
MATH 1001  Excursions in Mathematics (MATH)
3.0 cr; Prereq3 yrs high school math or placement exam or grade of at least C in GC 0731; fall, spring, every year
Breadth of mathematics, its nature/applications. Power of abstract reasoning.
MATH 1008  Trigonometry
2.67 cr; PrereqPlane geometry, two yrs high school algebra [or C or better in GC 0731]; AF or Aud, fall, spring, summer, every year
Analytic trigonometry, identities, equations, properties of trigometric functions, right/oblique triangles.
MATH 1031  College Algebra and Probability (MATH)
3.0 cr; Prereq3 yrs high school math or grade of at least C in GC 0731; Credit will not be granted if credit has been received for: 1051, 1151, 1155; fall, spring, every year
Algebra, analytic geometry explored in greater depth than is usually done in three years of high school mathematics. Additional topics from combinations, permutations, probability.
1031 Description
MATH 1038  College Algebra and Probability Submodule
1.0 cr; Prereq1051 or 1151 or 1155; AF or Aud, fall, spring, summer, every year
For students who need probability/permutations/combinations portion of 1031. Meets with 1031, has same grade/work requirements.
MATH 1051  Precalculus I
3.0 cr; Prereq3 yrs high school math or placement exam or grade of at least C in GC 0731; Credit will not be granted if credit has been received for: 1031, 1151; fall, spring, every year
Algebra, analytic geometry, exponentials, logarithms, beyond usual coverage found in threeyear high school mathematics program.
1051 Description
MATH 1131  Finite Mathematics (MATH)
3.0 cr; Prereq3 1/2 yrs high school math or grade of at least C in [1031 or 1051]; fall, spring, every year
Financial mathematics, probability, linear algebra, linear programming, Markov chains, some elementary computer programming.
MATH 1142  Short Calculus (MATH)
4.0 cr; =[MATH 1271, MATH 1281, MATH 1371, MATH 1571H]; Prereq3 1/2 yrs high school math or grade of at least C in [1031 or 1051]; fall, spring, summer, every year
Derivatives, integrals, differential equations, partial derivatives, maxima/minima of functions of several variables covered with less depth than full calculus. No trigonometry included.
1142 Description
MATH 1143  Introduction to Advanced Mathematics
4.0 cr; Prereq1142 or 1272 or 1372 or #; recommended especially for students in [social/biological sciences, business]; AF or Aud, fall
Topics that are covered in more depth in 2243 and 2263, plus probability theory. Matrices, eigenvectors, conditional probability, independence, distributions, basic statistical tools, linear regression. Linear differential equations and systems of differential equations. Multivariable differentiability and linearization.
MATH 1151  Precalculus II (MATH)
3.0 cr; Prereq3 1/2 yrs high school math or placement exam or grade of at least C in [1031 or 1051]; Credit will not be granted if credit has been received for: 1155; fall, spring, every year
Algebra, analytic geometry, trigonometry, complex numbers, beyond usual coverage found in threeyear high school mathematics program.
1151 Description
MATH 1155  Intensive Precalculus (MATH)
5.0 cr; Prereq3 yrs high school math or placement exam or grade of at least C in GC 0731; Credit will not be granted if credit has been received for: 1031, 1051, 1151; fall, spring, summer, every year
Algebra, analytic geometry, exponentials, logarithms, trigonometry, complex numbers, beyond usual coverage found in threeyear high school mathematics program. One semester version of 10511151.
1155 Description
MATH 1271  Calculus I (MATH)
4.0 cr; =[MATH 1142, MATH 1281, MATH 1371, MATH 1571H]; Prereq4 yrs high school math including trig or placement test or grade of at least C in 1151 or 1155; fall, spring, every year
Differential calculus of functions of a single variable. Introduction to integral calculus of a single variable, separable differential equations. Applications: maxmin, related rates, area, volume, arclength.
1271 Description
MATH 1272  Calculus II
4.0 cr; =[MATH 1252, MATH 1282, MATH 1372, MATH 1572H]; Prereq[1271 or equiv] with grade of at least C; fall, spring, summer, every year
Techniques of integration. Calculus involving transcendental functions, polar coordinates. Taylor polynomials, vectors/curves in space, cylindrical/spherical coordinates.
1272 Description
MATH 1281  Calculus with Biological Emphasis I (MATH)
4.0 cr; =[MATH 1142, MATH 1271, MATH 1371, MATH 1571H]; Prereq[[four yrs high school math including trigonometry] or [grade of at least C in [1151 or 1155]] or placement exam], [instr or @]; fall, every year
Differential calculus of singlevariable functions, basics of integral calculus. Applications emphasizing biological sciences.
MATH 1282  Calculus With Biological Emphasis II
4.0 cr; =[MATH 1252, MATH 1272, MATH 1372, MATH 1572H]; Prereq[1271 or 1281 or 1371] with grade of at least C; spring, every year
Techniques/applications of integration, differential equations/systems, matrix algebra, basics of multivariable calculus. Applications emphasizing biology.
MATH 1371  CSE Calculus I (MATH)
4.0 cr; =[MATH 1142, MATH 1271, MATH 1281, MATH 1571H]; PrereqCSE, background in [precalculus, geometry, visualization of functions/graphs], #; familiarity with graphing calculators recommended; fall, every year
Differentiation of singlevariable functions, basics of integration of singlevariable functions. Applications: maxmin, related rates, area, curvesketching. Emphasizes use of calculator, cooperative learning.
MATH 1372  CSE Calculus II
4.0 cr; =[MATH 1252, MATH 1272, MATH 1282, MATH 1572H]; PrereqCSE, grade of at least C in 1371; spring, every year
Techniques of integration. Calculus involving transcendental functions, polar coordinates, Taylor polynomials, vectors/curves in space, cylindrical/spherical coordinates. Emphasizes use of calculators, cooperative learning.
MATH 1461H  Honors Calculus IA for Secondary students (MATH)
2.0 cr; PrereqHigh school student, #; fall, every year
Accelerated sequence. Functions, parametric equations and polar coordinates, and vectors are presented using a geometric approach. Limits/continuity, derivates.
MATH 1462H  Honors Calculus IB for Secondary Students (MATH)
3.0 cr; PrereqHigh school student, #; spring, every year
Accelerated sequence. Differentiation, foundations of integration. Proofs, formal reasoning.
MATH 1471H  Honors Calculus I for Secondary Students (MATH)
5.0 cr; PrereqHigh school student, #; fall, every year
Differentiation/integration of singlevariable functions. Emphasizes concepts/explorations.
MATH 1472H  Honors Calculus II for Secondary Students
5.0 cr; Prereq1471H; fall, every year
Sequences/series, vector functions, differentiation in multivariable calculus. Introduction to firstorder systems of differential equations. Emphasizes concepts/explorations.
MATH 1473H  Honors Calculus IIA for Secondary Students
2.0 cr; Prereqhonors; fall, every year
Accelerated honors sequence for selected mathematically talented high school students. Introduction to linear methods and first order differential equations.
MATH 1474H  Honors Calculus IIB for Secondary Students
3.0 cr; Prereqhonors; spring, every year
Accelerated honors sequence for selected mathematically talented high school stduents. Multivariable calculus through differentiation. Focuses on proofs and formal reasoning.
MATH 1571H  Honors Calculus I (MATH)
4.0 cr [max 5.0 cr]; =[MATH 1142, MATH 1271, MATH 1281, MATH 1371]; PrereqCSE Honors office approval; fall, every year
Differential/integral calculus of functions of a single variable. Emphasizes hard problemsolving rather than theory.
MATH 1572H  Honors Calculus II
4.0 cr [max 5.0 cr]; =[MATH 1252, MATH 1272, MATH 1282, MATH 1372]; PrereqGrade of at least C in 1571, CSE Honors Office approval; parts of this sequence may be taken for cr by students who have taken nonhonors calc classes; fall, spring, every year
Continuation of 1571. Infinite series, differential calculus of several variables, introduction to linear algebra.
MATH 2001  Actuarial Science Seminar
1.0 cr; Prereq1272 or equiv; SN or Aud, spring, every year
Actuarial science as a subject and career. Guest lectures by actuaries. Resume preparation and interviewing skills. Review and practice for actuarial exams.
MATH 2066  Elementary Differential Equations
1.0  4.0 cr [max 4.0 cr]
Not taught: merely provides credit for transfer students who have taken a sophomorelevel differential equations class that does not contain enough linear algebra to qualify for credit for 2243.
MATH 2142  Elementary Linear Algebra
1.0  4.0 cr [max 1.0 cr]; AF or Aud
Not taught: merely provides credit for transfer students who have taken a sophomorelevel linear algebra course that does not contain enough differential equations to qualify for credit for 2243.
MATH 2243  Linear Algebra and Differential Equations
4.0 cr; =[MATH 2373]; Prereq1272 or 1282 or 1372 or 1572; fall, spring, summer, every year
Linear algebra: basis, dimension, matrices, eigenvalues/eigenvectors. Differential equations: firstorder linear, separable; secondorder linear with constant coefficients; linear systems with constant coefficients.
2243 Description
MATH 2263  Multivariable Calculus
4.0 cr; =[MATH 2374, MATH 2573H, MATH 3251]; Prereq1272 or 1372 or 1572; fall, spring, summer, every year
Derivative as a linear map. Differential/integral calculus of functions of several variables, including change of coordinates using Jacobians. Line/surface integrals. Gauss, Green, Stokes Theorems.
2263 Description
MATH 2283  Sequences, Series, and Foundations
3.0 cr; =[MATH 3283W]; Prereq& [2243 or 2263 or 2373 or 2374]; fall, spring, every year
Introduction to mathematical reasoning used in advanced mathematics. Elements of logic. Mathematical induction. Real number system. General, monotone, recursively defined sequences. Convergence of infinite series/sequences. Taylor's series. Power series with applications to differential equations. Newton's method.
MATH 2373  CSE Linear Algebra and Differential Equations
4.0 cr; =[MATH 2243]; Prereq[1272 or 1282 or 1372 or 1572], CSE; fall, spring, every year
Linear algebra: basis, dimension, eigenvalues/eigenvectors. Differential Equations: linear equations/systems, phase space, forcing/resonance, qualitative/numerical analysis of nonlinear systems, Laplace transforms. Emphasizes use of computer technology.
MATH 2374  CSE Multivariable Calculus and Vector Analysis
4.0 cr; =[MATH 2263, MATH 2573H, MATH 3251]; Prereq[1272 or 1282 or 1372 or 1572], CSE; fall, spring, every year
Derivative as a linear map. Differential/integral calculus of functions of several variables, including change of coordinates using Jacobians. Line/surface integrals. Gauss, Green, Stokes theorems. Emphasizes use of computer technology.
MATH 2473H  Honors Calculus III for Secondary Students
3.0 cr [max 5.0 cr]; Prereq1472H; fall, every year
Multivariable integration, vector analysis, nonhomogeneous linear equations, nonlinear systems of equations. Emphasizes concepts/explorations.
MATH 2474H  Advanced Topics for Secondary Students
3.0 cr; Prereq2473H; spring, every year
Topics may include linear algebra, combinatorics, advanced differential equations, probability/statistics, numerical analysis, dynamical systems, topology/geometry. Emphasizes concepts/explorations.
MATH 2573H  Honors Calculus III
4.0 cr [max 5.0 cr]; =[MATH 2263, MATH 2374, MATH 3251]; Prereq1572 or CSE Honors office approval; fall, spring, every year
Integral calculus of several variables. Vector analysis, including theorems of Gauss, Green, Stokes.
MATH 2574H  Honors Calculus IV
4.0 cr; Prereq[2573 or equiv], CSE Honors office approval; fall, spring, every year
Advanced linear algebra, differential equations. Additional topics as time permits.
MATH 2582H  Honors Calculus II: Advanced Placement
5.0 cr; Prereq?; AF or Aud, fall, every year
First semester of integrated three semester sequence covering infinite series, multivariable calculus (including vector analysis with Gauss, Green and Stokes theorems, linear algebra (with vector spaces), ODE, and introduction to complex analysis. Material is covered at a faster pace and at a somewhat deeper level than the regular honors sequence.
MATH 2583H  Honors Calc 3  Adv Placement
5.0 cr; Prereq2582H or #; AF or Aud
Second semester of threesemester sequence. Infinite series. Multivariable calculus including vector analysis with Gauss, Green, and Stokes theorems. Linear algebra (with vector spaces), ODE, and introduction to complex analysis. Material is covered at faster pace and deeper level than in regular honors sequence.
MATH 2999  Special Exam
1.0 cr; summer
Special exam.
MATH 3113  Topics in Elementary Mathematics I
4.0 cr; Prereq[Grade of at least C in 1031] or placement exam; fall, spring, summer, every year
Arithmetic/geometric sequences. Counting, building on techniques from college algebra. Graph theory. Integers, rational numbers; emphasizes aspects related to prime factorization. Modular arithmetic with applications.
MATH 3116  Topics in Elementary Math II: Short Course
2.0 cr; PrereqGrade of at least C in 3113; AF or Aud, fall, spring, summer, every year
Probability/Statistics, vector geometry, real/complex numbers. Meets during first half of semester only.
MATH 3118  Topics in Elementary Mathematics II
4.0 cr; PrereqGrade of at least C in 3113; fall, spring, every year
Probability/statistics, vector geometry, real/complex numbers, finite fields building on previously learned modular arithmetic, trees.
MATH 3283W  Sequences, Series, and Foundations: Writing Intensive (WI)
4.0 cr; =[MATH 2283]; Prereq& [2243 or 2263 or 2373 or 2374]; fall, spring, every year
Introduction to reasoning used in advanced mathematics courses. Logic, mathematical induction, real number system, general/monotone/recursively defined sequences, convergence of infinite series/sequences, Taylor's series, power series with applications to differential equations, Newton's method. Writingintensive component.
MATH 3584H  Honors Calculus IV: Advanced Placement
5.0 cr; Prereq[2583 or equiv], CSE Honors office approval
Advanced linear algebra, differential equations. Introduction to complex analysis.
MATH 3592H  Honors Mathematics I
5.0 cr; Prereq?; for students with mathematical talent; AF or Aud, fall, every year
First semester of threesemester sequence. Focuses on multivariable calculus at deeper level than regular calculus offerings. Rigorous introduction to sequences/series. Theoretical treatment of multivariable calculus. Strong introduction to linear algebra.
MATH 3593H  Honors Mathematics II
5.0 cr; Prereq3592H or #; AF or Aud, spring, every year
Second semester of threesemester sequence. Focuses on multivariable calculus at deeper level than regular calculus offerings. Rigorous introduction to sequences/series. Theoretical treatment of multivariable calculus. Strong introduction to linear algebra.
MATH 4005  Calculus Refresher
4.0 cr; Prereq?; AF or Aud
Review of firstyear calculus. Functions of one variable. Limits. Differentiation/integration of functions of one variable. Some applications, including maxmin, related rates. Volume and surface area of solids of revolution. Vectors/curves in the plane and in space.
MATH 4065  Theory of Interest
3.0 cr; Prereq1272 or 1372 or 1572; primarily for [mathematics, business] majors interested in actuarial science; fall, spring, every year
Time value of money. Annuities, sinking funds, bonds, similar items.
MATH 4113  Topics in Elementary Mathematics I
4.0 cr; Prereq[Grade of at least C in 1031] or placement exam; fall, spring, summer
Arithmetic/geometric sequences. Counting, building on techniques from college algebra. Graph Theory. Integers, rational numbers; emphasizes aspects related to prime factorization. Modular arithmetic with applications. Grading standard onethird higher than 3113.
MATH 4116  Topics in Elementary Math II: Short Course
2.0 cr; PrereqGrade of at least C in 4113; AF or Aud
Probability/Statistics, vector geometry, real/complex numbers. Meets during first half of semester only. Grading standard onethird higher than 3116.
MATH 4118  Topics in Elementary Mathematics II
4.0 cr; PrereqGrade of at least C in 4113; spring, every year
Probability/statistics, vector geometry, real/complex numbers, finitefields building on previously learned modular arithmetic, trees. Grading standard onethird higher than 3118.
MATH 4151  Elementary Set Theory
3.0 cr; PrereqOne soph math course or #; fall, every year
Basic properties of operations on sets, cardinal numbers, simply and wellordered sets, ordinal numbers, axiom of choice, axiomatics.
MATH 4152  Elementary Mathematical Logic
3.0 cr; =[MATH 5165]; Prereqone soph math course or #; spring, every year
Propositional logic. Predicate logic: notion of a first order language, a deductive system for first order logic, first order structures, Godel's completeness theorem, axiom systems, models of formal theories.
MATH 4242  Applied Linear Algebra
4.0 cr; =[MATH 4457]; Prereq2243 or 2373 or 2573; fall, spring, summer, every year
Systems of linear equations, vector spaces, subspaces, bases, linear transformations, matrices, determinants, eigenvalues, canonical forms, quadratic forms, applications.
MATH 4281  Introduction to Modern Algebra
4.0 cr; Prereq2283 or 3283 or #
Equivalence relations, greatest common divisor, prime decomposition,modular arithmetic, groups, rings, fields, Chinese remainder theorem,matrices over commutative rings, polynomials over fields.
MATH 4428  Mathematical Modeling
4.0 cr; Prereq2243 or 2373 or 2573; spring, every year
Modeling techniques for analysis/decisionmaking in industry. Optimization (sensitivity analysis, Lagrange multipliers, linear programming). Dynamical modeling (steadystates, stability analysis, eigenvalue methods, phase portraits, simulation). Probabilistic methods (probability/statistical models, Markov chains, linear regression, simulation).
MATH 4457  Methods of Applied Mathematics I
4.0 cr; =[MATH 4242]; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; fall, every year
Vector spaces, minimization principles, least squares approximation, orthogonal bases, linear functions, linear systems of ordinary differential equations. Applications include statics/dynamics of electrical circuits, mechanical structures. Stability/resonance, approximation/interpolation of data. Numerical methods and geometry.
MATH 4458  Methods of Applied Mathematics II
4.0 cr; Prereq4457; spring
Boundary value problems, partial differential equations, complex variables, dynamical systems, calculus of variations, numerical methods. Green's functions, delta functions, Fourier series/integrals, wavelets, conformal mapping, finite elements/differences. Applications: fluid/continuum mechanics, heat flow, signal processing, quantum mechanics.
MATH 4512  Differential Equations with Applications
3.0 cr; Prereq2243 or 2373 or 2573; fall, spring, every year
Laplace transforms, series solutions, systems, numerical methods, plane autonomous systems, stability.
MATH 4567  Applied Fourier Analysis
4.0 cr; Prereq2243 or 2373 or 2573; fall, spring, every year
Fourier series, integral/transform. Convergence. Fourier series, transform in complex form. Solution of wave, heat, Laplace equations by separation of variables. SturmLiouville systems, finite Fourier, fast Fourier transform. Applications. Other topics as time permits.
MATH 4603  Advanced Calculus I
4.0 cr; Prereq[2243 or 2373], [2263 or 2374] or 2574 or # ; fall, spring, summer every year
Axioms for the real numbers. Techniques of proof for limits, continuity, uniform convergence. Rigorous treatment of differential/integral calculus for singlevariable functions.
MATH 4604  Advanced Calculus II
4.0 cr; Prereq4603 or 5615 or # ; spring, every year
Sequel to MATH 4603. Topology of ndimensional Euclidian space. Rigorous treatment of multivariable differentiation and integration,
including chain rule, Taylor's Theorem, implicit function theorem, Fubini's Theorem, change of variables, Stokes' Theorem. Effective: Spring 2011.
MATH 4606  Advanced Calculus
4.0 cr; Prereq[2263 or 2374 or 2573], [2283 or 2574 or 3283 or #]; Credit will not be granted if credit has been received for:5615; fall, spring, summer, every year
Axioms for the real numbers. Techniques of proof for limit theorems, continuity, uniform convergence. Rigorous treatment of differential/integral calculus for single/multivariable functions.
MATH 4653  Elementary Probability
4.0 cr; Prereq[2263 or 2374 or 2573]; [2283 or 2574 or 3283] recommended; fall, spring, every year
Probability spaces, distributions of discrete/continuous random variables, conditioning. Basic theorems, calculational methodology. Examples of random sequences. Emphasizes problemsolving.
MATH 4707  Introduction to Combinatorics and Graph Theory
4.0 cr; Prereq2243, [2283 or 3283]; Credit will not be granted if credit has been received for: 5705, 5707; fall, spring, every year
Existence, enumeration, construction, algorithms, optimization. Pigeonhole principle, bijective combinatorics, inclusionexclusion, recursions, graph modeling, isomorphism. Degree sequences and edge counting. Connectivity, Eulerian graphs, trees, Euler.s formula, network flows, matching theory. Emphasizes mathematical induction as proof technique.
MATH 4990  Topics in Mathematics
1.0  4.0 cr [max 12.0 cr]; fall, spring, summer, every year
MATH 4991  Independent Study
1.0  4.0 cr [max 12.0 cr]; fall, spring, summer, every year
MATH 4992  Directed Reading
1.0  4.0 cr [max 12.0 cr]; fall, spring, summer, every year
MATH 4993  Directed Study
1.0  4.0 cr [max 12.0 cr]; fall, spring, summer, every year
MATH 4995  Senior Project for CLA
1.0 cr; Prereq2 sem of upper div math, ?; AF or Aud, fall, spring, summer, every year
Directed study. May consist of paper on specialized area of math or original computer program or other approved project. Covers some math that is new to student. Scope/topic vary with instructor.
MATH 4997W  Senior Project  Writing Intensive (WI)
1.0 cr [max 2.0 cr]; Prereq2 sem upper div math, ?; AF or Aud, fall, spring, summer, every year
Directed study. A 1015 page paper on a specialized area, including some math that is new to student. At least two drafts of paper given to instructor for feedback before final version. Student keeps journal of preliminary work on project. Scope/topic vary with instructor.
MATH 5067  Actuarial Mathematics I
4.0 cr; Prereq4065, [one sem [4xxx or 5xxx] [probability or statistics] course]; fall, every year
Future lifetime random variable, survival function. Insurance, life annuity, future loss random variables. Net single premium, actuarial present value, net premium, net reserves.
MATH 5068  Actuarial Mathematics II
4.0 cr; Prereq5067; spring, every year
Multiple decrement insurance, pension valuation. Expense analysis, gross premium, reserves. Problem of withdrawals. Regulatory reserving systems. Minimum cash values. Additional topics at instructor's discretion.
MATH 5075  Mathematics of Options, Futures, and Derivative Securities I
4.0 cr; PrereqTwo yrs calculus, basic computer skills; fall, every year
Mathematical background (e.g., partial differential equations, Fourier series, computational methods, BlackScholes theory, numerical methodsincluding Monte Carlo simulation). Interestrate derivative securities, exotic options, risk theory. First course of twocourse sequence.
MATH 5076  Mathematics of Options, Futures, and Derivative Securities II
4.0 cr; Prereq5075; AF or Aud, spring, every year
Mathematical background such as partial differential equations, Fourier series, computational methods, BlackScholes theory, numerical methods (including Monte Carlo simulation), interestrate derivative securities, exotic options, risk theory.
MATH 5165  Mathematical Logic I
4.0 cr; =[MATH 4152]; Prereq2283 or 3283 or Phil 5201 or CSci course in theory of algorithms or #; fall, every year
Theory of computability: notion of algorithm, Turing machines, primitive recursive functions, recursive functions, Kleene normal form, recursion theorem. Propositional logic.
MATH 5166  Mathematical Logic II
4.0 cr; Prereq5165; spring, every year
Firstorder logic: provability/truth in formal systems, models of axiom systems, Godel's completeness theorem. Godel's incompleteness theorem: decidable theories, representability of recursive functions in formal theories, undecidable theories, models of arithmetic.
MATH 5248  Cryptology and Number Theory
4.0 cr; Prereq2 sems soph math; fall, every year
Classical cryptosystems. Onetime pads, perfect secrecy. Public key ciphers: RSA, discrete log. Euclidean algorithm, finite fields, quadratic reciprocity. Message digest, hash functions. Protocols: key exchange, secret sharing, zeroknowledge proofs. Probablistic algorithms: pseudoprimes, prime factorization. Pseudorandom numbers. Elliptic curves.
MATH 5251  ErrorCorrecting Codes, Finite Fields, Algebraic Curves
4.0 cr; Prereq2 sems soph math; spring, every year
Information theory: channel models, transmission errors. Hamming weight/distance. Linear codes/fields, check bits. Error processing: linear codes, Hamming codes, binary Golay codes. Euclidean algorithm. Finite fields, BoseChaudhuriHocquenghem codes, polynomial codes, Goppa codes, codes from algebraic curves.
MATH 5285H  Honors: Fundamental Structures of Algebra I
4.0 cr; Prereq[2243 or 2373 or 2573], [2283 or 2574 or 3283]; fall, every year
Review of matrix theory, linear algebra. Vector spaces, linear transformations over abstract fields. Group theory, includingnormal subgroups, quotient groups, homomorphisms, class equation, Sylow's theorems. Specific examples: permutation groups, symmetry groups of geometric figures, matrix groups.
MATH 5286H  Honors: Fundamental Structures of Algebra II
4.0 cr; Prereq5285; fall, spring, every year
Ring/module theory, including ideals, quotients, homomorphisms,domains (unique factorization, euclidean, principal ideal), fundamental theorem for finitely generated modules over euclidean domains, Jordan canonical form. Introduction to field theory, including finite fields,algebraic/transcendental extensions, Galois theory.
MATH 5335  Geometry I
4.0 cr; Prereq[2243 or 2373 or 2573], [& 2263 or & 2374 or & 2574]; fall, every year
Advanced twodimensional Euclidean geometry from a vector viewpoint. Theorems/problems about triangles/circles, isometries, connections with Euclid's axioms. Hyperbolic geometry, how it compares with Euclidean geometry.
MATH 5336  Geometry II
4.0 cr; Prereq5335; spring, every year
Projective geometry, including: relation to Euclidean geometry, finitegeometries, fundamental theorem of projective geometry. NdimensionalEuclidean geometry from a vector viewpoint. Emphasizes N=3, including: polyhedra, spheres, isometries.
MATH 5345H  Honors: Introduction to Topology
4.0 cr; Prereq[2263 or 2374 or 2573], [& 2283 or & 2574 or & 3283]; fall, every year
Set theory. Euclidean/metric spaces. Basics of general topology, including compactness/connectedness.
MATH 5378  Differential Geometry
4.0 cr; Prereq[2263 or 2374 or 2573], [2243 or 2373 or 2574]; [2283 or 3283] recommended]; spring, every year
Basic geometry of curves in plane and in space, including Frenet formula, theory of surfaces, differential forms, Riemannian geometry.
MATH 5385  Introduction to Computational Algebraic Geometry
4.0 cr; Prereq[2263 or 2374 or 2573], [2243 or 2373 or 2574]; fall, every year
Geometry of curves/surfaces defined by polynomial equations. Emphasizes concrete computations with polynomials using computer packages, interplay between algebra and geometry. Abstract algebra presented as needed.
MATH 5445  Mathematical Analysis of Biological Networks
4.0 cr; PrereqLinear algebra, differential equations; spring, every year
Development/analysis of models for complex biological networks. Examples taken from signal transduction networks, metabolic networks, gene control networks, and ecological networks.
MATH 5447  Theoretical Neuroscience
4.0 cr; Prereq[2243 or 2373 or 2573], familiarity with some programming language; fall, every year
Nonlinear dynamical system models of neurons and neuronal networks. Computation by excitatory/inhibitory networks. Neural oscillations, adaptation, bursting, synchrony. Memory systems.
MATH 5467  Introduction to the Mathematics of Image and Data Analysis
4.0 cr; Prereq[2243 or 2373 or 2573], [2283 or 2574 or 3283 or #]; [[2263 or 2374], 4567] recommended; spring, every year
Background theory/experience in wavelets. Inner product spaces, operator theory, Fourier transforms applied to Gabor transforms, multiscale analysis, discrete wavelets, selfsimilarity. Computing techniques.
MATH 5481  Mathematics of Industrial Problems I
4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574], familiarity with some programming language; fall, every year
Topics in industrial math, including crystal precipitation, air quality modeling, electron beam lithography. Problems treated both theoretically and numerically.
MATH 5482  Mathematics of Industrial Problems II
4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574], familiarity with some programming language; spring
Topics in industrial math, including color photography, catalytic converters, photocopying.
MATH 5485  Introduction to Numerical Methods I
4.0 cr; Prereq[2243 or 2373 or 2573], familiarity with some programming language; fall, every year
Solution of nonlinear equations in one variable. Interpolation, polynomial approximation, numerical integration/differentiation, numerical solution of initialvalue problems.
MATH 5486  Introduction To Numerical Methods II
4.0 cr; Prereq5485; spring, every year
Direct/iterative methods for solving linear systems, approximation theory, methods for eigenvalue problems, methods for systems of nonlinear equations, numerical solution of boundary value problems for ordinary differential equations.
MATH 5487  Computational Methods for Differential and Integral Equations in Engineering and Science I
4.0 cr; Prereq4242
Numerical methods for elliptic partial differential equations, integral equations of engineering and science. Methods include finite element, finite difference, spectral, boundary integral.
MATH 5488  Computational Methods for Differential and Integral Equations in Engineering and Science II
4.0 cr; Prereq5487
Numerical methods for timedependent partial differential equations of engineering/science. Methods include finite element, finite difference, spectral, boundary integral. Applications to fluid flow, elasticity, electromagnetism.
MATH 5490  Topics in Applied Mathematics
4.0 cr [max 12.0 cr]; fall, spring, offered periodically
Topics vary by instructor. See class schedule.
MATH 5525  Introduction to Ordinary Differential Equations
4.0 cr; Prereq[2243 or 2373 or 2573], [2283 or 2574 or 3283]; fall, spring
Ordinary differential equations, solution of linear systems, qualitative/numerical methods for nonlinear systems. Linear algebra background, fundamental matrix solutions, variation of parameters, existence/uniqueness theorems, phase space. Rest points, their stability. Periodic orbits, PoincareBendixson theory, strange attractors.
MATH 5535  Dynamical Systems and Chaos
4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; fall, spring, every year
Dynamical systems theory. Emphasizes iteration of onedimensional mappings. Fixed points, periodic points, stability, bifurcations, symbolic dynamics, chaos, fractals, Julia/Mandelbrot sets.
MATH 5583  Complex Analysis
4.0 cr; =[00070]; Prereq2 sems soph math [including [2263 or 2374 or 2573], [2283 or 3283]] recommended; fall, spring, summer, every year
Algebra, geometry of complex numbers. Linear fractional transformations. Conformal mappings. Holomorphic functions. Theorems of Abel/Cauchy, power series. Schwarz' lemma. Complex exponential, trig functions. Entire functions, theorems of Liouville/Morera. Reflection principle. Singularities, Laurent series. Residues.
MATH 5587  Elementary Partial Differential Equations I
4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; fall, every year
Emphasizes partial differential equations w/physical applications, including heat, wave, Laplace's equations. Interpretations of boundary conditions. Characteristics, Fourier series, transforms, Green's functions, images, computational methods. Applications include wave propagation, diffusions, electrostatics, shocks.
MATH 5588  Elementary Partial Differential Equations II
4.0 cr [max 400.0 cr]; Prereq[[2243 or 2373 or 2573], [2263 or 2374 or 2574], 5587] or #; AF or Aud, spring, every year
Heat, wave, Laplace's equations in higher dimensions. Green's functions, Fourier series, transforms. Asymptotic methods, boundary layer theory, bifurcation theory for linear/nonlinear PDEs. Variational methods. Free boundary problems. Additional topics as time permits.
MATH 5594H  Honors Mathematics  Topics
4.0 cr [max 12.0 cr]; Prereq[3593H with grade of at least B, experience in writing proofs] or ?; intended for mathematicallytalented students with proven achievement in theoretical mathematics courses; AF or Aud
Topics vary depending on interests of instructor. Theoretical treatment of chosen topic.
MATH 5615H  Honors: Introduction to Analysis I
4.0 cr; Prereq[[2243 or 2373], [2263 or 2374], [2283 or 3283]] or 2574; fall, every year
Axiomatic treatment of real/complex number systems. Introduction to metric spaces: convergence, connectedness, compactness. Convergence of sequences/series of real/complex numbers, Cauchy criterion, root/ratio tests. Continuity in metric spaces. Rigorous treatment of differentiation of singlevariable functions, Taylor's Theorem.
MATH 5616H  Honors: Introduction to Analysis II
4.0 cr; Prereq5615; spring, every year
Rigorous treatment of RiemannStieltjes integration. Sequences/series of functions, uniform convergence, equicontinuous families, StoneWeierstrass Theorem, power series. Rigorous treatment of differentiation/integration of multivariable functions, Implicit Function Theorem, Stokes' Theorem. Additional topics as time permits.
MATH 5651  Basic Theory of Probability and Statistics
4.0 cr; Prereq[2263 or 2374 or 2573], [2243 or 2373]; [2283 or 2574 or 3283] recommended; Credit will not be granted if credit has been received for: Stat 4101, Stat 5101.; fall, spring, every year
Logical development of probability, basic issues in statistics. Probability spaces, random variables, their distributions/expected values. Law of large numbers, central limit theorem, generating functions, sampling, sufficiency, estimation.
MATH 5652  Introduction to Stochastic Processes
4.0 cr; Prereq5651 or Stat 5101; fall, spring, every year
Random walks, Markov chains, branching processes, martingales, queuing theory, Brownian motion.
MATH 5654  Prediction and Filtering
4.0 cr; Prereq5651 or Stat 5101; spring, every year
Markov chains, Wiener process, stationary sequences, OrnsteinUhlenbeck process. Partially observable Markov processes (hidden Markov models), stationary processes. Equations for general filters, Kalman filter. Prediction of future values of partially observable processes.
MATH 5705  Enumerative Combinatorics
4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2283 or 2374 or 2574 or 3283]; Credit will not be granted if credit has been received for: 4707; fall, spring, every year
Basic enumeration, bijections, inclusionexclusion, recurrence relations, ordinary/exponential generating functions, partitions, Polya theory. Optional topics include trees, asymptotics, listing algorithms, rook theory, involutions, tableaux, permutation statistics.
MATH 5707  Graph Theory and Nonenumerative Combinatorics
4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; [2283 or 3283 or experience in writing proofs] highly recommended; Credit will not be granted if credit has been received for: 4707; fall, spring, every year
Basic topics in graph theory: connectedness, Eulerian/Hamiltonian properties, trees, colorings, planar graphs, matchings, flows in networks. Optional topics include graph algorithms, Latin squares, block designs, Ramsey theory.
MATH 5711  Linear Programming and Combinatorial Optimization
4.0 cr; Prereq2 sems soph math [including 2243 or 2373 or 2573]; fall, spring, every year
Simplex method, connections to geometry, duality theory,sensitivity analysis. Applications to cutting stock, allocation of resources, scheduling problems. Flows, matching/transportationproblems, spanning trees, distance in graphs, integer programs, branch/bound, cutting planes, heuristics. Applications to traveling salesman, knapsack problems.
MATH 5900  Tutorial in Advanced Mathematics
1.0  6.0 cr [max 120.0 cr]; AF or Aud, fall, spring, summer, every year
Individually directed study.
MATH 5990  Topics in Mathematics
4.0 cr [max 12.0 cr]; fall, spring, offered periodically
Topics vary by instructor. See class schedule.
MATH 8001  Preparation for College Teaching
1.0 cr [max 3.0 cr]; Prereq! math grad student in good standing or #; SN or Aud, fall, spring, every year
New approaches to teaching/learning, issues in mathematics education, components/expectations of a college mathematics professor.
MATH 8141  Applied Logic
3.0 cr; AF or Aud, fall, spring
Applying techniques of mathematical logic to other areas of mathematics and computer science. Sample topics: complexity of computation, computable analysis, unsolvability of diophantine problems, program verification, database theory.
MATH 8142  Applied Logic
3.0 cr; AF or Aud, spring
Applying techniques of mathematical logic to other areas of mathematics, computer science. Complexity of computation, computable analysis, unsolvability of diophantine problems, program verification, database theory.
MATH 8151  Axiomatic Set Theory
3.0 cr; Prereq5166 or #; AF or Aud
Axiomatic development of basic properties of ordinal/cardinal numbers, infinitary combinatorics, well founded sets, consistency of axiom of foundation, constructible sets, consistency of axiom of choice and of generalized continuum hypothesis.
MATH 8152  Axiomatic Set Theory
3.0 cr; Prereq8151 or #; AF or Aud
Notion of forcing, generic extensions, forcing with finite partial functions, independence of continuum hypothesis, forcing with partial functions of infinite cardinalities, relationship between partial orderings and Boolean algebras, Booleanvalued models, independence of axiom of choice.
MATH 8166  Recursion Theory
3.0 cr; PrereqMath grad student or #; AF or Aud
Analysis of concept of computability, including various equivalent definitions. Primitive recursive, recursive, partial recursive functions. Oracle Turing machines. Kleene Normal Form Theorem. Recursive, recursively enumerable sets. Degrees of unsolvability. Arithmetic hierarchy.
MATH 8167  Recursion Theory
3.0 cr; Prereq8166; AF or Aud, spring
Sample topics: complexity theory, recursive analysis, generalized recursion theory, analytical hierarchy, constructive ordinals.
MATH 8172  Model Theory
3.0 cr; PrereqMath grad student or #; AF or Aud
Interplay of formal theories, their models. Elementary equivalence, elementary extensions, partial isomorphisms. LowenheimSkolem theorems, compactness theorems, preservation theorems. Ultraproducts.
MATH 8173  Model Theory
3.0 cr; Prereq8172 or #; AF or Aud
Types of elements. Prime models, homogeneity, saturation, categoricity in power. Forking.
MATH 8190  Topics in Logic
1.0  3.0 cr [max 12.0 cr]; AF or Aud, fall, spring
Offered for one year or one semester as circumstances warrant.
MATH 8201  General Algebra
3.0 cr; Prereq4xxx algebra or equiv or #; AF or Aud, fall, every year
Groups through Sylow, JordanH[o]lder theorems, structure of finitely generated Abelian groups. Rings and algebras, including Gauss theory of factorization. Modules, including projective and injective modules, chain conditions, Hilbert basis theorem, and structure of modules over principal ideal domains.
MATH 8202  General Algebra
3.0 cr; Prereq8201 or #; AF or Aud, spring, every year
Classical field theory through Galois theory, including solvable equations. Symmetric, Hermitian, orthogonal, and unitary form. Tensor and exterior algebras. Basic Wedderburn theory of rings; basic representation theory of groups.
MATH 8207  Theory of Modular Forms and LFunctions
3.0 cr; Prereq8202 or #; AF or Aud
Zeta and Lfunctions, prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, class number formulas; Riemann hypothesis; modular forms and associated Lfunction; Eisenstein series; Hecke operators, Poincar[e] series, Euler products; Ramanujan conjectures; Theta series and quadratic forms; waveforms and Lfunctions.
MATH 8208  Theory of Modular Forms and LFunctions
3.0 cr; Prereq8207 or #; AF or Aud
Applications of Eisenstein series: special values and analytic continuation and functional equations of Lfunctions. Trace formulas. Applications of representation theory. Computations.
MATH 8211  Commutative and Homological Algebra
3.0 cr; Prereq8202 or #; AF or Aud, fall
Selected topics.
MATH 8212  Commutative and Homological Algebra
3.0 cr; Prereq8211 or #; AF or Aud
Selected topics.
MATH 8245  Group Theory
3.0 cr; Prereq8202 or #; AF or Aud, fall, every year
Permutations, Sylow's theorems, representations of groups on groups, semidirect products, solvable and nilpotent groups, generalized Fitting subgroups, pgroups, coprime action on pgroups.
MATH 8246  Group Theory
3.0 cr; Prereq8245 or #; AF or Aud, fall, spring
Representation and character theory, simple groups, free groups and products, presentations, extensions, Schur multipliers.
MATH 8251  Algebraic Number Theory
3.0 cr; Prereq8202 or #; AF or Aud
Algebraic number fields and algebraic curves. Basic commutative algebra. Completions: padic fields, formal power series, Puiseux series. Ramification, discriminant, different. Finiteness of class number and units theorem.
MATH 8252  Algebraic Number Theory
3.0 cr; Prereq8251 or #; AF or Aud
Zeta and Lfunctions of global fields. Artin Lfunctions. HasseWeil Lfunctions. Tchebotarev density. Local and global class field theory. Reciprocity laws. Finer theory of cyclotomic fields.
MATH 8253  Algebraic Geometry
3.0 cr; Prereq8202 or #; AF or Aud, fall
Curves, surfaces, projective space, affine and projective varieties. Rational maps. Blowingup points. Zariski topology. Irreducible varieties, divisors.
MATH 8254  Algebraic Geometry
3.0 cr; Prereq8253 or #; AF or Aud, spring
Sheaves, ringed spaces, and schemes. Morphisms. Derived functors and cohomology, Serre duality. RiemannRoch theorem for curves, Hurwitz's theorem. Surfaces: monoidal transformations, birational transformations.
MATH 8270  Topics in Algebraic Geometry
1.0  3.0 cr [max 12.0 cr]; PrereqMath 8201, Math 8202; offered for one year or one semester as circumstances warrant; AF or Aud, fall, spring, every year
MATH 8271  Lie Groups and Lie Algebras
3.0 cr; Prereq8302 or #; AF or Aud, fall
Definitions and basic properties of Lie groups and Lie algebras; classical matrix Lie groups; Lie subgroups and their corresponding Lie subalgebras; covering groups; MaurerCartan forms; exponential map; correspondence between Lie algebras and simply connected Lie groups; BakerCampbellHausdorff formula; homogeneous spaces.
MATH 8272  Lie Groups and Lie Algebras
3.0 cr; Prereq8271 or #; AF or Aud, spring
Solvable and nilpotent Lie algebras and Lie groups; Lie's and Engels's theorems; semisimple Lie algebras; cohomology of Lie algebras; Whitehead's lemmas and Levi's theorem; classification of complex semisimple Lie algebras and compact Lie groups; representation theory.
MATH 8280  Topics in Number Theory
1.0  3.0 cr [max 12.0 cr]; Prereq#; offered for one year or one semester as circumstances warrant; AF or Aud
MATH 8300  Topics in Algebra
1.0  3.0 cr [max 12.0 cr]; PrereqGrad math major or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, every year
Selected topics.
MATH 8301  Manifolds and Topology
3.0 cr; Prereq[Some pointset topology, algebra] or #; AF or Aud, fall, every year
Classification of compact surfaces, fundamental group/covering spaces. Homology group, basic cohomology. Application to degree of a map, invariance of domain/dimension.
MATH 8302  Manifolds and Topology
3.0 cr; Prereq8301 or #; AF or Aud, spring, every year
Smooth manifolds, tangent spaces, embedding/immersion, Sard's theorem, Frobenius theorem. Differential forms, integration. Curvature, GaussBonnet theorem. Time permitting: de Rham, duality in manifolds.
MATH 8306  Algebraic Topology
3.0 cr; Prereq8301 or #; AF or Aud
Singular homology, cohomology theory with coefficients. EilenbergStenrod axioms, MayerVietoris theorem.
MATH 8307  Algebraic Topology
3.0 cr; Prereq8306 or #; AF or Aud
Basic homotopy theory, cohomology rings with applications. Time permitting: fibre spaces, cohomology operations, extraordinary cohomology theories.
MATH 8333  FTE: Master's
No description
MATH 8360  Topics in Topology
1.0  3.0 cr [max 12.0 cr]; Prereq8301 or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring
Selected topics.
MATH 8365  Riemannian Geometry
3.0 cr; Prereq8301 or basic pointset topology or #; AF or Aud, fall, every year
Riemannian metrics, curvature. Bianchi identities, GaussBonnet theorem, Meyers's theorem, CartanHadamard theorem.
MATH 8366  Riemannian Geometry
3.0 cr; Prereq8365 or #; AF or Aud, spring, every year
Gauss, Codazzi equations. Tensor calculus, Hodge theory, spinors, global differential geometry, applications.
MATH 8370  Topics in Differential Geometry
1.0  3.0 cr [max 12.0 cr]; Prereq8301 or 8365; offered for one yr or one sem as circumstances warrant; AF or Aud, fall, spring, every year
Current research in Differential Geometry.
MATH 8380  Topics in Advanced Geometry
1.0  3.0 cr [max 12.0 cr]; Prereq8301, 8365; AF or Aud, fall, spring
Current research.
MATH 8385  Calculus of Variations and Minimal Surfaces
3.0 cr; Prereq4xxx partial differential equations or #; AF or Aud
Comprehensive exposition of calculus of variations and its applications. Theory for onedimensional problems. Survey of typical problems. Necessary conditions. Sufficient conditions. Second variation, accessory eigenvalue problem. Variational problems with subsidiary conditions. Direct methods.
MATH 8386  Calculus of Variations and Minimal Surfaces
3.0 cr; Prereq8595 or #; AF or Aud
Theory of multiple integrals. Geometrical differential equations, i.e., theory of minimal surfaces and related structures (surfaces of constant or prescribed mean curvature, solutions to variational integrals involving surface curvatures), all extremals for variational problems of current interest as models for interfaces in real materials.
MATH 8387  Mathematical Modeling of Industrial Problems
3.0 cr; Prereq[5xxx numerical analysis, some computer experience] or #; AF or Aud, fall, every year
Mathematical models from physical, biological, social systems. Emphasizes industrial applications. Modeling of deterministic/probabilistic, discrete/continuous processes; methods for analysis/computation.
MATH 8388  Mathematical Modeling of Industrial Problems
3.0 cr; Prereq8597 or #; AF or Aud
Techniques for analysis of mathematical models. Asymptotic methods; design of simulation and visualization techniques. Specific computation for models arising in industrial problems.
MATH 8390  Topics in Mathematical Physics
1.0  3.0 cr [max 12.0 cr]; Prereq8601; offered for one yr or one sem as circumstances warrant; AF or Aud
Current research.
MATH 8401  Mathematical Modeling and Methods of Applied Mathematics
3.0 cr; Prereq4xxx numerical analysis and applied linear algebra or #; AF or Aud, fall, every year
Dimension analysis, similarity solutions, linearization, stability theory, wellposedness, and characterization of type. Fourier series and integrals, wavelets, Green's functions, weak solutions and distributions.
MATH 8402  Mathematical Modeling and Methods of Applied Mathematics
3.0 cr; Prereq8401 or #; AF or Aud, spring, every year
Calculus of variations, integral equations, eigenvalue problems, spectral theory. Perturbation, asymptotic methods. Artificial boundary conditions, conformal mapping, coordinate transformations. Applications to specific modeling problems.
MATH 8431  Mathematical Fluid Mechanics
3.0 cr; Prereq5xxx numerical analysis of partial differential equations or #; AF or Aud
Equations of continuity/motion. Kinematics. Bernoulli's theorem, stream function, velocity potential. Applications of conformal mapping.
MATH 8432  Mathematical Fluid Mechanics
3.0 cr; Prereq8431 or #
Plane flow of gas, characteristic method, hodograph method. Singular surfaces, shock waves, shock layers. Viscous flow, NavierStokes equations, exact solutions. Uniqueness, stability, existence theorems.
MATH 8441  Numerical Analysis and Scientific Computing
3.0 cr; Prereq[4xxx analysis, 4xxx applied linear algebra] or #; fall, every year
Approximation of functions, numerical integration. Numerical methods for elliptic partial differential equations, including finite element methods, finite difference methods, and spectral methods. Grid generation.
MATH 8442  Numerical Analysis and Scientific Computing
3.0 cr; Prereq8441 or #; 54775478 recommended for engineering and science grad students; spring, every year
Numerical methods for integral equations, parabolic partial differential equations, hyperbolic partial differential equations. Monte Carlo methods.
MATH 8444  FTE: Doctoral
No description
MATH 8445  Numerical Analysis of Differential Equations
3.0 cr; Prereq4xxx numerical analysis, 4xxx partial differential equations or #; AF or Aud, fall, every year
Finite element and finite difference methods for elliptic boundary value problems (e.g., Laplace's equation) and solution of resulting linear systems by direct and iterative methods.
MATH 8446  Numerical Analysis of Differential Equations
3.0 cr; Prereq8445 or #; AF or Aud, spring, every year
Numerical methods for parabolic equations (e.g., heat equations). Methods for elasticity, fluid mechanics, electromagnetics. Applications to specific computations.
MATH 8450  Topics in Numerical Analysis
1.0  3.0 cr [max 12.0 cr]; PrereqGrad math major or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, every year
Selected topics.
MATH 8470  Topics in Mathematical Theory of Continuum Mechanics
1.0  3.0 cr [max 12.0 cr]; AF or Aud, fall, spring
Offered for one year or one semester as circumstances warrant.
MATH 8501  Theory of Ordinary Differential Equations
3.0 cr; Prereq4xxx ODE or #; AF or Aud, fall, every year
Existence, uniqueness, continuity, and differentiability of solutions. Linear theory and hyperbolicity. Basics of dynamical systems. Local behavior near a fixed point, a periodic orbit, and a homoclinic or heteroclinic orbit. Perturbation theory.
MATH 8502  Dynamical Systems and Differential Equations
3.0 cr; Prereq8501 or #; AF or Aud, spring, every year
Selected topics: stable, unstable, and center manifolds. Normal hyperbolicity. Nonautonomous dynamics and skew product flows. Invariant manifolds and quasiperiodicity. Transversality and Melnikov method. Approximation dynamics. MorseSmale systems. Coupled oscillators and network dynamics.
MATH 8503  Bifurcation Theory in Ordinary Differential Equations
3.0 cr; Prereq8501 or #; AF or Aud
Basic bifurcation theory, Hopf bifurcation, and method averaging. Silnikov bifurcations. Singular perturbations. Higher order bifurcations. Applications.
MATH 8505  Applied Dynamical Systems and Bifurcation Theory I
3.0 cr; Prereq5525 or 8502 or #; AF or Aud
Static/Hopf bifurcations, invariant manifold theory, normal forms, averaging, Hopf bifurcation in maps, forced oscillations, coupled oscillators, chaotic dynamics, codimension 2 bifurcations. Emphasizes computational aspects/applications from biology, chemistry, engineering, physics.
MATH 8506  Applied Dynamical Systems and Bifurcation Theory II
3.0 cr; Prereq5587 or #; AF or Aud, fall
Background on analysis in Banach spaces, linear operator theory. LyapunovSchmidt reduction, static bifurcation, stability at a simple eigenvalue, Hopf bifurcation in infinite dimensions invariant manifold theory. Applications to hydrodynamic stability problems, reactiondiffusion equations, pattern formation, and elasticity.
MATH 8520  Topics in Dynamical Systems
1.0  3.0 cr [max 12.0 cr]; Prereq8502; AF or Aud, fall, spring
Current research.
MATH 8530  Topics in Ordinary Differential Equations
1.0  3.0 cr [max 3.0 cr]; Prereq8502; AF or Aud, fall, spring
Offered for one year or one semester as circumstances warrant.
MATH 8540  Topics in Mathematical Biology
1.0  3.0 cr [max 12.0 cr]; AF or Aud, fall, spring, every year
Offered for one year or one semester as circumstances warrant.
MATH 8571  Theory of Evolutionary Equations
3.0 cr; Prereq8502 or #; AF or Aud, fall, every year
Infinite dimensional dynamical systems, global attractors, existence and robustness. Linear semigroups, analytic semigroups. Linear and nonlinear reaction diffusion equations, strong and weak solutions, wellposedness of solutions.
MATH 8572  Theory of Evolutionary Equations
3.0 cr; Prereq8571 or #; AF or Aud, spring
Dynamics of NavierStokes equations, strong/weak solutions, global attractors. Chemically reacting fluid flows. Dynamics in infinite dimensions, unstable manifolds, center manifolds perturbation theory. Inertial manifolds, finite dimensional structures. Dynamical theories of turbulence.
MATH 8580  Topics in Evolutionary Equations
1.0  3.0 cr [max 12.0 cr]; Prereq8572 or #; offered for one yr or one semester as circumstances warrant; AF or Aud
MATH 8581  Applications of Linear Operator Theory
3.0 cr; Prereq4xxx applied mathematics or #; AF or Aud
Metric spaces, continuity, completeness, contraction mappings, compactness. Normed linear spaces, continuous linear transformations. Hilbert spaces, orthogonality, projections.
MATH 8582  Applications of Linear Operator Theory
3.0 cr; Prereq8581 or #; AF or Aud
Fourier theory. Selfadjoint, compact, unbounded linear operators. Spectral analysis, eigenvalueeigenvector problem, spectral theorem, operational calculus.
MATH 8583  Theory of Partial Differential Equations
3.0 cr; Prereq[Some 5xxx PDE, 8601] or #; AF or Aud, fall, every year
Classification of partial differential equations/characteristics. Laplace, wave, heat equations. Some mixed problems.
MATH 8584  Theory of Partial Differential Equations
3.0 cr; Prereq8583 or #; AF or Aud, spring, every year
Fundamental solutions/distributions, Sobolev spaces, regularity. Advanced elliptic theory (Schauder estimates, Garding's inequality). Hyperbolic systems.
MATH 8590  Topics in Partial Differential Equations
1.0  3.0 cr [max 3.0 cr]; Prereq8602; offered for one yr or one sem as circumstances warrant; AF or Aud, fall, spring, every year
Research topics.
MATH 8600  Topics in Advanced Applied Mathematics
1.0  3.0 cr [max 12.0 cr]; fall, spring, every year
Offered for one yr or one semester as circumstances warrant. Topics vary. For details, contact instructor.
MATH 8601  Real Analysis
3.0 cr; Prereq5616 or #; AF or Aud, fall, every year
Set theory/fundamentals. Axiom of choice, measures, measure spaces, Borel/Lebesgue measure, integration, fundamental convergence theorems, Riesz representation.
MATH 8602  Real Analysis
3.0 cr; Prereq8601 or #; AF or Aud, spring, every year
RadonNikodym, Fubini theorems. C(X). Lp spaces (introduction to metric, Banach, Hilbert spaces). StoneWeierstrass theorem. Basic Fourier analysis. Theory of differentiation.
MATH 8640  Topics in Real Analysis
3.0 cr [max 12.0 cr]; Prereq8602 or #; offered for one yr or one sem as circumstances warrant; AF or Aud
Current research.
MATH 8641  Spatial Ecology
3.0 cr; PrereqTwo semesters calculus, theoretical population ecology or four semesters more robust calculus, course in statistics or probability or #; SN or Aud
Introduction: role of space in population dynamics and interspecific interaction; includes single species and multispecies models, deterministic and stochastic theory, different modeling approaches, effects of implicit/explicit space on competition, pattern formation, stability diversity and invasion. Recent literature. Computer lab.
MATH 8651  Theory of Probability Including Measure Theory
3.0 cr; Prereq5616 or #; fall, every year
Probability spaces. Distributions/expectations of random variables. Basic theorems of Lebesque theory. Stochastic independence, sums of independent random variables, random walks, filtrations. Probability, moment generating functions, characteristic functions. Laws of large numbers.
MATH 8652  Theory of Probability Including Measure Theory
3.0 cr; Prereq8651 or #; spring, every year
Conditional distributions and expectations, convergence of sequences of distributions on real line and on Polish spaces, central limit theorem and related limit theorems, Brownian motion, martingales and introduction to other stochastic sequences.
MATH 8654  Fundamentals of Probability Theory and Stochastic Processes
3.0 cr; Prereq8651 or 8602 or #; spring
Review of basic theorems of probability for independent random variables; introductions to Brownian motion process, Poisson process, conditioning, Markov processes, stationary processes, martingales, super and submartingales, DoobMeyer decomposition.
MATH 8655  Stochastic Calculus with Applications
3.0 cr; Prereq8654 or 8659 or #; fall, every year
Stochastic integration with respect to martingales, Ito's formula, applications to business models, filtering, and stochastic control theory.
MATH 8659  Stochastic Processes
3.0 cr; Prereq8652 or #; fall, every year
Indepth coverage of various stochastic processes and related concepts, such as Markov sequences and processes, renewal sequences, exchangeable sequences, stationary sequences, Poisson point processes, Levy processes, interacting particle systems, diffusions, and stochastic integrals.
MATH 8660  Topics in Probability
1.0  3.0 cr [max 12.0 cr]; fall, spring, every year
Offered for one year or one semester as circumstances warrant.
MATH 8666  Doctoral PreThesis Credits
No description
MATH 8668  Combinatorial Theory
3.0 cr; AF or Aud, fall
Basic enumeration, including sets and multisets, permutation statistics, inclusionexclusion, integer/set partitions, involutions and Polya theory. Partially ordered sets, including lattices, incidence algebras, and Mobius inversion. Generating functions.
MATH 8669  Combinatorial Theory
3.0 cr; Prereq8668 or #; AF or Aud, spring, odd years
Further topics in enumeration, including symmetric functions, Schensted correspondence, and standard tableaux; nonenumerative combinatorics, including graph theory and coloring, matching theory, connectivity, flows in networks, codes, and extremal set theory.
MATH 8680  Topics in Combinatorics
1.0  3.0 cr [max 12.0 cr]; PrereqGrad math major or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, every year
Selected topics.
MATH 8701  Complex Analysis
3.0 cr; Prereq5616 or #; AF or Aud, fall, every year
Foundations of holomorphic functions of one variable; relation to potential theory, complex manifolds, algebraic geometry, number theory. Cauchy's theorems, Poisson integral. Singularities, series, product representations. Hyperbolic geometry, isometries. Covering surfaces, RiemannHurwitz formula. SchwarzChristoffel polygonal functions. Residues.
MATH 8702  Complex Analysis
3.0 cr; Prereq8701 or #; AF or Aud, spring, every year
Riemann mapping, uniformization, Dirichlet problem. Dirichlet principle, Green's functions, harmonic measures. Approximation theory. Complex analysis on tori (elliptic functions, modular functions, conformal moduli). Complex dynamical systems (Julia sets, Mandelbrot set).
MATH 8777  Thesis Credits: Master's
No description
MATH 8790  Topics in Complex Analysis
1.0  3.0 cr [max 12.0 cr]; Prereq8702 or #; offered for one yr or one sem as circumstances warrant; AF or Aud
Current research.
MATH 8801  Functional Analysis
3.0 cr; Prereq8602 or #; AF or Aud, fall, every year
Motivation in terms of specific problems (e.g., Fourier series, eigenfunctions). Theory of compact operators. Basic theory of Banach spaces (HahnBanach, open mapping, closed graph theorems). Frechet spaces.
MATH 8802  Functional Analysis
3.0 cr; Prereq8801 or #; AF or Aud, spring
Spectral theory of operators, theory of distributions (generalized functions), Fourier transformations and applications. Sobolev spaces and pseudodifferential operators. Cstar algebras (GelfandNaimark theory) and introduction to von Neumann algebras.
MATH 8888  Thesis Credit: Doctoral
No description
MATH 8990  Topics in Mathematics
1.0  6.0 cr [max 24.0 cr]; Prereq#; SN or Aud, fall, spring, every year
Readings, research.
MATH 8991  Independent Study
1.0  6.0 cr [max 24.0 cr]; Prereq#; SN or Aud, spring, summer, every year
Individually directed study.
MATH 8992  Directed Reading
1.0  6.0 cr [max 24.0 cr]; Prereq#; SN or Aud, fall, spring, every year
Individually directed reading.
MATH 8993  Directed Study
1.0  6.0 cr [max 24.0 cr]; Prereq#; SN or Aud, spring, every year
Individually directed study.
