Mathematics (MATH-UH)

This course will provide the basic mathematical toolkit needed for students who do not wish to pursue calculus but still need to be exposed to the mathematical concepts and techniques that are required to study elementary statistics and mathematical models in the social sciences. Emphasis will be placed on the understanding of important concepts and on developing analytical skills rather than just on computational skills, the use of algorithms, and the manipulation of formulae.

A fundamental understanding of mathematical functions is critical before engaging in the rigors of calculus. This course examines single variable functions, including their algebraic and geometric properties. By necessity, the course begins with an exploration of the following question: What is a function, and how can it be represented geometrically as a graph? The course delves into standard function manipulations and examines a range of mathematical functions, including polynomial, trigonometric, and exponential functions. The course further provides an in-depth study of trigonometric functions, trigonometric equations and trigonometric identities, including double angle and half angle formulae and their application. In addition, there is an introduction to the polar coordinates system and vectors in two and three dimensions. Basic sequences are also studied. By the end of the course, students will have a solid preparation for calculus, from the algebraic, geometric and analytic point of view.

Mathematics is a convenient and powerful language, providing a deep, unified framework for all scientific developments. All existing results from the three fundamental categories of mathematics - geometry, algebra, and analysis - can be formally expressed in terms of set theory, predicates, quantifiers, and logical connectives. This course explores the axiomatic method, some elements of logic and formal languages, and set theory. In addition, the system of real numbers and all other fundamental number systems can be firmly established on the ground of natural numbers; therefore, the course introduces elementary arithmetic and the universal method of constructing new objects from already known sets by means of equivalence relations. Abstract concepts are introduced through basic but fundamental and universal examples or problems, carefully chosen as illuminants of broader ideas and sources of new theoretical and practical applications.

This course presents the basic principles of calculus by examining functions and their derivatives and integrals with a special emphasis placed on the utilitarian nature of the subject material. Since the derivative measures the instantaneous rate of change of a function and the definite integral measures the total accumulation of a function over an interval, these two ideas form the basis for nearly all mathematical formulas in science, engineering, economics, and other fields. This course also provides instruction in how to model situations in order to solve problems. Applications include graphing, and maximizing and minimizing functions. In addition to two weekly lectures, students attend a weekly recitation focused on applications. Placement into Calculus with Applications is decided by discussion with mentors and the results of a mathematics placement examination. This course focuses on the needs of students in science and engineering.

This course presents the foundations of calculus by examining functions and their derivatives and integrals with a special emphasis placed on the utilitarian nature of the subject material. Since the derivative measures the instantaneous rate of change of a function and the definite integral measures the total accumulation of a function over an interval, these two ideas form the basis for nearly all mathematical formulas in science, engineering, economics, and other fields. This course also provides instruction in how to model situations in order to solve problems. Applications include graphing, and maximizing and minimizing functions. In addition to two weekly lectures, students attend a weekly recitations focused on applications. Placement into Calculus with Applications is decided by discussion with mentors and the results of a mathematics placement examination. This course focuses on the needs of students in economics.

This course explores functions of several variables and has applications to science and engineering. Specific topics include: vectors in the plane and space; partial derivatives with applications; double and triple integrals; spherical and cylindrical coordinates; surface and line integrals; and divergence, gradient, and curl. In addition, the theorems of Gauss and Stokes are rigorously introduced.

This course explores functions of several variables and has applications to science and engineering as well as economics. This special course for those majoring in economics includes: vectors in the plane and space; partial derivatives with applications; Lagrange multipliers; constrained and unconstrained optimization; double and triple integrals; spherical and cylindrical coordinates; surface and line integrals.

In many applications of mathematics, a response of some systems is nearly a linear function of the input. These linear systems, which arise in elasticity, in electrical engineering, and in economics for example, involve linear equations in many unknowns. The associated matrix algebra is a rich and beautiful field of mathematics. It is also central to the analysis of linear ordinary and partial differential equations. The material in this course places emphasis on theorems and proofs, and includes systems of linear equations, Gaussian elimination, matrices, determinants, Cramer's rule, vectors, vector spaces, basis and dimension, linear transformations, eigenvalues, eigenvectors, and quadratic forms.

In many applications of engineering and mathematics, a response of systems is a linear function of the input. These linear systems, which arise in elasticity, in circuits, and in numerical simulations, for example, involve linear equations in many unknowns. The associated matrix algebra is a rich field of mathematics. It is also central to the analysis and numerical solutions of linear ordinary and partial differential equations used in engineering and other fields. The topics include practical approaches to systems of linear equations, matrices, determinants, Gaussian elimination, basis and dimension, linear transformations, eigenvalues and eigenvectors.

Ordinary differential equations arise in virtually all fields of engineering. Newton's equations of motion, the rate equations of chemical reactions, the currents flowing in electric circuits, all can be expressed as ordinary differential equations. The course studies first- and second- order equations.

Ordinary differential equations arise in virtually all fields of applied mathematics. Newton's equations of motion, the rate equations of chemical reactions, the currents flowing in electric circuits, all can be expressed as ordinary differential equations. The solutions of these equations usually evolve a combination of analytic and numerical methods. The course studies first- and second-order equations, solutions using infinite series, Laplace transforms, linear systems, numerical methods.

Most real world phenomena include non-deterministic or non-deterministically predictable features. The course is designed to provide an introduction to the mathematical treatment of such aspects, acquainting the students with both probability and statistics. The course includes: mathematical definition of probability; combinatorics; finite, discrete and continuous probabilities of single and joint random variables; law of large numbers and normal approximation; sampling; estimation; testing of hypotheses; correlation and regression.

Algebra is a part of every field of mathematics, and has applications in the discrete systems of computer science. Fractions, together with their familiar laws of addition, multiplication, and division, provide an example of algebra. The complex numbers form another. This course introduces more general algebras and their properties and applications. Topics considered in this course include groups, homomorphisms, automorphisms and permutation groups. Rings, ideals and quotient rings, Euclidean rings, and polynomial rings are also considered.

Real Analysis builds a more rigorous foundation for calculus and prepares the way for more advanced courses. The emphasis is on the careful formulation of the concepts of calculus, and the formulation and proof of key theorems. The goal is to understand the need for and the nature of a mathematical proof. The course studies the real number system, the convergence of sequences and series, functions of one real variable, continuity, connectedness, compactness, and metric spaces.

Often, the most difficult task of the applied mathematician is the formulation of an analyzable model in the face of a perplexing phenomenon or data set. This course gives students an introduction to all aspects of this process. The basic tools of modeling, including dimensional analysis, asymptotic approximations, qualitative analysis of differential equations, elementary probability, are presented. They are then used, sometimes in combination with very simple numerical simulations, to give a mathematical formulation of and to analyze several self-contained examples, taken mostly from the natural sciences. Students will also be asked to develop or analyze a model of their own choice, in a field of their interest.

The research seminar provides students with an overview of the diverse multidisciplinary research areas that have captured the interest and fascination of NYUAD mathematicians. Through exposure to NYUAD faculty research, students identify areas of interest for their own capstone research and develop and write an in-depth research proposal over the course of the semester. The final capstone proposal is due at the end of the seminar so that students can begin the Capstone Project in Mathematics 1. Students who have chosen Mathematics as their secondary major do not need to complete the research seminar course; students must instead take two courses in their chosen elective which are not research courses.

This course is a continuation of the study of algebraic structures started in Abstract Algebra 1. The notion of rings and fields are thoroughly studied, as well as polynomials over rings such as the ring of integers. This course develops ideas to prepare the students to study Galois theory, one of the most important theories in algebra. The topics include Euclidean domains, principal ideal domains, unique factorization domains, imaginary and real quadratic number fields, extension fields and roots of polynomials, constructions with straight edge and compass, and elements of Galois theory.

The second part of the analysis series is devoted to the calculus of functions of several variables. The transition from a single variable to many variables involves important new concepts, which are essential to understanding applications to the natural world. The course entails a rigorous study of functions of several variables, limits and continuity, differentiable functions, the implicit function theorem, Riemann integral, Stokes formula and an introduction to Lebesgue integration.

The course provides an introduction to rigorous probability theory using measure theory. The necessary notions of measure theory are introduced as needed, and the proofs of essential introductory results of measure theory and probability theory are presented in detail. The main topics of the course include the definition and construction of probability spaces and random variables, limit theorems, conditional expectations, martingales, Markov chains and stochastic processes.

Cryptology existed since the ancient times with the art of ciphers and codes and has now evolved to become a branch of mathematics and computer science that develops the tools and the algorithms underpinning its two divisions: cryptography (development of protocols for the security, authentication and safety of communication) and cryptanalysis (the science of breaking cryptosystems). Before the modern era, cryptography was concerned solely with data confidentiality. For that purpose, classical cryptosystems involving symmetric-key algorithms have been developed. In recent decades, the field has been revolutionized with the invention of asymmetric-key encryption algorithms such as public-key ciphers and later expanded beyond confidentiality concerns to include techniques for user authentication, data integrity checking, data origin authentication, digital signatures, interactive proofs, and secure computation, amongst others. Modern cryptography hence relied on a great deal of fascinating mathematics arising from elementary number theory and the more recent advances expanded to include arithmetic algebraic geometry and algebraic number theory.

Many laws of physics are formulated as differential equations or partial differential equations, e.g. the propagation of sound waves, the diffusion of a gas, and the flow of a fluid. These equations are usually nonlinear and the study of their dynamical properties (long time behavior, changes of properties of solutions, …) turns out to be very difficult. The goal of this course is to study some simple aspects of dynamical systems and chaos. Applications to physics, chemistry, biology, and population dynamics are given. In particular, the course will involve the study of many examples coming from physics, biology and engineering. The examples studied will depend on the interests of the students and their majors.

The course explores how mathematical problems can be analyzed and solved using numerical methods. As such, the subject has very broad applications in applied mathematics, physics, engineering, finance, and life sciences. Topics covered in this course include numerical algorithms for solving linear equations and nonlinear systems of equations, least squares problems, eigenvalue problems, interpolation, numerical quadrature, optimization, and differential equations. Theory and practical examples are combined to study these topics.

Many laws of physics are formulated as partial differential equations, including the propagation of sound waves, the diffusion of a gas, and the flow of a fluid. This course discusses the simplest examples of such laws as embodied in the wave equation, the diffusion equation, and Laplace's equation. The course also discusses nonlinear conservation laws and the theory of shock waves. Applications to physics, chemistry, biology, and population dynamics are discussed.

Complex analysis, also known as the theory of functions of a complex variable, is the branch of mathematical analysis devoted to complex valued functions of complex variable. It is further used in other branches of mathematics, including algebraic geometry and number theory, and also has diverse applications in science and engineering: fluid dynamics, elasticity, nuclear and electrical engineering, to name just a few examples. The geometrical content of analysis in the complex plane is especially appealing. Topics covered include: complex numbers and complex functions; differentiation and the Cauchy-Riemann equations, Cauchy's theorem, and the Cauchy integral formula; singularities, residues, Taylor and Laurent series; fractional linear transformations and conformal mappings.

This course is a transition from vector calculus to differential geometry, the study of curved spaces. The course plan is to move from a study of extrinsic geometry of curves and surfaces in space, familiar from multivariable calculus, to the intrinsic geometry of manifolds. This includes the study of tangent spaces and vector fields and the concept of Riemannian manifolds and leading to explicit characterizations of metrics, connections, and curvatures. Computational tools will include tensor algebra and differential forms. Using these, derivatives on manifolds and integration on manifolds will generalize the corresponding notions from multivariable calculus. Further topics also include the Euler characteristic, The Gauss-Bonnet theorem, symmetry, homogeneous spaces, and applications such as Electromagnetism and General Relativity.

The senior Capstone Experience in Mathematics requires students to engage in a long-term, mentored learning experience that culminates in a piece of research. The specific project is developed during the Research Seminar in Mathematics. During the Capstone Project, the proposed work comes to fruition in the form of a research paper along the lines of those found in leading journals in mathematics.

Continuation of Capstone Project in Mathematics 1 (MATH-UH 4000) culminating with students presenting their work.

This course is a basic introduction to topology, with a balance between point-set topology, geometric topology, and algebraic topology. The concept of a topological space is introduced and some of its more important properties, like connectedness and compactness, are studied. Then the main focus is on topological surfaces with the aim of establishing the fundamental classification theorem for compact surfaces, connecting to the Euler characteristic. After developing the foundations and the geometric intuition, computational algebraic aspects such as homology are introduced. Further classification uses homotopy, the fundamental group, and covering spaces. The concepts are illustrated in various applications, including the Brouwer Fixed Point Theorem.