University of Minnesota
School of Mathematics
School of Mathematics         Part of a three-valent tree made fractal-ish.  
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Courses List

MATH 1001 - Excursions in Mathematics (MATH)
3.0 cr; Prereq-3 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; Prereq-Plane geometry, two yrs high school algebra [or C or better in GC 0731]; A-F 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; Prereq-3 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; Prereq-1051 or 1151 or 1155; A-F 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; Prereq-3 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 three-year high school mathematics program.
  1051 Description

MATH 1131 - Finite Mathematics (MATH)
3.0 cr; Prereq-3 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]; Prereq-3 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; Prereq-1142 or 1272 or 1372 or #; recommended especially for students in [social/biological sciences, business]; A-F 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; Prereq-3 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 three-year high school mathematics program.
  1151 Description

MATH 1155 - Intensive Precalculus (MATH)
5.0 cr; Prereq-3 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 three-year high school mathematics program. One semester version of 1051-1151.
  1155 Description

MATH 1271 - Calculus I (MATH)
4.0 cr; =[MATH 1142, MATH 1281, MATH 1371, MATH 1571H]; Prereq-4 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: max-min, related rates, area, volume, arc-length.
  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 single-variable 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]; Prereq-CSE, background in [precalculus, geometry, visualization of functions/graphs], #; familiarity with graphing calculators recommended; fall, every year
Differentiation of single-variable functions, basics of integration of single-variable functions. Applications: max-min, related rates, area, curve-sketching. Emphasizes use of calculator, cooperative learning.

MATH 1372 - CSE Calculus II
4.0 cr; =[MATH 1252, MATH 1272, MATH 1282, MATH 1572H]; Prereq-CSE, 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; Prereq-High 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; Prereq-High 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; Prereq-High school student, #; fall, every year
Differentiation/integration of single-variable functions. Emphasizes concepts/explorations.

MATH 1472H - Honors Calculus II for Secondary Students
5.0 cr; Prereq-1471H; fall, every year
Sequences/series, vector functions, differentiation in multivariable calculus. Introduction to first-order systems of differential equations. Emphasizes concepts/explorations.

MATH 1473H - Honors Calculus IIA for Secondary Students
2.0 cr; Prereq-honors; 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; Prereq-honors; 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]; Prereq-CSE Honors office approval; fall, every year
Differential/integral calculus of functions of a single variable. Emphasizes hard problem-solving rather than theory.

MATH 1572H - Honors Calculus II
4.0 cr [max 5.0 cr]; =[MATH 1252, MATH 1272, MATH 1282, MATH 1372]; Prereq-Grade of at least C- in 1571, CSE Honors Office approval; parts of this sequence may be taken for cr by students who have taken non-honors 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; Prereq-1272 or equiv; S-N 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 sophomore-level 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]; A-F or Aud
Not taught: merely provides credit for transfer students who have taken a sophomore-level 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]; Prereq-1272 or 1282 or 1372 or 1572; fall, spring, summer, every year
Linear algebra: basis, dimension, matrices, eigenvalues/eigenvectors. Differential equations: first-order linear, separable; second-order linear with constant coefficients; linear systems with constant coefficients.
  2243 Description

MATH 2263 - Multivariable Calculus
4.0 cr; =[MATH 2374, MATH 2573H, MATH 3251]; Prereq-1272 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]; Prereq-1472H; 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; Prereq-2473H; 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]; Prereq-1572 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-?; A-F 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; Prereq-2582H or #; A-F or Aud
Second semester of three-semester 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; Prereq-Grade of at least C- in 3113; A-F 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; Prereq-Grade 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. Writing-intensive 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; A-F or Aud, fall, every year
First semester of three-semester 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; Prereq-3592H or #; A-F or Aud, spring, every year
Second semester of three-semester 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-?; A-F or Aud
Review of first-year calculus. Functions of one variable. Limits. Differentiation/integration of functions of one variable. Some applications, including max-min, 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; Prereq-1272 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 one-third higher than 3113.

MATH 4116 - Topics in Elementary Math II: Short Course
2.0 cr; Prereq-Grade of at least C- in 4113; A-F or Aud
Probability/Statistics, vector geometry, real/complex numbers. Meets during first half of semester only. Grading standard one-third higher than 3116.

MATH 4118 - Topics in Elementary Mathematics II
4.0 cr; Prereq-Grade 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 one-third higher than 3118.

MATH 4151 - Elementary Set Theory
3.0 cr; Prereq-One soph math course or #; fall, every year
Basic properties of operations on sets, cardinal numbers, simply and well-ordered sets, ordinal numbers, axiom of choice, axiomatics.

MATH 4152 - Elementary Mathematical Logic
3.0 cr; =[MATH 5165]; Prereq-one 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]; Prereq-2243 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; Prereq-2283 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; Prereq-2243 or 2373 or 2573; spring, every year
Modeling techniques for analysis/decision-making in industry. Optimization (sensitivity analysis, Lagrange multipliers, linear programming). Dynamical modeling (steady-states, 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; Prereq-4457; 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; Prereq-2243 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; Prereq-2243 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. Sturm-Liouville 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 single-variable functions.

MATH 4604 - Advanced Calculus II
4.0 cr; Prereq-4603 or 5615 or # ; spring, every year
Sequel to MATH 4603. Topology of n-dimensional 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-/multi-variable 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 problem-solving.

MATH 4707 - Introduction to Combinatorics and Graph Theory
4.0 cr; Prereq-2243, [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, inclusion-exclusion, 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; Prereq-2 sem of upper div math, ?; A-F 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]; Prereq-2 sem upper div math, ?; A-F or Aud, fall, spring, summer, every year
Directed study. A 10-15 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; Prereq-4065, [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; Prereq-5067; 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; Prereq-Two yrs calculus, basic computer skills; fall, every year
Mathematical background (e.g., partial differential equations, Fourier series, computational methods, Black-Scholes theory, numerical methods--including Monte Carlo simulation). Interest-rate derivative securities, exotic options, risk theory. First course of two-course sequence.

MATH 5076 - Mathematics of Options, Futures, and Derivative Securities II
4.0 cr; Prereq-5075; A-F or Aud, spring, every year
Mathematical background such as partial differential equations, Fourier series, computational methods, Black-Scholes theory, numerical methods (including Monte Carlo simulation), interest-rate derivative securities, exotic options, risk theory.

MATH 5165 - Mathematical Logic I
4.0 cr; =[MATH 4152]; Prereq-2283 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; Prereq-5165; spring, every year
First-order 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; Prereq-2 sems soph math; fall, every year
Classical cryptosystems. One-time pads, perfect secrecy. Public key ciphers: RSA, discrete log. Euclidean algorithm, finite fields, quadratic reciprocity. Message digest, hash functions. Protocols: key exchange, secret sharing, zero-knowledge proofs. Probablistic algorithms: pseudoprimes, prime factorization. Pseudo-random numbers. Elliptic curves.

MATH 5251 - Error-Correcting Codes, Finite Fields, Algebraic Curves
4.0 cr; Prereq-2 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, Bose-Chaudhuri-Hocquenghem 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; Prereq-5285; 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 two-dimensional 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; Prereq-5335; spring, every year
Projective geometry, including: relation to Euclidean geometry, finitegeometries, fundamental theorem of projective geometry. N-dimensionalEuclidean geometry from a vector viewpoint. Emphasizes N=3, including: polyhedra, spheres, isometries.

MATH 5345 - 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; Prereq-Linear 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, multi-scale analysis, discrete wavelets, self-similarity. 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 initial-value problems.

MATH 5486 - Introduction To Numerical Methods II
4.0 cr; Prereq-5485; 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; Prereq-4242
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; Prereq-5487
Numerical methods for time-dependent partial differential equations of engineering/science. Methods include finite element, finite difference, spectral, boundary integral. Applications to fluid flow, elasticity, electromagnetism.

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, Poincare-Bendixson 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 one-dimensional mappings. Fixed points, periodic points, stability, bifurcations, symbolic dynamics, chaos, fractals, Julia/Mandelbrot sets.

MATH 5583 - Complex Analysis
4.0 cr; =[00070]; Prereq-2 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 #; A-F 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 mathematically-talented students with proven achievement in theoretical mathematics courses; A-F 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 single-variable functions, Taylor's Theorem.

MATH 5616H - Honors: Introduction to Analysis II
4.0 cr; Prereq-5615; spring, every year
Rigorous treatment of Riemann-Stieltjes integration. Sequences/series of functions, uniform convergence, equicontinuous families, Stone-Weierstrass 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; Prereq-5651 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; Prereq-5651 or Stat 5101; spring, every year
Markov chains, Wiener process, stationary sequences, Ornstein-Uhlenbeck 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, inclusion-exclusion, 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 Non-enumerative 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; Prereq-2 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]; A-F or Aud, fall, spring, summer, every year
Individually directed study.

MATH 8001 - Preparation for College Teaching
1.0 cr [max 3.0 cr]; Prereq-! math grad student in good standing or #; S-N 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; A-F 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; A-F 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; Prereq-5166 or #; A-F 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; Prereq-8151 or #; A-F 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, Boolean-valued models, independence of axiom of choice.

MATH 8166 - Recursion Theory
3.0 cr; Prereq-Math grad student or #; A-F 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; Prereq-8166; A-F or Aud, spring
Sample topics: complexity theory, recursive analysis, generalized recursion theory, analytical hierarchy, constructive ordinals.

MATH 8172 - Model Theory
3.0 cr; Prereq-Math grad student or #; A-F or Aud
Interplay of formal theories, their models. Elementary equivalence, elementary extensions, partial isomorphisms. Lowenheim-Skolem theorems, compactness theorems, preservation theorems. Ultraproducts.

MATH 8173 - Model Theory
3.0 cr; Prereq-8172 or #; A-F 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]; A-F or Aud, fall, spring
Offered for one year or one semester as circumstances warrant.

MATH 8201 - General Algebra
3.0 cr; Prereq-4xxx algebra or equiv or #; A-F or Aud, fall, every year
Groups through Sylow, Jordan-H[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; Prereq-8201 or #; A-F 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 L-Functions
3.0 cr; Prereq-8202 or #; A-F or Aud
Zeta and L-functions, prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, class number formulas; Riemann hypothesis; modular forms and associated L-function; Eisenstein series; Hecke operators, Poincar[e] series, Euler products; Ramanujan conjectures; Theta series and quadratic forms; waveforms and L-functions.

MATH 8208 - Theory of Modular Forms and L-Functions
3.0 cr; Prereq-8207 or #; A-F or Aud
Applications of Eisenstein series: special values and analytic continuation and functional equations of L-functions. Trace formulas. Applications of representation theory. Computations.

MATH 8211 - Commutative and Homological Algebra
3.0 cr; Prereq-8202 or #; A-F or Aud, fall
Selected topics.

MATH 8212 - Commutative and Homological Algebra
3.0 cr; Prereq-8211 or #; A-F or Aud
Selected topics.

MATH 8245 - Group Theory
3.0 cr; Prereq-8202 or #; A-F or Aud, fall, every year
Permutations, Sylow's theorems, representations of groups on groups, semi-direct products, solvable and nilpotent groups, generalized Fitting subgroups, p-groups, co-prime action on p-groups.

MATH 8246 - Group Theory
3.0 cr; Prereq-8245 or #; A-F 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; Prereq-8202 or #; A-F or Aud
Algebraic number fields and algebraic curves. Basic commutative algebra. Completions: p-adic fields, formal power series, Puiseux series. Ramification, discriminant, different. Finiteness of class number and units theorem.

MATH 8252 - Algebraic Number Theory
3.0 cr; Prereq-8251 or #; A-F or Aud
Zeta and L-functions of global fields. Artin L-functions. Hasse-Weil L-functions. Tchebotarev density. Local and global class field theory. Reciprocity laws. Finer theory of cyclotomic fields.

MATH 8253 - Algebraic Geometry
3.0 cr; Prereq-8202 or #; A-F or Aud, fall
Curves, surfaces, projective space, affine and projective varieties. Rational maps. Blowing-up points. Zariski topology. Irreducible varieties, divisors.

MATH 8254 - Algebraic Geometry
3.0 cr; Prereq-8253 or #; A-F or Aud, spring
Sheaves, ringed spaces, and schemes. Morphisms. Derived functors and cohomology, Serre duality. Riemann-Roch 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]; Prereq-Math 8201, Math 8202; offered for one year or one semester as circumstances warrant; A-F or Aud, fall, spring, every year

MATH 8271 - Lie Groups and Lie Algebras
3.0 cr; Prereq-8302 or #; A-F 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; Maurer-Cartan forms; exponential map; correspondence between Lie algebras and simply connected Lie groups; Baker-Campbell-Hausdorff formula; homogeneous spaces.

MATH 8272 - Lie Groups and Lie Algebras
3.0 cr; Prereq-8271 or #; A-F 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; A-F or Aud

MATH 8300 - Topics in Algebra
1.0 - 3.0 cr [max 12.0 cr]; Prereq-Grad math major or #; offered as one yr or one sem crse as circumstances warrant; A-F or Aud, fall, spring, every year
Selected topics.

MATH 8301 - Manifolds and Topology
3.0 cr; Prereq-[Some point-set topology, algebra] or #; A-F 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; Prereq-8301 or #; A-F or Aud, spring, every year
Smooth manifolds, tangent spaces, embedding/immersion, Sard's theorem, Frobenius theorem. Differential forms, integration. Curvature, Gauss-Bonnet theorem. Time permitting: de Rham, duality in manifolds.

MATH 8306 - Algebraic Topology
3.0 cr; Prereq-8301 or #; A-F or Aud
Singular homology, cohomology theory with coefficients. Eilenberg-Stenrod axioms, Mayer-Vietoris theorem.

MATH 8307 - Algebraic Topology
3.0 cr; Prereq-8306 or #; A-F or Aud
Basic homotopy theory, cohomology rings with applications. Time permitting: fibre spaces, cohomology operations, extra-ordinary cohomology theories.

MATH 8333 - FTE: Master's
No description

MATH 8360 - Topics in Topology
1.0 - 3.0 cr [max 12.0 cr]; Prereq-8301 or #; offered as one yr or one sem crse as circumstances warrant; A-F or Aud, fall, spring
Selected topics.

MATH 8365 - Riemannian Geometry
3.0 cr; Prereq-8301 or basic point-set topology or #; A-F or Aud, fall, every year
Riemannian metrics, curvature. Bianchi identities, Gauss-Bonnet theorem, Meyers's theorem, Cartan-Hadamard theorem.

MATH 8366 - Riemannian Geometry
3.0 cr; Prereq-8365 or #; A-F 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]; Prereq-8301 or 8365; offered for one yr or one sem as circumstances warrant; A-F 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]; Prereq-8301, 8365; A-F or Aud, fall, spring
Current research.

MATH 8385 - Calculus of Variations and Minimal Surfaces
3.0 cr; Prereq-4xxx partial differential equations or #; A-F or Aud
Comprehensive exposition of calculus of variations and its applications. Theory for one-dimensional 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; Prereq-8595 or #; A-F 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 #; A-F 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; Prereq-8597 or #; A-F 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]; Prereq-8601; offered for one yr or one sem as circumstances warrant; A-F or Aud
Current research.

MATH 8401 - Mathematical Modeling and Methods of Applied Mathematics
3.0 cr; Prereq-4xxx numerical analysis and applied linear algebra or #; A-F or Aud, fall, every year
Dimension analysis, similarity solutions, linearization, stability theory, well-posedness, 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; Prereq-8401 or #; A-F 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; Prereq-5xxx numerical analysis of partial differential equations or #; A-F 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; Prereq-8431 or #
Plane flow of gas, characteristic method, hodograph method. Singular surfaces, shock waves, shock layers. Viscous flow, Navier-Stokes 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; Prereq-8441 or #; 5477-5478 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; Prereq-4xxx numerical analysis, 4xxx partial differential equations or #; A-F 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; Prereq-8445 or #; A-F 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]; Prereq-Grad math major or #; offered as one yr or one sem crse as circumstances warrant; A-F 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]; A-F or Aud, fall, spring
Offered for one year or one semester as circumstances warrant.

MATH 8501 - Theory of Ordinary Differential Equations
3.0 cr; Prereq-4xxx ODE or #; A-F 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; Prereq-8501 or #; A-F 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. Morse-Smale systems. Coupled oscillators and network dynamics.

MATH 8503 - Bifurcation Theory in Ordinary Differential Equations
3.0 cr; Prereq-8501 or #; A-F 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; Prereq-5525 or 8502 or #; A-F or Aud
Static/Hopf bifurcations, invariant manifold theory, normal forms, averaging, Hopf bifurcation in maps, forced oscillations, coupled oscillators, chaotic dynamics, co-dimension 2 bifurcations. Emphasizes computational aspects/applications from biology, chemistry, engineering, physics.

MATH 8506 - Applied Dynamical Systems and Bifurcation Theory II
3.0 cr; Prereq-5587 or #; A-F or Aud, fall
Background on analysis in Banach spaces, linear operator theory. Lyapunov-Schmidt reduction, static bifurcation, stability at a simple eigenvalue, Hopf bifurcation in infinite dimensions invariant manifold theory. Applications to hydrodynamic stability problems, reaction-diffusion equations, pattern formation, and elasticity.

MATH 8520 - Topics in Dynamical Systems
1.0 - 3.0 cr [max 12.0 cr]; Prereq-8502; A-F or Aud, fall, spring
Current research.

MATH 8530 - Topics in Ordinary Differential Equations
1.0 - 3.0 cr [max 3.0 cr]; Prereq-8502; A-F 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]; A-F 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; Prereq-8502 or #; A-F 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, well-posedness of solutions.

MATH 8572 - Theory of Evolutionary Equations
3.0 cr; Prereq-8571 or #; A-F or Aud, spring
Dynamics of Navier-Stokes 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]; Prereq-8572 or #; offered for one yr or one semester as circumstances warrant; A-F or Aud

MATH 8581 - Applications of Linear Operator Theory
3.0 cr; Prereq-4xxx applied mathematics or #; A-F 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; Prereq-8581 or #; A-F or Aud
Fourier theory. Self-adjoint, compact, unbounded linear operators. Spectral analysis, eigenvalue-eigenvector problem, spectral theorem, operational calculus.

MATH 8583 - Theory of Partial Differential Equations
3.0 cr; Prereq-[Some 5xxx PDE, 8601] or #; A-F 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; Prereq-8583 or #; A-F 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]; Prereq-8602; offered for one yr or one sem as circumstances warrant; A-F 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; Prereq-5616 or #; A-F 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; Prereq-8601 or #; A-F or Aud, spring, every year
Radon-Nikodym, Fubini theorems. C(X). Lp spaces (introduction to metric, Banach, Hilbert spaces). Stone-Weierstrass theorem. Basic Fourier analysis. Theory of differentiation.

MATH 8640 - Topics in Real Analysis
3.0 cr [max 12.0 cr]; Prereq-8602 or #; offered for one yr or one sem as circumstances warrant; A-F or Aud
Current research.

MATH 8641 - Spatial Ecology
3.0 cr; Prereq-Two semesters calculus, theoretical population ecology or four semesters more robust calculus, course in statistics or probability or #; S-N 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; Prereq-5616 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; Prereq-8651 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; Prereq-8651 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 sub-martingales, Doob-Meyer decomposition.

MATH 8655 - Stochastic Calculus with Applications
3.0 cr; Prereq-8654 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; Prereq-8652 or #; fall, every year
In-depth 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 Pre-Thesis Credits
No description

MATH 8668 - Combinatorial Theory
3.0 cr; A-F or Aud, fall
Basic enumeration, including sets and multisets, permutation statistics, inclusion-exclusion, 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; Prereq-8668 or #; A-F or Aud, spring, odd years
Further topics in enumeration, including symmetric functions, Schensted correspondence, and standard tableaux; non-enumerative 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]; Prereq-Grad math major or #; offered as one yr or one sem crse as circumstances warrant; A-F or Aud, fall, spring, every year
Selected topics.

MATH 8701 - Complex Analysis
3.0 cr; Prereq-5616 or #; A-F 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, Riemann-Hurwitz formula. Schwarz-Christoffel polygonal functions. Residues.

MATH 8702 - Complex Analysis
3.0 cr; Prereq-8701 or #; A-F 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]; Prereq-8702 or #; offered for one yr or one sem as circumstances warrant; A-F or Aud
Current research.

MATH 8801 - Functional Analysis
3.0 cr; Prereq-8602 or #; A-F 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 (Hahn-Banach, open mapping, closed graph theorems). Frechet spaces.

MATH 8802 - Functional Analysis
3.0 cr; Prereq-8801 or #; A-F or Aud, spring
Spectral theory of operators, theory of distributions (generalized functions), Fourier transformations and applications. Sobolev spaces and pseudo-differential operators. C-star algebras (Gelfand-Naimark 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-#; S-N or Aud, fall, spring, every year
Readings, research.

MATH 8991 - Independent Study
1.0 - 6.0 cr [max 24.0 cr]; Prereq-#; S-N or Aud, spring, summer, every year
Individually directed study.

MATH 8992 - Directed Reading
1.0 - 6.0 cr [max 24.0 cr]; Prereq-#; S-N or Aud, fall, spring, every year
Individually directed reading.

MATH 8993 - Directed Study
1.0 - 6.0 cr [max 24.0 cr]; Prereq-#; S-N or Aud, spring, every year
Individually directed study.