Calculus IV (15 MATH 254), Introduction to Ordinary Differential Equations (15 MATH 355), and Introduction to Abstract Mathematics (15 MATH 357).
Text:
Rosenlicht, Introduction to Analysis
Gordon, Real Analysis - A First Glance, 2nd Edition
15 MATH 504
Ordered sets, the real field, the complex field, Euclidean space, finite, countable and uncountable sets, metric spaces, compact sets, convergent sequences of numbers, Cauchy sequences, upper and lower limits, Bolzano-Weierstrass theorem, series, the number e, convergence tests for series, absolute convergence, addition and multiplication of series, rearrangements. Aut. Qtr.
15 MATH 505
Limits and continuity of functions, continuity and compactness, connectedness and continuity, discontinuities, monotone functions, derivatives, the Mean Value theorem, l'Hopital's rule, higher order derivatives, Taylor's theorem, Riemann-Stieltjes integral, integration and differentiation of vector-valued functions, rectifiable curves. Win. Qtr.
15 MATH 506
Uniform convergence for sequences and series of functions, equi-continuous families of functions, the Stone-Weierstrass theorem, functions of several variables. Spr. Qtr.
ABSTRACT ALGEBRA I, II, III
15 MATH 511, 512, 513
3 UG or GR CR ea. qtr.
Prerequisite:
Linear Algebra II (15 MATH 352), Introduction to Abstract Mathematics (15 MATH 357). Sequence may be started with either 511 or 512 (i.e. 511 is not a prerequisite for 512; however, 512 is a prerequisite for 513).
Text:
Lang, Linear Algebra, 3rd Edition. (Text for 15 MATH 511 only)
15 MATH 511
Advanced Linear Algebra: Abstract vector spaces, determinants, eigenvalues and eigenvectors, algebra of linear transformations, canonical forms including triangular, Jordan and rational forms. Aut. Qtr.
Text:
Goodman, Algebra: Abstract and Concrete (Text for 15 MATH 512 and 15 MATH 513)
15 MATH 512
Definition and basic properties of groups, subgroups, permutation groups, direct products, isomorphisms, homomorphisms, normal subgroups and factor groups. Win. Qtr.
15 MATH 513
Selected topics in number theory. Binary relations and binary operations. Definitions and basic properties of rings and fields, integral domain, quotient fields, quotient rings and ideals, factorization of polynomials over fields, unique factorization domains, Euclidean domains, Gaussian integers, extension fields, algebraic extensions, geometric constructions, finite fields. Spr. Qtr.
NUMERICAL ANALYSIS I, II, III
15 MATH 514, 515, 516
3 UG or GR CR ea. qtr.
Prerequisite:
Calculus IV (15 MATH 254); Differential Equations (15 MATH 273) or Introduction to Ordinary Differential Equations (15 MATH 355); Matrix Methods (15 MATH 276) or Linear Algebra II (15 MATH 352); a working knowledge of some programming language.
Chapters 1, 4, 5. Introduction to a floating point arithmetic, roundoff error, error propagation.Solution of non-linear equations by bisection, secant, regula-falsi, and Newton methods with emphasis on error analysis and utility of computations. Polynomial interpolation, error bounds and the Runge phenomenon. Cubic spline interpolation and extremal properties. Orthogonal polynomials and least squares approximation.Computer applications. Aut. Qtr.
15 MATH 515
Chapters 2, 4. Gauss elimination, pivoting strategies. Error analysis and vector norms. Iterative methods for linear systems including Jacobi and Gauss-Seidel methods. Eigenvalue-eigenvector computations by power, inverse power, and Rayleigh quotient methods. Householder transformations, Hessenberg matrices and the Q-R method. The singular value decomposition and least squares problems. Computer applications. Win. Qtr.
15 MATH 516
Chapters 6, 7, 8. Numerical differentiation. Newton-Cotes and Gaussian quadrature, Romberg integration, FFT, Adaptive quadrature. Numerical methods for initial value ordinary differential equations including methods of Runge-Kutta type and predictor-corrector methods. Stability, consistency, and convergence are analyzed. Finite difference methods for two-point boundary value problems. Decent methods for optimization problems. Computer applications. Spr. Qtr.
APPLIED MATHEMATICS PRACTICUM
15 MATH 517, 518, 519
3 UG or GR CR ea. qtr.
Prerequisite:
Calculus IV (15 MATH 254), Differential Equations (15 MATH 273), and computer programming experience.
Text:
TBA
15 MATH 517
Techniques in applied mathematics; ordinary and partial differential equations, numerical methods, perturbation techniques, modeling. Under the guidance of the instructor, teams of students solve problems from industry, government, etc. and present reports on their findings. Offered variable quarters.
15 MATH 518
A continuation of 15 MATH 517.
15 MATH 519
A continuation of 15 MATH 518.
MATHEMATICAL STATISTICS I, II, III
15 MATH 521, 522, 523
3 UG or GR CR ea. qtr.
Prerequisite:
Calculus IV (15 MATH 254) and Probability and Statistics I (15 MATH 361).
Text:
Hogg, McKean, and Craig, Introduction to Mathematical Statistics, 6 th Edition.
15 MATH 521
Chapters 1, 2, 3 (through 3.4). Random variables, probability distribution functions, mathematical expectation, inequalities, moment-generating functions, transformation of variables, marginal and conditional distributions, independence, binomial, Poisson, Gamma and normal distributions.
Aut. Qtr.
15 MATH 522
Chapters 3 (starting 3.5), 4, 5. Multivariate Normal, t- and F- distributions, sampling distributions: , order statistics, , distribution of sample mean and sample variance, stochastic convergence, central limit theorem, , , confidence intervals, hypothesis testing, chi-square tests, Monte Carlo methods, bootstarp. Win. Qtr.
15 MATH 523
Chapters 6, 7,8. , Uniformly most powerful tests, likelihood ratio tests, sufficient statistics, Rao-Blackwell theorem, exponential family , Rao-Cramer bound , sequential tests, minimax and classification procedure. Spr. Qtr.
LINEAR PROGRAMMING I, II
15 MATH 524, 525
3 UG or GR CR ea. qtr.
Prerequisite:
Calculus IV (15 MATH 254); Linear Algebra II (15 MATH 352)
Text:
TBA
15 MATH 524
The simplex method (initialization, iteration, termination, sensitivity), the revised simplex method, duality, complementary slackness, the transportation problem, applications. Win. Qtr.
15 MATH 525
The transshipment problem, caterer problem, networks, max flow/min cut, matching problems, primal dual algorithm, Ford-Fulkerson algorithm, integer programming (cutting planes and branch and bound), interior point methods (ellipsoid method, Karmarkar’s method), applications. Spr. Qtr.
NON-LINEAR OPTIMIZATION
15 MATH 526
3 UG or GR CR
Prerequisite:
Calculus IV (15 MATH 254)
Text:
TBA
Methods of unconstrained optimization, the steepest descent method, Newton’s Method, conjugate direction methods, quasi-Newton and variable metric methods, theory and methods of constrained penalty methods. Spr. Qtr.
APPLIED STATISTICAL INFERENCE
15 MATH 531
3 UG or GR CR
Prerequisite:
Calculus IV (15 MATH 254) and Linear Algebra II (15 MATH 352)
Text:
Milton and Arnold, Introduction to Probability and Statistics, 4th Edition.
Quick review of probability distributions. Inferences about population means and variance. Aut., Sum. Qtrs.
APPLIED REGRESSION ANALYSIS
15 MATH 532
3 UG or GR CR
Prerequisite:
Applied Statistical Inference (15 MATH 531) or Probability and Statistics I and II (15 MATH 361, 362)
Text:
Neter, Applied Linear Statistical Models; SAS System for Linear Models, 3rd Edition.
Correlation and multiple regression.One-way ANOVA and multiple comparisons. Projects using SAS packages. Win., Sum. Qtrs.
ANALYSIS OF VARIANCE
15 MATH 533
3 UG or GR CR
Prerequisite:
Applied Regression Analysis (15 MATH 532)
Text:
Neter, Applied Linear Statistical Models; SAS System for Linear Models, 3rd Edition.
ANOVA for some standard experimental designs and unbalanced designs. Repeated measures and the analysis of covariance. Spr. Qtr.
SAS PROGRAMMING
15 MATH 534
3 UG or GR CR
Prerequisite:
Applied Regression Analysis (15 MATH 532) ~ can be taken concurrently.
Text:
Delwiche & Slaughter, The Little SAS Book
Carpenter, CarpenterÕs Guide to the SAS Macro
This course will study various aspects of the SAS statistical package from a programming language perspective. It will emphasize the SAS data steps including the infile, input, merge, set, do-loop, if-then commands, etc. SAS mathematical, statistical, and data functions are discussed, as well as learning to write MACROs and how to do extensive matrix computations using PROC IML, also PROC INSIGHT, and the high resolution graphics procedures. The concentration is on programming issues rather than on statistical procedures; however, several statistical procedures are discussed and illustrated. Win., Sum. Qtrs.
APPLIED STATISTICS USING S-PLUS
15 MATH 535
3 UG or GR CR
Prerequisite:
Probability and Statistics (15 MATH 361, 362, 363) or Applied Statistical Inference (15 MATH 531) or Applied Regression Analysis (15 MATH 532) or Analysis of Variance (15 MATH 533).
Text:
TBA
To obtain and enhance statistical analysis and programming skills using S-Plus. Various modern techniques in linear statistical modeling, write statistical functions, create graphs. Spr. Qtr.
PROBABILISTIC ASPECTS OF FINANCIAL MODELING
15 MATH 540
3 UG or GR CR
Prerequisite:
Probability & Statistics (15 MATH 361) or Mathematical Statistics I (15 MATH 521). Applied Probability and Stochastic Processes (15 MATH 577) recommended.
Text:
TBA
An introduction to the mathematical theory behind discrete and continuous time financial models. Covers martingales, martingales measures, change of measure, martingale representation, and Black Scholes formula.
COMPUTATIONAL FINANCIAL MATHEMATICS I, II, III
15 MATH 541, 542, 543
3 UG or GR CR ea qtr.
Prerequisite:
Calculus IV (15 MATH 254), Differential Equations (15 MATH 273), Matrix Methods (15 MATH 276), Probability & Statistics (15 MATH 361) or equivalent courses. 15 MATH 541 is a prerequisite for 15 MATH 542; 15 MATH 542 is a prerequisite for 15 MATH 543.
Text:
Stojanovic, Computational Financial Mathematics Using Mathematica
15 MATH 541
Symbolic and numerical solutions of ODEs, Brownian motion, stochastic calculus, Black Scholes formula, computer lab using Mathematica. Aut. Qtr.
15 MATH 542
Stock market statistics, Bayesian and non-Bayesian estimates, implied volatility, numerical PDEs, optimal control of PDEs. Computer lab using Mathematica. Win. Qtr.
15 MATH 543
American options, optimal stopping, Dupire PDE, portfolio rules, portfolio optimization, computer lab using Mathematica. Spr. Qtr.
NUMBER THEORY
15 MATH 551
3 UG or GR CR ea. qtr.
Prerequisite:
Intro. To Algebra I, II (15 MATH 401, 402)
Text:
Joseph Silverman, A Friendly Introduction to Number Theory, 2nd Edition
Heat equation, separation of variables, LaPlace equation, Fourier series, vibrating strings, and membranes.
15 MATH 554
Sturm-Liouville problems. PDE with at least three independent variables, Green’s functions, non-homogenous problem, Fourier transform, characterization. Win., Spr. Qtrs.
APPLIED LINEAR ALGEBRA I, II
15 MATH 555, 556
3 UG or GR CR ea. qtr.
Prerequisite:
Linear Algebra I (15 MATH 351) or Matrix Methods (15 MATH 276)
Text:
J.L. Goldberg, Matrix Theory with Applications
15 MATH 555
Gaussian elimination, triangular factorization, band matrices, linear independence, computation of column space and nullspace of a matrix, orthogonality and geometry of Rn projections onto subspaces, least squares approximation, the pseudo-inverse. Aut., Sum. Qtrs
15 MATH 556
Stability of linear differential and difference equations, the Spectral Theorem for symmetric matrices, positive definite matrices, the generalized eigenvalue problem, the Rayleigh quotient and minimax principles. Win., Sum. Qtrs.
SCIENTIFIC PROGRAMMING WITH MATLAB
15 MATH 560
3 UG or GR CR
Prerequisite:
Calculus IV (15 MATH 254), Linear Algebra (15 MATH 351, 352) or Differential Equations (15 MATH 273)
Text:
TBA
Applications of scientific programming with MATLAB to calculus, linear algebra, or differential equations. Spr. Qtr.
NUMERICAL METHODS IN APPLIED MATHEMATICS
15 MATH 561
3 UG or GR CR
Prerequisite:
Calculus IV (15 MATH 254), Differential Equations (15 MATH 273), and
Matrix Methods (15 MATH 276)
Text:
G.D. Smith, Numerical Solution of Partial Differential Equations, 3rd Edition
Methodology and ideas behind numerical schemes, focusing on finite difference and finite element methods applied to problems in elasticity, fluid dynamics, heat conduction, groundwater flow, and wave propagation.
Aut. Qtr.
TIME SERIES
15 MATH 571
3 UG or GR CR
Prerequisite:
Probability and Statistics II (15 MATH 362) or Mathematical Statistics (15 MATH 522) or any course ont regression.
Text:
Shumway and Stoffer, Time Series Analysis and Its Applications
Estimation and use of the autocorrelation function (ACF) and partial autocorrelation function (PACF); linear stationary models, including autoregressive (AR), moving average (MA), and ARIMA models; model identification, estimation, and forecasting; spectrum and periodgram of stationary processes. Techniques will be illustrated using computer software on real time series data.
RELIABILITY - SURVIVAL ANALYSIS
15 MATH 572
3 UG or GR CR
Prerequisite:
Mathematical Statistics (15 MATH 522) or a course on statistical inference.
Text:
Lee & Wang, Statistical Methods for Survival Data Analysis
Allison, Survival Analysis Using the SAS System
Topics in applied life data analysis including reliability analysis (as in engineering fields) and survival analysis (as in medical and actuarial fields.) Survival and hazard functions, life table and product limits estimates, exponential, Weibull, and other parametric models. Censored data, co-variate models (parametric, non-parametric, semi-parametric), maximum likelihood methods. Examples given and analyzed using PROC LIFETEST, LIFEREG, PHGLM, etc. in SAS. Spr. Qtr.
APPLIED BAYESIAN ANALYSIS
15 MATH 573
3 UG or GR CR
Prerequisite:
Mathematical Statistics III (15 MATH 523) or equivalent course on statistical inference.
Text:
Gelman, Carlin, Stern, Rubin, Bayesian Data Analysis (subject to change)
Basic principles of Bayesian inference, including the concepts of prior and posterior distributions. Choice of prior distribution. Bayesian inference in one-parameter and two-parameter distributions where closed form answers are possible. Bayesian inference using (a) direct simulation and Monte Carlo, (b) Markov Chain Monte Carlo (MCMC), and their applications. Hierarchical models and applications. Testing point null hypothesis and the related issues in model selection and comparison, as time permits. Spr. Qtr.
NON-PARAMETRIC STATISTICS
15 MATH 573
3 UG or GR CR
Prerequisite:
Mathematical Statistics (15 MATH 523) or Probability and Statistics II (15 MATH 362) and consent of instructor.
Text:
Hollander and Wolfe, Non-parametric Statistical Methods, 1999.
One- and two-sample location problems. Wilcoxion statistics, rank tests, one- and two-way layout tests for independence, linear rank statistics, Kolmogorov test. Aut. Qtr.
ROBUST STATISTICS
15 MATH 575
3 UG or GR CR
Prerequisite:
Probability and Statistics II (15 MATH 362) or equivalent.
Text:
Hoaglin, Mosteller, Tukey, Understanding Robust and Exploratory Data Analysis.
Methods of data analysis that are used when a sample is not assumed to have come from a normal distribution. Classical methods of inference and estimation, while optimal with "normal" data are highly sensitive to arbitrarily small amounts of contamination in the sample. Theoretical, applied, and computational aspects of robustness will be covered. Topics may include Monte Carlo adaptive estimation, jackknifing, and bootstrapping. Win. Qtr.
TOPICS IN APPLIED STATISTICS
15 MATH 576
3 UG or GR CR
Prerequisite:
Mathematical Statistics (15 MATH 523) or permission of instructor.
Text:
TBA
This course covers selected topics in applied statistics, depending on the area of specialty of the instructor. Win., Sum. Qtrs.
APPLIED PROBABILITY & STOCHASTIC PROCESSES I, II
15 MATH 577, 578
3 UG or GR CR ea. qtr.
Prerequisite:
Calculus IV (15 MATH 254) and Probability & Statistics I (15 MATH 361)
Text:
Karlin and Taylor, Introduction to Stochastic Modeling
15 MATH 577
Basic elements of probability theory and stochastic processes, Markov chains, the Poisson process. Win. Qtr.
15 MATH 578
Additional topics from the theory of stochastic processes, plus applications. Spr. Qtr.
MATH AND MATHEMATICA
15 MATH 580
3 UG or GR CR
Prerequisite:
Calculus IV (15 MATH 254), Linear Algebra (15 MATH 352) or Matrix Methods (15 MATH 276) and Differential Equations (15 MATH 273 or 355). No prior knowledge of programming or Mathematica is required.
Text:
NO BOOK NEEDED
Projects using Mathematica. Aut., Sum. Qtrs.
INTEGRAL EQUATIONS
15 MATH 582
3 UG or GR CR
Prerequisite:
Advanced Calculus I, II (15 MATH 504, 505) or permission from instructor.
Text:
Jerri, Introduction to Integral Equations.
Finite rank kernels, Fredholm’s alternative, operators on Banach spaces, and application to Neumann series, resolvent for small values of the parameter. Operators on Hilbert spaces and application to the Hilbert-Schmidt theory of integral equations with symmetric kernels. Spectral theorem for compact operators and eigenvalue expansions. Spr. Qtr.
CALCULUS OF VARIATIONS
15 MATH 583
3 UG or GR CR
Prerequisite:
Linear Algebra II (15 MATH 352) or Matrix Methods (15 MATH 276); Calculus IV (15 MATH 254); Differential Equations (15 MATH 273) or Introduction to Ordinary Differential Equations (15 MATH 355).
Text:
Troutman, Variational Calculus with Elementary Convexity.
Euler-Lagrange equations, transversals, application to mechanics of particles and continua, integral constraints and application to isoperimetric problem, algebraic constraints and application to geodesics on surfaces, Hamilton-Jacobi method, solutions in bounded regions, differential constraints, Jacobi's sufficient condition. Win. Qtr.
COMBINATORICS
15 MATH 584
3 UG or GR CR
Prerequisite:
Matrix Methods (15 MATH 276) or Linear Algebra I (15 MATH 351).
Text:
Brualdi, Introductory Combinatorics.
Introduction to the theory and practice of enumeration, the Pigeonhole principle, permutations and combinations, binomial coefficients, inclusion-exclusion principle, recurrence relations, generating functions. Spr. Qtr.
GRAPH THEORY
15 MATH 588
3 UG or GR CR
Prerequisite:
Linear Algebra I (15 MATH 351) or Matrix Methods (15 MATH 276).
Text:
Bondy, Murty, Graph Theory with Applications.
Fundamental concepts of graphs and directed graphs, trees, connectivity, factorization, covering and packing, line graphs, planarity, traversability, colorability. Spr. Qtr.
The complex number system, elementary analytic functions and power series, conformal mapping and linear fractional transformations, Cauchy's Integral Theorem, Cauchy's Integral Formula, local properties of analytic functions, Schwarz's Lemma, calculus of residues, the Schwarz reflection principle, normal families, Riemann Mapping Theorem, harmonic functions, Dirichlet problem, entire and Meromorphic functions. Aut. ,Win., Spr. Qtrs.
GENERAL TOPOLOGY I, II, III
15 MATH 604, 605, 606
4 UG or GR CR ea. qtr.
Prerequisite:
Advanced Calculus III (15 MATH 506).
Text:
Munkres, Topology, 2nd Edition.
The topics to be covered include topologies, bases, subspaces, continuity, compactness and paracompactness, connectedness, some separation axioms, product spaces, quotient spaces, the compact-open topology, homotopy, the fundamental group, the Seifert – Van Kampen theorem, covering space theory (the lifting theorem, the group of Deck transformations, classification of covering spaces), smooth manifolds, the tangent bundle, regular values, the smooth approximation theorem, surfaces, homology, the Eilenberg-Steenrod axioms, the Euler characteristic, universal coefficient and Kunneth theorems, cohomology, products, Poincare duality, as well as additional topics chosen by the instructor. Aut., Win., Spr. Qtrs.
REAL ANALYSIS I, II, III
15 MATH 607, 608, 609
4 UG or GR CR ea. qtr.
Prerequisite:
Advanced Calculus III (15 MATH 506).
Text:
Royden, Real Analysis, 4th Edition.
Elementary set theory, Axiom of Choice, elementary topology. Lebesgue Measure and integration on the real line. Abstract measure and integration theory, product measures and Fubini's theorem, absolute continuity and the Radon-Nikodym theorem, signed measures and decomposition theorems, integration on locally compact spaces, Lp-spaces and the Riesz Representation Theorem. Elementary theory of topological vector spaces, normed spaces and Hilbert spaces, elementary theory of continuous linear operators. Aut., Win., Spr. Qtrs.
ALGEBRAIC STRUCTURES I, II, III
15 MATH 610, 611, 612
4 UG or GR CR ea. qtr.
Prerequisite:
Introduction to Abstract Algebra (15 MATH 513) or permission of instructor.
Text:
Dummit, Abstract Algebra, 2nd Edition
15 MATH 610
Group theory: Sylow's theorems, Fundamental Theorem of abelian groups, Jordan-Holder theorems, and solvable groups. Modules: Free modules and Zorn’s Lemma. Modules over PID. Categories: Products, co-products, and free objects. Aut. Qtr.
15 MATH 611
Fields: Algebraic and transcendental extensions, algebraic closure, Galois theory of finite extensions, and solvability by radicals. Win. Qtr.
LINEAR MODELS AND MULTIVARIATE ANALYSIS I, II, III
15 MATH 613, 614, 615
4 UG or GR CR ea. qtr.
Prerequisite:
Applied Linear Algebra (15 MATH 555, 556) or an equivalent course which covers the contents of 15 MATH 555 and positive definite matrices; Applied Statistical Inference (15 MATH 531), Applied Regression Analysis (15 MATH 532), Analysis of Variance (15 MATH 533); Mathematical Statistics (15 MATH 523), or equivalent courses.
Text:
S.R. Searle, Linear Models
15 MATH 613
Review of linear algebra, matrix theory, multivariate normal distribution, central and non-central chi-square, t and F-distributions, quadratic forms, best linear unbiased estimators (BLUE). Theory of linear models. The full rank and non-full rank models, multiple linear regression, one-way ANOVA. Aut. Qtr.
Text:
G.A. Milliken and D.E. Johnson, Analysis of Messy Data
15 MATH 614
Applications of the theory developed in Linear Models I. Selected topics from experimental design. Two way ANOVA, fixed, nested and random effects. Analysis of covariance. Split plot, and split-split plot designs, repeated measures, mixed models. Analysis using SAS.Win. Qtr.
Text:
R.A. Johnson and D.W. Wichern, Applied Multivariate Analysis
15 MATH 615
Continuation of topics from experimental design. Topics in Multivariate analysis. Multivariate T-tests, MANOVA, principal components, factor analysis, etc. Analysis using SAS Spr. Qtr.
ORDINARY DIFFERENTIAL EQUATIONS I, II, III
15 MATH 616, 617, 618
4 UG or GR CR ea. qtr.
Prerequisite:
Advanced Calculus (15 MATH 506).
Text:
Hirsch and Smale, Differential Equations, Dynamical Systems and Linear Algebra.
Sanchez, Ordinary Differential Equations and Stability Theory: An Introduction.
Existence and uniqueness for initial value problems. Linear systems. Linearization, plane systems, stability. Aut., Win., Spr. Qtrs.
MATHEMATICAL LOGIC I, II, III
15 MATH 621, 622, 623
4 UG or GR CR ea. qtr.
Prerequisite:
Intro to Abstract Algebra II (15 MATH 512) or Automata & Formal Lang I (ECES 670) or permission of instructor.
Text:
Enderton, A Mathematical Introduction to Logic.
Formal systems (first order). Proof of theoretic and model theoretic techniques and interconnections. Compactness and completeness theorems. Non-Standard models with applications to analysis. Peano arithmetic and set theory as illustrations of important first order systems. Aut., Win., Spr. Qtrs.
DYNAMICAL SYSTEMS I, II, III
15 MATH 624, 625, 626
4 UG or GR CR ea. qtr.
Prerequisite:
Advanced Calculus (15 MATH 506).
Text:
Hirsch & Smale, Differential Equations, Dynamical Systems, and Linear Algebra
Maps of the interval, period doubling to chaos, symbolic dynamics, Smale's horseshoe example, homoclinic orbits, bifurcation, Julia sets. Aut., Win., Spr. Qtrs.
PARTIAL DIFFERENTIAL EQUATIONS I, II, III
15 MATH 627, 628, 629
4 UG or GR CR ea. qtr.
Prerequisite:
Advanced Calculus (15 MATH 506).
Text:
Evans, Partial Differential Equations, 1998.
15 MATH 627
Transport equations, initial value problem; Laplace equation, fundamental solution, mean-value formulas, Green’s function; Heat equation, fundamental solution, strong maximum principle; Wave equations, solution by spherical means, energy methods; Nonlinear first-order PDE, characteristics. Aut. Qtr.
15 MATH 628
Holder spaces; Sobolev spaces; Approximation by smooth functions; Extensions; Traces; Sovolev inequalities; Compact embedding; Other spaces of functions; Elliptical equations, existence of weak solutions; Regularity; Maximum principles; Eigenfunctions and eigenvalues. Win. Qtr.
15 MATH 629
Existence of weak solutions for second-order parabolic equations, maximum principles; Galerkin approximations; Fixed point methods; Method of subsolutions and supersolutions; Semigroup theory. Spr. Qtr.
ADVANCED THEORY OF STATISTICS I, II, III
15 MATH 631, 632, 633
4 UG or GR CR ea. qtr.
Prerequisite:
Mathematical Statistics (15 MATH 523) and Advanced Calculus (15 MATH 506).
Text:
Bickel and Doksum, Mathematical Statistics.
This sequence is a continuation of Mathematical Statistics (15 MATH 521, 522, 523).
15 MATH 627
Review of probability theory, distribution theory, sufficient statistics, efficiency of estimators, maximum likelihood estimators, large sample theory, consistency, asymptotic efficiency, confidence intervals and testing. Aut. Qtr.
15 MATH 628
Elements of decision theory (unbiased estimation, admissibility, inadmissibility), Bayesian analysis, minimax estimators, invariant estimator, Bayes and minimax tests; likelihood ratio tests. Win. Qtr.
15 MATH 629
Topics selected from: uniformly most powerful tests, general linear hypotheses, multiple decision problems, sequential analysis, density estimation, empirical processes, etc. Spr. Qtr.
PROBABILITY THEORY I, II, III
15 MATH 634, 635, 636
4 UG or GR CR ea. qtr.
Prerequisite:
Mathematical Statistics (15 MATH 523) and Advanced Calculus (15 MATH 506).
Text:
Patrick Billingsley, Probability and Measure, 3rd Edition
15 MATH 634
Measure theory and Lebesgue integration theory (brief), probability measures, random variables, expectation laws of large numbers, Borel-Cantelli Lemmas, Zero-one laws, Glivenko-Cantelli Theorem, applications. Aut. Qtr
15 MATH 635
Weak convergence, characteristic functions, Central limit theorem, law of iterated logarithm, other limit theorems for independent and dependent sequences, conditional probability, conditional expectation. Win. Qtr.
15 MATH 636
Topics selected from martingales, Brownian motion process, invariance principle, and other material from stochastic processes. Spr. Qtr
ANALYTICAL METHODS I, II, III
15 MATH 701, 702, 703
3 GR CR ea. qtr. Credits may not be applied toward a degree in Mathematics.
Prerequisite:
Calculus IV (15 MATH 254) and Differential Equations (15 MATH 273).
First order differential equations. Linear differential equations of second and higher order. Equations with constant coefficients, Euler method of undetermined coefficients, variation of parameters. Fuchs-Frobenious method for linear second-order equations, application to Bessel functions. Laplace transforms (Ch. 1, 2, 4, 5.) Aut. Qtr.
15 MATH 702
Linear algebra, Gaussian elimination, inverse matrices, determinants, diagonalization. Application to quadratic forms and to systems of linear differential equations (Ch. 6, 7.) Vector differential calculus, double and triple integrals, line integrals, potential, surface integrals, Green theorem, Stokes theorem, Gauss theorem (Ch. 8, 9.) Win. Qtr.
15 MATH 703
Fourier analysis: Fourier series, Fourier transforms. Sturm-Liouville problems. Partial Differential Equations: Separation of variables. Heat equation, wave equation, Laplace equation. Double Fourier series, Fourier-Bessel series. Application of Laplace transforms (Ch. 10, 11, parts of 5.) Spr. Qtr.
MEASURE THEORETIC CALCULUS I, II, III
15 MATH 704, 705, 706
4 GR CR ea. qtr.
Prerequisite:
Real Analysis (15 MATH 607, 608).
Text:
Gariepy and Evans, Measure Theory and Fine Properties of Functions.
Jacobians, the area formula, Coarea Formula, Sobolev functions, Sobolev inequalities,capacity, quasi-continuity, BV functions. Win. Qtr
15 MATH 706
Coarea Formula for BV functions, Isoperimetric Inequality, the reduced boundary, Gauss-Green Theorem, Lp differentiability, Whitney’s Extension Theorem, approximation by C1 functions. Spr. Qtr.
ADVANCED NUMERICAL ANALYSIS I, II, III
15 MATH 710, 711, 712
4 UG or GR CR ea. qtr.
Prerequisite:
Numerical Analysis (15 MATH 516) or equivalent experience.
Text:
TBA
Topics to be chosen from: numerical solution of ordinary differential equations, numerical solution of partial differential equations; variational methods, finite elements; computational algebra, fast Fourier Transform. These and other topics to be included are dependent on the instructor's choice. Aut., Win., Spr. Qtrs.
STATISTICAL CONSULTING
15 MATH 720, 721, 722, 723
3 GR CR ea. qtr.
Prerequisite:
Mathematical Statistics I, II, III (15 MATH 521, 522, 523) AND Applied Statistical Inference (15 MATH 531), Applied Regression Analysis (15 MATH 532), Analysis of Variance (15 MATH 533).
Text:
TBA
Students enrolled in this course will participate in the statistical consulting mission of the Statistical Consulting Laboratory of the Department of Mathematical Sciences. Under the guidance of the director(s) of the Lab, students will typically work in teams of two on projects brought to the lab by other researchers from on- and off-campus. Students will be expected to interact with these researchers. A significant amount of class time will be devoted to learning new statistical techniques necessary for particular projects, as well as developing consulting and presentation skills.
MAT Courses
The following courses are offered for the M.A.T. Program and are offered only during the summer term.
TECHNOLOGY IN MATHEMATICS
15 MATH 747
3 GR CR
Graphical, symbolic, and numerical computation with applications using a variety of mathematical hardware and software.
ANALYSIS I & II
15 MATH 751, 752
4 GR CR ea. qtr.
Theory of calculus of one variable. Analysis I: Continuity and differentiability. Analysis II: Riemann integral and infinite series.
GEOMETRY I & II
15 MATH 755, 756
4 GR CR ea. qtr.
First term: Axiomatic geometry, both neutral and Euclidean. Second term: Transformational geometry. use of Geometer's sketchpad will be an integral part of the courses.
PROBABILITY & STATISTICS I & II
15 MATH 757, 758
3 GR CR ea. qtr.
First term: Discrete and continuous random variables and their distributions. Expected value and variance. Joint distributions and covariance. Second term: central limit theorem, estimation and hypothesis testing, linear regression, analysis of variance.
NUMBER THEORY
15 MATH 761
4 GR CR
Congruences, divisibility, primes, number-theoretic functions, number bases and applications.
MODERN ALGEBRA
15 MATH 762
4 GR CR
The theory of rings and fields with emphasis on the algebra of polynomials.
M.A.T. PROJECT I
15 MATH 798
2 GR CR
Preparation and presentation of the MAT project. Summer quarter only.
M.A.T. PROJECT II
15 MATH 799
2 GR CR
Preparation and presentation of the MAT project. Summer quarter only.
MATRICES
15 MATH 801
3 GR CR
Topic Matrices and linear transformations in 2 and 3 dimensions and related topics.
MATHEMATICAL MODELS
15 MATH 802
3 GR CR
Model building and the mathematical analysis of models.
HISTORICAL TOPICS IN MATHEMATICS
15 MATH 808
3 GR CR
Historical perspectives on subjects in secondary mathematics.
