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Twin Cities Courses

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MATHEMATICS (MATH)
College of Science and Engineering; UMR Chancellor's Office
Mathematics, Sch of
 
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 PSTL 731 or 732]; fall, spring, every year)
Introduction to the breadth and nature of mathematics and the power of abstract reasoning, with applications to topics that are relevant to the modern world, such as voting, fair division of assets, patterns of growth, and opinion polls.



MATH 1031 - College Algebra and Probability (MATH)
(3.0 cr; =[MATH 1051, PSTL 1006]; Prereq-3 yrs high school math or satisfactory score on placement exam or grade of at least C- in [PSTL 731 or PSTL 732]; fall, spring, summer, every year)
Graphs of equations/functions, transformations of graphs; linear, quadratic, polynomial, and rational functions, with applications; inverses/composition of functions; exponential and logarithmic functions, with applications; basic probability rules, conditional probability, binomial probabilities.



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 (MATH)
(3.0 cr; =[PSTL 1006, MATH 1031]; Prereq-3 yrs of high school math or satisfactory score on placement test or grade of at least C- in [PSTL 731 or PSTL 732]; fall, spring, summer, every year)
Graphs of equations/functions, transformations of graphs; linear, quadratic, polynomial, and rational functions, with applications; zeroes of polynomials; inverses and composition of functions; exponential and logarithmic functions, with applications; coverage beyond that found in the usual 3 years of high school math.



MATH 1142 - Short Calculus (MATH)
(4.0 cr; Prereq-Satisfactory score on placement test or grade of at least C- in [1031 or 1051]; fall, spring, summer, every year)
A streamlined one-semester tour of differential and integral calculus in one variable, and differential calculus in two variables. No trigonometry/does not have the same depth as MATH 1271-1272. Formulas and their interpretation and use in applications.



MATH 1151 - Precalculus II (MATH)
(3.0 cr; =[MATH 1155]; Prereq-Satisfactory score on placement exam or grade of at least C- in [1031 or 1051]; fall, spring, summer, every year)
Properties of trigonometric functions and their inverses, including graphs and identities, with applications; polar coordinates, equations, graphs; complex numbers, complex plane, DeMoivre's Theorem; conic sections; systems of linear equations and inequalities, with applications; arithmetic and geometric sequences and series.



MATH 1155 - Intensive Precalculus (MATH)
(5.0 cr; =[MATH 1151]; Prereq-3 yrs high school math or satisfactory score on placement exam or grade of at least C- in [PSTL 731 or PSTL 732]; fall, spring, every year)
Graphs of equations and functions; polynomial and rational functions; inverses and composition of functions; exponentials and logarithms; trig functions, graphs, identities; polar coordinates; complex numbers; systems of linear equations; arithmetic, geometric sequences, series; applications.



MATH 1241 - Calculus and Dynamical Systems in Biology (MATH)
(4.0 cr; Prereq-[4 yrs high school math including trig or satisfactory score on placement test or grade of at least C- in [1151 or 1155]], CBS student; fall, spring, every year)
Differential/integral calculus with biological applications. Discrete/continuous dynamical systems. Models from fields such as ecology/evolution, epidemiology, physiology, genetic networks, neuroscience, and biochemistry.



MATH 1271 - Calculus I (MATH)
(4.0 cr; =[MATH 1281, MATH 1371, MATH 1471, MATH 1571H]; Prereq-4 yrs high school math including trig or satisfactory score on placement test or grade of at least C- in [1151 or 1155]; fall, spring, summer, every year)
Differential calculus of functions of a single variable, including polynomial, rational, exponential, and trig functions. Applications, including optimization and related rates problems. Single variable integral calculus, using anti-derivatives and simple substitution. Applications may include area, volume, work problems.



MATH 1272 - Calculus II
(4.0 cr; =[MATH 1472, MATH 1252, MATH 1372, MATH 1282, 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.



MATH 1371 - CSE Calculus I (MATH)
(4.0 cr; =[MATH 1281, MATH 1271, MATH 1471, MATH 1571H]; Prereq-CSE or pre-bioprod & biosys engn (PRE), background in [precalculus, geometry, visualization of functions/graphs], #; familiarity with graphing calculators recommended; fall, spring, every year)
Differentiation of single-variable functions, basics of integration of single-variable functions. Applications: max-min, related rates, area, curve-sketching. Use of calculator, cooperative learning.



MATH 1372 - CSE Calculus II
(4.0 cr; =[MATH 1472, MATH 1252, MATH 1282, MATH 1572H, MATH 1272]; Prereq-Grade of at least C- in [1371 or equiv], CSE or pre-Bioprod/Biosys Engr; spring, every year)
Techniques of integration. Calculus involving transcendental functions, polar coordinates, Taylor polynomials, vectors/curves in space, cylindrical/spherical coordinates. Use of calculators, cooperative learning.



MATH 1474 - Honors Calculus IIB for Secondary Students
(3.0 cr; Prereq-1473H; spring, every year)
Accelerated honors sequence. Linear Algebra from geometric viewpoint. First-order systems of differential equations.



MATH 1571H - Honors Calculus I (MATH)
(4.0 cr; =[MATH 1281, MATH 1371, MATH 1271, MATH 1471]; Prereq-Honors student and permission of University Honors Program; A-F only, 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; =[MATH 1472, MATH 1252, MATH 1372, MATH 1282, MATH 1272]; Prereq-1571H, honors student, permission of University Honors Program; A-F only, 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, MATH 2574H, MATH 2471]; Prereq-[1272 or 1282 or 1372 or 1572] w/grade of at least C-; 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.



MATH 2263 - Multivariable Calculus
(4.0 cr; =[MATH 2573H, MATH 3251, MATH 2374, MATH 2473]; Prereq-[1272 or 1372 or 1572] w/grade of at least C-; fall, spring, summer, every year)
Derivative as linear map. Differential/integral calculus of functions of several variables, including change of coordinates using Jacobians. Line/surface integrals. Gauss, Green, Stokes Theorems.



MATH 2283 - Sequences, Series, and Foundations
(3.0 cr; =[MATH 3283W]; Prereq-[&2243 or &2263 or &2373 or &2374] w/grade of at least C-; fall, spring, every year)
Mathematical reasoning. 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 2574H, MATH 2243, MATH 2471]; Prereq-[1272 or 1282 or 1372 or 1572] w/grade of at least C-, CSE or pre-Bio Prod/Biosys Engr; 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. Use of computer technology.



MATH 2374 - CSE Multivariable Calculus and Vector Analysis
(4.0 cr; =[MATH 2573H, MATH 3251, MATH 2473, MATH 2263]; Prereq-[1272 or 1282 or 1372 or 1572] w/grade of at least C-, CSE or pre-Bioprod/Biosys Engr; fall, spring, every year)
Derivative as linear map. Differential/integral calculus of functions of several variables, including change of coordinates using Jacobians. Line/surface integrals. Gauss, Green, Stokes theorems. Use of computer technology.



MATH 2474 - 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; =[MATH 3251, MATH 2374, MATH 2473, MATH 2263]; Prereq-Math 1572H, honors student and permission of University Honors Program; A-F only, fall, every year)
Integral calculus of several variables. Vector analysis, including theorems of Gauss, Green, Stokes.



MATH 2574H - Honors Calculus IV
(4.0 cr; =[MATH 2373, MATH 2243, MATH 2471]; Prereq-2573H, honors student and permission of University Honors Program; A-F only, spring, every year)
Advanced linear algebra, differential equations. Additional topics as time permits.



MATH 3113 - Topics in Elementary Mathematics I
(4.0 cr; Prereq-Grade of at least C- in [1031 or PSTL 1006] or placement exam; fall, 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] w/grade of at least C-; 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], IT 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 only, 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 4065 - Theory of Interest
(4.0 cr; Prereq-1272 or 1372 or 1572; A-F only, fall, spring, every year)
Time value of money, compound interest and general annuities, loans, bonds, general cash flows, basic financial derivatives and their valuation. Primarily for students who are interested in actuarial mathematics.



MATH 4067W - Actuarial Mathematics in Practice (WI)
(3.0 cr; Prereq-4065, ACCT 2050, ECON 1101, ECON 1102; A-F only, spring, every year)
Real world actuarial problems that require integration of mathematical skills with knowledge from other disciplines such as economics, statistics, and finance. Communication and interpersonal skills are enhanced by teamwork/presentations to the practitioner actuaries who co-instruct.



MATH 4113 - Topics in Elementary Mathematics I
(4.0 cr; Prereq-[Grade of at least C- in 1031] or placement exam; fall, spring, summer, offered periodically)
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, finite fields building on previously learned modular arithmetic, trees. Grading standard one-third higher than 3118.



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 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; =[01072]; 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; =[01776]; Prereq-4603 or 5615 or #; spring, every year)
Sequel to MATH 4603. Topology of n-dimensional Euclidean 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.



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]; 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. 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)
TBD



MATH 4993 - Directed Study
(1.0 - 4.0 cr [max 12.0 cr]; fall, spring, summer, every year)
TBD



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, including normal 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, finite geometries, fundamental theorem of projective geometry. N-dimensional Euclidean 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]; A-F only, fall, every year)
Rigorous introduction to general topology. Set theory, Euclidean/metric spaces, compactness/connectedness. May include Urysohn metrization, Tychonoff theorem or fundamental group/covering spaces.



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 2574; 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 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. Methods for solving linear systems, eigenvalue problems, systems of nonlinear equations.



MATH 5486 - Introduction To Numerical Methods II
(4.0 cr; Prereq-5485; spring, every year)
Numerical integration/differentiation. Numerical solution of initial-value problems, boundary value problems for ordinary differential equations, partial differential equations.



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, offered periodically)
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; 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, fall, offered periodically)
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]; 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/transportation problems, 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 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; 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, offered periodically)
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, offered periodically)
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, offered periodically)
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 periodically)
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, offered periodically)
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, offered periodically)
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, offered periodically)
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, offered periodically)
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, offered periodically)
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, offered periodically)
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
(1.0 cr; Prereq-Master's student, adviser and DGS consent; No Grade, fall, spring, summer, every year)
(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, offered periodically)
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, offered periodically)
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
(1.0 cr; Prereq-Doctoral student, adviser and DGS consent; No Grade, fall, spring, summer, every year)
(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 periodically)
Offered for one year or one semester as circumstances warrant.



MATH 8501 - Differential Equations and Dynamical Systems I
(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 - Differential Equations and Dynamical Systems II
(3.0 cr; Prereq-8501 or #; A-F or Aud, spring, every year)
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, offered periodically)
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, offered periodically)
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 periodically)
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, offered periodically)
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, offered periodically)
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
(1.0 - 6.0 cr [max 12.0 cr]; Prereq-Doctoral student who has not passed prelim oral; no required consent for 1st/2nd registrations, up to 12 combined cr; % for 3rd/4th registrations, up to 24 combined cr; doctoral student admitted before summer 2007 may register up to four times, up to 60 combined cr; No Grade, fall, spring, summer, every year)
TBD



MATH 8668 - Combinatorial Theory
(3.0 cr; A-F or Aud, fall, offered periodically)
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 academic 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
(1.0 - 18.0 cr [max 50.0 cr]; Prereq-Max 18 cr per semester or summer; 10 cr total required [Plan A only]; No Grade, fall, spring, summer, every year)
(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, offered periodically)
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
(1.0 - 24.0 cr [max 100.0 cr]; Prereq-Max 18 cr per semester or summer; 24 cr required; No Grade, fall, spring, every year)
(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-#; fall, 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.



MATH 8994 - Topics at the IMA
(1.0 - 3.0 cr [max 6.0 cr]; fall, spring, every year)
Current research at IMA.



 
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