## Courses

**Courses primarily for undergraduates:**

(4-0) Cr. 0. F.S.

For students who do not have adequate facility with topics from high school algebra or do not meet the algebra admission requirement. The course is divided into tracks of one- and two-semester lengths. For most students a diagnostic exam will determine which track must be taken. Students will receive a grade in MATH 25 or MATH 30 respectively depending on the level of material covered. Satisfactory completion of MATH 30 is recommended for students planning to take MATH 140, MATH 143, MATH 145, MATH 150, or MATH 151, while MATH 25 is sufficient for MATH 104, MATH 105, MATH 195, STAT 101 or STAT 105. Students must complete MATH 30 to remove a deficiency in the algebra admission requirement. Topics include signed numbers, polynomials, rational and radical expressions, exponential and logarithmic expressions, and equations.
Offered on a satisfactory-fail basis only.

(4-0) Cr. 0. F.S.

Students should initially enroll in MATH 10. See description of MATH 10.
Offered on a satisfactory-fail basis only.

(4-0) Cr. 0. F.S.

Students should initially enroll in MATH 10. See description of MATH 10.
Offered on a satisfactory-fail basis only.

(1-0) Cr. 1. F.S.

For new majors. Campus resources and opportunities available to students. Careers and programs of study in mathematics. Mathematical reasoning, culture and resources. Description of main branches of mathematics.
Offered on a satisfactory-fail basis only.

(3-0) Cr. 3. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry*

Permutations, combinations, probability, expected value, and applications.
Either MATH 104 or MATH 150 may be counted toward graduation, but not both.

(3-0) Cr. 3. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry.*

Introduction to contemporary mathematics with an emphasis on use of mathematics to solve real world problems. Typical topics are the mathematics of voting, methods of fair division and apportionment, and elementary game theory.

(3-0) Cr. 3. F.S.

Inquiry-based approach to mathematics, emphasizing the art, history, and beauty of the subject. Typical topics are mathematics from art, music, puzzles, patterns, and reasoning.

(3-1) Cr. 3. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry; or MATH 30.*

Coordinate geometry, quadratic and polynomial equations, functions, graphing, rational functions, exponential and logarithmic functions, inverse functions, quadratic inequalities, systems of linear equations. Prepares students for MATH 160.
Students in the College of Liberal Arts and Sciences may not count MATH 140 toward the General Education Requirements.

(4-0) Cr. 4. F.S.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry; or MATH 140.*

Preparation for MATH 160, MATH 165, and MATH 181. Functions, graphing, basic trigonometry, logarithms, exponentials. Emphasis on co-variational reasoning. Students in the College of Liberal Arts and Sciences may not count MATH 143 toward General Education Requirements.
Only one of MATH 143 and MATH 145 may count toward graduation.

(3-0) Cr. 3. F.S.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry; or minimum of C- in MATH 140.*

Mathematical ideas regarding the conception of space. General trigonometry, with an emphasis on the calculation of lengths, areas, and angles. The Law of Sines and the Law of Cosines. Polar, cylindrical, and spherical coordinate systems. Conic sections and quadric surfaces.
Students in the College of Liberal Arts and Sciences may not count MATH 145 toward the General Education Requirements. Only one of MATH 143 and MATH 145 may count toward graduation.

(2-1) Cr. 3. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry*

Linear equations and inequalities, matrix algebra, linear programming, discrete probability.
Either MATH 104 or MATH 150 may be counted toward graduation, but not both.

(2-1) Cr. 3. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry*

Differential calculus, applications to max-min problems, integral calculus and applications. Will not serve as prerequisite for MATH 265 or MATH 266.
Only one of MATH 151, MATH 160, the sequence MATH 165-MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(4-0) Cr. 4. F.S.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of geometry; or MATH 143; or MATH 140*

Analytic geometry, derivatives and integrals of elementary functions, simple differential equations, and applications. Will not serve as a prerequisite for MATH 265 or MATH 266.
Only one of MATH 151, MATH 160, the sequence MATH 165-MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(4-0) Cr. 4. F.S.SS.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of geometry, 1 semester of trigonometry; or MATH 143*

Differential calculus, applications of the derivative, introduction to integral calculus.
Only one of MATH 151 or MATH 160 or the sequence MATH 165-MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(4-0) Cr. 4. F.S.SS.

*Prereq: Minimum of C- in MATH 165 or high math placement scores*

Integral calculus, applications of the integral, infinite series, parametric curves and polar coordinates.
Only one of MATH 151, MATH 160, the sequence MATH 165-MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(4-0) Cr. 4. F.

*Prereq: Permission of instructor and MATH 165 or high math placement scores*

Integral calculus, applications of the integral, infinite series, parametric curves, and polar coordinates. Additional material of a theoretical, conceptual, computational, or modeling nature. Some of the work may require more ingenuity than is required for MATH 166. Preference will be given to students in the University Honors Program.
Only one of MATH 151 or MATH 160, the sequence MATH 165-MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(4-0) Cr. 4. F.S.

*Prereq: Satisfactory performance on placement assessment, 2 years of high school algebra, 1 year of high school geometry, 1 semester of trigonometry; or MATH 143*

Exponential and logarithm functions, difference equations, derivatives, and applications of the derivative. Examples taken from biology.
Only one of MATH 151, MATH 160, the sequence MATH 165- MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(4-0) Cr. 4. S.

*Prereq: MATH 181*

Integration, first and second order differential equations, applications of the definite integral, introduction to multivariable calculus. Examples taken from biology.
Only one of MATH 151, MATH 160, the sequence MATH 165-MATH 166, or the sequence MATH 181-MATH 182 may be counted towards graduation.

(2-2) Cr. 3. F.S.

*Prereq: Satisfactory performance on placement assessment, 2 years high school algebra, 1 year of high school geometry, enrollment in elementary education or early childhood education*

Whole number operations through analysis of properties, theoretical and hands-on models, mathematical analysis of elementary students’ thinking; standard and non-standard algorithms; structure of the decimal system; linear measurement; two- and three-dimensional geometric shapes and spatial sense; number theory; algebra as it relates to elementary curricula/teaching profession.
Students in the College of Liberal Arts and Sciences may not count MATH 195 toward General Education Requirements.

(2-2) Cr. 3. F.S.

*Prereq: Minimum of C- in MATH 195 and enrollment in elementary education or early childhood education.*

Integer, fraction and decimal operations through analysis of properties, theoretical and hands-on models, mathematical analysis of elementary students’ thinking; standard and non-standard algorithms; two- and three-dimensional measurement; probability and statistics; proportional reasoning; algebra as it relates to elementary curricula/teaching profession.

(3-0) Cr. 3. F.S.SS.

*Prereq: 2 semesters of calculus*

Systems of linear equations, determinants, vector spaces, linear transformations, orthogonality, least-squares methods, eigenvalues and eigenvectors. Emphasis on applications and techniques.
Only one of MATH 207 and MATH 317 may be counted toward graduation.

(4-0) Cr. 4. F.S.

*Prereq: Permission of the instructor; and MATH 166 or MATH 166H*

Analytic geometry and vectors, differential calculus of functions of several variables, multiple integrals, vector calculus. Additional material of a theoretical, conceptual, computational, or modeling nature. Some of the work may require more ingenuity than is required in MATH 265. Preference will be given to students in the University Honors Program.

(3-0) Cr. 3. F.S.SS.

*Prereq: Minimum of C- in MATH 166 or MATH 166H*

Solution methods for ordinary differential equations. First order equations, linear equations, constant coefficient equations. Eigenvalue methods for systems of first order linear equations. Introduction to stability and phase plane analysis.

(1-0) Cr. 1. F.S.SS.

*Prereq: MATH 266*

Laplace transforms and series solutions to ordinary differential equations.
Together, MATH 266 and MATH 268 are the same as MATH 267.

(1-0) Cr. 1. F.S.SS.

*Prereq: Familiarity with ordinary differential equations of first and second order, permission of department.*

Systems portion of MATH 266 and MATH 267: Eigenvalue methods for systems of first order linear equations. Introduction to stability and phase plane analysis. For students supplementing transfer courses in differential equations in order to earn credit in MATH 266 or 267.
Students with credit in MATH 266 or MATH 267 may not earn credit in MATH 269.

Cr. 1-3. Repeatable.

*Prereq: Permission of the instructor.*

Independent study.

Cr. 1-3. Repeatable.

*Prereq: Permission of the instructor.*

Independent study.

(2-2) Cr. 3. F.

*Prereq: Enrollment in elementary education and minimum of C- in MATH 196*

Mathematical reasoning and topics in Euclidean and non-Euclidean geometry, including transformations, congruence, and similarity; exploration of probability with simulations; linearity and connections to Calculus; fractals and fractal dimension.

(3-0) Cr. 3. S.

*Prereq: MATH 301*

Theory of rings and fields. Introduction to Galois theory. Emphasis on writing proofs.

(3-0) Cr. 3. F.

*Prereq: MATH 166 or MATH 166H; MATH 201 or experience with proofs*

Enumeration strategies involving permutations, combinations, partitions, binomial coefficients, inclusion-exclusion principle, recurrence relations, generating functions. Additional topics selected from probability, algebraic combinatorics, and applications.

(3-0) Cr. 3. S.

*Prereq: MATH 166 or MATH 166H; MATH 201 or experience with proofs*

Structure and extremal properties of graphs. Topics are selected from: trees, networks, colorings, paths and cycles, connectivity, planarity, directed graphs, matchings, Ramsey theory, forbidden structures, enumeration, applications.

(4-0) Cr. 4. F.S.

*Prereq: Credit or enrollment in MATH 201*

Systems of linear equations, determinants, vector spaces, inner product spaces, linear transformations, eigenvalues and eigenvectors. Emphasis on writing proofs and results.
Only one of MATH 207 and MATH 317 may be counted toward graduation.

(3-0) Cr. 3. Alt. F., offered even-numbered years.

*Prereq: MATH 301*

Set theory, metric spaces, topological spaces, continuity, connectedness, homeomorphisms, compactness, and topological invariants. Examples from surfaces, knots, and various abstract objects. Emphasis on writing proofs.

(Cross-listed with STAT). (3-0) Cr. 3. F.S.

*Prereq: MATH 265 (or MATH 265H)*

Probability; distribution functions and their properties; classical discrete and continuous distribution functions; multivariate probability distributions and their properties; moment generating functions; simulation of random variables and use of the R statistical package.
Credit for both STAT 341 and STAT 447 may not be applied toward graduation.

(Cross-listed with STAT). (3-0) Cr. 3. F.S.

*Prereq: STAT 341; MATH 207 or MATH 317*

Transformations of random variables; sampling distributions; confidence intervals and hypothesis testing; theory of estimation and hypothesis tests; linear model theory; use of the R statistical package for simulation and data analysis.

(3-0) Cr. 3. S.

*Prereq: MATH 265*

Functions of a complex variable, including differentiation, integration and series expansions, residues, evaluation of integrals, conformal mapping.

(3-0) Cr. 3. F.

*Prereq: MATH 265*

Vector and matrix programming and graphing in MATLAB for scientific applications. Polynomial interpolation and approximation. Systems of linear equations and numerical linear algebra. Numerical differentiation and integration. Newton methods for solving nonlinear equations and optimization in one and several variables. Fast Fourier transform. Emphasis on effective use of mathematical software and understanding of its strengths and limitations.

Cr. R. Repeatable, maximum of 2 times. F.S.SS.

*Prereq: Permission of the department cooperative education coordinator; junior classification*

Required of all cooperative education students. Students must register for this course prior to commencing each work period.

(Dual-listed with MATH 507). (3-0) Cr. 3. F.

*Prereq: MATH 317; or MATH 207 and experience writing proofs*

Advanced topics in applied linear algebra including eigenvalues, eigenvalue localization, singular value decomposition, symmetric and Hermitian matrices, nonnegative and stochastic matrices, matrix norms, canonical forms, matrix functions. Applications to mathematical and physical sciences, engineering, and other fields.

(3-0) Cr. 3. F.S.SS.

*Prereq: Minimum of C- in MATH 201*

A careful development of calculus of functions of one real variable: real number properties, sequences and series, limits, continuity, differentiation, and integration.

(Cross-listed with COM S). (3-0) Cr. 3. S.

*Prereq: MATH 301 or MATH 207 or MATH 317 or COM S 230*

Propositional and predicate logic. Topics selected from Horn logic, equational logic, resolution and unification, foundations of logic programming, reasoning about programs, program specification and verification, model checking and binary decision diagrams, temporal logic and modal logic.

(3-0) Cr. 3. S.

*Prereq: MATH 435*

Continuation of Euclidean geometry with topics from elliptic, projective, or hyperbolic geometry. Emphasis on analytic methods.

(3-0) Cr. 3. S.

*Prereq: MATH 265; STAT 101 or 104 or 105 or 201 or 226.*

Applications of mathematical methods to problems in finance. Lagrange Multiplier Method, applications to mean-variance portfolio selection and utility maximization, binomial asset pricing model. Binary Martingales, Optional Stopping Theorem, Central Limit Theorem, applications to financial derivative pricing.

(Cross-listed with COM S). (3-0) Cr. 3. S.

*Prereq: MATH 265 and either MATH 266 or MATH 267; knowledge of a programming language*

First order Euler method, high order Runge-Kutta method, and multistep method for solving ordinary differential equations. Finite difference and finite element methods for solving partial differential equations. Local truncation error, stability, and convergence for finite difference method. Numerical solution space, polynomial approximation, and error estimate for finite element method.

Cr. 1-3. Repeatable, maximum of 9 credits.

*Prereq: Permission of instructor.*

No more than 9 credits of Math 490 or Math 490H may be counted toward graduation.

Cr. 1-3. Repeatable, maximum of 9 credits.

*Prereq: Permission of the instructor.*

No more than 9 credits of Math 490 or 490H may be counted toward graduation.

Cr. 2-3.

Writing and presenting a formal mathematics paper. Upon approval by the department, the paper will satisfy the departmental advanced English requirement.

Cr. arr. Repeatable, maximum of 9 credits.

*Prereq: Permission of instructor*

Topics of current interest.

(Cross-listed with C I). (3-0) Cr. 3. F.

*Prereq: 15 credits in college mathematics. If in a teacher licensure program, concurrent enrollment in C I 426 or C I 526.*

Develop an understanding of instructional planning, lesson implementation, and assessment in grades 5-12 mathematics, with a focus on reform-based mathematics, equity, and conceptual understanding.

**Courses primarily for graduate students, open to qualified undergraduates:**

(3-0) Cr. 3. F.

*Prereq: MATH 265 and (MATH 207 or MATH 317)*

A development of the real numbers. Study of metric spaces, completeness, compactness, sequences, and continuity of functions. Differentiation and integration of real-valued functions, sequences of functions, limits and convergence, equicontinuity.

(3-0) Cr. 3. F.

*Prereq: MATH 302*

Algebraic systems and their morphisms, with emphasis on groups and rings.

(Dual-listed with MATH 407). (3-0) Cr. 3. F.

*Prereq: MATH 317; or MATH 207 and experience writing proofs*

Advanced topics in applied linear algebra including eigenvalues, eigenvalue localization, singular value decomposition, symmetric and Hermitian matrices, nonnegative and stochastic matrices, matrix norms, canonical forms, matrix functions. Applications to mathematical and physical sciences, engineering, and other fields.

(3-0) Cr. 3. S.

*Prereq: MATH 515*

Metric spaces, topological spaces, compactness, abstract theory of measure and integral, differentiation of measures, Banach spaces.

(3-0) Cr. 3. S.

*Prereq: MATH 481 or MATH 561*

Finite difference methods for partial differential equations. Methods for elliptic equations; explicit and implicit methods for parabolic and hyperbolic equations; stability, accuracy, and convergence theory, including von Neumann analysis, modified equations, and the Courant-Friedrichs-Lewy condition.

(3-0) Cr. 3. F.

*Prereq: MATH 414 or MATH 501*

Techniques of classical and functional analysis with applications to differential equations and integral equations. Vector spaces, metric spaces, Hilbert and Banach spaces, Sobolev spaces and other function spaces, contraction mapping theorem, distributions, Fourier series and Fourier transform, linear operators, spectral theory of differential and integral operators, Green's functions and boundary value problems, weak solutions of partial differential equations and variational methods, calculus in Banach spaces and applications.

(Cross-listed with COM S, CPR E). (3-0) Cr. 3. S.

*Prereq: CPR E 308 or MATH 481; experience in scientific programming; knowledge of FORTRAN or C*

Introduction to parallelization techniques and numerical methods for distributed memory high performance computers. A semester project in an area related to each student’s research interests is required.

(Cross-listed with CPR E, INFAS). (3-0) Cr. 3. Alt. S., offered even-numbered years.

*Prereq: E E 524 or MATH 317 or MATH 407 or COM S 330*

Basic principles of covert communication, steganalysis, and forensic analysis for digital images. Steganographic security and capacity, matrix embedding, blind attacks, image forensic detection and device identification techniques. Related material on coding theory, statistics, image processing, pattern recognition.

(1-0) Cr. 1. SS.

*Prereq: Enrollment in the Master of School Mathematics program or professional studies in education*

Research studies in mathematics learning and teaching, exemplary practices in mathematics education, and current state and national trends in the mathematics curriculum in grades K-12. Students in MSM take each of 540A, 540B, and 540C. Topics are offered on a 3-year cycle. A. Assessment, equity, and teaching of statistics. Offered SS 2017. B. Geometry and discrete mathematics, and problem solving. Offered SS 2018. C. Teaching of analysis, algebra, and the use of technology. Offered SS 2016.

(1-0) Cr. 1.

*Prereq: Enrollment in the Master of School Mathematics program or professional studies in education*

Research studies in mathematics learning and teaching, exemplary practices in mathematics education, and current state and national trends in the mathematics curriculum in grades K-12. Topics are offered on a 3-year cycle. Offered SS 2017.

**MATH 540B: Seminar in Mathematics Education: Geometry and discrete mathematics, and problem solving.**

(1-0) Cr. 1.

*Prereq: Enrollment in the Master of School Mathematics program or professional studies in education*

Research studies in mathematics learning and teaching, exemplary practices in mathematics education, and current state and national trends in the mathematics curriculum in grades K-12. Offered on a 3-year cycle. Offered SS 2018.

(1-0) Cr. 1.

*Prereq: Enrollment in the Master of School Mathematics program or professional studies in education*

Research studies in mathematics learning and teaching, exemplary practices in mathematics education, and current state and national trends in the mathematics curriculum in grades K-12. Topics are offered on a 3-year cycle. Offered SS 2016.

(4-0) Cr. 4.

*Prereq: 3 semesters of calculus and enrollment in the master of school mathematics program*

Offered on a 3-year cycle, offered SS. 2016. The fundamental concepts of calculus which are critical to the effective understanding of the material in first year calculus. Emphasis is on a constructivist approach to learning, cooperative groups, problem solving, and use of technology.

(2-2) Cr. 3.

*Prereq: 3 semesters in calculus or concurrent enrollment in 545 and enrollment in the master of school mathematics program*

Offered on a 3- year cycle, offered SS. 2016. The use of technology in secondary mathematics with an emphasis on the exploration, creation, and implementation of algorithms.

(4-0) Cr. 4.

*Prereq: Enrollment in the master of school mathematics program*

Offered on a 3-year cycle, offered SS. 2018. Applications of graph theory, game theory, voting theory, recursion, combinatorics, and algebraic structures. Issues in integrating discrete topics into the secondary curriculum. Use of the computer to explore discrete mathematics.

(3-0) Cr. 3.

*Prereq: MATH 435 or equivalent and enrollment in the master of school mathematics program*

Offered on a 3-year cycle, offered SS. 2018. A study of geometry with emphasis on metrics, the group of isometries, and the group of similarities. Specific spaces studied normally include the Euclidean plane, the 2-sphere, projective 2-space, and hyperbolic geometry. Emphasis on analytical methods. Incorporation of geometry software.

(Cross-listed with STAT). (3-0) Cr. 3. F.

*Prereq: STAT 542*

Markov chains on discrete spaces in discrete and continuous time (random walks, Poisson processes, birth and death processes) and their long-term behavior. Optional topics may include branching processes, renewal theory, introduction to Brownian motion.

(3-0) Cr. 3. S.

*Prereq: MATH 317*

Numerical linear algebra including LU factorization, QR factorization, linear least squares, singular value decompositions, eigenvalue problems, and iterative methods for large linear systems.

(3-0) Cr. 3. S.

*Prereq: MATH 265 and one of MATH 317, 507, 510*

Theory and methods for constrained and unconstrained optimization. Steepest-descent, conjugate gradient, Newton and quasi-Newton, line search and trust-region, first and second order necessary and sufficient conditions, linear, quadratic and general nonlinear programming.

(3-0) Cr. 3. F.

*Prereq: MATH 317 or MATH 507 or MATH 510*

Algorithms for linear programming, integer and combinatorial optimization. Linear programming, duality theory, simplex algorithm; the solution of the shortest-path, minimum spanning tree, max-flow/min-cut, minimum cost flow, maximum matching, and traveling salesman problems; integer linear programming, branch-and-bound, local and global search algorithms; matroids and greedy algorithms.

(Cross-listed with AER E, E E, M E). (3-0) Cr. 3. F.

*Prereq: E E 324 or AER E 331 or MATH 415; and MATH 207*

Linear algebra review. Least square method and singular value decomposition. State space modeling of linear continuous-time systems. Solution of linear systems. Controllability and observability. Canonical description of linear equations. Stability of linear systems. State feedback and pole placements. Observer design for linear systems.

(Cross-listed with AER E, E E, M E). (3-0) Cr. 3. S.

*Prereq: E E 577*

Linear vs nonlinear systems. Phase plane analysis. Bifurcation and center manifold theory. Lyapunov stability. Absolute stability of feedback systems. Input-output stability. Passivity theory and feedback linearization. Nonlinear control design techniques.

Cr. arr. Repeatable.

(0.5-0) Cr. 0.5. F.

Fall semester orientation seminar. Required for graduate students in Mathematics and Applied Mathematics. Topics include teaching at the university level and communication of mathematics.
Offered on a satisfactory-fail basis only.

(0.5-0) Cr. 0.5. S.

Spring semester orientation seminar. Required for graduate students in Mathematics and Applied Mathematics. Topics include teaching at the university level and communication of mathematics.
Offered on a satisfactory-fail basis only.

Cr. arr. Repeatable.

Cr. arr.

**Courses for graduate students:**

(3-0) Cr. 3. Alt. F., offered odd-numbered years.

*Prereq: MATH 504*

Model theory of propositional and predicate logic, the Soundness Theorem, the Compactness Theorem, the Goedel-Henkin Completeness Theorem, the Incompleteness Theorem, computability theory. As time permits: modal and temporal logic, set theory (the continuum hypothesis). Emphasis on the relationship between `provable' and `true' and the relationship between `computable' and `definable'.

(3-0) Cr. 3. Alt. F., offered even-numbered years.

*Prereq: MATH 504*

Combinatorial designs and Latin squares. Construction methods including finite fields. Error-correcting codes. Adjacency matrices and algebraic combinatorics.

(3-0) Cr. 3. Alt. S., offered odd-numbered years.

*Prereq: MATH 504*

Ordered sets and lattices. Generating functions. Moebius inversion and other enumeration methods.

(3-0) Cr. 3. Alt. S., offered even-numbered years.

*Prereq: MATH 607*

Study of extremal graph problems and methods. Topics include canonical Ramsey theory, generalizations of Turan’s theorem, Szemeredi's regularity lemma, random graph theory.

Cr. arr.

(3-0) Cr. 3. Alt. F., offered even-numbered years.

*Prereq: MATH 504*

First semester of full-year course. Subalgebras, homomorphisms, congruence relations, and direct products. Lattices and closure operators. Varieties and quasivarieties of algebras, free algebras, Birkhoff's theorems, clones, Mal'cev conditions. Advanced topics.

(3-0) Cr. 3. Alt. F., offered odd-numbered years.

*Prereq: MATH 504*

Categories and functors and their applications.

(3-0) Cr. 3. Alt. S., offered even-numbered years.

*Prereq: MATH 504*

Representations of algebraic structures. Content varies by semester.

(3-0) Cr. 3. Alt. S., offered odd-numbered years.

*Prereq: MATH 501 or MATH 515*

Topics selected from: Geometry of curves and surfaces. Manifolds, coordinate systems. Tangent and cotangent vectors, vector fields. Tensors, differential forms, Riemannian metrics. Connections, covariant differentiation, curvature tensors. Applications to physics and engineering.

(3-0) Cr. 3. Alt. F., offered odd-numbered years.

*Prereq: MATH 515*

Fundamental theory of normed linear spaces and algebras, such as topology and continuity, duality and spectral theory, emphasizing aspects that provide a framework for the study of the spectrum of an operator, analytic function theory, and modern operator theory.

(Cross-listed with STAT). (3-0) Cr. 3. F.

*Prereq: MATH 414 or MATH 501 or equivalent course.*

Sequences and set theory; Lebesgue measure, measurable functions. Absolute continuity of functions, integrability and the fundamental theorem of Lebesgue integration. General measure spaces, probability measure, extension theorem and construction of Lebesgue-Stieljes measures on Euclidean spaces. Measurable transformations and random variables, induced measures and probability distributions. General integration and expectation, Lp-spaces and integral inequalities. Uniform integrability and absolute continuity of measures. Probability densities and the Radon-Nikodym theorem. Product spaces and Fubini-Tonelli theorems.

(Cross-listed with STAT). (3-0) Cr. 3. S.

*Prereq: STAT 641, or STAT 543 and MATH 515.*

Probability spaces and random variables. Kolmogorov's consistency theorem. Independence, Borel-Cantelli lemmas and Kolmogorov's 0 - 1 Law. Comparing types of convergence for random variables. Sums of independent random variables, empirical distributions, weak and strong laws of large numbers. Convergence in distribution and its characterizations, tightness, characteristic functions, central limit theorems and Lindeberg-Feller conditions. Conditional probability and expectation. Discrete parameter martingales and their properties and applications.

(Cross-listed with STAT). (3-0) Cr. 3. S.

Weak convergence. Random walks and Brownian motion. Martingales. Stochastic integration and Ito's Formula. Stochastic differential equations and applications.

(Cross-listed with PHYS). (3-0) Cr. 3. S.

Modeling of the dynamics of complex systems on multiple scales: Classical and dissipative molecular dynamics, stochastic modeling and Monte-Carlo simulation; coarse grained nonlinear dynamics, interface propagation and spatial pattern formation.

(3-0) Cr. 3. S.

*Prereq: MATH 655*

Sobolev spaces, general theory of second order linear elliptic, parabolic and hyperbolic partial differential equations, first order linear hyperbolic systems, variational methods, fixed point methods.

(3-0) Cr. 3. Alt. F., offered even-numbered years.

*Prereq: MATH 516 or MATH 520 or MATH 561 or MATH 656*

Weak and variational formulations of elliptic problems; weak derivatives and Sobolev spaces; Lax-Milgram theorem, Bramble-Hilbert lemma; examples of finite element spaces; polynomial approximation theory; error estimates for finite element methods; implementation issues; mixed finite element methods for Stokes problems; applications.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. 3. Repeatable.

Cr. arr. Repeatable.