2017  Spring Semester
GRADUATE COURSE SPRING 2017
This schedule is subject to changes. Please contact the Course Instructor for confirmation.
Course  Sec # 
Course Title  Course Day & Time  Rm #  Instructor 
Math 4309  15672  Mathematical Biology  TuTh, 2:304 p.m.  M 115  R. Azevedo 
12060/13695 
Introduction to Real Analysis II  TuTh, 1011:30 a.m.  F 154  D. Labate  
Math 4351  21457  Differential Geometry II  MW, 12:30 p.m.  SW 219  M. Ru 
Math 4355  21456  Mathematics of Signal Representation  TuTh, 12:30 p.m.  CAM 101  D. Labate 
Math 4364  19420  Intro. to Numerical Analysis in Scientific Computing  MW, 45:30 p.m.  CBB 118  T. Pan 
Math 4365  17384  Numerical Methods for Differential Equations  MW, 12:30 p.m.  SW 423  J. He 
14513/13696 
Advanced Linear Algebra I  TuTh, 2:304 p.m.  CBB 104  E. Kao  
18470/18471 
Advanced Linear Algebra I (online)  Online  Online  J. Morgan  
12061/13697 
Advanced Linear Algebra II  TuTh, 2:304 p.m.  F 154  D. Wagner  
Math 4380  12062  A Mathematical Introduction to Options 
TuTh, 12:30 p.m.  CAM 103  I. Timofeyev 
Math 4389  12063  Survey of Undergraduate Mathematics  MWF 910 a.m.  SEC 201/Hybrid  M. Almus 
Course  Section  Course Title  Course Day & Time  Instructor 
Math 5330  13515  Abstract Algebra  Arrange (online course)  K. Kaiser 
Math 5332  12089  Differential Equations  Arrange (online course)  G. Etgen 
Math 5386  15302  Regression and Linear Models  Arrange (online course)  C. Peters 
Math 5397  26816  Dynamical Systems  Arrange (online course)  A. Török 
Course 
Section  Course Title  Course Day & Time  Rm #  Instructor 
Math 6303  12096  Modern Algebra II  MWF, 1011 a.m.  AH 2  M. Tomforde 
Math 6308  13696  Advanced Linear Algebra I  TuTh, 2:304 p.m.  CBB 104  E. Kao 
Math 6308  18471  Advanced Linear Algebra I (online)  Online  Online  J. Morgan 
Math 6309  13697  Advanced Linear Algebra II  MWF, Noon1 p.m.  F 154  D. Wagner 
Math 6313  13695  Introduction to Real Analysis  TuTh, 1011:30 a.m.  F 154  D. Labate 
Math 6321  12113  Theory of Functions of a Real Variable  MWF, 11 a.m.Noon  AH 2  M. Kalantar 
Math 6353  21449  Complex Analysis & Geo II  MW, 12:30 p.m.  AH 301  G. Heier 
Math 6361  13699  Applicable Analysis  TuTh, 45:30 p.m.  AH 11  G. Auchmuty 
Math 6367  12114  Optimization Theory  TuThu, 11:30 a.m.1 p.m.  SW 221  R. Glowinski 
Math 6371  12115  Numerical Analysis  MW, 12:30 p.m.  SEC 203  Y. Kuznetsov 
Math 6373  21450  Automatic Learning & Data Mining  TuTh, 11:30 a.m.1 p.m.  CAM 103  R. Azencott 
Math 6378  17464  Basic Scientific Computing  TuTh, 12:30 p.m.  AH 301  R. Sanders 
Math 6383  12116  Probability Statistics  TuTh, 1011:30 a.m.  SW 423  W. Fu 
Math 6395  21452  Analytic Functions, Hardy Spaces and Operator Function Theory  MWF, Noon1 p.m.  AH 2  D. Blecher 
Math 7321  21453  Functional Analysis  TuTh, 12:30 p.m.  M 104  B. Bodmann 
Math 7326  21454  Dynamical Systems  MWF, 11 a.m.Noon  AH 301  V. Climenhaga 
Math 7350  12176  Geometry of Manifolds  MW, 45:30 p.m.  AH 10  W. Ott 
Course Details
SENIOR UNDERGRADUATE COURSES
Math 4309 (15672)  Mathematical Biology 

Prerequisites:  
Text(s):  A Biologist's Guide to Mathematical Modeling in Ecology and Evolution by Sarah P. Otto and Troy Day; ISBN13:9780691123448 
Description: 
Topics in mathematical biology, epidemiology, population models, models of genetics and evolution, network theory, pattern formation, and neuroscience. Students may not receive credit for both MATH 4309 and BIOL 4309. 
<< back to top >>
Math 4332 (12060)  Introduction to Real Analysis II


Prerequisites:  MATH 4331 or consent of instructor 
Text(s):  Real Analysis with Real Applications  Edition: 1; Allan P. Donsig, Allan P. Donsig; ISBN: 9780130416476 
Description: 
Further development and applications of concepts from MATH 4331. Topics may vary depending on the instructor's choice. Possibilities include: Fourier series, pointset topology, measure theory, function spaces, and/or dynamical systems. 
<< back to top >>
Math 4351 (21457)  Differential Geometry II


Prerequisites: 
MATH 4350. 
Text(s):  Instructor's notes will be provided. 
Description: 
Continuation of the study of Differential Geometry from MATH 4350. Holonomy and the GaussBonnet theorem, introduction to hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature, abstract surfaces (2D Riemannian manifolds). 
<< back to top >>
Math 4355 (21456)  Mathematics of Signal Representation


Prerequisites: 
MATH 2433 and six additional hours of 30004000 level Mathematics 
Text(s): 
A First Course in Wavelets with Fourier Analysis  Edition: 2 by Albert Boggess, Francis J. Narcowich, ISBN13: 9780470431177 
Description: 
Fourier series of realvalued functions, the integral Fourier transform, timeinvariant linear systems, bandlimited and timelimited signals, filtering and its connection with Fourier inversion, Shannon’s sampling theorem, discrete and fast Fourier transforms, relationship with signal processing. 
<< back to top >>
Math 4364 (19420) Numerical Analysis in Scientific Computing


Prerequisites: 
MATH 3331 and COSC 1410 or equivalent or consent of instructor. Instructor's Prerequisite Notes: 1. MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics) 2. Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple. 
Text(s): 
Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, BrooksCole Publishers, ISBN:9780538733519 
Description:  This is an one semester course which introduces core areas of numerical analysis and scientific computing along with basic themes such as solving nonlinear equations, interpolation and splines fitting, curve fitting, numerical differentiation and integration, initial value problems of ordinary differential equations, direct methods for solving linear systems of equations, and finitedifference approximation to a twopoints boundary value problem. This is an introductory course and will be a mix of mathematics and computing. 
<< back to top >>
Math 4365 (17384)  Numerical Methods for Differential Equations


Prerequisites:  MATH 3331, or equivalent, and three additional hours of 3000–4000 level Mathematics. 
Text(s):  TITLE:TBA, AUTHOR:TBA, ISBN:TBA 
Description:  Numerical differentiation and integration, multistep and RungeKutta methods for ODEs, finite difference and finite element methods for PDEs, iterative methods for linear algebraic systems and eigenvalue computation. 
<< back to top >>
Math 4377 (14513)  Advanced Linear Algebra I


Prerequisites:  MATH 2331 or equivalent, and three additional hours of 3000–4000 level Mathematics. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description: 
Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors. Additional Notes: This is a proofbased course. It will cover Chapters 14 and the first two sections of Chapter 5. Topics include systems of linear equations, vector spaces and linear transformations (developed axiomatically), matrices, determinants, eigenvectors and diagonalization. 
<< back to top >>
Math 4377 (18470)  Advanced Linear Algebra I (Online)


Prerequisites:  MATH 2331 or equivalent, and six additional hours of 3000–4000 level Mathematics. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description:  Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors. 
<< back to top >>
Math 4378 (12061)  Advanced Linear Algebra II


Prerequisites:  MATH 4377 
Text(s):  Linear Algebra, Fourth Edition, by S.H. Friedberg, A.J Insel, L.E. Spence,Prentice Hall, ISBN 0130084514; 9780130084514 
Description: 
Similarity of matrices, diagonalization, Hermitian and positive definite matrices, normal matrices, and canonical forms, with applications. Instructor's Additional notes: This is the second semester of Advanced Linear Algebra. I plan to cover Chapters 5, 6, and 7 of textbook. These chapters cover Eigenvalues, Eigenvectors, Diagonalization, CayleyHamilton Theorem, Inner Product spaces, GramSchmidt, Normal Operators (in finite dimensions), Unitary and Orthogonal operators, the Singular Value Decomposition, Bilinear and Quadratic forms, Special Relativity (optional), Jordan Canonical form. 
<< back to top >>
<< back to top >>
Math 4380 (12062)  A Mathematical Introduction to Options  
Prerequisites:  MATH 2433 and MATH 3338. 
Text(s):  An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation  Edition: 1; Desmond Higham; 9780521547574 
Description:  Arbitragefree pricing, stock price dynamics, callput parity, BlackScholes formula, hedging, pricing of European and American options. 
<< back to top >>
<< back to top >>
Math 4389 (12063)  Survey of Undergraduate Mathematics  
Prerequisites:  MATH 3330, MATH 3331, MATH 3333, and three hours of 4000level Mathematics. 
Text(s):  Instructor will use her own notes 
Description:  A review of some of the most important topics in the undergraduate mathematics curriculum. 
<< back to top >>
ONLINE GRADUATE COURSES
<< back to top >>
MATH 5330 (13515)  Abstract Algebra


Prerequisites:  Graduate standing. 
Text(s): 
Abstract Algebra , A First Course by Dan Saracino. Waveland Press, Inc. ISBN 0881336653 
Description: 
Groups, rings and fields; algebra of polynomials, Euclidean rings and principal ideal domains. Does not apply toward the Master of Science in Mathematics or Applied Mathematics. Other Notes: This course is meant for students who wish to pursue a Master of Arts in Mathematics (MAM). Please contact me in order to find out whether this course is suitable for you and/or your degree plan. Notice that this course cannot be used for MATH 3330, Abstract Algebra. 
<< back to top >>
MATH 5332 (12089)  Differential Equations


Prerequisites:  Graduate standing. MATH 5331. 
Text(s):  TBA 
Description: 
Linear and nonlinear systems of ordinary differential equations; existence, uniqueness and stability of solutions; initial value problems; higher dimensional systems; Laplace transforms. Theory and applications illustrated by computer assignments and projects. Applies toward the Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees. 
<< back to top >>
<< back to top >>
MATH 5386 (15302)  Regression and Linear Models


Prerequisites:  Graduate standing. Two semesters of calculus, one semester of linear algebra, and MATH 5385, or consent of instructor. 
Text(s):  Introduction to Linear Regression Analysis  Edition:5; Montgomery, Peck, Vining; ISBN: 9780470542811; Wiley 
Description: 
Simple and multiple linear regression, linear models, inferences from the normal error model, regression diagnostics and robust regression, computing assignments with appropriate software. Applies toward Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees. Note: This course is VEE approved for the regression component only. Approval Code: 445811008. For more information on VEE approved courses, click here. 
<< back to top >>
GRADUATE COURSES
<< back to top >>
MATH 6303 (12096)  Modern Algebra II


Prerequisites:  Graduate standing. MATH 4333 or MATH 4378 or consent of instructor 
Text(s): 
Abstract Algebra, 3rd Edition by David S. Dummit and Richard M. Foote. ISBN13: 9780471433347 ISBN10: 0471433349 
Description:  Topics from the theory of groups, rings, fields, and modules. 
<< back to top >>
MATH 6308 (14458)  Advanced Linear Algebra I


Prerequisites:  Graduate standing. MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description: 
Transformations, eigenvalues and eigenvectors. Additional Notes: This is a proofbased course. It will cover Chapters 14 and the first two sections of Chapter 5. Topics include systems of linear equations, vector spaces and linear transformations (developed axiomatically), matrices, determinants, eigenvectors and diagonalization. 
<< back to top >>
MATH 6308 (20438)  Advanced Linear Algebra I (online)


Prerequisites:  Graduate standing. MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors. 
Text(s):  Linear Algebra  Edition: 4; Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence; ISBN: 9780130084514 
Description:  Transformations, eigenvalues and eigenvectors. An expository paper or talk on a subject related to the course content is required 
<< back to top >>
MATH 6309 (13697)  Advanced Linear Algebra II


Prerequisites:  Graduate standing and MATH 6308 
Text(s):  Linear Algebra, Fourth Edition, by S.H. Friedberg, A.J Insel, L.E. Spence,Prentice Hall, ISBN 0130084514; 9780130084514 
Description:  Similarity of matrices, diagonalization, hermitian and positive definite matrices, canonical forms, normal matrices, applications. An expository paper or talk on a subject related to the course content is required. 
<< back to top >>
MATH 6313 (13695) Introduction to Real Analysis II


Prerequisites:  Graduate standing and MATH 6312. 
Text(s):  TBA 
Description:  Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals. An expository paper or talk on a subject related to the course content is required. 
<< back to top >>
MATH 6321 (12113)  Theory of Functions of a Real Variable


Prerequisites: 
Graduate standing. MATH 4332 or consent of instructor. Instructor's Prerequisite Notes: MATH 6320 
Text(s): 
Primary (Required): Real Analysis: Modern Techniques and Their Applications, Gerald Folland (2nd edition); ISBN: 9780471317166. Supplementary (Recommended): Real Analysis for Graduate Students, Richard F. Bass, (2nd edition); ISBN: 9781481869140 
Description: 
Lebesque measure and integration, differentiation of real functions, functions of bounded variation, absolute continuity, the classical Lp spaces, general measure theory, and elementary topics in functional analysis. Instructor's Additional Notes: Math 6321 is the second course in a twosemester sequence intended to introduce the theory and techniques of modern analysis. The core of the course covers elements of functional analysis, Radon measures, elements of harmonic analysis, the Fourier transform, distribution theory, and Sobolev spaces. Additonal topics will be drawn from potential theory, ergodic theory, and the calculus of variations. 
<< back to top >>
<< back to top >>
MATH 6353 (21449)  Complex Analysis & Geo II


Prerequisites:  Graduate standing. Math 6352 or consent of instructor. 
Text(s): 
 Principles of Algebraic Geometry  Edition: 1, Author: Phillip Griffiths, Joseph Harris; ISBN: 9780471050599 (recommended)  Positivity in Algebraic Geometry I: Classical Setting: Line Bundles and Linear Series, Author: R.K. Lazarsfeld; ISBN: 9783540225331 (recommended) 
Description:  Idea sheaves with its applications and advanced techniques in transdental algebraic geometry. 
<< back to top >>
MATH 6361 (13699)  Applicable Analysis


Prerequisites:  Graduate standing. MATH 4332 or consent of instructor. 
Text(s): 
The instructor will provide lecture notes on the material. A reference text is L.D. Berkowitz, Convexity and Optimization in Rn, WileyInterscience 2002.

Description: 
This course provides an introduction to the mathematical analysis of finite dimensional optimization problems. Topics to be studied include the existence of, and the extremality conditions that hold at, solutions of constrained and unconstrained optimaization problems. Elementary theory of convex sets, functions and constructions from convex analysis will be introduced and used. Concepts include subgradients, conjugate functions and some duality theory. Specific problems to be studied include energy and least squares methods for solving linear equations, important inequalities, eigenproblems and some nonlinear programming problems from applications.

<< back to top >>
MATH 6367 (12114) Optimization Theory


Prerequisites:  Graduate standing. MATH 4331 and MATH 4377. 
Text(s): 
Instructor will provide notes. R. Glowinski, J.L. Lions, JW He, Exact and Approximate Controllability for Distributed Systems: A Numerical Approach, Cambridge University Press, New York, NY, 2008. ISBN: 9780521885720 (recommended) 
Description:  Constrained and unconstrained finite dimensional nonlinear programming, optimization and EulerLagrange equations, duality, and numerical methods. Optimization in Hilbert spaces and variational problems. EulerLagrange equations and theory of the second variation. Application to integral and differential equations. 
<< back to top >>
MATH 6371 (12115)  Numerical Analysis


Prerequisites:  Graduate standing. 
Text(s):  Numerical Mathematics (Texts in Applied Mathematics), 2nd Ed., V.37, Springer, 2010. By A. Quarteroni, R. Sacco, F. Saleri. ISBN: 9783642071010 
Description:  Ability to do computer assignments. Topics selected from numerical linear algebra, nonlinear equations and optimization, interpolation and approximation, numerical differentiation and integration, numerical solution of ordinary and partial differential equations. 
<< back to top >>
MATH 6373 (21450)  Automatic Learning & Data Mining


Prerequisites:  Graduate standing. Probability & Statistics 
Text(s):  Instructor will provide his own notes. 
Description: 
Automatic Learning of unknown functional relationships Y = F(X) between an output Y and highdimensional inputs X , involves algorithms dedicated to the intensive analysis of large "training sets" of N "examples" of inputs/outputs pairs (Xn,Yn ), with n= 1…N to discover efficient "blackboxes" approximating the unknown function X>F(X). Automatic learning was first applied to emulate intelligent tasks involving complex patterns identification, in artificial vision, face recognition, sounds identification, speech understanding, handwriting recognition, texts classification and retrieval, etc. Automatic learning has now been widely extended to the analysis of high dimensional biological data sets in proteomics and genes interactions networks, as well as to smart mining of massive data sets gathered on the Internet. The course will study major machine learning algorithms derived from Positive Definite Kernels and their associated SelfReproducing Hilbert spaces. We will study the implementation, performances, and drawbacks of Support Vector Machines classifiers, Kernel based Non Linear Clustering, Kernel based Non Linear Regression, Kernel PCA. We will explore connections between kernel based learning and Dictionary Learning as well as Artificial Neural Nets with emphasis on key conceptual features such as generalisation capacity. We will present classes of Positive Definite Kernels designed to handle the long "string descriptions" of proteins involved in genomics and proteomics. The course will focus on understanding key concepts through their mathematical formalization, as well as on computerized algorithmic implementation and intensive testing on actual data sets 
<< back to top >>
<< back to top >>
MATH 6378 (17464)  Basic Scientific Computing


Prerequisites:  Graduate standing. MATH 4364 and MATH 4365 or equivalent, and either COSC 1304 or COSC 2101 or equivalents. 
Text(s):  Instructor will provide his own notes. 
Description:  A projectoriented course in fundamental techniques for high performance scientific computation. Hardware architecture and floating point performance, code design, data structures and storage techniques related to scientific computing, parallel programming techniques, applications to the numerical solution of problems such as algebraic systems, differential equations and optimization. Data visualization. 
<< back to top >>
MATH 6383 (12116)  Probability Statistics


Prerequisites: 
Graduate standing. MATH 3334, MATH 3338 and MATH 4378. Instructor's Prerequisites:TBA 
Text(s): 
Recommended Text: John A. Rice : Mathematical Statistics and Data Analysis, 3^{rd} editionBrooks / Cole, 2007. ISBN13: 9780534399429. Reference Texts: P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2^{nd} ed. 1999 Chapman Hall/CRC. ISBN: 9780412317606 Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson, Generalized Linear Models: with Applications in Engineering and the Sciences, 2^{nd} ed. Wiley, 2010. ISBN: 9780470454633. 
Description: 
A survey of probability theory, probability models, and statistical inference. Includes basic probability theory, stochastic processes, parametric and nonparametric methods of statistics. Instructor's Description: This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced statistical theory and techniques in modelling data of various types, including continuous, binary, counts and others. The selected topics will include basic probability distributions, likelihood function and parameter estimation, hypothesis testing, regression models for continuous and categorical response variables, variable selection methods, model selection, large sample theory, shrinkage models, ANOVA and some recent advances. 
<< back to top >>
MATH 6395 (21452)  Analytic Functions, Hardy Spaces and Operator Function Theory


Prerequisites:  Graduate Standing. Some parts of the Real Variables sequence would be helpful, e.g. Math 6320 
Text(s):  Banach Spaces of Analytic Functions (Dover Books on Mathematics), by Kenneth Hoffman; ISBN: 9780486458748. Instructor will also provide some typed notes, drawn from several texts. 
Description:  Brief description: We will start with some important theorems in complex analysis related to normal families of analytic functions. We then will study the basic theory of the disk algebra and the important theory of Hardy spaces (which we have not taught at UH for some years). We will follow Hoffman's book closely here. In the second half of the course we will discuss some operator function theory e.g. related to the invariant subspace problem (Beurling's theorem and generalizations). We will also discuss abstract operator algebras on a Hilbert space and their theory, and connections to noncommutative function theory. The course will end with a choice of student projects depending on what they are each interested in, for example a treatment of noncommutative integration and noncommutative Hardy spaces. 
<< back to top >>
MATH 7321 (21453)  Functional Analysis


Prerequisites: 
Graduate standing. 
Text(s):  Textbook: 
Description:  Linear topological spaces, Banach and Hilbert spaces, duality, and spectral analysis. 
<< back to top >>
MATH 7326 (21454)  Dynamical Systems


Prerequisites: 
Graduate standing. Math 6320 or equivalent background in measure theory. Some familiarity with smooth manifolds would be useful but will not be assumed. 
Text(s):  Textbook: Katok and Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge, ISBN13: 9780521575577, ISBN10: 0521575575 Additional reference text (not required): Rufus Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, 2008 (2nd Revised Edition), Springer Lecture Notes in Mathematics #470, ISBN 9783540776055 
Description:  This course will give an introduction to the theory of dynamical systems, with particular emphasis on those systems displaying hyperbolic (chaotic) behavior. After a general overview, we will describe the key properties of uniformly hyperbolic systems, including structural stability and finite Markov partitions. Then we will explain how tools from thermodynamics can be used to deduce statistical properties of the system, especially for the "physically relevant" SinaiRuelleBowen measure. Finally, we will give a brief overview of the more physically realistic class of nonuniformly hyperbolic systems, including the multiplicative ergodic theorem, Pesin theory, and countablestate Markov codings. 
<< back to top >>
MATH 7350 (12176)  Geometry of Manifolds


Prerequisites:  Graduate standing. MATH 6342. 
Text(s):  Introduction to Smooth Manifolds (Graduate Texts in Mathematics, Vol. 218) 2nd Edition, John Lee, ISBN13: 9781441999818; ISBN10: 1441999817 
Description:  Math 7350 is an introduction to the theory of differentiable manifolds. Topics include vector bundles, embedding theory, tensors, integration on manifolds, flows, elements of Lie theory, and Riemannian metrics. 