

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; Prereq3 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]; Prereq3 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; Prereq1051 or 1151 or 1155; AF or Aud, fall, spring, summer, every year) For students who need probability/permutations/combinations portion of 1031. Meets with 1031, has same grade/work requirements.
MATH
1051
 Precalculus I
(MATH)
(3.0 cr; =[PSTL 1006, MATH 1031]; Prereq3 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; PrereqSatisfactory score on placement test or grade of at least C in [1031 or 1051]; fall, spring, summer, every year) A streamlined onesemester 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 12711272. Formulas and their interpretation and use in applications.
MATH
1151
 Precalculus II
(MATH)
(3.0 cr; =[MATH 1155]; PrereqSatisfactory 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]; Prereq3 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]; Prereq4 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 antiderivatives 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]; PrereqCSE or prebioprod & biosys engn (PRE), background in [precalculus, geometry, visualization of functions/graphs], #; familiarity with graphing calculators recommended; fall, spring, every year) Differentiation of singlevariable functions, basics of integration of singlevariable functions. Applications: maxmin, related rates, area, curvesketching. Use of calculator, cooperative learning.
MATH
1372
 CSE Calculus II
(4.0 cr; =[MATH 1472, MATH 1252, MATH 1282, MATH 1572H, MATH 1272]; PrereqGrade of at least C in [1371 or equiv], CSE or preBioprod/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; Prereq1473H; spring, every year) Accelerated honors sequence. Linear Algebra from geometric viewpoint. Firstorder systems of differential equations.
MATH
1571H
 Honors Calculus I
(MATH)
(4.0 cr; =[MATH 1281, MATH 1371, MATH 1271, MATH 1471]; PrereqHonors student and permission of University Honors Program; AF only, fall, every year) Differential/integral calculus of functions of a single variable. Emphasizes hard problemsolving rather than theory.
MATH
1572H
 Honors Calculus II
(4.0 cr; =[MATH 1472, MATH 1252, MATH 1372, MATH 1282, MATH 1272]; Prereq1571H, honors student, permission of University Honors Program; AF 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; Prereq1272 or equiv; SN or Aud, spring, every year) Actuarial science as a subject and career. Guest lectures by actuaries. Resume preparation and interviewing skills. Review and practice for actuarial exams.
MATH
2066
 Elementary Differential Equations
(1.0  4.0 cr [max 4.0 cr]) Not taught: merely provides credit for transfer students who have taken a sophomorelevel differential equations class that does not contain enough linear algebra to qualify for credit for 2243.
MATH
2142
 Elementary Linear Algebra
(1.0  4.0 cr [max 1.0 cr]; AF or Aud) Not taught: merely provides credit for transfer students who have taken a sophomorelevel linear algebra course that does not contain enough differential equations to qualify for credit for 2243.
MATH
2243
 Linear Algebra and Differential Equations
(4.0 cr; =[MATH 2373, 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: firstorder linear, separable; secondorder 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 preBio 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 preBioprod/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; Prereq2473H; spring, every year) Topics may include linear algebra, combinatorics, advanced differential equations, probability/statistics, numerical analysis, dynamical systems, topology/geometry. Emphasizes concepts/explorations.
MATH
2573H
 Honors Calculus III
(4.0 cr; =[MATH 3251, MATH 2374, MATH 2473, MATH 2263]; PrereqMath 1572H or Math 2574H, honors student and permission of University Honors Program; AF 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]; PrereqMath 1572H or Math 2573H, honors student and permission of University Honors Program; AF only, spring, every year) Advanced linear algebra, differential equations. Additional topics as time permits.
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. Writingintensive 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; AF only, fall, every year) First semester of threesemester sequence. Focuses on multivariable calculus at deeper level than regular calculus offerings. Rigorous introduction to sequences/series. Theoretical treatment of multivariable calculus. Strong introduction to linear algebra.
MATH
3593H
 Honors Mathematics II
(5.0 cr; Prereq3592H or #; AF or Aud, spring, every year) Second semester of threesemester sequence. Focuses on multivariable calculus at deeper level than regular calculus offerings. Rigorous introduction to sequences/series. Theoretical treatment of multivariable calculus. Strong introduction to linear algebra.
MATH
4065
 Theory of Interest
(4.0 cr; Prereq1272 or 1372 or 1572; AF 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; Prereq4065, ACCT 2050, ECON 1101, ECON 1102; AF 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 coinstruct.
MATH
4151
 Elementary Set Theory
(3.0 cr; PrereqOne soph math course or #; fall, every year) Basic properties of operations on sets, cardinal numbers, simply and wellordered sets, ordinal numbers, axiom of choice, axiomatics.
MATH
4152
 Elementary Mathematical Logic
(3.0 cr; =[MATH 5165]; Prereqone soph math course or #; spring, every year) Propositional logic. Predicate logic: notion of a first order language, a deductive system for first order logic, first order structures, Godel's completeness theorem, axiom systems, models of formal theories.
MATH
4242
 Applied Linear Algebra
(4.0 cr; =[MATH 4457]; Prereq2243 or 2373 or 2573; fall, spring, summer, every year) Systems of linear equations, vector spaces, subspaces, bases, linear transformations, matrices, determinants, eigenvalues, canonical forms, quadratic forms, applications.
MATH
4281
 Introduction to Modern Algebra
(4.0 cr; Prereq2283 or 3283 or #) Equivalence relations, greatest common divisor, prime decomposition,
modular arithmetic, groups, rings, fields, Chinese remainder theorem,
matrices over commutative rings, polynomials over fields.
MATH
4428
 Mathematical Modeling
(4.0 cr; Prereq2243 or 2373 or 2573; spring, every year) Modeling techniques for analysis/decisionmaking in industry. Optimization (sensitivity analysis, Lagrange multipliers, linear programming). Dynamical modeling (steadystates, stability analysis, eigenvalue methods, phase portraits, simulation). Probabilistic methods (probability/statistical models, Markov chains, linear regression, simulation).
MATH
4512
 Differential Equations with Applications
(3.0 cr; Prereq2243 or 2373 or 2573; fall, spring, every year) Laplace transforms, series solutions, systems, numerical methods, plane autonomous systems, stability.
MATH
4567
 Applied Fourier Analysis
(4.0 cr; Prereq2243 or 2373 or 2573; fall, spring, every year) Fourier series, integral/transform. Convergence. Fourier series, transform in complex form. Solution of wave, heat, Laplace equations by separation of variables. SturmLiouville systems, finite Fourier, fast Fourier transform. Applications. Other topics as time permits.
MATH
4603
 Advanced Calculus I
(4.0 cr; =[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 singlevariable functions.
MATH
4604
 Advanced Calculus II
(4.0 cr; =[01776]; Prereq4603 or 5615 or #; spring, every year) Sequel to MATH 4603. Topology of ndimensional 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
4653
 Elementary Probability
(4.0 cr; Prereq[2263 or 2374 or 2573]; [2283 or 2574 or 3283] recommended; fall, spring, every year) Probability spaces, distributions of discrete/continuous random variables, conditioning. Basic theorems, calculational methodology. Examples of random sequences. Emphasizes problemsolving.
MATH
4707
 Introduction to Combinatorics and Graph Theory
(4.0 cr; Prereq2243, [2283 or 3283]; fall, spring, every year) Existence, enumeration, construction, algorithms, optimization. Pigeonhole principle, bijective combinatorics, inclusionexclusion, recursions, graph modeling, isomorphism. Degree sequences and edge counting. Connectivity, Eulerian graphs, trees, Euler's formula, network flows, matching theory. 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; Prereq2 sem of upper div math, %; AF or Aud, fall, spring, summer, every year) Directed study. May consist of paper on specialized area of math or original computer program or other approved project. Covers some math that is new to student. Scope/topic vary with instructor.
MATH
4997W
 Senior project (Writing Intensive)
(WI)
(1.0 cr [max 2.0 cr]; Prereq2 sem upper div math, %; AF or Aud, fall, spring, summer, every year) Directed study. A 1015 page paper on a specialized area, including some math that is new to student. At least two drafts of paper given to instructor for feedback before final version. Student keeps journal of preliminary work on project. Scope/topic vary with instructor.
MATH
5067
 Actuarial Mathematics I
(4.0 cr; Prereq4065, [one sem [4xxx or 5xxx] [probability or statistics] course]; fall, every year) Future lifetime random variable, survival function. Insurance, life annuity, future loss random variables. Net single premium, actuarial present value, net premium, net reserves.
MATH
5068
 Actuarial Mathematics II
(4.0 cr; Prereq5067; spring, every year) Multiple decrement insurance, pension valuation. Expense analysis, gross premium, reserves. Problem of withdrawals. Regulatory reserving systems. Minimum cash values. Additional topics at instructor's discretion.
MATH
5075
 Mathematics of Options, Futures, and Derivative Securities I
(4.0 cr; PrereqTwo yrs calculus, basic computer skills; fall, every year) Mathematical background (e.g., partial differential equations, Fourier series, computational methods, BlackScholes theory, numerical methodsincluding Monte Carlo simulation). Interestrate derivative securities, exotic options, risk theory. First course of twocourse sequence.
MATH
5076
 Mathematics of Options, Futures, and Derivative Securities II
(4.0 cr; Prereq5075; AF or Aud, spring, every year) Mathematical background such as partial differential equations, Fourier series, computational methods, BlackScholes theory, numerical methods (including Monte Carlo simulation), interestrate derivative securities, exotic options, risk theory.
MATH
5165
 Mathematical Logic I
(4.0 cr; =[MATH 4152]; Prereq2283 or 3283 or Phil 5201 or CSci course in theory of algorithms or #; fall, every year) Theory of computability: notion of algorithm, Turing machines, primitive recursive functions, recursive functions, Kleene normal form, recursion theorem. Propositional logic.
MATH
5166
 Mathematical Logic II
(4.0 cr; Prereq5165; spring, every year) Firstorder logic: provability/truth in formal systems, models of axiom systems, Godel's completeness theorem. Godel's incompleteness theorem: decidable theories, representability of recursive functions in formal theories, undecidable theories, models of arithmetic.
MATH
5248
 Cryptology and Number Theory
(4.0 cr; Prereq2 sems soph math; fall, every year) Classical cryptosystems. Onetime pads, perfect secrecy. Public key ciphers: RSA, discrete log. Euclidean algorithm, finite fields, quadratic reciprocity. Message digest, hash functions. Protocols: key exchange, secret sharing, zeroknowledge proofs. Probablistic algorithms: pseudoprimes, prime factorization. Pseudorandom numbers. Elliptic curves.
MATH
5251
 ErrorCorrecting Codes, Finite Fields, Algebraic Curves
(4.0 cr; Prereq2 sems soph math; spring, every year) Information theory: channel models, transmission errors. Hamming weight/distance. Linear codes/fields, check bits. Error processing: linear codes, Hamming codes, binary Golay codes. Euclidean algorithm. Finite fields, BoseChaudhuriHocquenghem codes, polynomial codes, Goppa codes, codes from algebraic curves.
MATH
5285H
 Honors: Fundamental Structures of Algebra I
(4.0 cr; Prereq[2243 or 2373 or 2573], [2283 or 2574 or 3283]; fall, every year) Review of matrix theory, linear algebra. Vector spaces, linear transformations over abstract fields. Group theory, 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; Prereq5285; fall, spring, every year) Ring/module theory, including ideals, quotients, homomorphisms,
domains (unique factorization, euclidean, principal ideal), fundamental theorem for finitely generated modules over euclidean domains, Jordan canonical form. Introduction to field theory, including finite fields,
algebraic/transcendental extensions, Galois theory.
MATH
5335
 Geometry I
(4.0 cr; Prereq[2243 or 2373 or 2573], [& 2263 or & 2374 or & 2574]; fall, every year) Advanced twodimensional Euclidean geometry from a vector viewpoint. Theorems/problems about triangles/circles, isometries, connections with Euclid's axioms. Hyperbolic geometry, how it compares with Euclidean geometry.
MATH
5336
 Geometry II
(4.0 cr; Prereq5335; spring, every year) Projective geometry, including: relation to Euclidean geometry, finite
geometries, fundamental theorem of projective geometry. Ndimensional
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]; AF 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; PrereqLinear algebra, differential equations; spring, every year) Development/analysis of models for complex biological networks. Examples taken from signal transduction networks, metabolic networks, gene control networks, and ecological networks.
MATH
5447
 Theoretical Neuroscience
(4.0 cr; Prereq2243 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, multiscale analysis, discrete wavelets, selfsimilarity. 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; Prereq5485; spring, every year) Numerical integration/differentiation. Numerical solution of initialvalue 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, PoincareBendixson theory, strange attractors.
MATH
5535
 Dynamical Systems and Chaos
(4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; fall, spring, every year) Dynamical systems theory. Emphasizes iteration of onedimensional mappings. Fixed points, periodic points, stability, bifurcations, symbolic dynamics, chaos, fractals, Julia/Mandelbrot sets.
MATH
5583
 Complex Analysis
(4.0 cr; =[00070]; Prereq2 sems soph math [including [2263 or 2374 or 2573], [2283 or 3283]] recommended; fall, spring, summer, every year) Algebra, geometry of complex numbers. Linear fractional transformations. Conformal mappings. Holomorphic functions. Theorems of Abel/Cauchy, power series. Schwarz' lemma. Complex exponential, trig functions. Entire functions, theorems of Liouville/Morera. Reflection principle. Singularities, Laurent series. Residues.
MATH
5587
 Elementary Partial Differential Equations I
(4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; fall, every year) Emphasizes partial differential equations w/physical applications, including heat, wave, Laplace's equations. Interpretations of boundary conditions. Characteristics, Fourier series, transforms, Green's functions, images, computational methods. Applications include wave propagation, diffusions, electrostatics, shocks.
MATH
5588
 Elementary Partial Differential Equations II
(4.0 cr; Prereq[[2243 or 2373 or 2573], [2263 or 2374 or 2574], 5587] or #; AF or Aud, spring, every year) Heat, wave, Laplace's equations in higher dimensions. Green's functions, Fourier series, transforms. Asymptotic methods, boundary layer theory, bifurcation theory for linear/nonlinear PDEs. Variational methods. Free boundary problems. Additional topics as time permits.
MATH
5594H
 Honors Mathematics  Topics
(4.0 cr [max 12.0 cr]; Prereq[3593H with grade of at least B, experience in writing proofs] or %; intended for mathematicallytalented students with proven achievement in theoretical mathematics courses; AF or Aud, 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 singlevariable functions, Taylor's Theorem.
MATH
5616H
 Honors: Introduction to Analysis II
(4.0 cr; Prereq5615; spring, every year) Rigorous treatment of RiemannStieltjes integration. Sequences/series of functions, uniform convergence, equicontinuous families, StoneWeierstrass Theorem, power series. Rigorous treatment of differentiation/integration of multivariable functions, Implicit Function Theorem, Stokes' Theorem. Additional topics as time permits.
MATH
5651
 Basic Theory of Probability and Statistics
(4.0 cr; Prereq[2263 or 2374 or 2573], [2243 or 2373]; [2283 or 2574 or 3283] recommended; Credit will not be granted if credit has been received for: Stat 4101, Stat 5101.; fall, spring, every year) Logical development of probability, basic issues in statistics. Probability spaces, random variables, their distributions/expected values. Law of large numbers, central limit theorem, generating functions, sampling, sufficiency, estimation.
MATH
5652
 Introduction to Stochastic Processes
(4.0 cr; Prereq5651 or Stat 5101; fall, spring, every year) Random walks, Markov chains, branching processes, martingales, queuing theory, Brownian motion.
MATH
5654
 Prediction and Filtering
(4.0 cr; Prereq5651 or Stat 5101; spring, every year) Markov chains, Wiener process, stationary sequences, OrnsteinUhlenbeck process. Partially observable Markov processes (hidden Markov models), stationary processes. Equations for general filters, Kalman filter. Prediction of future values of partially observable processes.
MATH
5705
 Enumerative Combinatorics
(4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2283 or 2374 or 2574 or 3283]; fall, spring, every year) Basic enumeration, bijections, inclusionexclusion, recurrence relations, ordinary/exponential generating functions, partitions, Polya theory. Optional topics include trees, asymptotics, listing algorithms, rook theory, involutions, tableaux, permutation statistics.
MATH
5707
 Graph Theory and Nonenumerative Combinatorics
(4.0 cr; Prereq[2243 or 2373 or 2573], [2263 or 2374 or 2574]; [2283 or 3283 or experience in writing proofs] highly recommended; Credit will not be granted if credit has been received for: 4707; fall, spring, every year) Basic topics in graph theory: connectedness, Eulerian/Hamiltonian properties, trees, colorings, planar graphs, matchings, flows in networks. Optional topics include graph algorithms, Latin squares, block designs, Ramsey theory.
MATH
5711
 Linear Programming and Combinatorial Optimization
(4.0 cr; Prereq2 sems soph math [including 2243 or 2373 or 2573]; fall, spring, every year) Simplex method, connections to geometry, duality theory,
sensitivity analysis. Applications to cutting stock, allocation of resources, scheduling problems. Flows, matching/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]; AF or Aud, fall, spring, summer, every year) Individually directed study.
MATH
5990
 Topics in Mathematics
(4.0 cr [max 12.0 cr]; fall, spring, offered periodically) Topics vary by instructor. See class schedule.
MATH
8001
 Preparation for College Teaching
(1.0 cr; PrereqMath grad student in good standing or #; SN or Aud, fall, spring, every year) New approaches to teaching/learning, issues in mathematics education, components/expectations of a college mathematics professor.
MATH
8141
 Applied Logic
(3.0 cr; AF or Aud, fall, spring, 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; AF 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; Prereq5166 or #; AF or Aud) Axiomatic development of basic properties of ordinal/cardinal numbers, infinitary combinatorics, well founded sets, consistency of axiom of foundation, constructible sets, consistency of axiom of choice and of generalized continuum hypothesis.
MATH
8152
 Axiomatic Set Theory
(3.0 cr; Prereq8151 or #; AF or Aud) Notion of forcing, generic extensions, forcing with finite partial functions, independence of continuum hypothesis, forcing with partial functions of infinite cardinalities, relationship between partial orderings and Boolean algebras, Booleanvalued models, independence of axiom of choice.
MATH
8166
 Recursion Theory
(3.0 cr; PrereqMath grad student or #; AF or Aud) Analysis of concept of computability, including various equivalent definitions. Primitive recursive, recursive, partial recursive functions. Oracle Turing machines. Kleene Normal Form Theorem. Recursive, recursively enumerable sets. Degrees of unsolvability. Arithmetic hierarchy.
MATH
8167
 Recursion Theory
(3.0 cr; Prereq8166; AF or Aud, spring, offered periodically) Sample topics: complexity theory, recursive analysis, generalized recursion theory, analytical hierarchy, constructive ordinals.
MATH
8172
 Model Theory
(3.0 cr; PrereqMath grad student or #; AF or Aud) Interplay of formal theories, their models. Elementary equivalence, elementary extensions, partial isomorphisms. LowenheimSkolem theorems, compactness theorems, preservation theorems. Ultraproducts.
MATH
8173
 Model Theory
(3.0 cr; Prereq8172 or #; AF or Aud) Types of elements. Prime models, homogeneity, saturation, categoricity in power. Forking.
MATH
8190
 Topics in Logic
(1.0  3.0 cr [max 12.0 cr]; AF or Aud, fall, spring, offered periodically) Offered for one year or one semester as circumstances warrant.
MATH
8201
 General Algebra
(3.0 cr; Prereq4xxx algebra or equiv or #; AF or Aud, fall, every year) Groups through Sylow, JordanH[o]lder theorems, structure of finitely generated Abelian groups. Rings and algebras, including Gauss theory of factorization. Modules, including projective and injective modules, chain conditions, Hilbert basis theorem, and structure of modules over principal ideal domains.
MATH
8202
 General Algebra
(3.0 cr; Prereq8201 or #; AF or Aud, spring, every year) Classical field theory through Galois theory, including solvable equations. Symmetric, Hermitian, orthogonal, and unitary form. Tensor and exterior algebras. Basic Wedderburn theory of rings; basic representation theory of groups.
MATH
8207
 Theory of Modular Forms and LFunctions
(3.0 cr; Prereq8202 or #; AF or Aud) Zeta and Lfunctions, prime number theorem, Dirichlet's theorem on primes in arithmetic progressions, class number formulas; Riemann hypothesis; modular forms and associated Lfunction; Eisenstein series; Hecke operators, Poincar[e] series, Euler products; Ramanujan conjectures; Theta series and quadratic forms; waveforms and Lfunctions.
MATH
8208
 Theory of Modular Forms and LFunctions
(3.0 cr; Prereq8207 or #; AF or Aud) Applications of Eisenstein series: special values and analytic continuation and functional equations of Lfunctions. Trace formulas. Applications of representation theory. Computations.
MATH
8211
 Commutative and Homological Algebra
(3.0 cr; Prereq8202 or #; AF or Aud, fall, offered periodically) Selected topics.
MATH
8212
 Commutative and Homological Algebra
(3.0 cr; Prereq8211 or #; AF or Aud) Selected topics.
MATH
8245
 Group Theory
(3.0 cr; Prereq8202 or #; AF or Aud, fall, every year) Permutations, Sylow's theorems, representations of groups on groups, semidirect products, solvable and nilpotent groups, generalized Fitting subgroups, pgroups, coprime action on pgroups.
MATH
8246
 Group Theory
(3.0 cr; Prereq8245 or #; AF or Aud, fall, spring, offered periodically) Representation and character theory, simple groups, free groups and products, presentations, extensions, Schur multipliers.
MATH
8251
 Algebraic Number Theory
(3.0 cr; Prereq8202 or #; AF or Aud) Algebraic number fields and algebraic curves. Basic commutative algebra. Completions: padic fields, formal power series, Puiseux series. Ramification, discriminant, different. Finiteness of class number and units theorem.
MATH
8252
 Algebraic Number Theory
(3.0 cr; Prereq8251 or #; AF or Aud) Zeta and Lfunctions of global fields. Artin Lfunctions. HasseWeil Lfunctions. Tchebotarev density. Local and global class field theory. Reciprocity laws. Finer theory of cyclotomic fields.
MATH
8253
 Algebraic Geometry
(3.0 cr; Prereq8202 or #; AF or Aud, fall, offered periodically) Curves, surfaces, projective space, affine and projective varieties. Rational maps. Blowingup points. Zariski topology. Irreducible varieties, divisors.
MATH
8254
 Algebraic Geometry
(3.0 cr; Prereq8253 or #; AF or Aud, spring, offered periodically) Sheaves, ringed spaces, and schemes. Morphisms. Derived functors and cohomology, Serre duality. RiemannRoch theorem for curves, Hurwitz's theorem. Surfaces: monoidal transformations, birational transformations.
MATH
8270
 Topics in Algebraic Geometry
(1.0  3.0 cr [max 12.0 cr]; PrereqMath 8201, Math 8202; offered for one year or one semester as circumstances warrant; AF or Aud, fall, spring, every year)
MATH
8271
 Lie Groups and Lie Algebras
(3.0 cr; Prereq8302 or #; AF or Aud, fall, 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; MaurerCartan forms; exponential map; correspondence between Lie algebras and simply connected Lie groups; BakerCampbellHausdorff formula; homogeneous spaces.
MATH
8272
 Lie Groups and Lie Algebras
(3.0 cr; Prereq8271 or #; AF or Aud, spring, 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]; AF or Aud, fall, spring, offered periodically) Various topics in Number Theory.
MATH
8300
 Topics in Algebra
(1.0  3.0 cr [max 12.0 cr]; PrereqGrad math major or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, every year) Selected topics.
MATH
8301
 Manifolds and Topology
(3.0 cr; Prereq[Some pointset topology, algebra] or #; AF or Aud, fall, every year) Classification of compact surfaces, fundamental group/covering spaces. Homology group, basic cohomology. Application to degree of a map, invariance of domain/dimension.
MATH
8302
 Manifolds and Topology
(3.0 cr; Prereq8301 or #; AF or Aud, spring, every year) Smooth manifolds, tangent spaces, embedding/immersion, Sard's theorem, Frobenius theorem. Differential forms, integration. Curvature, GaussBonnet theorem. Time permitting: de Rham, duality in manifolds.
MATH
8306
 Algebraic Topology
(3.0 cr; Prereq8301 or #; AF or Aud) Singular homology, cohomology theory with coefficients. EilenbergStenrod axioms, MayerVietoris theorem.
MATH
8307
 Algebraic Topology
(3.0 cr; Prereq8306 or #; AF or Aud) Basic homotopy theory, cohomology rings with applications. Time permitting: fibre spaces, cohomology operations, extraordinary cohomology theories.
MATH
8333
 FTE: Master's
(1.0 cr; PrereqMaster'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]; Prereq8301 or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, offered periodically) Selected topics.
MATH
8365
 Riemannian Geometry
(3.0 cr; Prereq8301 or basic pointset topology or #; AF or Aud, fall, every year) Riemannian metrics, curvature. Bianchi identities, GaussBonnet theorem, Meyers's theorem, CartanHadamard theorem.
MATH
8366
 Riemannian Geometry
(3.0 cr; Prereq8365 or #; AF or Aud, spring, every year) Gauss, Codazzi equations. Tensor calculus, Hodge theory, spinors, global differential geometry, applications.
MATH
8370
 Topics in Differential Geometry
(1.0  3.0 cr [max 12.0 cr]; Prereq8301 or 8365; offered for one yr or one sem as circumstances warrant; AF or Aud, fall, spring, every year) Current research in Differential Geometry.
MATH
8380
 Topics in Advanced Geometry
(1.0  3.0 cr [max 12.0 cr]; Prereq8301, 8365; AF or Aud, fall, spring, offered periodically) Current research.
MATH
8385
 Calculus of Variations and Minimal Surfaces
(3.0 cr; Prereq4xxx partial differential equations or #; AF or Aud) Comprehensive exposition of calculus of variations and its applications. Theory for onedimensional problems. Survey of typical problems. Necessary conditions. Sufficient conditions. Second variation, accessory eigenvalue problem. Variational problems with subsidiary conditions. Direct methods.
MATH
8386
 Calculus of Variations and Minimal Surfaces
(3.0 cr; Prereq8595 or #; AF or Aud) Theory of multiple integrals. Geometrical differential equations, i.e., theory of minimal surfaces and related structures (surfaces of constant or prescribed mean curvature, solutions to variational integrals involving surface curvatures), all extremals for variational problems of current interest as models for interfaces in real materials.
MATH
8387
 Mathematical Modeling of Industrial Problems
(3.0 cr; Prereq[5xxx numerical analysis, some computer experience] or #; AF or Aud, fall, every year) Mathematical models from physical, biological, social systems. Emphasizes industrial applications. Modeling of deterministic/probabilistic, discrete/continuous processes; methods for analysis/computation.
MATH
8388
 Mathematical Modeling of Industrial Problems
(3.0 cr; Prereq8597 or #; AF or Aud) Techniques for analysis of mathematical models. Asymptotic methods; design of simulation and visualization techniques. Specific computation for models arising in industrial problems.
MATH
8390
 Topics in Mathematical Physics
(1.0  3.0 cr [max 12.0 cr]; Prereq8601; offered for one yr or one sem as circumstances warrant; AF or Aud) Current research.
MATH
8401
 Mathematical Modeling and Methods of Applied Mathematics
(3.0 cr; Prereq4xxx numerical analysis and applied linear algebra or #; AF or Aud, fall, every year) Dimension analysis, similarity solutions, linearization, stability theory, wellposedness, and characterization of type. Fourier series and integrals, wavelets, Green's functions, weak solutions and distributions.
MATH
8402
 Mathematical Modeling and Methods of Applied Mathematics
(3.0 cr; Prereq8401 or #; AF or Aud, spring, every year) Calculus of variations, integral equations, eigenvalue problems, spectral theory. Perturbation, asymptotic methods. Artificial boundary conditions, conformal mapping, coordinate transformations. Applications to specific modeling problems.
MATH
8431
 Mathematical Fluid Mechanics
(3.0 cr; Prereq5xxx numerical analysis of partial differential equations or #; AF or Aud) Equations of continuity/motion. Kinematics. Bernoulli's theorem, stream function, velocity potential. Applications of conformal mapping.
MATH
8432
 Mathematical Fluid Mechanics
(3.0 cr; Prereq8431 or #) Plane flow of gas, characteristic method, hodograph method. Singular surfaces, shock waves, shock layers. Viscous flow, NavierStokes equations, exact solutions. Uniqueness, stability, existence theorems.
MATH
8441
 Numerical Analysis and Scientific Computing
(3.0 cr; Prereq[4xxx analysis, 4xxx applied linear algebra] or #; fall, every year) Approximation of functions, numerical integration. Numerical methods for elliptic partial differential equations, including finite element methods, finite difference methods, and spectral methods. Grid generation.
MATH
8442
 Numerical Analysis and Scientific Computing
(3.0 cr; Prereq8441 or #; 54775478 recommended for engineering and science grad students; spring, every year) Numerical methods for integral equations, parabolic partial differential equations, hyperbolic partial differential equations. Monte Carlo methods.
MATH
8444
 FTE: Doctoral
(1.0 cr; PrereqDoctoral student, adviser and DGS consent; No Grade, fall, spring, summer, every year) (No description)
MATH
8445
 Numerical Analysis of Differential Equations
(3.0 cr; Prereq4xxx numerical analysis, 4xxx partial differential equations or #; AF or Aud, fall, every year) Finite element and finite difference methods for elliptic boundary value problems (e.g., Laplace's equation) and solution of resulting linear systems by direct and iterative methods.
MATH
8446
 Numerical Analysis of Differential Equations
(3.0 cr; Prereq8445 or #; AF or Aud, spring, every year) Numerical methods for parabolic equations (e.g., heat equations). Methods for elasticity, fluid mechanics, electromagnetics. Applications to specific computations.
MATH
8450
 Topics in Numerical Analysis
(1.0  3.0 cr [max 12.0 cr]; PrereqGrad math major or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, every year) Selected topics.
MATH
8470
 Topics in Mathematical Theory of Continuum Mechanics
(1.0  3.0 cr [max 12.0 cr]; AF or Aud, fall, spring, offered periodically) Offered for one year or one semester as circumstances warrant.
MATH
8501
 Differential Equations and Dynamical Systems I
(3.0 cr; Prereq4xxx ODE or #; AF or Aud, fall, every year) Existence, uniqueness, continuity, and differentiability of solutions. Linear theory and hyperbolicity. Basics of dynamical systems. Local behavior near a fixed point, a periodic orbit, and a homoclinic or heteroclinic orbit. Perturbation theory.
MATH
8502
 Differential Equations and Dynamical Systems II
(3.0 cr; Prereq8501 or #; AF 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. MorseSmale systems. Coupled oscillators and network dynamics.
MATH
8503
 Bifurcation Theory in Ordinary Differential Equations
(3.0 cr; Prereq8501 or #; AF or Aud) Basic bifurcation theory, Hopf bifurcation, and method averaging. Silnikov bifurcations. Singular perturbations. Higher order bifurcations. Applications.
MATH
8505
 Applied Dynamical Systems and Bifurcation Theory I
(3.0 cr; Prereq5525 or 8502 or #; AF or Aud) Static/Hopf bifurcations, invariant manifold theory, normal forms, averaging, Hopf bifurcation in maps, forced oscillations, coupled oscillators, chaotic dynamics, codimension 2 bifurcations. Emphasizes computational aspects/applications from biology, chemistry, engineering, physics.
MATH
8506
 Applied Dynamical Systems and Bifurcation Theory II
(3.0 cr; Prereq5587 or #; AF or Aud, fall, offered periodically) Background on analysis in Banach spaces, linear operator theory. LyapunovSchmidt reduction, static bifurcation, stability at a simple eigenvalue, Hopf bifurcation in infinite dimensions invariant manifold theory. Applications to hydrodynamic stability problems, reactiondiffusion equations, pattern formation, and elasticity.
MATH
8520
 Topics in Dynamical Systems
(1.0  3.0 cr [max 12.0 cr]; Prereq8502; AF or Aud, fall, spring, offered periodically) Current research.
MATH
8530
 Topics in Ordinary Differential Equations
(1.0  3.0 cr [max 3.0 cr]; Prereq8502; AF or Aud, fall, spring, offered 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]; AF or Aud, fall, spring, every year) Offered for one year or one semester as circumstances warrant.
MATH
8571
 Theory of Evolutionary Equations
(3.0 cr; Prereq8502 or #; AF or Aud, fall, every year) Infinite dimensional dynamical systems, global attractors, existence and robustness. Linear semigroups, analytic semigroups. Linear and nonlinear reaction diffusion equations, strong and weak solutions, wellposedness of solutions.
MATH
8572
 Theory of Evolutionary Equations
(3.0 cr; Prereq8571 or #; AF or Aud, spring, offered periodically) Dynamics of NavierStokes equations, strong/weak solutions, global attractors. Chemically reacting fluid flows. Dynamics in infinite dimensions, unstable manifolds, center manifolds perturbation theory. Inertial manifolds, finite dimensional structures. Dynamical theories of turbulence.
MATH
8580
 Topics in Evolutionary Equations
(1.0  3.0 cr [max 12.0 cr]; Prereq8572 or #; offered for one yr or one semester as circumstances warrant; AF or Aud)
MATH
8581
 Applications of Linear Operator Theory
(3.0 cr; Prereq4xxx applied mathematics or #; AF or Aud) Metric spaces, continuity, completeness, contraction mappings, compactness. Normed linear spaces, continuous linear transformations. Hilbert spaces, orthogonality, projections.
MATH
8582
 Applications of Linear Operator Theory
(3.0 cr; Prereq8581 or #; AF or Aud) Fourier theory. Selfadjoint, compact, unbounded linear operators. Spectral analysis, eigenvalueeigenvector problem, spectral theorem, operational calculus.
MATH
8583
 Theory of Partial Differential Equations
(3.0 cr; Prereq[Some 5xxx PDE, 8601] or #; AF or Aud, fall, every year) Classification of partial differential equations/characteristics. Laplace, wave, heat equations. Some mixed problems.
MATH
8584
 Theory of Partial Differential Equations
(3.0 cr; Prereq8583 or #; AF or Aud, spring, every year) Fundamental solutions/distributions, Sobolev spaces, regularity. Advanced elliptic theory (Schauder estimates, Garding's inequality). Hyperbolic systems.
MATH
8590
 Topics in Partial Differential Equations
(1.0  3.0 cr [max 3.0 cr]; Prereq8602; offered for one yr or one sem as circumstances warrant; AF or Aud, fall, spring, every year) Research topics.
MATH
8600
 Topics in Advanced Applied Mathematics
(1.0  3.0 cr [max 12.0 cr]; fall, spring, every year) Offered for one yr or one semester as circumstances warrant. Topics vary. For details, contact instructor.
MATH
8601
 Real Analysis
(3.0 cr; Prereq5616 or #; AF or Aud, fall, every year) Set theory/fundamentals. Axiom of choice, measures, measure spaces, Borel/Lebesgue measure, integration, fundamental convergence theorems, Riesz representation.
MATH
8602
 Real Analysis
(3.0 cr; Prereq8601 or #; AF or Aud, spring, every year) RadonNikodym, Fubini theorems. C(X). Lp spaces (introduction to metric, Banach, Hilbert spaces). StoneWeierstrass theorem. Basic Fourier analysis. Theory of differentiation.
MATH
8640
 Topics in Real Analysis
(3.0 cr [max 12.0 cr]; Prereq8602 or #; offered for one yr or one sem as circumstances warrant; AF or Aud) Current research.
MATH
8641
 Spatial Ecology
(3.0 cr; PrereqTwo semesters calculus, theoretical population ecology or four semesters more robust calculus, course in statistics or probability or #; SN or Aud) Introduction: role of space in population dynamics and interspecific interaction; includes single species and multispecies models, deterministic and stochastic theory, different modeling approaches, effects of implicit/explicit space on competition, pattern formation, stability diversity and invasion. Recent literature. Computer lab.
MATH
8651
 Theory of Probability Including Measure Theory
(3.0 cr; Prereq5616 or #; fall, every year) Probability spaces. Distributions/expectations of random variables. Basic theorems of Lebesque theory. Stochastic independence, sums of independent random variables, random walks, filtrations. Probability, moment generating functions, characteristic functions. Laws of large numbers.
MATH
8652
 Theory of Probability Including Measure Theory
(3.0 cr; Prereq8651 or #; spring, every year) Conditional distributions and expectations, convergence of sequences of distributions on real line and on Polish spaces, central limit theorem and related limit theorems, Brownian motion, martingales and introduction to other stochastic sequences.
MATH
8654
 Fundamentals of Probability Theory and Stochastic Processes
(3.0 cr; Prereq8651 or 8602 or #; spring, 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 submartingales, DoobMeyer decomposition.
MATH
8655
 Stochastic Calculus with Applications
(3.0 cr; Prereq8654 or 8659 or #; fall, every year) Stochastic integration with respect to martingales, Ito's formula, applications to business models, filtering, and stochastic control theory.
MATH
8659
 Stochastic Processes
(3.0 cr; Prereq8652 or #; fall, every year) Indepth coverage of various stochastic processes and related concepts, such as Markov sequences and processes, renewal sequences, exchangeable sequences, stationary sequences, Poisson point processes, Levy processes, interacting particle systems, diffusions, and stochastic integrals.
MATH
8660
 Topics in Probability
(1.0  3.0 cr [max 12.0 cr]; fall, spring, every year) Offered for one year or one semester as circumstances warrant.
MATH
8666
 Doctoral PreThesis Credits
(1.0  6.0 cr [max 12.0 cr]; PrereqDoctoral 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; AF or Aud, fall, offered periodically) Basic enumeration, including sets and multisets, permutation statistics, inclusionexclusion, integer/set partitions, involutions and Polya theory. Partially ordered sets, including lattices, incidence algebras, and Mobius inversion. Generating functions.
MATH
8669
 Combinatorial Theory
(3.0 cr; Prereq8668 or #; AF or Aud, spring, odd academic years) Further topics in enumeration, including symmetric functions, Schensted correspondence, and standard tableaux; nonenumerative combinatorics, including graph theory and coloring, matching theory, connectivity, flows in networks, codes, and extremal set theory.
MATH
8680
 Topics in Combinatorics
(1.0  3.0 cr [max 12.0 cr]; PrereqGrad math major or #; offered as one yr or one sem crse as circumstances warrant; AF or Aud, fall, spring, every year) Selected topics.
MATH
8701
 Complex Analysis
(3.0 cr; Prereq5616 or #; AF or Aud, fall, every year) Foundations of holomorphic functions of one variable; relation to potential theory, complex manifolds, algebraic geometry, number theory. Cauchy's theorems, Poisson integral. Singularities, series, product representations. Hyperbolic geometry, isometries. Covering surfaces, RiemannHurwitz formula. SchwarzChristoffel polygonal functions. Residues.
MATH
8702
 Complex Analysis
(3.0 cr; Prereq8701 or #; AF or Aud, spring, every year) Riemann mapping, uniformization, Dirichlet problem. Dirichlet principle, Green's functions, harmonic measures. Approximation theory. Complex analysis on tori (elliptic functions, modular functions, conformal moduli). Complex dynamical systems (Julia sets, Mandelbrot set).
MATH
8777
 Thesis Credits: Master's
(1.0  18.0 cr [max 50.0 cr]; PrereqMax 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]; Prereq8702 or #; offered for one yr or one sem as circumstances warrant; AF or Aud) Current research.
MATH
8801
 Functional Analysis
(3.0 cr; Prereq8602 or #; AF or Aud, fall, every year) Motivation in terms of specific problems (e.g., Fourier series, eigenfunctions). Theory of compact operators. Basic theory of Banach spaces (HahnBanach, open mapping, closed graph theorems). Frechet spaces.
MATH
8802
 Functional Analysis
(3.0 cr; Prereq8801 or #; AF or Aud, spring, offered periodically) Spectral theory of operators, theory of distributions (generalized functions), Fourier transformations and applications. Sobolev spaces and pseudodifferential operators. Cstar algebras (GelfandNaimark theory) and introduction to von Neumann algebras.
MATH
8888
 Thesis Credit: Doctoral
(1.0  24.0 cr [max 100.0 cr]; PrereqMax 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#; SN 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#; SN or Aud, fall, spring, every year) Individually directed reading.
MATH
8993
 Directed Study
(1.0  6.0 cr [max 24.0 cr]; Prereq#; SN or Aud, spring, every year) Individually directed study.
MATH
8994
 Topics at the IMA
(1.0  3.0 cr [max 6.0 cr]; fall, spring, every year) Current research at IMA.






