Math 4332/6313

12556/14457

Math 4377/6308

15306/14458

Math 4377/6308

20437/20438

Math 4378/6309

12557/14459 

MATH 3331 and BIOL 3306 or consent of instructor.

Instructor's Prerequisite Notes:

Linear Algebra (MATH 2331) and Differential Equations (MATH 3321 or MATH 3331)

Mathematical Models in Biology by Leah Edelstein-Keshet (2005); ISBN-13:978-0898715545

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.

Additional Instructor's notes: 

This course introduces and analyzes a variety of mathematical models of biological systems at the molecular, cellular, and population levels. Applications to enzyme kinetics, population dynamics, gene expression, epidemiology, and neuroscience will all be discussed. Studying these systems will require mathematical techniques for dynamical systems, stochastic processes, pattern formation, and matrix analysis.

Further development and applications of concepts from MATH 4331. Topics may vary depending on the instructor's choice. Possibilities include: Fourier series, point-set topology, measure theory, function spaces, and/or dynamical systems.

MATH 3331 or equivalent, and three additional hours of 3000–4000 level Mathematics.

Initial and boundary value problems, waves and diffusions, reflections, boundary values, Fourier series.

Additional Notes: A partial differential equation (PDE) is an equation that states a relationship between a function of two or more independent variables and the partial derivatives of this function with respect to these independent variables. PDEs arise in all fields of engineering and science. Most physical processes are governed by PDEs.

In this one-semester introductory course for PDEs we will consider three basic classes (elliptic, parabolic and hyperbolic) of PDEs and discuss two types of physical problem (equilibrium and propagation ones). Topics covered in class will include but not limited to the following: initial and boundary value problems; waves and diffusions; separations of variables; Dirichlet, Neumann and Robin boundary conditions; Fourier series; harmonic functions.

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.

Numerical Analysis (9th edition), by R.L. Burden and J.D. Faires, Brooks-Cole Publishers, ISBN:9780538733519

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 finite-difference approximation to a two-points boundary value problem. This is an introductory course and will be a mix of mathematics and computing.

Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors.

Additional Notes: This is a proof-based course. It will cover Chapters 1-4 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.

Abstract Algebra , A First Course by Dan Saracino. Waveland Press, Inc. ISBN 0-88133-665-3
(You can use the first edition. The second edition contains additional chapters that cannot be covered in this course.)

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. 

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.

A survey of the concepts of limit, continuity, differentiation and integration for functions of one variable and functions of several variables; selected applications. Applies toward the Master of Arts in Mathematics degree; does not apply towards the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

Additional Notes: This course is an introduction to Analysis. It will cover limit, continuity, differentiation and integration for functions of one variable and functions of several variables, and some selected applications. More precisely, it will cover the textbook from the chapter 3 to the chapter 7 (skip the section 15 and the section 24).

On-line course is taught through Blackboard Learn, visit
https://accessuh.uh.edu/login.php for information on obtaining ID and password.

Homework: Homework will be submitted through Blackboard Learn by pdf file. The deadline for each homework assignment can be found in Blackboard Learn. No late homework assignments accepted.

Exams: There are two exams. The mid-term exam, and the comprehensive final exam. The dates are to be dertermined 

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: 4458-11008. For more information on VEE approved courses, click here.

TBA

Emphasis on canonical forms and finite dimensional spectral theory.

Transformations, eigenvalues and eigenvectors.

Additional Notes: This is a proof-based course. It will cover Chapters 1-4 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.

Graduate standing. MATH 4332 or consent of instructor.

Instructor's Prerequisite Notes: MATH 6320

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

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 two-semester 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.

Graduate standing. MATH 4331 or consent of instructor. 

Instructor's Prerequisite Notes: Math 6322 or consent of instructor.

Geometry of the complex plane, mappings of the complex plane, integration, singularities, spaces of analytic functions, special function, analytic continuation, and Riemann surfaces. Additional Notes: This course is an introduction to complex analysis. This two semester course will cover the theory of holomorphic functions, residue theorem, harmonic and subharmonic functions, Schwarz's lemma, Riemann mapping theorem, Casorati-Weterstrass theorem, infinite product, Weierstrass' (factorization) theorem, little and big Picard Theorems and compact Riemann surfaces theory.

Graduate standing.  MATH 3334, MATH 3338 and MATH 4378, or consent of instructor.

Instructor's Prerequisites: Two years of Calculus, Math 6308 Advanced Linear Algebra I, Math 5386 Regression and Linear Models, andMath 6382 Probability and Statistics or equivalent. 

Recommended Text: John A. Rice : Mathematical Statistics and Data Analysis, 3^rd editionBrooks / Cole, 2007. ISBN-13: 978-0-534-39942-9. 

Reference Texts: 

-P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2^nd ed. 1999 Chapman Hall/CRC

-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: 978-0-470-45463-3. 

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.

"Arbitrage Theory in Continuous Time" (3rd Edition), Tomas Björk, Oxford University Press, 2009. ISBN: 9780199574742

"Stochastic Calculus for Finance II: Continuous-Time Models," Steven Shreve, Springer, 2004. ISBN: 9780387401010

No textbook. Course notes will be distributed.

We will discuss various classes of C*-algebras constructed from dynamical systems, and examine the interplay that exists between the analysis, algebra, and dynamics. The course will start with an introduction to symbolic dynamics and discrete dynamical systems, and afterward we will construct classes of C*-algebras (e.g., Cuntz-Krieger algebras, graph C*-algebras, crossed products by the integers) that are intimately related to the dynamical systems.

Graduate standing and consent of instructor.

Instructor's Prerequisite Notes: MATH 4360 or 6370. Basic knowledge in numerical analysis and scientific computing. 

Numerical Methods for Conservation Laws, by Randall LeVeque; ISBN: 9783764327231

The first part of the course is to introduce mathematical theory for hyperbolic conservation laws that arise in many applications such as traffic flow, gas dynamics and fluid dynamics. The second part of the course is on advanced numerical methods for solving hyperbolic equations.

Recommended:

-Handbook of Stochastic Methods: For Physics, Chemistry and Natural Sciences, C.W. Gardiner

-Interacting Particle Systems (Classics in Mathematics), Thomas M. Liggett

-Multiscale Methods: Averaging and Homogenization (Texts in Applied Mathematics), G.A. Pavliotis & Andrew Stuart

Graduate standing and consent of instructor.

Additional Prerequisite Notes:It is strongly encouraged that students have a background in Linear Algebra and Real Analysis. Prior knowledge of Matlab is not required but the course will require writing simple routines in Matlab or an equivalent language.

- A basis theory primer, by C. Heil, 2011

- A wavelet tour of signal processing, by S. Mallat, Third edition, 2009

- A First Course on Wavelets , by E. Hernandez and G. Weiss, 1996.

- (papers and notes)

We live in a data-intensive age which is bringing significant changes in the process of scientific discovery. During the last decade, sparsity has emerged as a leading theme in connection with the goal to produce faster and simpler algorithms for a wide range of signal processing applications. By enabling to accurately approximate data/functions in a certain class using a relatively small number of nonzero coefficients, sparse representations have the power to reveal the essential information we are looking for in the data. Therefore, sparsity implies not only data compression. Understanding the sparsity of a given data type entails a precise knowledge of the modelling and approximation of that data type. This knowledge is essential to design highly efficient algorithms for a tasks such as classification, denoising, interpolation, and segmentation. Multiscale techniques based on wavelets and their generalizations have emerged in the last decade as the most successful approach for sparse signal representations, as testified, for example, by their use in the new FBI fingerprint database and in JPEG2000, the new standard for image compression. Multiscale techniques were also extended beyond the traditional setting of physical spaces allowing for the efficient analysis of general structures, such as manifolds, graphs and point clouds in Euclidean space.

The aim of this course is to provide the mathematical tools to understand multiscale representations starting from the setting of traditional wavelets up to more advanced and emerging constructions such as curvelets, shearlets and diffusion wavelets. Several applications of these ideas will be presented.

Tentative list of topics:

- Orthonormal bases and frames. A basic problem in mathematics and engineering is to represent a function or a signal as superposition of elementary components. I will introduce the theory of frames and show that it provides the general framework to address this problem. Orthonormal bases are a special example of frames.

- Elements of Fourier analysis: I will briefly review Fourier analysis, including Fourier series and Fourier transforms.

- Wavelet construction and Multiresolution Analysis: The first wavelet basis, the Haar basis, was discovered in 1909 before wavelet theory was born. Unfortunately, the elements of this basis are not continuous. The success of the wavelet theory is due to the ability to construct a variety of wavelet bases with very nice mathematical properties such as smoothness, compact support, vanish moments, etc. Multiresolution analysis is a general method for constructing wavelet bases with prescribed properties.

- Sparse compression and approximation theory: One striking feature of wavelets is their ability to represent function with discontinuities. I will introduce linear and nonlinear approximations and discuss the approximation properties of wavelets and their generalizations (curvelets, shearlets, bandlets).

- Multiscale analysis on high-dimensional data: Multiscale analysis of random walks on graphs and applications to analysis of high-dimensional data sets.

- Modern signal processing: Multiscale methods and wavelets appear today in state-of-the-art signal processing applications, including analysis and diagnostics, quantization and compression, transmission and storage, noise reduction and removal. I will present some applications to data/image analysis. Additional applications will be further explored by the students as individual or group projects.

Graduate Standing and consent of instructor. 

Additional Prerequisite Notes: Students should have previous familiarity (at the undergraduate level) with random variables and probability distributions

No single textbook.

Reading assignments will be a  small set of specific chapters extracted from the following reference texts:

"The Elements of Statistical Learning, Data Mining", Friedman, Hastie,Tibshirani; ISBN: 978-0387848570

"Kernel Methods in Computational Biology", B. Schölkopf, K. Tsuda, J.-P. Vert; ISBN:978-0262195096

"Introduction to Support Vector Machines", N. Cristianini,  J. Shawe-Taylor; ISBN: 978-0521780193

Automatic Learning of unknown functional relationships Y = F(X) between an output  Y and high-dimensional 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 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 Self-Reproducing 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 these techniques 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 very long "string descriptions" of  proteins involved in genomics and proteomics.

Emphasis will be on understanding key concepts and their mathematical formalization,  with a strong focus on algorithmic implementation and testing on  actual data sets.

Homework assignments will involve implementation and reports on several applied projects. Students will be assumed to be able to use either  Matlab or equivalent scientific softwares. Final exam will involve the in depth reading of one scientific paper and giving a public lecture on the paper.

Manifolds and tangent bundles, submanifolds and imbeddings, integral manifolds, triangulation of manifolds, connections and holonomy; Riemannian geometry, surface theory, Morse theory, and G-structures.

1.) An Introduction to Analysis of Financial Data with R, by Ruey S. Tsay, 2012, ISBN: 9780470890813

2.) Multivariate Time Series Analysis: With R and Financial Applications, by Ruey S. Tsay, 2013, Wiley, ISBN: 9781118617908