MAT 100 Introductory Mathematics
This course stresses elementary mathematics and basic quantitative knowledge at the pre-calculus level. Students understand and work effectively with real numbers, algebraic expressions, polynomials, equations, and functions. Students learn how to present a real-life problem in mathematical terms and model social and scientific phenomena. This course provides a broad-based mathematical knowledge to build upon in quantitative reasoning courses as well as in applied and specialized courses in business and the social and natural sciences. The requirement of Introductory Mathematics may also be satisfied upon admission by designated scores on the SAT exam or a placement examination during the first semester at AUBG. Cr. 3 (6 ECTS Cr.). Offered every semester.
MAT 102 Finite Mathematics
This course provides students with basic knowledge and primary skills from several important mathematical areas, including linear algebra (linear systems and matrices), linear programming, logic (truth sets and Venn diagrams), probability theory, counting principles, and applications to probability. The study of Markov chains at the end becomes an attractive application of all ideas and techniques considered earlier. Gen. Ed.: Quantitative Reasoning. Prerequisite: MAT 100 or equivalent. Cr. 3 (6 ECTS Cr.). Offered every semester.
MAT 103 Calculus I
This course develops (primarily on technical and intuitive level and with only minor references to deeper points like completeness) the initial notions and skills of analysis in the real line—limits and continuity; derivatives (the problem of “rates of change”) and curve sketching; integrals (the “area” or “accumulation” problem) and techniques of integration—with the fundamental theorem of calculus linking the two main problems. Gen. Ed.: Quantitative Reasoning. Prerequisite: MAT 100 or equivalent. Cr. 3 (6 ECTS Cr.). Offered every semester.
MAT 104 Calculus II
This course aims to develop and extend the methods and techniques of Calculus I. Topics discussed include inverse functions, logarithmic and exponential functions, inverse trigonometric functions, L’Hospital’s rule and applications, integration techniques, improper integrals, parametric curves and polar coordinates, infinite sequences and series, power series, representation of functions as sums of power series, Taylor and Maclaurin series, and polynomials. Gen. Ed.: Quantitative Reasoning. Prerequisite: MAT 103 or equivalent. Cr. 3 (6 ECTS Cr.). Offered every spring.
MAT 105 Elementary Linear Algebra and Analytical Geometry
This course offers a general view of some vital ideas and techniques in the field beginning with a discussion of systems of linear equations (the natural source of the subject) and proceeding to the important techniques of matrices, matrix operations, and determinants. An illustration of the general concepts in plane and space geometry helps students to cultivate their intuition and interpretative skills, and an elementary introduction to general vector spaces, linear transformations, and eigenvalue problems initiates students into this powerful technique. Gen. Ed.: Quantitative Reasoning. Prerequisite: MAT 100 or equivalent. Cr. 3 (6 ECTS Cr.). Offered every semester.
MAT 201 Mathematical Statistics
This course offers a general view of some important ideas and techniques in probability theory and mathematical statistics, including random variables and probability distribution functions, expectations, moment generating functions, limit theorems, sampling distributions, the principle of estimation, and hypothesis testing. Prerequisites: MAT 104 and STA 105. Cr. 3 (6 ECTS Cr.). Offered every spring.
MAT 205 Introduction to Abstract Algebra
This course offers an introduction to basic algebraic structures like groups, rings, integral domains, and fields. This course discusses fundamental structure theorems for factorization and discusses applications of general results to some specific and very important objects, such as symmetric groups, ring of integers, polynomial rings, and matrix rings. This course also covers splitting fields and roots of a polynomial, and polynomials with integer, rational, real, and complex coefficients. Prerequisite: MAT 105. Cr. 3 (6 ECTS Cr.). Offered every fall.
MAT 212 Calculus III – Multivariate Calculus and Geometry
This course extends techniques of calculus in two and three dimensions. Topics covered include vectors and geometry of space, quadratic surfaces, space curves, and cylindrical and spherical coordinates. Also included are partial derivatives and extreme value problems for functions of several variables, Lagrange multipliers, double and triple integrals, and iterated integrals and applications. Prerequisites: MAT 103 and MAT 104. Cr. 3 (6 ECTS Cr.). Offered every fall.
MAT 213 Introduction to Differential Equations
This course introduces a variety of solution methods for ordinary differential equations: first-order equations, second-order equations (solution space, base of solutions, Wronskian), power series method, Laplace transform, and system of linear equations. Prerequisite: MAT 103. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 214 Numerical Analysis
This course introduces students to the basic concepts and techniques in the field, including methods for the solution of equations in one variable, polynomial approximation, spline approximation and interpolation, numerical differentiation and integration, and initial value problems for ordinary differential equations. Prerequisite: MAT 104. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 225 Advanced Linear Algebra
This course offers an extended view of the basic concepts of general vector spaces, fundamental structure theorems for linear maps, and eigenvalue technique. It covers spectral theorems for symmetric, Hermitian, and unitary maps (and matrices) and application to quadratic and Hermitian forms. Triangulation and Jordan canonical forms are also discussed. Prerequisite: MAT 102 or MAT 105. Cr. 3 (6 ECTS Cr.). Offered every spring.
MAT 305 Topics in Abstract Algebra
An advanced course with an emphasis on learning to understand, construct, and present proofs. The following topics are included: groups and group action, Sylow theorems, the free group, generators and relations, the Todd-Coxeter algorithm, ring theory, Hilbert’s Nullstellensatz, unique factorization domains, Noetherian rings, modules, free modules, generators and relations, Hilbert basis theorem, the structure theorem for abelian groups, fields, algebraic and transcendental elements, algebraically closed fields, and the fundamental theorem of algebra. As an application, this course suggests either an introduction to Galois theory or introduction to commutative and noncommutative Groebner basis. This course also requires an accompanying weekly seminar. Prerequisites: MAT 105 and MAT 205. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 313 Calculus IV with Differential Geometry
This course extends techniques from Calculus III by studying scalar and vector fields in n-dimensional spaces and operations on them. The notions of line and surface integrals are introduced, and Green’s, Stokes’ and Gauss’s theorems and their applications are discussed. Starting with parametrized surfaces in R3, this course introduces the concepts of embedded manifolds, tangent spaces, and tangent bundles as well as Gauss curvature for two-dimensional surfaces. The notion of differential forms on manifolds is developed, and the general Stokes’ theorem for forms is formulated at the end of the course. Prerequisites: MAT 105 and MAT 212. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 314 Complex Analysis
This course provides an introduction to analytic functions of one complex variable and their basic properties and applications. The material includes complex numbers, connectedness in the complex plane, conformal mappings, holomorphic functions and Cauchy’s integral formulas, Liouville’s theorem, mean value property and maximum modulus principle, Taylor and Laurent expansions, analytic functions and analytic continuation principle, as well as residue theorem and evaluation of integrals by the method of residues. Prerequisites: MAT 105 and MAT 212. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 315 Real Analysis
Analysis and geometry are at the roots of such basic areas of mathematics as general topology, geometric topology, differential geometry, functional analysis, measure theory, probability theory, dynamical systems, and differential equations, to name a few. This course introduces students to set theory, general topology, metric spaces, measure theory, Lebesgue integration, and function spaces. Though the basic structure of analysis was set in the nineteenth and the beginning of the twentieth century, we will explore such up-to-date applications as analysis of fractals or applications to financial calculus through some of the projects. Prerequisites: MAT 104 and MAT 105. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 317 Dynamical Systems
This course provides an excellent example of the application of abstract mathematics. The study of the time evolution of mathematical models of real-world phenomena from economics, computer science, biology, ecology, engineering, finance, physics, etc., applies methods and techniques from geometry, topology, differential and difference equations, measure theory, etc. Moreover, the use of computer algebra systems such as MatLab allows for the detailed development of non-trivial models of concrete dynamical systems. This course is an introduction to discrete and continuous dynamical systems. The goal is to provide a set of tools that can be used to understand such systems from a qualitative and quantitative perspective. Possible topics will include linear and nonlinear phase portraits, limit sets (fixed points, orbits, etc.), stability, bifurcations, chaos, fractals, etc. Concepts and methods from geometry, topology, and analysis will be introduced along the way. Prerequisites: MAT 105 and MAT 212. Cr. 3 (6 ECTS Cr.). Offered once every two years.
MAT 421 Galois Theory
Galois theory, in its many manifestations, is a central topic in modern mathematics. The powerful idea of Galois correspondence can be generalized to apply to such diverse topics as algebraic number theory, differential equations, algebraic topology, mathematical physics, theoretical computer science. This course will discuss the problem of solutions of polynomial equations both in explicit terms and in terms of abstract algebraic structures. We shall study the relation between roots and coefficients of a polynomial: elementary symmetric functions; complex roots of unity, and solutions by radicals of cubic and quartic equations; the characteristic of a field and the prime subfield; field extensions and characterization of finite normal extensions as splitting fields; the structure and construction of finite fields; the Galois group and the Galois correspondence; radical field extensions; solvable groups and solvability by radicals of equations. Prerequisites: MAT 105 and MAT 205. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 431 Introduction to Lie Algebras
Lie Algebras are mathematical objects which, besides being of interest in their own right, elucidate problems in several areas in mathematics. Lie algebras and Lie groups have high degree influence on the present day mathematics, theoretical physics, and recently in computer science, including machine learning. The classification of the finite-dimensional complex Lie algebras is a beautiful piece of applied linear algebra. This course aims to introduce Lie algebras, develop some of the techniques for studying them, and describe parts of the classification mentioned above, especially the parts concerning root systems and Dynkin diagrams. This course is at the advanced undergraduate level, with an emphasis on learning to understand, construct, and present proofs. Exposure to abstract algebra will be an advantage. Prerequisite: MAT 105. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 451 Mathematical Finance
This course introduces the Black-Scholes model – a model which is the backbone of derivatives trading, a multitrillion-dollar industry. In answering the financial question, “What is the fair price of an option?” we will have to introduce a rather involved mathematical machinery. The BS model is founded on two important assumptions: the principle of no-arbitrage and the assumption that prices follow a random-walk/Brownian motion, i.e., that prices satisfy a diffusion equation. We begin with the conceptually simpler discrete time approach (binary trees) and time permitting we extend to continuous time (Brownian motion, stochastic differential equations – the Black-Scholes equation, stochastic integration). Much of the mathematics of BSM is based on probability theory. No prior knowledge of finance is necessary. Prerequisite: MAT 201 or ECO 310. Cr. 3 (6 ECTS Cr.). Offered irregularly.
MAT 471 Category Theory
This course will introduce students to category theory, which though very abstract and general is readily applicable to many sciences, most notably computer science. We could vaguely define it as the study of the “algebra of composition of functions.” The change of perspective – study not the mathematical objects in isolation but the functions between them, is of profound importance. This change allows us to view the overall structure of mathematical theories and their interactions. We will concentrate on universal properties such as (co)limits and adjunctions. Examples will be drawn both from mathematics (e.g., linear algebra) and from the sciences (e.g., functional programming, automata theory, quantum mechanics, etc.). Exposure to Abstract Algebra or Real Analysis or Theory of Computation or Haskell will be helpful, but more important is enthusiasm and readiness to learn. Prerequisites: MAT 104 and MAT 105. Cr. 3. (6 ECTS Cr.). Offered irregularly.
MAT 491/492 Senior Thesis I and II
A senior thesis may be arranged by qualifying students with a faculty advisor for ambitious research programs that cover one or two semesters. Prerequisite: declared MAT major. Cr. 3 (6 ECTS Cr.). Offered every semester as contracted.
PHY 110 Mechanics and Thermodynamics
This course introduces some of the basic laws and principles of classical mechanics, thermodynamics, and statistical physics with an emphasis on how they can be used to explain important natural phenomena or technological developments. This course discusses important turning points in the history of physics and includes in-class physics experiments and laboratory exercises. Gen. Ed.: Scientific Investigation. Prerequisite: MAT 100 or equivalent. Cr. 4 (8 ECTS Cr.). Offered every semester.
PHY 120 Electromagnetism, Relativity, and Quantum Physics
This course includes the study of vibrations and waves, electricity and magnetism, relativity, quantum, and nuclear physics. Emphasis is placed on in-class demonstrations and experiments, and laboratory exercises are included. Gen. Ed.: Scientific Investigation. Prerequisite: MAT 100 or equivalent. Cr. 4 (8 ECTS Cr.). Offered every semester.
PHY 160 Astronomy
This course starts with the subject and history of astronomy, the motion of celestial bodies, the laws of motion and gravity, electromagnetic waves, and stellar spectra. Some aspects of classical and relativistic mechanics are involved. Next, the Solar System is studied – formation and properties of the Sun, the planets, moons, asteroids, and comets. Properties of the planets are explained using basic physics. Past and present explorations of planets, asteroids, and comets are discussed. Then the course focuses on stars - location, structure, properties, energy production, classification, formation, evolution, and death. The beauty and diversity of interstellar matter are revealed. The properties of stellar remnants and star corpses (white dwarfs, neutron stars, and black holes) are explained. At the end, the formation and the general properties of the Universe are discussed. These properties include the Milky Way galaxy, types and properties of other galaxies, Hubble’s law and expansion of the Universe, quasars and active galaxies, the Big Bang model, and the evolution and fate of the Universe. Gen. Ed.: Scientific Investigation. Prerequisite: MAT 100 or equivalent. Cr. 3 (6 ECTS Cr.). Offered every semester.
PHY 210 Classical Mechanics
This course starts with an overview of Newtonian mechanics with emphasis on the theoretical and mathematical foundations of the subject. Conservation laws are studied and applied to solve problems for conservative systems. This discussion is followed by variational calculus and Lagrangian mechanics for systems with constraints. This course ends with an overview of Hamiltonian mechanics and the least action principle. Prerequisites: MAT 103 and PHY 110. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 220 Theory of Electromagnetism
This course is an introduction to the theory of electricity and magnetism and its mathematical description, connecting electric and magnetic phenomena. Topics include electrostatics, magnetic fields, electromagnetic induction, DC and AC circuits, and the electromagnetic properties of matter. Maxwell’s equations in their integral and differential form are studied. This course concludes with an overview of the relativistic formulation of electrodynamics. Prerequisites: MAT 212 and PHY 120. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 230 Quantum Physics
This course is an introduction to quantum physics, the history of its discovery and creation, the basic quantum effects (and experiments demonstrating them), the mathematical formalism of quantum theory, and the applications to information, communication, and computation sciences. We will follow a modern approach, motivated by deep conceptual problems, which takes the viewpoint that quantum effects, such as entanglement, are an information resource for communications and computations. This viewpoint is causing a revival of the interest in quantum theory where now the emphasis is on its information content. We will concentrate mostly on finite systems; thus linear algebra and elementary probability theory will suffice. We will introduce states, observables, quantum dynamics, entanglement, no-cloning, etc., and their applications to cryptography, quantum communication and computing, etc. Gen. Ed.: Scientific Investigation. Prerequisite: MAT 105. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 260 Physical Electronics
This course gives basic knowledge of the physics of semiconductors. Main types of semiconductor devices and their properties are studied. Some typical electronic circuits, their use and characteristics, are discussed. Students will also become familiar with using modern electronic instruments for measurement and data collection. Prerequisite: MAT 100 or equivalent. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 310 Thermodynamics and Statistical Mechanics
Thermodynamics describes phenomena and concepts typical of huge systems (e.g., temperature, entropy, work, heat) while statistical mechanics provides a bridge from the micro to the macro description of such systems (via micro-canonical, canonical, and grand canonical probability distributions). These concepts and methods are of central importance in physics but also in a wide range of other disciplines such as chemistry, material science, biology, ecology, engineering, complex systems, energy economics and policies, etc. This course will cover the laws of thermodynamics, thermodynamic potentials, Boltzmann statistics, quantum statistics, etc. Prerequisites: MAT 212 and PHY 110. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 320 Advanced Quantum Physics
The goal of this course is starting from the quantum description of states and dynamics and applying techniques such as perturbation theory, variational or semiclassical analysis, mean field theory, etc. to apply quantum theory to the study of the structure and transformations of matter. A typical example is the description of metals as a gas of electrons in a crystal lattice. Some of the topics covered will be three-dimensional Schrodinger equation and angular momentum, bound and scattering states, quantum tunneling, identical particles, etc. This course is essential for such applied courses as PHY 420 Condensed Matter Physics and PHY 460 Materials Science. Prerequisites: MAT 212 and PHY 230. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 350 Cosmology and Astrophysics
Cosmology is the study of the evolution of the Universe from the Big Bang to the formation of galaxies and stars. This course will introduce students to the observational data and its interpretation in the present standard model of the Universe. Some of the topics that will be covered are cosmic dynamics, dark matter and energy, cosmic microwave background, the inflation period in the evolution, nucleosynthesis, and the formation of structures (galaxies, etc.). Prerequisite: MAT 212. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 420 Condensed Matter Physics
Condensed matter physics is an important area of current research and serves as the basis for modern electronic technology and materials science. This course starts with the structure of solids, lattice dynamics, and phonons. The electron theory of solids is described and applied to explain the properties of metals, semiconductors, dielectrics, and superconductors. Magnetic properties, optical properties, and elementary excitations in solids (plasmons, polarons, and excitons) are studied. Modern topics, including nanocrystals and photonic crystals, are discussed. Prerequisites: PHY 230 and PHY 310. Cr. 3 (6 ECTS Cr.). Offered every semester.
PHY 430 Quantum Field Theory
Quantum Field Theory (QFT) studies the quantization of systems with infinitely many degrees of freedom. It is the foundation of the physics of elementary particle (the standard model) and condensed matter. Techniques developed in QFT such as the path integral, perturbation theory, quantization of gauge fields, and renormalization group, are used in many areas in and outside physics, for example in mathematical finance. Prerequisites: PHY 210 and PHY 230. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 440 Gravity and General Relativity
This course is an introduction to General Relativity – Einstein’s geometric theory of gravity. This course begins with a review of Special Relativity emphasizing the geometric aspects of the Lorentz transformations in Minkowski space-time. Using Einstein’s equivalence principle, we develop the concept of curved space-time and explain how gravity is the effect of this curving. Then we introduce the relevant mathematical tools to treat curved spaces and present the Einstein-Hilbert equation, which links the curvature to the mass (energy) density. At the end, we discuss specific solutions of Einstein-Hilbert’s equation, such as Schwarzschild’s metric. Prerequisites: MAT 212 and PHY 120. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 460 Materials Science
Materials Science studies how the (microscopic) structure and the (macroscopic) properties of materials are related. The structure is determined by the quantum mechanical binding of the (sub)atomic constituents. Utilizing quantum theory and statistical physics one passes from the microstructure of a material to its macroscopic properties (mechanical, thermal, electromagnetic, optical, etc.). Materials Science plays a key role in the development of nanotechnology, quantum technology, cutting edge medical technologies, as well as more traditional fields in machine, civil, electrical and electronic, chemical engineering. Prerequisite: PHY 420. Cr. 3 (6 ECTS Cr.). Offered irregularly.
PHY 491/492 Senior Thesis I/II
A senior thesis may be arranged by qualifying students with a faculty advisor for ambitious research programs that cover one or two semesters. Prerequisite: declared Physics major. Cr. 3 (6 ECTS Cr.). Offered every semester as contracted.
For other courses satisfying the General Education requirements for Scientific Investigation, see Physics.
SCI 150 Principles of Biology
Biology touches our lives every day. Whether we are concerned about the health of our own bodies or the health of the planet, an understanding of the basic principles of biology is important. This course explores some of the fundamental and unifying concepts of modern biology. Topics covered may include cell structure and processes, genetics, evolution, biodiversity, animal and plant form and function, and ecology. The interconnections within the natural world, along with biology’s relevance to everyday life, will be highlighted. Biology is an extremely diverse and complex discipline, and an introductory course can only explore a thin slice of this diversity and complexity. Although this is an introductory course, it will provide enough depth and rigor to equip students to make scientifically informed evaluations of biological issues confronting contemporary society. Furthermore, it is hoped that after completing this course, students will have a greater appreciation of the wonders of the natural world. Gen. Ed.: Scientific Investigation. Cr. 3 (6 ECTS Cr.). Offered every semester.
SCI 160 Introduction to Environmental Science
This course gives students a basic understanding of the scientific aspects of environmental issues, thus enabling them to engage in current environmental debates more intelligently. More specifically, this course explores the functions and services provided by healthy ecosystems and humanity’s impact on the natural world. This course discusses from a scientific viewpoint some of the major threats to the world’s ecosystems (such as overpopulation, pollution, biodiversity loss, climate change, and overexploitation). Students will also learn how the scientific method is applied to the study of these problems as society seeks solutions. Gen. Ed.: Scientific Investigation. Cr. 3 (6 ECTS Cr.). Offered every fall.
STA 105 Statistics
This course is designed to give students the ability to interpret results drawn from data. It serves students’ needs in business, economics, and other social sciences so that they can make sense of studies and surveys. At the end of the course, students will gain experience to communicate effectively using statistical ideas and concepts. Both descriptive and inferential methods will be presented with sufficient theory to assure understanding of the material. Cr. 3 (6 ECTS Cr.). Offered every semester.