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 3331 or equivalent, and three additional hours of 3000-4000 level Mathematics. Previous exposure to Partial Differential Equations (Math 3363) is recommended.

"Partial Differential Equations: An Introduction (second edition)," by Walter A. Strauss, published by Wiley, ISBN-13 978-0470-05456-7  

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

Instructor's Description: will cover the first 6 chapters of the textbook. See the departmental course description.

MATH 2433 and six additional hours of 3000-4000 level Mathematics.  

Calculus of Functions of Several Variables

Description: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, The Codazzi and Gauss Equations, Covariant Differentiation, Parallel Translation.

Instructor's 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 3331 and COSC 1410 or equivalent. (2017—2018 Catalog)

MATH 3331 or MATH 3321 or equivalent, and three additional hours of 3000-4000 level Mathematics (2018—2019 Catalog)

*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 3331 and COSC 1410 or equivalent. (2017—2018 Catalog)

MATH 3331 or MATH 3321 or equivalent, and three additional hours of 3000-4000 level Mathematics (2018—2019 Catalog)

(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

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 

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

Instructor's Description: The course covers the following topics: vector spaces, subspaces, linear combinations,systems of linear equations, linear dependence and linear independence, bases and dimension,linear transformations, null spaces, ranges, matrix rank, matrix inverse and invertibility,determinants and their properties, eigenvalues and eigenvectors, diagonalizability.

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

Instructor's Description: The course covers the following topics: vector spaces, subspaces, linear combinations, systems of linear equations, linear dependence and linear independence, bases and dimension, linear transformations, null spaces, ranges, rank, inverse and invertibility, determinants, eigenvalues and eigenvectors.

Instructor's Prerequisite: MATH 3339

- An Introduction to Statistical Learning (with Applications in R), by James, Witten, Hastie, and Tibshirani

- Neural Networks with R, by G. Ciaburro, B. Venkateswaran, ISBN: 9781788397872

Description: Linear and logistic regression, fit quality assessment, model validation, decision trees, bootstrap, random forests, neural networks.

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

Graduate standing.

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%

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

(see the description for more prerequisite details)

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

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

Required Text: Abstract Algebra by David S. Dummit and Richard M. Foote, ISBN: 9780471433347

This book is encyclopedic with good examples and it is one of the few books that includes material for all of the four main topics we will cover: groups, rings, field, and modules.  While some students find it difficult to learn solely from this book, it does provide a nice resource to be used in parallel with class notes or other sources.

Catalog Prerequisite: Graduate standing, MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors.

Instructor's Prerequisite: MATH 2331, or equivalent, and a minimum of three semester hours of 3000-4000 level Mathematics.

Catalog Description: An expository paper or talk on a subject related to the course content is required.

Instructor's Description: The course covers the following topics: vector spaces, subspaces, linear combinations,systems of linear equations, linear dependence and linear independence, bases and dimension,linear transformations, null spaces, ranges, matrix rank, matrix inverse and invertibility,determinants and their properties, eigenvalues and eigenvectors, diagonalizability.

Catalog Prerequisite: Graduate standing, MATH 2331 and a minimum of 3 semester hours transformations, eigenvalues and eigenvectors.

Instructor's Prerequisite: MATH 2331, or equivalent, and a minimum of three semester hours of 3000-4000 level Mathematics.

Graduate standing and MATH 3334.

In depth knowledge of Math 3325 and Math 3333 is required.

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.

Graduate Standing and MATH 4331. In depth knowledge of Math 3333 required.

 No textbook required. Lecture notes provided

This is a year-long course. The first semester part 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

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

link to text

Catalog Description: Point-set topology: compactness, connectedness, quotient spaces, separation properties, Tychonoff’s theorem, the Urysohn lemma, Tietze’s theorem, and the characterization of separable metric spaces

Instructor's Description: Topology is a foundational pillar supporting the study of advanced mathematics. It is an elegant subject with deep links to algebra and analysis. We will study general topology as well as elements of algebraic topology (the fundamental group and homology theories).

Though traditionally viewed as a pure subject, algebraic topology has experienced a renaissance in recent years with the emergence of applied algebraic topology. To wit, SIAM has recently launched a new journal on applied algebra and geometry.

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

Graduate standing and MATH 4331 and MATH 4377

Students are expected to have a good grounding in basic real analysis and linear algebra.

"Convex Optimization", Stephen Boyd, Lieven Vandenberghe, Cambridge University Press, ISBN: 9780521833783 (This text is available online. Speak to the instructor for more details)

Catalog Prerequisites:Graduate standing. MATH 4364 and MATH 4365 or equivalent, and either COSC 1304 or COSC 2101 or equivalents, or consent of instructor

Instructor Prerequisites: Knowledge of C and/or Fortran.

Instructor's notes will be provided.

Recommended text/Optional: Elementary Numerical Analysis, ISBN: 9780471433378

Description: A project-oriented 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.

Instructor's Description: Fundamental techniques in high performance scientific computation. Hardware architecture and floating point performance. Pointers and dynamic memory allocation. Data structures and storage techniques related to numerical algorithms. Parallel programming techniques. Code design. Applications to numerical algorithms for the solution of systems of equations, differential equations and optimization. Data visualization. This course also provides an introduction to computer programming issues and techniques related to large scale numerical computation.

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.

Graduate standing.

Instructor's prerequisites: Linear Algebra, basics of Probability, basics of Functional Analysis

-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

Description: May be repeated with approval of chair.

Instructor's description: Quantum Information Theory is the study of the information processing tasks (such as storing, manipulating, transmitting, encoding, decoding) that can be accomplished using quantum mechanical systems. The course will start with the introduction to quantum mechanics, and will briefly cover basic aspects of quantum information theory, such as quantum algorithms, error-correction, complexity theory, quantum entropy, entanglement theory, noisy and noiseless quantum channels. Besides rigorously going over these topics, we will informally discuss applications of quantum information and quantum mechanics to the real world, as well as their appearance in the real world and popular culture. The work load will include a weekly list of practice problems and weekly research assignment on a general topic. Active participation is class is required. No quizzes, exams or a final will be given.

Graduate standing.  

This course is focused on the derivation and analysis of the kinetic equations governing the evolution of interacting multi-agent systems. The following are the examples of kinetic PDEs that will considered in this course: backward and forward Kolmogorov equations, Boltzmann-type equations for agent systems interacting through binary collisions, linear kinetic equations for agent-backgound interation, Vlassov and Vlassov-Poisson equations for mean-field type interactions.

Kinetic equations of these types appear in diverse applications. We consider the examples of multi-agent systems that arise in Classical Physics (radiative transfer, rarefied gas dynamics), Life Sciences (birth and death processes, growth of mutant cells, self-organized systems and swarming models), Economics and Social Sciences (models for wealth distribution, trading goods and opinion formation).

Catalog Description: This course is part of a two semester sequence covering the main results in functional analysis, includingHilbert spaces, Banach spaces, topological vector spaces such as distributions, and linear operators on thesesspaces.

Instructor's Description: You will be asked from time to time (maybe once every week or two) to make a presentation at the board from the typed notes. If you need help with this please ask. We will be starting with a little topology - so you might glance through any good basic book on topology to familiarize yourself with `compactness' , `locally compact', continuous functions between topological spaces, the basic theory of metric spaces. The reason I review some topology is to familiarize the students with the use of `nets'(=generalized sequences) in topology.

We will mostly avoid measure theory, so don't be too concerned if you lack that background. After each chapter we will schedule a problem solving workshop, based on the homework assigned for that chapter.

Final grade is aproximately based on a total score of 200 points consisting of homework (100 points) and class presentations (100 points). The instructor may change this at his discretion.

The first semester will be a general presentation, of the basic facts in Linear Analysis, Banach space theory, and Hilbert spaces and compact operators. The second semester will be a more technical development of the theory of linear operators on Hilbert space. We will also cover topics which the students request. We will probably only make it through section III in the first semester.

Outline of the 2-semester syllabus:

I. General topology.

A very short section emphasizing nets.

II. Banach Space Theory.

Normed spaces and their operators. Review of basic principles like the Hahn-Banach theorem (pillars of functional analysis). Banach space theory. Dual spaces. Locally convex and weak topologies.

III. Hilbert spaces.

Deeper facts about Hilbert spaces. Bounded linear operators on Hilbert spaces. Compact operators and thir spectral theorem.

IV. Algebras and spectral theory.

Banach algebras. The Gelfand transform. Stone-Weierstrass. The spectral theorem. The Fourier Transform for locally compact groups.

V. Special topics/Students requests.