nikodym
theorem, bounded variation, Lebesgue's differentiation theorem,
F.T.C. for Lebesgue integral, density, approximate continuity.

Prerequisite: Linear algebra and undergraduate analysis

Description: Analysis on manifolds, Riemannian geometry, and
topics selected by the instructor.

Prerequisite: Linear algebra and numerical analysis

Description: Computational problems of linear algebra: linear
systems and the eigen-problem. Error analysis. Various algorithms:
Givens, Jacobi, Householder for Hermitian matrices and L-R, Q-R for
the non-Hermitian case as well as Jacobi-type algorithms.

Prerequisite: MTH
632

Description: Fourier series and integrals, convergence and
summability, theorems on Fourier coefficients, uniqueness
properties.

Prerequisite: Introductory differential equations and advanced
calculus (or introductory real analysis)

Description: Existence theorems, linear and nonlinear differential
equations, regular and singular boundary value problems, stability
theory of linear and nonlinear systems. Liapunov's second method.
Geometric theory of differential equations in the plane.

Prerequisite: Advanced calculus or introductory real and complex
analysis

Description: Fredholm theory; Hilbert-Schmidt theorem; singular
integral equations; Wiener-Hopf equation. Applications of integral
equations to potential theory.

Prerequisite: Advanced calculus or introductory real analysis,
or permission of instructor

Description: The Cauchy problem for partial differential
equations, classification of second order linear partial
differential equations, properties of solutions for elliptic,
parabolic and hyperbolic equations, existence of solutions for
elliptic partial differential equations. Topics from Fourier and
Laplace transforms, potential theory, Green's functions, integral
equations, Sobolev spaces, and Schwartz distributions.

Prerequisite: MTH
414 or MTH
514

Description: Primitive recursive functions, general recursive
functions and partial recursive functions (Kleene's normal form
theorem, enumeration theorem, recursive theorem, etc. ). In the
second half more advanced topics will be discussed as decided by
the students and the instructor.

Prerequisite: MTH
414 or MTH
514

Description: Formal systems for intuitionistic predicate calculus
and arithmetic and their metatheory. Various systems for
intuitionistic analysis described and compared.

Prerequisite: MTH
619-620, or permission of instructor

Description: Topics selected by the instructor. These may include:
tensor product, exterior product, the existence of determinants,
bilinear forms, Witt's theorem, Clifford algebra, special theorems,
representations of finite groups, characters, theorems of Brauer,
Commutative algebra, finitely generated modules over Dedekind
domains (the classical ideal theory), dimensions of rings and
modules. Hilbert's theorem on syzygies, the finite dimensionality
of regular local rings.

Prerequisite: MTH
625-626

Description: Topics to be chosen from: Boundary behavior of
analytic functions, founded analytic functions, conformal mapping;
Riemann surfaces; Potential theory and Nevanlinna theory.

Prerequisite: MTH
625-626

Description: Topics include fundamental properties of holomorphic
functions, complex analytic manifolds, integral representations,
Cousin problems.

Prerequisite: MTH
519 and MTH
627-628 or consent of instructor

Description:Topics selected by the instructor. They are usually
more specialized topics from certain area of topology such as
algebraic topology, differential topology, geometric topology, low
dimensional topology, quantum topology, etc.

Prerequisite: MTH
430 or consent of instructor

Description: Continued fractions, Perron's modular function,
Minkowski's linear forms theorem, badly approximable numbers,
approximations to algebraic numbers, inhomogeneous approximations
and Kronecker's theorem, uniform distribution and Weyl's criterion,
irregularities of distribution.

Prerequisite: MTH
420 and MTH
430, or consent of instructor

Description: Principal ideal rings, modules over principal ideal
rings, integral rings extensions, algebraic field extension, norm,
trace, discriminant, Noetherian rings, Dedekind rings. Algebraic
number fields: finiteness of the class number, Dirichlet unit
theorem, splitting of prime ideals in an extension field,
ramification. Galois extensions of number fields. Topics in
quadratic, cubic, and cyclotomic fields.

Prerequisite: MTH 534-MTH 625 or equivalent

Description: Banach spaces, summability, Banach limits, uniform
boundedness, interior mapping theorem, graphs, Hahn-Banach theorem,
Lp spaces, C[a,b], finite dimensional, weak and weak* topology.
Alaoglu theorem, reflexivity and weak compactness theory. Hilbert
spaces, spectral theorem for self-adjoint operators, linear
topological vector spaces.

Prerequisite: Consent of instructor

Description: Commutative ring theory (including integral
dependence, local rings, valuation rings, formal power series).
Algebraic varieties with specialization to curves and surfaces,
Riemann-Roch theorem.

Prerequisite: MTH 519 and MTH
635-636 or consent of instructor.

Description: Topics selected by the instructor. They are usually
more specialized topics from certain area of geometry such as
algebraic geometry, differential geometry, Reimannian geometry,
hyperbolic geometry, symplectic geometry, etc.

Description: Writing and submission of thesis or dissertation under the supervision of the major professor.

Description: Permission of department and instructor required.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.

Prerequisite: Consent of instructor.