MATH 1110 Probability and Statistics I
Probability and Statistics I
Elementary probability theory with applications; random variables; distributions including a thorough discussion of the binomial, and normal distributions; central limit theorem; histograms; sampling distributions; confidence intervals; tests of hypotheses; linear models; regression and correlation; chisquare test; nonparametric statistics. 1110 is a prerequisite for 1120. These courses do not count toward the Mathematics B.S. requirement in SSE. Students may receive credit for only one of MATH 1110, 1140 or 1230.
Prerequistites: High school algebra.
credit hours: 3
MATH 1140 Statistics for Business
Statistics for Business
An introductory statistics course for BSM students using MSExcel. Includes confidence intervals and hypothesis tests for one and two populations and introduction to linear regression. Extensive coverage of data collection and analysis as needed to evaluate statistical results and to make good decisions in business. In comparison to Math 1110, the course spends more time on statistical inference problems, less on probability. This course does not count toward the Mathematics B.S. requirement. Students may receive credit for only one of MATH 1110, 1140 or 1230.
Prerequistites: High school algebra.
credit hours: 4
MATH 1150 Long Calculus
Long Calculus
The material of Calculus 1210 is covered in two semesters, with diversions for topics in algebra, trigonometry, complex numbers as the need for these topics arises. Mathematics 1150 is a prerequisite for 1160. Students finishing the course sequence 11501160 may continue with 1220 or any other course having Calculus 1201 as a prerequisite. The combination of 1150 and 1160 may count as one course toward the B.S. degree requirement.
credit hours: 3
MATH 1160 Long Calculus
Long Calculus
The material of Calculus 1210 is covered in two semesters, with diversions for topics in algebra, trigonometry, complex numbers as the need for these topics arises. Mathematics 1150 is a prerequisite for 1160. Students finishing the course sequence 11501160 may continue with 1220 or any other course having Calculus 1201 as a prerequisite. The combination of 1150 and 1160 may count as one course toward the B.S. degree requirement.
Prerequistites: MATH 1150.
credit hours: 3
MATH 1210 Calculus I
Calculus I
Functions and their graphs, limits and continuity, derivatives and applications of derivatives, and introduction to the integral.
Prerequistites: High school algebra, geometry, and trigonometry.
credit hours: 4
MATH 1220 Calculus II
Calculus II
Integration; exponential, logarithmic, and trigonometric functions; techniques of integration; mean value theorem; Taylor's Theorem and Taylor series; and infinite series.
Prerequistites: Grade of at least C in MATH 1160 or 1210.
credit hours: 4
MATH 1230 Statistics for Scientists
Statistics for Scientists
The objective of this course is to provide a practical overview of the statistical methods and models most likely to be encountered by scientists in practical research applications. Students will learn statistical concepts by generating and analyzing stochastic datasets using the Minitab software package. Specific topics that will be covered in this course include discrete and continuous distributions, sampling methods, and descriptive statistics, the Central Limit Theorem and its applications, estimation methods, confidence intervals, hypothesis testing, linear regression, and Analysis of Variance . Students may receive credit for only one of MATH 1110, 1140 or 1230. Only MATH 1230 counts towards the B.S. degree.
Prerequistites: MATH 1210 or permission of instructor.
credit hours: 3
MATH 1310 Consolidated Calculus
Consolidated Calculus
A combined course in Calculus I and II for students with a background in Calculus I.
Notes: Students receive credit for both this course and 1210 if they receive a B or higher. Students may not receive credit for both 1310 and 1220.
Prerequistites: A score of 3 or higher on the AB or BC Calculus AP test or permission of the mathematics department undergraduate coordinator.
credit hours: 4
MATH 2170 Discrete Mathematics
Discrete Mathematics
An introduction to the concepts and techniques of discrete mathematics including set theory, mathematical induction, graphs, trees, ordered sets, Boolean algebras, and the basic laws of combinatorics.
Prerequistites: MATH 1220 or 1310.
credit hours: 3
MATH 2210 Calculus III
Calculus III
A basic course in differential and integral calculus of several variables. Vectors in the plane and space. Vector functions, derivatives, arc length, curvature. Functions of several variables: continuity, partial derivatives, chain rule, gradient, optimization, Lagrange multipliers. Double and triple integrals: change of variables, polar coordinates, cylindrical and spherical coordinates, surface area. Vector fields: gradient, curl, divergence, line and surface integrals, Green's, Stokes', and Divergence theorems.
Prerequistites: MATH 1220 or 1310.
credit hours: 4
MATH 2240 Introduction to Applied Mathematics
Introduction to Applied Mathematics
An introduction to the techniques of applied mathematics. The emphasis will be on the mathematical modeling by differential equations of a variety of applications in the natural sciences. Numerical and graphical techniques for finding both quantitative and qualitative information about solutions will be discussed and implemented on the computer. No programming experience is assumed.
Notes: Students may not receive credit for both 2240 and 4240.
Prerequistites: MATH 1220 or 1310.
credit hours: 4
MATH 3050 Real Analysis I
Real Analysis I
Introduction to analysis. Real numbers, limits, continuity, uniform continuity, sequences and series, compactness, convergence, Riemann integration. An indepth treatment of the concepts underlying calculus.
Prerequistites: MATH 2210.
credit hours: 3
MATH 3070 Introduction to Probability
Introduction to Probability
An introduction to probability theory. Counting methods, conditional probability and independence. Discrete and continuous distributions, expected value, joint distributions and limit theorems. Prepares student for future work in probability and statistics.
Prerequistites: MATH 2210 or equivalent.
credit hours: 3
MATH 3080 Introduction to Statistical Inference
Introduction to Statistical Inference
Basics of statistical inference. Sampling distributions, parameter estimation, hypothesis testing, optimal estimates and tests. Maximum likelihood estimates and likelihood ratio tests. Data summary methods and categorical data analysis. Analysis of variance and introduction to linear regression.
Prerequistites: MATH 2210, MATH 3070.
credit hours: 3
MATH 3090 Linear Algebra
Linear Algebra
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.
Prerequistites: MATH 2210.
credit hours: 4
MATH 3110 Abstract Algebra I
Abstract Algebra I
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.
Prerequistites: MATH 2210.
credit hours: 3
MATH 3130 Special Topics in Mathematics
Special Topics in Mathematics
Courses offered by visiting professors or permanent faculty. For description, consult department.
credit hours: 3
MATH 3140 Experimental Mathematics
Experimental Mathematics
The exploration of Mathematical tools in Symbolic Languages. Examples are taken from calculus, differential equations, and linear algebra.
Prerequistites: MATH 1210, 1220, 2210.
credit hours: 3
MATH 3200 Combinatorics
Combinatorics
Basics of combinatorics with emphasis on problem solving. Provability, pigeonhole principle, mathematical induction. Counting techniques, generating functions, recurrence relations, Polya's counting formula, a theorem of Ramsey.
Prerequistites: MATH 1210, 1220, and either 2210 or 3090 or approval of instructor.
credit hours: 3
MATH 3250 Theory of Computation
Theory of Computation
Introduction to the theory of computation: Formal languages, finite automata and regular languages, deterministic and nondeterministic computation, context free grammars, languages, pushdown automata, turning machines, undecidable problems, recursion theorem, computational complexity and NPcompleteness.
Prerequistites: MATH 2170 or equivalent.
credit hours: 3
MATH 3260 Advanced Algorithms
Advanced Algorithms
Students who have taken neither MATH 2170 nor MATH 3200 require the permission of the instructor. A study of important algorithms (including searching and sorting, graph/network algorithms, and algorithms in number theory) and algorithm design techniques (including greedy, recursive, and probabilistic algorithms). Covers the analysis of algorithms (including worstcase and averagecase analysis) and discussions of complexity classes for decision and enumeration problems (including P, NP, #P, PSPACE).
Prerequistites: MATH 3050 or 3110 or 3200.
credit hours: 3
MATH 3280 Introduction to Theory of Information
Introduction to Theory of Information
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.
Prerequistites: MATH 3050 or 3090 and familiarity with discrete probability.
credit hours: 3
MATH 3310 Scientific Computing I
Scientific Computing I
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. Singlestep and multistep methods for initialvalue problems for ordinary differential equations, variable step size. Current algorithms and software.
Prerequistites: MATH 2210, 2240, or 4240.
credit hours: 3
MATH 3650 Number Theory and Applications
Number Theory and Applications
The subject of number theory is one of the oldest in mathematics. The course will cover some basic material and describe interesting applications. One of the recurrent themes is the realization that mathematics that was developed usually for its own sake, has found applications in many unexpected problems. Some of the topics covered in the class are Pythagorean triples, prime numbers, divisibility and the highest common divisor, linear diophantine equations, congruences, roundrobin tournaments and perpetual calendars, multiple functions, perfect numbers, primitive roots, pseudorandom numbers, decimal fractions and continued fractions, quadratic reciprocity.
credit hours: 3
MATH 3980 Seminar in Mathematics (Capstone)
Seminar in Mathematics (Capstone)
Under faculty guidance, students will select a topic in current mathematical research, write an expository article on that topic, and give an oral presentation. This seminar is required of all mathematics majors who are not doing an Honors Project within the department.
Notes: Completion of 3980 and 3990 fulfills the college intensivewriting requirement.
Meets capstone requirement.
Prerequistites: MATH 3050, 3090, and two additional courses at the 3000level or above.
credit hours: 1
MATH 3990 Seminar in Mathematics (Capstone)
Seminar in Mathematics (Capstone)
Under faculty guidance, students will select a topic in current mathematical research, write an expository article on that topic, and give an oral presentation. This seminar is required of all mathematics majors who are not doing an Honors Project within the department.
Notes: Completion of 3980 and 3990 fulfills the college intensivewriting requirement.
Meets capstone requirement.
Prerequistites: MATH 3050, 3090, and two additional courses at the 3000level or above.
credit hours: 3
MATH 4060 Real Analysis II
Real Analysis II
An indepth treatment of multivariable calculus. Extends the material covered in Mathematics 2210. Chain rule, inverse and implicit function theorems, Riemann integration in Euclidean nspace, GaussGreenStokes theorems, applications.
Prerequistites: MATH 3050 and 3090.
credit hours: 3
MATH 4120 Abstract Algebra II
Abstract Algebra II
Abstract vector spaces, quotient spaces, linear transformations, dual spaces, determinants. Solvable groups. Field extensions, Galois theory, solvability of equations by radicals.
Prerequistites: MATH 3090 and 3110.
credit hours: 3
MATH 4210 Differential Geometry
Differential Geometry
Theory of plane and space curves including arc length, curvature, torsion, Frenet equations, surfaces in threedimensional space. First and second fundamental forms, Gaussian and mean curvature, differentiable mappings of surfaces, curves on a surface, special surfaces.
Prerequistites: MATH 3050 and 3090.
credit hours: 3
MATH 4240 Ordinary Differential Equations
Ordinary Differential Equations
Review of linear algebra, firstorder equations (models, existence, uniqueness, Euler method, phase line, stability of equilibria), higherorder linear equations, Laplace transforms and applications, power series of solutions, linear firstorder, systems (autonomous systems, phase plane), application of matrix normal forms, linearization and stability of nonlinear systems, bifurcation, Hopf bifurcation, limit cycles, PoincareBendixson theorem, partial differential equations (symmetric boundaryvalue problems on an interval, eigenvalue problems, eigenfunction expansion, initialvalue problems in 1D).
Notes: Students may not receive credit for both 2240 and 4240.
Prerequistites: MATH 3090.
credit hours: 3
MATH 4250 Mathematical Foundations of Computer Security
Mathematical Foundations of Computer Security
This course studies the mathematics underlying computer security, including both public key and symmetric key cryptography, cryptoprotocols 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 analysis of secure information flow.
Prerequistites: Calculus, MATH 2170 and MATH 3110 or permission of instructor.
credit hours: 3
MATH 4300 Complex Analysis
Complex Analysis
The complex number system, complex integration and differentiation, conformal mapping, Cauchy's theorem, calculus of residues.
Prerequistites: MATH 3050.
credit hours: 3
MATH 4410 Topology
Topology
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.
Prerequistites: MATH 3050.
credit hours: 3
MATH 4411 Introduction to Algebraic Topology with Applications
Introduction to Algebraic Topology with Applications
An introduction to algebraic topology with perspectives on applications to sensor networks, target detection and learning theory. Elementary algebraic topology: fundamental group, simplicial complexes, homology, long exact sequences, excision, Lefschetz fixed point theorem, persistent homology. Applications to coverage in sensor networks, deSilvaGhrist criterium, target enumeration.
credit hours: 3
MATH 4470 Analytical Methods of Applied Mathematics
Analytical Methods of Applied Mathematics
Derivations of transport, heat/reactiondiffusion, wave, Poisson's equations; wellposedness; characteristics methods 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 SturmLiouville eigenexpansions; method of separation of variables, frequencies of wave equations, stable and unstable modes, long time behavior of heat equations; deltafunction, fundamental solution of Laplace equation, Newton potential; Green's function and Poisson formula; Dirichlet Principle.
Prerequistites: MATH 2210 and 2240 or 4240.
credit hours: 3
MATH 4780 Introduction to Concurrency
Introduction to Concurrency
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.
Prerequistites: MATH 2170 and MATH 3100 or approval of instructor.
credit hours: 3
MATH 4900 Advanced Topics in Mathematics
Advanced Topics in Mathematics
This course covers a variety of advanced topics in mathematics and exposes students to recent developments not available in other parts of the mathematics curriculum. Topics covered will vary from semester to semester. Recent topics offered include Knot Theory and 3Manifolds, Algebraic Combinatorics, Cardiac Modeling, Number Theory.
Notes: Each section will have the specific topic listed as a subtitle and will have specific prerequisites at the 3000level or above. It meets in conjunction with graduate level courses MATH 77107790. Students may receive credit for MATH 4900 more than once, when the topics covered are distinct.
Prerequistites: Approval of instructor.
credit hours: 3
MATH 4910 Independent Studies
Independent Studies
No more than four hours of 49104920 may be counted toward satisfying the major requirements.
Prerequistites: Approval of the department.
credit hours: 13
MATH 4920 Independent Studies
Independent Studies
No more than four hours of 49104920 may be counted toward satisfying the major requirements.
Prerequistites: Approval of the department.
credit hours: 3
MATH 6020 Mathematical Statistics
Mathematical Statistics
Thorough review of key distributions for probability and statistics, including the multivariate calculus needed to develop them. Full derivation of sampling distribution. Classical principles of inference including best tests and estimations. Methods of finding tests and estimators. Introduction to Bayesian estimators.
Prerequistites: MATH 3070 and 2210.
credit hours: 3
MATH 6030 Stochastic Processes
Stochastic Processes
Markov processes, Poisson processes, queueing models, introduction to Brownian Motion.
Prerequistites: MATH 3070
credit hours: 3
MATH 6040 Linear Models
Linear Models
Overview of multivariate analysis, theory of least squares linear regression, regression diagnostics, introduction to generalized linear models with emphasis on logistic regression. The student will complete several extended data analysis assignments using SAS, SPlus, or R.
Prerequistites: MATH 3010 and 3090 or equivalent.
credit hours: 3
MATH 6080 Introduction to Statistical Inference
Introduction to Statistical Inference
Basics of Statistical inference. Sampling distributions, parameter estimation, hypothesis testing, optimal estimates and tests. Maximum likelihood estimates and likelihood ratio tests. Data summary methods, categorical data analysis. Analysis of variance and introduction to linear regression.
Prerequistites: MATH 2210, MATH 3070.
credit hours: 3
MATH 6240 Ordinary Differential Equations
Ordinary Differential Equations
Review of linear algebra, firstorder equations (models, existence, uniqueness, Euler method, phase line, stability of equilibria), higherorder linear equations, Laplace transforms and applications, power series of solutions, linear firstorder, systems (autonomous systems, phase plane), application of matrix normal forms, linearization and stability of nonlinear
systems, bifurcation, Hopf bifurcation, limit cycles, PoincareBendixson theorem, partial differential equations (symmetric boundaryvalue problems on an interval, eigenvalue problems, eigenfunction expansion, initialvalue problems in 1D). Students may not receive credit for both 2240 and 4240.
Prerequistites: MATH 3090.
credit hours: 3
MATH 6250 Mathematical Foundations of Computer Security
Mathematical Foundations of Computer Security
This course studies the mathematics underlying computer security, including both public key and symmetric key cryptography, cryptoprotocols 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 analysis of secure information flow.
Prerequistites: Calculus, MATH 2170 and MATH 3110 or permission of instructor.
credit hours: 3
MATH 6350 Optimization
Optimization
Constrained and unconstrained nonlinear optimization; Linear programming, combinatorial optimization as time allows. Emphasis is on realistic problems whose solution requires computers, using Maple or Mathematica.
Prerequistites: MATH 3090 or equivalent.
credit hours: 3
MATH 6370 Time Series Analyis
Time Series Analyis
This course provides an introduction to time series analysis at the graduate level. The course is about modeling based on three main families of techniques: (i) the classical decomposition into trend, seasonal and noise components; (ii) ARIMA processes and the Box and Jenkins methodology; (iii) Fourier analysis. If time permits, other possible topics include state space modeling and fractional processes. The course is focused on the theory, but some key examples and applications are also covered and implemented in the software package R.
Prerequistites: One course from MATH 6020/7240, MATH 6040/7260 or MATH 7360; one course from MATH 7550, MATH 6050/3050 or MATH 6710/7210. Exceptions to these prerequisites may be granted by permission of the instructor.
credit hours: 3
MATH 6380 Introduction to Theory of Information
Introduction to Theory of Information
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.
Corequisites: MATH 3050 or 3090 and familiarity with discrete probability.
credit hours: 3
MATH 6510 Topology I and II
Topology I and II
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. EilenbergSteenrod axioms. Computational techniques, including long exact sequences. MayerVietoris sequences, excision, and cellular chain complexes. Introduction to singular cohomology.
Prerequistites: MATH 3050 and 4060.
credit hours: 3
MATH 6520 Topology I and II
Topology I and II
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. EilenbergSteenrod axioms. Computational techniques, including long exact sequences. MayerVietoris sequences, excision, and cellular chain complexes. Introduction to singular cohomology.
Prerequistites: MATH 3050 and 4060.
credit hours: 3
MATH 6550 Differential Geometry I
Differential Geometry I
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 MaurerCartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.
credit hours: 3
MATH 6560 Differential Geometry II
Differential Geometry II
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 MaurerCartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.
credit hours: 3
MATH 6610 Algebra I
Algebra I
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, JordanHölder, 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.
Prerequistites: MATH 3090 and 3110.
credit hours: 3
MATH 6620 Algebra II
Algebra II
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, JordanHölder 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.
Prerequistites: MATH 3090 and 3110.
credit hours: 3
MATH 6650 Differential Equations I
Differential Equations I
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.
credit hours: 3
MATH 6660 Differential Equations II
Differential Equations II
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.
credit hours: 3
MATH 6710 Analysis I
Analysis I
Lebesgue measure on R. Measurable functions (including Lusin's and Egoroff's theorems). The Lebesgue integral. Monotone and dominated convergence theorems. RadonNikodym 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. HahnBanach, open mapping, and uniform boundedness theorems. Hilbert space. Representation of linear functionals. Completeness and compactness. Compact operators, integral equations, applications to differential equations, selfadjoint operators, unbounded operators.
Prerequistites: MATH 3050, 3090, and 4060.
credit hours: 3
MATH 6720 Analysis II
Analysis II
Lebesgue measure on R. Measurable functions (including Lusin's and Egoroff's theorems). The Lebesgue integral. Monotone and dominated convergence theorems. RadonNikodym 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. HahnBanach, open mapping, and uniform boundedness theorems. Hilbert space. Representation of linear functionals. Completeness and compactness. Compact operators, integral equations, applications to differential equations, selfadjoint operators, unbounded operators.
Prerequistites: MATH 3050, 3090, and 4060.
credit hours: 3
MATH 6750 Computation I, II
Computation I, II
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.
credit hours: 3
MATH 6760 Computation I, II
Computation I, II
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.
credit hours: 3
MATH 6810 Applied Mathematics I
Applied Mathematics I
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.
Notes: Mathematics 6510, 6520, 6550, 6560, 6610, 6620, 6650, 6660, 6710, 6720, 6750, 6760, 6810, 6820 are particularly recommended for students planning to do graduate work in mathematics.
Prerequistites: MATH 3050, 3090, 3470, and 4060.
credit hours: 3
MATH 6820 Applied Mathematics II
Applied Mathematics II
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.
Notes: Mathematics 6510, 6520, 6550, 6560, 6610, 6620, 6650, 6660, 6710, 6720, 6750, 6760, 6810, 6820 are particularly recommended for students planning to do graduate work in mathematics.
Prerequistites: MATH 3050, 3090, 3470, and 4060.
credit hours: 3
MATH 6840 Numerical Methods in Partial Differential Equations
Numerical Methods in Partial Differential Equations
This course will present a detailed analysis of the methods for numerically approximating the solution of ordinary and partial differential equations typically encountered in applications from engineering and physics. Mathematical theory, practical implementation and applications will be emphasized equally. Typical applications to be discussed include population dynamics, particle dynamics, waves, diffusion processes.
Prerequistites: MATH 3310 and 3470 or approval of instructor.
credit hours: 3
MATH 7010 Topology I
Topology I
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. EilenbergSteenrod axioms. Computational techniques, including long exact sequences. MayerVietoris sequences, excision, and cellular chain complexes. Introduction to singular cohomology.
Prerequistites: Math 3050 and 4060.
credit hours: 3
MATH 7020 Topology II
Topology II
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. EilenbergSteenrod axioms. Computational techniques, including long exact sequences. MayerVietoris sequences, excision, and cellular chain complexes. Introduction to singular cohomology.
Prerequistites: Math 3050 and 4060.
credit hours: 3
MATH 7110 Algebra I
Algebra I
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, JordanHölder , 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.
Prerequistites: Math 3090 and 3110.
credit hours: 3
MATH 7120 Algebra II
Algebra II
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, JordanHölder , 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.
Prerequistites: Math 3090 and 3110.
credit hours: 3
MATH 7210 Analysis I
Analysis I
credit hours: 3
MATH 7220 Analysis II
Analysis II
credit hours: 3
MATH 7240 Mathematical Statistics
Mathematical Statistics
Consists of Math 6020 and additional meetings and readings to cover advanced limit theorems and foundations of mathematical statistics.
Prerequistites: Math 6070, 6080 and 7210 or permission of the instructor.
credit hours: 3
MATH 7260 Linear Models
Linear Models
Review of linear algebra pertinent to least squares regression. Review of multivariate normal, chisquare, t, F distributions. Classical theory of linear regression and related inference. Regression diagnostics. Extensive practice in data analysis.
Prerequistites: Math 3070/6070, 3080/6080.
Corequisites: Math 309 or approval of instructor.
credit hours: 3
MATH 7310 Applied Mathematics I
Applied Mathematics I
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.
credit hours: 3
MATH 7320 Applied Mathematics II
Applied Mathematics II
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.
credit hours: 3
MATH 7350 Scientific Computing I
Scientific Computing I
Introduction to numerical analysis: wellposedness and condition number, stability and convergence of numerical methods, a priori and aposteriori analysis, source 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.
Prerequistites: MATH 3310 or MATH 73107320.
credit hours: 3
MATH 7360 Data Analysis
Data Analysis
This course covers the statistical analysis of datasets using R software package. The R environment, which is 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.
credit hours: 3
MATH 7370 Time Series Analysis
Time Series Analysis
This course provides an introduction to time series analysis at the graduate level. The course is about modeling based on three main families of techniques: (i) the classical decomposition into trend, seasonal and noise components; (ii) ARIMA processes and the Box and Jenkins methodology; (iii) Fourier analysis. If time permits, other possible topics include state space modeling and fractional processes. The course is focused on the theory, but some key examples and applications are also covered and implemented in the software package R.
Prerequistites: One course from MATH 6020/7240, MATH 6040/7260 or MATH 7360; one course from MATH 7550, MATH 6050/3050 or MATH 6710/7210. Excep
credit hours: 3
MATH 7420 Literature Seminar
Literature Seminar
credit hours: 3
MATH 7510 Differential Geometry I
Differential Geometry I
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 MaurerCartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.
credit hours: 3
MATH 7520 Differential Geometry II
Differential Geometry II
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 MaurerCartan equation. Isometries, principal bundles, symmetric spaces, Kähler geometry.
credit hours: 3
MATH 7530 Partial Differential Equations I
Partial Differential Equations I
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. L2 theory for second order elliptic equations, existence via LaxMilgram Theorem, Fredholm alternative, a brief introduction to L2 estimates, Harnack inequality, eigenexpansion. L2 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 Lp 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, RankineHugoniot jump condition, uniqueness issue, entropy condition, Riemann problem for Burger's equation, psystems.
Prerequistites: MATH 3050, 4060, 4470/6470/731000, 7210 and 7220 or by instructor's approval.
credit hours: 3
MATH 7540 Partial Differential Equations II
Partial Differential Equations II
A brief introduction to elliptic and parabolic regularity theory, the L^p and Schauder estimates. Nonlinear elliptic equations, variational methods, methods of upper and lower solutions, fixed point method, bifurcation method. Nonlinear parabolic equations, global existence, stability of steady states, traveling wave solutions. Conservation laws, RankineHugonoit jump condition, uniqueness issue,, entropy condition, Reimann problem for Burger's equation and psystems.
Prerequistites: MATH 7530 or by instructor's approval.
credit hours: 3
MATH 7550 Probability and Statistics I
Probability and Statistics I
Various types of convergence, independent increments, 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: NeymanPearson lemmas, general linear models, analysis of variances and covariance, regression. Introduction to time series, sampling design, and Bayesian theory.
credit hours: 3
MATH 7560 Probability and Statistics II
Probability and Statistics II
Various types of convergence, independent increments, 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: NeymanPearson lemmas, general linear models, analysis of variances and covariance, regression. Introduction to time series, sampling design, and Bayesian theory.
credit hours: 3
MATH 7570 Scientific Computation I
Scientific Computation I
Floating point arithmetic (limitations and pitfalls). Numerical linear algebra, solving linear system by direct and iterative methods, eigenvalue problems, singular value decompositions, numerical integrations, interpolations. Unconstrained optimization.
Prerequistites: MATH 7350.
credit hours: 3
MATH 7580 Scientific Computation II
Scientific Computation II
Numerical ODE, both initial and boundary value problems. Numerical PDE. Introduction to fluid dynamics and other areas of application.
Prerequistites: MATH 7350 and 7570.
credit hours: 3
MATH 7710 Special Topics
Special Topics
credit hours: 3
MATH 7790 Special Topics
Special Topics
credit hours: 3
MATH 7800 Seminar in Mathematics
Seminar in Mathematics
credit hours: 3
MATH 7980 Reading and Research
Reading and Research
credit hours: 3
MATH 9990 Dissertation Research
Dissertation Research
credit hours: 3
MATH H4990 Honors Thesis
Honors Thesis
Thesis may serve to satisfy part of the departmental honors requirements.
Prerequistites: Approval of the department.
credit hours: 3
MATH H5000 Honors Thesis
Honors Thesis
Thesis may serve to satisfy part of the departmental honors requirements.
Prerequistites: Approval of the department.
credit hours: 3
