An Introduction to Stochastic Modeling" by Mark Pinsky, Samuel Karlin. Academic Press, Fourth Edition.
ISBN-10: 9780123814166
ISBN-13: 978-0123814166

We study the theory and applications of stochastic processes. Topics include discrete-time and continuous-time Markov chains, Poisson process, branching process, Brownian motion. Considerable emphasis will be given to applications and examples.

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, fixed points and the contraction mapping principle, differentiation and integration.

MATH 2433 and MATH 2331. Math 2433 Calculus of Functions of Several Variables

Instructor's notes will be provided.

Description: This year-long course will introduce the theory of the geometry of curves and surfaces in three-dimensional space using calculus techniques, exhibiting the interplay between local and global quantities.

Topics include: curves in the plane and in space, global properties of curves and surfaces in three dimensions, the first fundamental form, curvature of surfaces, Gaussian curvature and the Gaussian map, geodesics, minimal surfaces, Gauss' Theorem Egregium, Gauss-Bonnet theorem etc.

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

*Ability to do computer assignments in FORTRAN, C, Matlab, Pascal, Mathematica or Maple.

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 2331 , or equivalent, and six additional hours of 3000-4000 level Mathematics.

Conditioning and stability of linear systems, matrix factorizations, direct and iterative methods for solving linear systesm, computing eigenvalues and eigenvectors, introduction to linear and nonlinear optimization 

Linear Algebra Using MATLAB, Selected material from the text Linear Algebra and Differential Equations Using Matlab by Martin Golubitsky and Michael Dellnitz)
The text  will made available to enrolled students free of charge.

Software: Scientific Note Book (SNB) 5.5  (available through MacKichan Software, http://www.mackichan.com/)

Syllabus: Chapter 1 (1.1, 1.3, 1.4), Chapter 2 (2.1-2.5), Chapter 3 (3.1-3.8), Chapter 4 (4.1-4.4), Chapter 5 (5.1-5.2, 5.4-5-6), Chapter 6 (6.1-6.4), Chapter 7 (7.1-7.4), Chapter 8 (8.1)

Project: Applications of linear algebra to demographics. To be completed by the end of the semester as part of the final.

Course Description: Solving Linear Systems of Equations, Linear Maps and Matrix Algebra, Determinants and Eigenvalues, Vector Spaces, Linear Maps, Orthogonality, Symmetric Matrices, Spectral Theorem

Students will also learn how to use the computer algebra portion of SNB for completing the project. 

Homework: Weekly assignments to be emailed as SNB file.

There will be two tests and a Final.

Grading: Tests count for 90% (25+25+40), HW 10%

A set of notes on curves and surfaces will be written and distributed by Dr. Ru

The course will be an introduction to the study of Differential Geometry-one of the classical (and also one of the more appealing) subjects of modern mathematics. We will primarily concerned with curves in the plane and in 3-space, and with surfaces in 3-space. We will use multi-variable calculus, linear algebra, and ordinary differential equations to study the geometry of curves and surfaces in R3. Topics include: Curves in the plane and in 3-space, curvature, Frenet frame, surfaces in 3-space, the first and second fundamental form, curvatures of surfaces, Gauss's theorem egrigium, geodesics, Gauss-Bonnet theorem, and minimal surfaces.

Instructor's lecture notes.

This course is an introduction to complex analysis. It will cover the theory of holomorphic functions, Cauchy theorem and Cauchy integral formula, residue theorem, harmonic and subharmonic functions, and other topics.

On-line course is taught through Blackboard Learn, visit http://www.uh.edu/webct/ for information on obtaining ID and password.

The course will be based on my notes.

In each week, some lecture notes will be posted in Blackboard Learn, including homework assignment.

Homework will be turned in by the required date through Blackboard Learn. It must be in pdf file. There are two exams. Homework and test problems are mostly computational in nature.

MATH 2331 and minimum of 3 semester hours of 3000 level mathematics.

This first course in the sequence Math 4331-4332 provides a solid introduction to deeper properties of the real numbers, continuous functions, differentiability and integration needed for advanced study in mathematics, science and engineering. It is assumed that the student is familiar with the material of Math 3333, including an introduction to the real numbers, basic properties of continuous and differentiable functions on the real line, and an ability to do epsilon-delta proofs.

Topics: Open and closed sets, compact and connected sets, convergence of sequences, Cauchy sequences and completeness, properties of continuous functions, fixed points and the contraction mapping principle, differentiation and integration.

Required: Instructor's lecture notes

Recommended Texts:
• Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale (available at Amazon or in the library)
• Ordinary Differential Equations by V. I. Arnold, M.I.T press, 1998 (paperback)
• Geometrical Methods in the Theory of Ordinary Differential Equations by V. I. Arnold, Springer Verlag, 2nd Edition 1988.
• Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields by J. Guckenheimer and P. Holmes (Applied Mathematical Sciences Vol 42) Springer Verlag.
• Mathematical Methods of Classical Mechanics by V. I. Arnold, Springer Verlag, 2nd

*Special notes: It is not necessary to buy anything of the references, though Ordinary Differential Equations by V. I. Arnold and Differential Equations, Dynamical Systems and Linear Algebra by M. Hirsch and S. Smale would be most useful to own. The books are useful for reference and lecture notes will be based on these texts and a variety of sources.

This course is an introduction to differential equations. We cover linear theory: existence and uniqueness for autonomous and non-autonomous equations; stability analysis; stable and unstable manifolds and Floquet theory. We will also cover topics such as quasiperiodic motion; normal form theory; perturbation theory and classical mechanics.

Assessment: There will be one midterm (worth 20 points), a final exam (30 points) as well as 2 to 4 take-home problem sheets (to make up 50 points in total).

(Recommended) Topology, A First Course, J. R. Munkres, Second Edition, Prentice-Hall Publishers.

V. Runde A taste of topology , Springer Universitext (paperback, inexpensive).

Topology is the perfect course to take as a first year graduatestudent, since it does not contain too much material, or material that is too sophisticated(the typed notes for the course are about 47 pages). It is a central and  fundamental course and one which graduate students usually enjoy very much!

Thetopic is basically point-set topology, we will discuss a little algebraic topology at the end. We begin by discussing a little set theory, the basic definitions of topology and basis, and go on to discuss separation properties, compactness, connectedness, nets, continuity, local compactness, Urysohn's lemma, local compactness, Tietze's theorem, the characterization of separable metric spaces, paracompactness, partitions of unity, and basic constructions such as subspaces, quotients, and products and the Tychonoff theorem. At the end we will discuss a little algebraic topology, like simple connectedness and the fundamental group. 

You do not need a textbook, although I recommend the Munkres or the Rundebooks. You are expected to read the classnotes carefully each week, line by line, and bring to me the things you don't understand there. Classnotes will be put on the web. You are also expected to do most of the homework sets, and turn in selected homework problems for grading.

You are encouraged to work with others, form study groups, and so on, however copied turned in homework will not help you assimilate the material, and will not be graded. The final grade is aproximately based on a total score of 300 points consisting of homework and other assignments (100 points), a semester test (100 points), and a final exam (100 points). That is, the final exam will count as much as the semester test; and it may be a take-home exam due a different date than the officially scheduled time listed on the UH website for the final exam for this course. In the final week of the semester you will be given time to work on a project, which you will be able to choose.

Graduate Standing. Math 6322-6323, or equivalent, or consent of instructor

Positivity in Algebraic Geometry I, by Lazarsfeld. ISBN: 9783540225331 (not required)

Principles of Algebraic Geometry, by Griffiths-Harris. ISBN: 9780471050599 (not required)

J.K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, (2005). ISBN: 9789812705433

 A.W. Naylor and G.R. Sell, Linear Operator Theory in Engineering and Science,  Springer. ISBN: 9780387950013

Introduction to Nonlinear Optimization Theory, Algorithms, and Applications with MATLAB; by Amir Beck, SIAM. ISBN: 9781611973648

Recommended Texts :
- A First Look at Rigorous Probability Theory by Jeffrey Rosenthal, 2000..
- An Intermediate Course in Probability Theory by Allan Gut, Springer 2009 (any edition)

Review of Undergraduate Probability:
- A First Course in Probability, 6th Edit. by Sheldon Ross, 2002, Prentice Hall

Complementary Texts for further reading:
- Probability: theory and Examples, 3rd Edit., Richard Durrett, Duxbury Press
- An Introduction to Probability Theory and Its Applications, Vol 1,  by William Feller
- Probability by Leo Breiman, 1968, Addison-Wesley 

General Background (A).
(1) Combinatorial analysis and axioms of probability
(2) Elementary random variables theory: expectation, variance, moments, distribution
function, probability density functions, impact of change of variable on density functions
(3) Major discrete probability distributions: Bernoulli, Binomial, Poisson, Geometric
Major continuous probability distributions: Uniform, Normal, Exponential
(4) Basic Modelling Applications
(5) Conditional probability: Bayes formula, Independence, Conditional Expectation,
Conditional density function, Conditional Probability distribution, Independent identically distributed random variables
(6) Joint distributions, joint density functions, marginal distributions, marginal densities, covariances and correlation coefficients
(7) Moment generating functions, Characteristic functions,

Measure theory (B).
(1) Elementary measure theory : Boolean algebras, probability spaces , continuity of probabilities, Borel-Cantelli lemma, Chebychevs inequality,
(2) Convergence of random variables: Almost sure convergence, Convergence in distribution, Law of Large Numbers, Central Limit theorem

Markov chains and random walks (C).
Markov chain theory for finite or countable state spaces
(1) Markov property and Transition matrix, Irreducibility
(2) First hitting times, Transience, Recurrence ,
(3) Stationary distributions : existence theorems and computation
(4) Random walks on Z and Z2 as Markov chains; Gambler's ruin problem

Introduction to Mathematical Finance: Discrete-time Models, by Stanley Pliska, Blackwell, 1997.

1) Y.Efendiev, Th. Y. Hou, Multiscale finite element methods, Springer, 2009

2) M.A.Olshanskii, Lecture notes  on multigrid  methods. (Available free online).

The course surveys and studies the main concepts and recent advances in multiscale finite  element methods (FEM) and multilevel algebraic solvers. A broad range of scientific and engineering problems involve multiple scales. Large disparities in the modeled scales appear in virtually all areas of modern sciences engineering, for example, composite materials, porous media, turbulent transport and so on. The direct numerical solution of multiple scale problems is difficult even with the advent of supercomputers. Multiscale FEM are designed to capture the mutiscale structure of the solution. The notion of different scales is also of key importance for multilevel/multigrid methods to solve large systems of algebraic equationsarising in many applications. Pioneered in the 70s multigrid soon become a crucial ingredient in engineering software and is known to be among a few methods to provide an optimal complexity in terms of arithmetic operations per unknown. Nowadays, every researcher working with the numerical solution of partial differential equations should at least be familiar with this powerful approach. This course introduces to multigrid methods and their applications in computational physics. Applications will be considered to basic PDEs as well as to various fluids models, and Maxwell equations.

- Linear Algebra: matrices, eigenvectors and eigenvalues, diagonalization, determinants, quadratic forms associated to symetric matrices

- Basic Probability and Statistics: random variables, independence, probability distributions, CDF’s and densities, histograms, quantiles, correlations and covariances

- Hilbert Spaces: background knowledge  and basic examples 

No single textbook.  Reading assignments will be a  small set of scientific papers. Some background information and concepts will be referred to the following book: 

The Elements of Statistical Learning, Data Mining:   Freedman, Hastie, Tibshirani 

Summary: Automatic learning has  been widely extended to the analysis of high dimensional data sets  and in particular to smart data mining of massive data sets of time series. The course will focus first on applications of machine learning to automated clustering, modeling, and  forecasting of time series in multiple application domains where data are naturally indexed by time : economic and financial data, multi-sensors recordings, brain activity recordings, climate data recordings, etc.

We will present  and study key techniques such as entropy based distances, kernel based non linear clustering,  spectral compression  of networks dynamics, kernel based nonlinear forecasting, reconstruction of networks connctivity. Emphasis will be on understanding importants  concepts and their mathematical formalization,  with a strong focus on software implementation of algorithms  and intensive testing on  actual or simulated  data sets.

Homework and exams:  Homework assignments will mostly involve computational implementation on several applied projects. Students will have  to use either  Matlab or R or equivalent scientific softwares. Projects Reports will have to be typed (using LaTeX or Word scientific). Two midterm exams will be held in class (1h30 without notes or books), and will require only answering conceptual or theoretical questions, with no mathematical proof requested . Final exam will involve in depth reading  of one scientific paper and preparation of a set of slides. Actual presentation of the slides will not be requested

Final grade = 20% final  + 10% midterm1 + 10% midterm2   + 60% homeworks

This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced topics of design and analysis of experiments with applications to biological studies, public health and industries. The selected topics will include review of linear regressions, completely randomlized design, randomized block design, factorial designs, etc.. The instructor reserves the right to exclude certain topics from the textbook and add other topics not covered in the textbook.

Grading. Final grades will be based on class attendance and in-class discussion (10%), assignment (30%), midterm exams (30% each), the final research project (written report,30%).

R software
R is a open source statistical analysis software, and can be downloaded for free
At http://www.r-project.org/ .
SAS is an industry-standard software that are used and recommended by pharmaceutical companies for analzing clinical trial data for reports to the FDA.

Instructor's notes will be provided.

Randomness and variability which are fundamental features of nearly all biological systems. In this course we will use the theory of probability and stochastic processes to develop models of biological systems. We will discuss models of chemical reactions, gene regulatory networks, epidemics, and neuronal networks. Students taking the course should be comfortable with undergraduate probability, multivariate calculus, differential equations and linear algebra. Topics to be covered include: a review of probability, including numerical techniques for generating random samples, Markov processes with discrete and continuous space variables, diffusion processes, Wiener and Ornstein-Uhlenbeck processes, point processes, Gillespie’s algorithm and other algorithms for simulating stochastic processes and their application in biology, statistical analysis of time series, power spectra of random processes. A portion of the course will be devoted to numerical simulations of stochastic systems using MATLAB.

Instructor's notes

This class is designed for graduate students who would like to learn python programming with an emphasis on scientific computing. First, we will discuss standard python constructions. Then, we will discuss data structures which are relevant for scientific computing. We will also discuss various computing algorithms (e.g. ODE integration and computation of statistical quantities, optimization, data processing and machine learning) and implementation in python. Background mathematical material on computational algorithms will be given in class. We will also discuss some computer science algorithms (sorting, graph search, etc.) and implementations.

Functional Analysis (2nd Edition) by Walther Rudin, McGraw Hill, 1991. ISBN: 9780070542365

Functional analysis combines two fundamental branches of mathematics: analysis and linear algebra. Limitingarguments from analysis become essential in order to resolve questions from linear algebra ininfinite-dimensional spaces. In addition, there are close connections between algebraic and topologicalproperties in such spaces that deepen our understanding even in the finite dimensional case.

Topics covered in this first part of the course sequence include: Topological vector spaces (linear mappings,metrizability, bounded operators and continuity, seminorms and local convexity, quotient spaces); Completeness(Baire category and the Banach-Steinhaus theorem, an application to Fourier series, open mapping theorem,closed graph theorem);Convexity (Hahn-Banach theorem, weak topology and separation theorems,compactness and duality, subspaces and quotients); Spectral theory (Banach algebras and their representation,commutativity, resolutions of the identity, spectral theorem,eigenvalues of normal operators, positivity); Distributions (linear functionals on topological vector spaces,working with distributions localization theorems).

The grade will be based on notes prepared by the students.

D. Braess; Finite Elements. Theory, Fast Solvers and Applications in Solid Mechanics. 3rd Edition.Cambridge Univ. Press, Cambridge, 2007.

C. Brenner and L. Ridgway Scott; The Mathematical Theory of Finite Element Methods. 3rd Edition. Springer, New York, 2008

1. Financial Modeling with Jump Processes, Second Edidition, By Rama Cont and
Peter Tankov, Chapman and Hall, 2016.

2. Levy Processes in Finance: Pricing Financial Derivatives, by Wim Schoutens,
Wiley, 2003.

This course is about using Levy processes in modelling financial time series and valuation of contingenct claims.  We start with a revew of diffusion processes, jump processes, point processes, and random measures.  we cover Ito calculus for semimartingales and Girsanov measure transformation.  We then expore the building Levy processes, parameter estimation and valuation of financial derivatives.  We also address issues relating to pricing and hedging in incomplete markets.