MATH 3331 and BIOL 3306 or consent of instructor.

Required texts: A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Sarah P. Otto and Troy Day; (2007, Princeton University Press)
ISBN-13:9780691123448

Reference texts: (excerpts will be provided)

• 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.

Introduction to basic concepts, results, methods, and applications of graph theory.

Additional Description: How does information propagate between friends and acquaintances on social media? How do diseases spread, and when do epidemics start? How should we design power grids to avoid failures, and systems of roads to optimize traffic flow? These questions can be addressed using network theory . Students in the course will develop a sound knowledge of the basics of graph theory, as well as some of the computational tools used to address the questions above. Course topics include basic structural features of networks, generative models of networks, centrality, random graphs, clustering, and dynamical processes on graphs.

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.

Continuation of the study of Differential Geometry from MATH 4350. Holonomy and the Gauss-Bonnet theorem, introduction to hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature, abstract surfaces (2D Riemannian manifolds).

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

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

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.

Selected topics in Mathematics. May be repeated with approval of chair.

Course Content:

A. Tensor analysis: Coordinate system construction, nonorthogonal coordinate systems, covariant and contravariant vector components, rotations and orthogonal transformations, coordinate system transformations, Cartesian tensors, general tensors, and higher rank tensors, metric tensors, covariant derivatives, Christoffel symbols, applications of tensors in mechanics, electromagnetic theory, special relativity, quantum mechanics, and general relativity.

B. Complex variable algebra review, complex differentiation and Cauchy-Riemann equations, complex integration and Cauchy’s theorem, Cauchy integral formulas, Taylor and Laurent series

C. Residue theorem, evaluation of integrals, branch cuts, conformal mapping and applications, applications to Fourier transforms and Green functions

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.

MATH 2331, In depth knowledge of Math 3331 (Differential Equations) or Math 3321 (Engineering Mathematics)

(see the description for more prerequisite details)

The students in this online section will be introduced to topics in scientific computing, including numerical solutions to nonlinear equations, numerical differentiation and integration, numerical solutions of systems of linear equations, least squares solutions and multiple regression, numerical solutions of nonlinear systems of equations, numerical optimization, numerical solutions to discrete dynamical systems, and numerical solutions to initial value problems and boundary value problems. Computations in this course will primarily be illustrated directly in an Excel spreadsheet, or via VBA programming, but students who prefer to do their computations using Matlab, Julia, Python or some other programming language are also welcome. For students who want to do their computing in Excel, there will be tutorials associated with the use of Excel, and programming in VBA. Students who decide to use Excel are expected to have access and basic familiarity with Excel, but they are not expected to know advanced spreadsheet functionality or have programming experience with VBA. Students will not be tested over Excel or VBA, and students using Matlab, Julia or Python will also receive some help materials.

Graduate standing. Notice: This course belongs to the group IV. Applied Math,which meets the requirement for MA degree. Students must submit a general petition to count this course towards the Applied Math requirement for the MA degree.

Lecture Notes will be provided

•  Introduction
• Least Square Regression
• Regularization and Cross Validation
• Bias-Variance Trade-off
• Bayesian Analysis
• Logistic regression
• Support Vector Machines
• Convex Optimization
• Ensemble Learning
• Clustering and k-NN Learning
• Robustness, Outlier Detection and Sparseness
• Dimensionality Reduction
• Artificial Neural Network
• Convolutional Neural Network
• Fisher’s Linear Discriminant and Neural

Without requiring have any previous computer background, the students should be able to learn MATLAB and Python to write codes for algorithms in each section.

Graduate standing. Three semesters of Calculus or consent of instructor. Basic knowledge of ODE's is helpful, but not required. 

Notice: This course belongs to the group IV. Applied Math,which meets the requirement for MA degree. Students must submit a general petition to count this course towards the Applied Math requirement for the MA degree.

Steven H. Strogatz: Nonlinear Dynamics and Chaos (with Applications to Physics, Biology, Chemistry, and Engineering) Second Edition, 2014.

Print ISBN: 9780813349107
E-book ISBN: 9780813349114

We will discuss applications of nonlinear dynamics, following the book by Strogatz. Topics that will be considered include (for more details, check the book's table of contents): an introduction to Ordinary Differential Equations (ODE's), one-dimensional ODE's and their bifurcations; two-dimensional ODE's (linear case, limit cycles and the Poincare-Bendixson Theorem, the Hopf bifurcation), chaotic systems (logistic family, Lorenz equations, Henon map). For visualization we will use tools that do not require programming, with the option to additionally run/write Matlab or Python code.

 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.

Transformations, eigenvalues and eigenvectors. An expository paper or talk on a subject related to the course content is required

Syllabus

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 4331

There is no prescribed text and other texts that may be of interest include some material from E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol IIA, Springer, and chapter 7 of L. C. Evans, Partial Differential Equations, American Math.Society. Handouts for some background material will be provided.

Course Description:
Existence and uniqueness theory in partial differential equations; generalized solutions and convergence of approximate solutions to partial differential systems.

Additional Description: This course will provide an introduction to the analysis of parabolic initial boundary value problems and related evolution (time-dependent) equations. It is independent of Math 6326, and involves quite different considerations although it requires a similar background. The prerequisite is a good knowledge of multivariable calculus, the basic constructions of Hilbert and Banach spaces and some linear operator theory as covered in the first semester of Applicable analysis, M6360


The emphasis is on the use of Hilbert space methods and weak formulations of the problems, especially Galerkin methods for the approximation and representation of solutions of the problems. Some examples from applications will be treated including various diffusion problems that arise in a wide variety of applications including biology, chemistry, engineering and finance. Issues regarding the asymptotics of the solution, dependence on boundary conditions and the approximation of solutions will be treated.

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.

Linear models, loglinear models, hypothesis testing, sampling, modeling and testing of multivariate data, dimension reduction.

 - 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.

Speak to the instructor for textbook information.

Artificial neural networks for automatic classification and prediction. Training and testing of multi-layers perceptrons. Basic Deep Learning methods. Applications to real data will be studied via multiple projects.

None

Upon completion of the course,students will be able to apply Finite Difference,Finite Volume and Finite Element methods for the numerical solution of elliptic and parabolic partial differential equations.
The course consits in three major parts.In the begining of the course, we will discuss the differential and variational formulations of the most typical boundary value problems for the diffusion and convection- diffusion equations.In the second part of the course,a systematic description of finite difference,finite volume and nodal finite element methods will be given. In the third part, We consider the most important variants of the mixed finite element method which currently is very popular in many applications.Finally,we will study explicit and implicit dinite difference methods for the time dependent diffusion and convection- diffusion equations with applications in geosciences and fluid mechanics.

Speak to the instructor for textbook information.

Speak to the instructor for the course description.

Speak to the instructor for textbook information.

The course presents comprehensive introduction to information visualization and thus, provides the students with necessary background for visual representation and analytics of complex data. The course will cover topics on design strategies, techniques to display multidimensional information structures, and exploratory visualization tools.

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

Primary Text: The Heston Model and Its Extensions in Matlab and C#, by Fabrice Douglas Rouch, Wiley, 2013.

Supplementary Text: Arbitrage Theory in Continuous Time, 3rd edition, by Tomas Bjork, Oxford University Press, 2009.

Stochastic calculus, Brownian motion, change of measures, Martingale representation theorem, pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities, single-factor and multi-factor HJM models, and models involving jump diffusion and mean reversion.

Additional Description: The course is an introduction to continuous-time models in finance. We use the stochastic volatility model of Heston as the principal paradigm and choose Fourier transform and its variants as the tools for pricing. We introduce stochastic calculus, Brownian motion, Levy processes, change of measures, martingale ans semi-martingale and the notion of time change of a stochastic process. We then apply these ideas in pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities. The use of MATLAB is expected.

Detailed syllabus (PDF)

Graduate standing. Linear algebra, probability, statistics, or consent of instructor. Students must be in the Statistics and Data Science, MS Program

Speak to the instructor for textbook information.

Longitudinal data and correlated data analysis, growth-curve models, mixed effects models, correlation structure, analysis of time-to-event data, hazard and survival functions, Kaplan-Meier estimate, log-rank test.

Graduate standing.  Linear algebra, probability, statistics, or consent of instructor. Students must be in the Statistics and Data Science, MS Program

Speak to the instructor for textbook information.

Estimation of allele frequency, Hardy-Weinberg equilibrium, testing on differentially expressed genes, multiple comparison.

Graduate standing.

TBA

Graduate standing.

TBD

- Lecture notes will be provided to you every class.

For personal reference, you may look at the following books:
-Michael Nielsen and Isaac Chuang, "Quantum computation and quantum information", Cambridge university press, 2010
-Mark Wilde, "From Classical to Quantum Shannon Theory" arXiv:1106.1445

You do not need to purchase either of these books. I can recommend any additional books if you request it, which you may borrow from my office.

Course Overview:

During the course we aim to cover the following topics:
-Basics of quantum mechanics
-Universal quantum computation
-Quantum teleportation and other protocols
-Quantum error-correction
-Noisy quantum channels
-Quantum complexity
-Quantum algorithms

Graduate standing. Students must be in the Statistics and Data Science, MS Program

TBA

Machine Learning refers to a set of methods designed to extract information from data with the goal to make predictions or perform various types of decisions. This area has witnessed a remarkable growth during the last decade as machine learning is central to the development of intelligent systems and the analysis of massive and complex data found in science or social media. Machine learning algorithms currently enable systems such as Siri, the Google self driving car, or PathAI for medical diagnostics.

This course is an introduction to the theoretical foundations of machine learning and will be focused on the underlying mathematical concepts needed to understand the methods used in modern data science, without neglecting the algorithmic and computational aspects of the subject.

Topics of the course include Support Vector Machines, Reproducing Kernel Hilbert Spaces, the Vapnik-Chervonenkis theory, concentration inequalities, dimensionality reduction and spectral clustering.

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 neglected, 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.

Course Description: Differentiable Manifolds, tangent space, tangent bundle, vector bundle, Riemannian metric, connections, curvature, completeness geodesics, Jacobi fields, spaces of constant curvature, and comparison theorems.

Additional Description: This course is an introduction to the theory of smooth manifolds, with an emphasis on their geometry. The first third of the course will cover the basic definitions and examples of smooth manifolds, smooth maps, tangent spaces, and vector fields. Later in the semester we will use Euclidean, spherical, and hyperbolic geometry to introduce the notion of a Riemannian metric; we will study parallel transport, geodesics, the exponential map, and curvature. Other topics will include Lie theory and differential forms, including exterior differentiation and Stokes theorem.

The textbook highlights connections of this theory to dynamical systems; these may be mentioned in lectures but are not the focus of this course: this is first and foremost a course in differential geometry, which is oriented towards the associated preliminary exam.