MATH 3331 and BIOL 3306 or consent of instructor.

• An Introduction to Systems Biology, 2/e, U. Alon (an excellent, recently updated text on the “design principles of biological circuits”)

• Random Walks in Biology, H.C. Berg (a classic introduction to the applicability of diffusive processes and the Reynolds number at the cellular scale)

• Mathematical Models in Biology, L. Edelstein-Keshet (a systematic development of discrete, continuous, and spatially distributed biological models)

• Nonlinear Dynamics and Chaos, S.H. Strogatz (a very readable introduction to phase-plane analysis and bifurcation theory in dynamical systems with an emphasis on visual thinking; contains numerous applications in biology)

• Thinking in Systems, D.H. Meadows (a lay introduction to control systems and analyzing parts-to-whole relationships, their organizational principles, and sensitivity in their design)

• Adaptive Control Processes: A Guided Tour, R. Bellman (a classic, more technical introduction to self-regulating systems, feedback control, decision processes, and dynamic programming)

Catalog 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.

Instructor's description: An introduction to mathematical methods for modeling biological dynamical systems. This course will survey canonical models of biological systems using the mathematics of calculus, differential equations, logic, matrix theory, and probability.

Applications will span several spatial orders-of-magnitude, from the microscopic (sub-cellular), to the mesoscopic (multi-cellular tissue and organism) and macroscopic (population-level: ecological, and epidemiological) scales. Specific applications will include biological-signaling diffusion, enzyme kinetics, genetic feedback networks, population dynamics, neuroscience, and the dynamics of infectious diseases. Optional topics (depending on schedule and student interest) may be chosen from such topics as: game theory, artificial intelligence and learning, language processing, economic multi-agent modeling, Turing systems, information theory, and stochastic simulations.

The course will be taught from two complementary perspectives:
(1) critical analysis of biological systems’ modeling using applicable mathematical tools, and
(2) a deeper understanding of mathematical theory, illustrated through biological applications.

Relevant mathematical theory for each course section will be reviewed from first principles, with an emphasis on bridging abstract formulations to practical modeling techniques and dynamical behavior prediction.

The course will include some programming assignments, to be completed in Matlab or Python programming languages (available free through UH Software and public domain, respectively). However, advanced programming techniques are not required, and resources for introduction to these languages will be provided.

Theory and applications for such statistical learning techniques as linear and logistic regression, classification and regression trees, random forests, neutral networks. Other topics might include: fit quality assessment, model validation, resampling methods. R Statistical programming will be used throughout the course.

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.

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

Differential forms in R^n (particularly R^2 and integration, the intrinsic theory of surfaces through differential forms, the Gauss-Bonnet theorem, Stokes’ theorem, manifolds, Riemannian metric and curvature. Other topics at discretion of instructor.

ODEs as models for systems in biology, physics, and elsewhere; existence and uniqueness of solutions; linear theory; stability of solutions; bifurcations in parameter space; applications to oscillators and classical mechanics.

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

Catalog Description: Root finding, interpolation and approximation, numerical differentiation and integration, numerical linear algebra, numerical methods for differential equations.

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

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

Catalog Description: Root finding, interpolation and approximation, numerical differentiation and integration, numerical linear algebra, numerical methods for differential equations.

Instructor's 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 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.

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, Cayley-Hamilton Theorem, Inner Product spaces, Gram-Schmidt, Normal Operators (in finite dimensions), Unitary and Orthogonal operators, the Singular Value Decomposition, Bilinear and Quadratic forms, Special Relativity (optional), Jordan Canonical form.

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.

Proportionality and geometric similarity, empirical modeling with multiple regression, discrete dynamical systems, differential equations, simulation and optimization. Computing assignments require only common spreadsheet software and VBA programming.

Data collection and types of data, descriptive statistics, probability, estimation, model assessment, regression, analysis of categorical data, analysis of variance. Computing assignments using a prescribed software package (e.g., R or Matlab) will be given. Applies toward the Master of Arts in Mathematics degree; does not apply toward Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

 Graduate standing. MATH 4333 or MATH 4378 

Additional Prerequisites: students should be comfortable with basic measure theory, groups rings and fields, and point-set topology

No textbook is required.

Topics from the theory of groups, rings, fields, and modules. 

Additional Description: This is primarily a course about analysis on topological groups. The aim is to explain how many of the techniques from classical and harmonic analysis can be extended to the setting of locally compact groups (i.e. groups possessing a locally compact topology which is compatible with their algebraic structure). In the first part of the course we will review basic point set topology and introduce the concept of a topological group. The examples of p-adic numbers and the Adeles will be presented in detail, and we will also spend some time discussing SL_2(R). Next we will talk about characters on topological groups, Pontryagin duality, Haar measure, the Fourier transform, and the inversion formula. We will focus on developing details in specific groups (including those mentioned above), and applications to ergodic theory and to number theory will be discussed.

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 for Graduate Students, Richard F. Bass

Supplementary (Recommended): Real Analysis: Modern Techniques and Their Applications, Gerald Folland (2nd edition); ISBN: 9780471317166.

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 3334, MATH 3338 or MATH 3339, and MATH 4378.  Students must be in the Statistics and Data Science, MS Program

Speak to the instructor for textbook information.

Solvability of finite dimensional, integral, differential, and operator equations, contraction mapping principle, theory of integration, Hilbert and Banach spaces, and calculus of variations.

•  D.P. Bertsekas; Dynamic Programming and Optimal Con- trol, Vol. I, 4th Edition. Athena Scientific, 2017, ISBN-10: 1-886529-43-4
• J.R. Birge and F.V. Louveaux; Introduction to Stochastic Programming. Springer, New York, 1997, ISBN: 0-387-98217-

Constrained and unconstrained finite dimensional nonlinear programming, optimization and Euler-Lagrange equations, duality, and numerical methods. Optimization in Hilbert spaces and variational problems. Euler-Lagrange equations and theory of the second variation. Application to integral and differential equations.

Additional Description: This course consists of two parts. The first part is concer- ned with an introduction to Stochastic Linear Programming (SLP) and Dynamic Programming (DP). As far as DP is concerned, the course focuses on the theory and the appli- cation of control problems for linear and nonlinear dynamic systems both in a deterministic and in a stochastic frame- work. Applications aim at decision problems in finance. In the second part, we deal with continuous-time systems and optimal control problems in function space with em- phasis on evolution equations.

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

Reference Texts:

-P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2nd ed. 1999 Chapman Hall/CRC. ISBN: 978-0412317606

-Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson, Generalized Linear Models: with Applications in Engineering and the Sciences, 2nd ed. Wiley, 2010. ISBN: 978-0-470-45463-3.

Catalog 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

Reference texts: 

• "Review of Lebesgue Measure and Integration" by Christopher Heil
• "Banach and Hilbert Space Review" by Christopher Heil

This is a course of mathematics exploring foundational and theorical concepts underlying the development and applications of intelligent systems and deep learning algorithms. The emphasis of the course will be theoretical aspects. The aim of the course is to provide the necessary background to start a graduate research project in this emerging area of investigation.

Topics of the course include: statistical learning theory, Support Vector Machines, geometry of high-dimensional data, manifold learning, dimensionality reduction, expressive power of neural networks, generalization in neural networks, convolutional neural networks.

This is class is targeted to graduate students interested in mastering theoretical tools underlying machine learning and data science.

Even though algorithmic aspects of the topics will not be ignored and exploration of algorithmic issues will be assigned for individual or group projects, this course will not duplicate existing courses on machine learning or data science offered in the Computer Science Department that are focused on algorithmic implementation and computation

No particular textbook is required. The following lists several good references for various topics related to this course (which go far beyond the material covered in class).

References for Bayesian (statistical) inverse problems and uncertainty quantification are:

• Statistical and Computational Inverse Problems by J. Kaipio and E. Somersalo. Springer 2005.
• Computational Uncertainty Quantification for Inverse Problems by J. Bardsley. SIAM 2018.
• Introduction to Uncertainty Quantification by T. J. Sullivan. Springer 2015.
• An Introduction to Data Analysis and Uncertainty Quantification for Inverse Problems by L. Tenorio. SIAM 2017.

References for inverse problems in general as well as optimization algorithms are:

• Computational Methods for Inverse Problems by C. R. Vogel, SIAM 2002.
• Discrete Inverse Problems: Insight and Algorithms by P. C. Hansen, SIAM 2010.
• An Introduction to the Mathematical Theory of Inverse Problems by A. Kirsch, Springer 2011.
• Inverse Problem Theory by A. Tarantola, SIAM 2005.
• Numerical Optimization by J. Nocedal and S. J. Wright. Springer 2006.

• (The second half of) "Topology, A First Course," J. R. Munkres.
• A. Hatcher, Algebraic topology (free online at the authors website)

The course will begin with reviewing the fundamental group, and will cover much of the second half of Munkres’ book (which  contains many important and beautiful topics), with additions from other books such as Hatcher’s Algebraic topology. We emphasize the many striking applications. Special requests will be honored if possible.