Undergraduate Course Descriptions

Prerequisites:

High school mathematics or permission of the department.

Description:

This course serves as preparation for MATH-UA 120 Discrete Mathematics, MATH-UA 121 Calculus I, MATH-UA 131 Mathematics for Economics I, and MATH-UA 140 Linear Algebra.  Topics include: intermediate algebra and trigonometry; algebraic, exponential, logarithmic, and trigonometric functions and their graphs.

Prerequisites:

One of the following:

1. SAT score of 670 or higher on mathematics portion March 2016 and later
2. SAT score of 650 or higher on mathematics portion before March 2016
3. ACT/ACTE Math score of 30 or higher
4. AB score of 3 or higher
5. BC score of 3 or higher
6. A level Maths score of C or higher (anyone who took Further Maths should contact the math department as it varies depending on the exam board)
7. AS level Maths score of B or higher
8. Completion of Algebra, Trigonometry, and Functions (MATH-UA 009) with a grade of C or higher
9. Passing placement exam

IB Prerequisites for 2021 - 2027

• IB Analysis and Approaches HL score of 5
• IB Applications and Interpretations HL score of 5
• IB Analysis and Approaches SL score of 7

IB Prerequisites for 2014 - 2020

• IB Mathematics HL score of 5
• IB Mathematics SL score of 6 or higher
• IB Mathematical Studies SL score of 7

Description:

A first course in discrete mathematics. Sets, algorithms, induction. Combinatorics. Graphs and trees. Combinatorial circuits. Logic and Boolean algebra.

Prerequisites:

One of the following:

1. SAT score of 670 or higher on mathematics portion March 2016 and later
2. SAT score of 650 or higher on mathematics portion before March 2016
3. ACT/ACTE Math score of 30 or higher
4. AB score of 3 or higher
5. BC score of 3 or higher
6. A level Maths score of C or higher (anyone who took Further Maths should contact the math department as it varies depending on the exam board)
7. AS level Maths score of B or higher
8. Completion of Algebra, Trigonometry, and Functions (MATH-UA 009) with a grade of C or higher
9. Passing placement exam

IB Prerequisites for MATH-UA 121 Calculus I 2021 - 2027

• IB Analysis and Approaches HL score of 5
• IB Applications and Interpretations HL score of 5
• IB Analysis and Approaches SL score of 7

IB Prerequisites for MATH-UA 121 Calculus I 2014 - 2020

• IB Mathematics HL score of 5
• IB Mathematics SL score of 6 or higher
• IB Mathematical Studies SL score of 7

Refer to the Calculus information page for more information.

Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Description:

Derivatives, antiderivatives, and integrals of functions of one real variable. Trigonometric, inverse trigonometric, logarithmic and exponential functions. Applications, including graphing, maximizing and minimizing functions. Areas and volumes.

Prerequisites:

Passing MATH-UA 121 Calculus I with a grade of C or better, an AB or a BC of 4 or higher, A level Maths of B or higher, IB Analysis and Approaches HL score of 6 (students entering 2021 - 2027), IB Applications and Interpretations HL score of 6 (students entering 2021 - 2027), IB Mathematics HL score of 6 or higher (no Topic 9) (students entering 2014 - 2020), or passing a placement test.

Refer the the Calculus information page for more information. 

Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Description:

Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.

Prerequisites:

Passing MATH-UA 122 Calculus II with a grade of C or higher, BC of 5, IB Analysis and Approaches HL score of 7 (students entering 2021 - 2027), IB Mathematics HL score of 6 or higher (with Topic 9) (students entering 2014 - 2020), IB Further Mathematics HL score of 6 or higher (students entering 2014 - 2020), or passing placement test. (anyone who took Further Maths should contact the math department as it varies depending on the exam board)

Refer to the Calculus information page for more information. 

Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Description:

Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.

Prerequisites:

A- in MATH-UA 122 or equivalent, 5 on the AP Calculus BC, IB Analysis and Approaches HL score of 7 (students entering 2021 - 2027), IB Mathematics HL score of 6 or higher (with Topic 9) (students entering 2014 - 2020), IB Further Mathematics HL score of 6 or higher (students entering 2014 - 2020).

Description:

Similar to MATH-UA 123 Calculus III, but at a faster pace and deeper level. Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes. Students interested in an honors mathematics degree especially encouraged to consider this course.

Prerequisites:

One of the following:

1. SAT score of 670 or higher on mathematics portion March 2016 and later
2. SAT score of 650 or higher on mathematics portion before March 2016
3. ACT/ACTE Math score of 30 or higher
4. AB score of 3 or higher
5. BC score of 3 or higher
6. A level Maths score of C or higher (anyone who took Further Maths should contact the math department as it varies depending on the exam board)
7. AS level Maths score of B or higher
8. Completion of Algebra, Trigonometry, and Functions (MATH-UA 009) with a grade of C or higher
9. Passing placement exam

IB Prerequisites 2021 - 2027

• IB Analysis and Approaches HL score of 5
• IB Applications and Interpretations HL score of 5
• IB Analysis and Approaches SL score of 7

IB Prerequisites for 2014 - 2020

• IB Mathematics HL score of 5
• IB Mathematics SL score of 6 or higher
• IB Mathematical Studies SL score of 7

Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Description:

This course is only open to Economics Majors and prospective majors. If an Economics Major decides to double or joint major in Math these courses will replace Calculus I - III. Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Elements of calculus and linear algebra are important to the study of economics. This class is designed to provide the appropriate tools for study in the policy concentration. Examples and motivation are drawn from important topics in economics. Topics covered include derivatives of functions of one and several variables; interpretations of the derivatives; convexity; constrained and unconstrained optimization.

Prerequisites:

Completion of MATH-UA 131 Math for Economics I with a grade of C or higher, or passing departmental placement exam. 

Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Description:

A continuation of Mathematics for Economics I. Matrix algebra; eigenvalues; Ordinary differential equations and stability analysis, multivariable integration and (possibly) dynamic optimization.  

Prerequisites:

MATH-UA 132 Mathematics for Economics II with a grade of C or higher. 

Students cannot take and apply both Calculus courses and Math for Economics courses towards their degree requirements.

Description:

This course is only open to Economics Majors and prospective majors. If an Economics Major decides to double major in Math these courses will replace Calculus I - III.

Further topics in vector calculus. Vector spaces, matrix analysis. Linear and nonlinear programming with applications to game theory. This course will provide economics students who have taken MATH-UA 131 Mathematics for Economics I and MATH-UA 132 Mathematics for Economics II with the tools to take higher-level mathematics courses.

Prerequisites:

One of the following:

1. SAT score of 670 or higher on mathematics portion March 2016 and later
2. SAT score of 650 or higher on mathematics portion before March 2016
3. ACT/ACTE Math score of 30 or higher
4. AB score of 3 or higher
5. BC score of 3 or higher
6. A level Maths score of C or higher (anyone who took Further Maths should contact the math department as it varies depending on the exam board)
7. AS level Maths score of B or higher
8. Completion of Algebra, Trigonometry, and Functions (MATH-UA 009) with a grade of C or higher
9. Passing placement exam

IB Prerequisites for 2021 - 2027

• IB Analysis and Approaches HL score of 5
• IB Applications and Interpretations HL score of 5
• IB Analysis and Approaches SL score of 7

IB Prerequisites for 2014 - 2020

• IB Mathematics HL score of 5
• IB Mathematics SL score of 6 or higher
• IB Mathematical Studies SL score of 7

Description:

Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer's rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms.

Prerequisites:

A grade of C or higher in MATH-UA 121 Calculus I or MATH-UA 212 Math for Economics II (for Economics majors) and PHYS-UA 11 General Physics.

Description:

In this course, students will learn how to do computer simulations of such phenomena as orbits (Kepler problem and N-body problem), epidemic and endemic disease (including evolution in response to the selective pressure of a malaria), musical stringed instruments (piano, guitar, and violin), and traffic flow in a city (with lights, breakdowns, and gridlock at corners). The simulations are based on mathematical models, numerical methods, and Matlab programming techniques that will be taught in class. The use of animations (and sound where appropriate) to present the results of simulations will be emphasized.

Prerequisites:

One of the following:

• SAT score of 670 or higher on mathematics portion March 2016 and later
• SAT score of 650 or higher on mathematics portion before March 2016
• ACT/ACTE Math score of 30 or higher
• AB score of 3 or higher
• BC score of 3 or higher
• A level Maths score of C or higher (anyone who took Further Maths should contact the math department as it varies depending on the exam board)
• AS level Maths score of B or higher
• Completion of Algebra, Trigonometry, and Functions (MATH-UA 009) with a grade of A- or higher
• Passing placement exam

IB Prerequisites 2021 - 2027

• IB Analysis and Approaches HL score of 5
• IB Applications and Interpretations HL score of 5
• IB Analysis and Approaches SL score of 7

IB Prerequisites 2014 - 2020

• IB Mathematics HL score of 5
• IB Mathematics SL score of 6 or higher
• IB Mathematical Studies SL score of 7

Description:

Prerequisites:

MATH-UA 121 Calculus I (or equivalent) or MATH-UA 212 Math for Economics II (for Economics majors), with a grade of B- or better, though completion of MATH-UA 123 Calculus III (multivariate calculus) or MATH-UA 213 Math for Economics III (for Economics majors) is preferred and recommended. Students should also have some familiarity with introductory physics (even at the advanced high school level).

Description:

An introduction to the dynamical processes that drive the circulation of the atmosphere and ocean, and their interaction. This is the core of climate science. Lectures will be guided by consideration of observations and experiments, but the goal is to develop an understanding of the unifying principles of planetary fluid dynamics. Topics include the global energy balance, convection and radiation (the greenhouse effect), effects of planetary rotation (the Coriolis force), structure of the atmospheric circulation (the Hadley cell and wind patterns), structure of the oceanic circulation (wind-driven currents and the thermohaline circulation), climate and climate variability (including El Nino and anthropogenic warming).

Prerequisites:

A grade of C or higher in MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) Suggested: PHYS-UA 106 Mathematical Physics

Description:

Fluid dynamics is the branch of physics that describes motions of fluids as varied as the flow of blood in the human body, the flight of an insect or the motions of weather systems on Earth. The course introduces the key concepts of fluid dynamics: the formalism of continuum mechanics, the conservation of mass, energy and momentum in a fluid, the Euler and Navier-Stokes equations, viscosity and vorticity. These concepts are applied to study classic problems in fluid dynamics, such as potential flow around a cylinder, the Stokes flow, the propagation of sound and gravity waves and the onset of instability in shear flow.

Prerequisites:

None

Description:

Among the topics to be covered are: the axioms of set theory; Boolean operations on sets; set-theoretic representation of relations, functions and orderings; the natural numbers; theory of transfinite cardinal and ordinal numbers; the axiom of choice and its equivalents; and the foundations of analysis. If time permits we may also consider some more advanced topics, such as large cardinals or the independence results.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) with a grade of C or better and/or the equivalent, and MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra with a grade of C or better and/or the equivalent. Not open to students who have taken MATH-UA 235 Probability and Statistics.

Note: This course is intended for math majors and other students with a strong interest in mathematics. It requires fluency in topics such as multi-variable integration and therefore a grade of B or better in MATH-UA 123 or MATH-UA 213 (or the equivalent) is strongly recommended.

Description:

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.

Prerequisites:

MATH-UA 233 Theory of Probability with a grade of C or better and/or the equivalent. Not open to students who have taken MATH-UA 235 Probability and Statistics.

Description:

An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences.

Prerequisites:

MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) with a grade of C or better and/or the equivalent. Not open to students who have taken MATH-UA 233 Theory of Probability and/or MATH-UA 234 Mathematical Statistics.

Description:

A combination of MATH-UA 233 Theory of Probability and MATH-UA 234 Mathematical Statistics at a more elementary level, so as to afford the student some acquaintance with both probability and statistics in a single term. In probability: mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; law of large numbers and the normal approximation; application to coin-tossing, radioactive decay, etc. In statistics: sampling; normal and other useful distributions; testing of hypotheses; confidence intervals; correlation and regression; applications to scientific, industrial, and financial data.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) with a grade of B+ or better and/or the equivalent, and MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra with a grade of B+ or better and/or the equivalent, and MATH-UA 120 Discrete Math with a grade of B+ or better and/or the equivalent.. Not open to students who have taken MATH-UA 235 Probability and Statistics. While B+ or higher is the standard requirement for this course, the department will consider petitions if you are on the borderline of that requirement.

Description:

The aim of this class is to introduce students to probability theory, with a greater emphasis on rigor, more material, and a faster pace than the Theory of Probability class. The material will include discrete and continuous probability, and the most fundamental limit theorems (law of large numbers and Central Limit Theorem). Students will be made familiar with the classical models, computations on densities, and convergence to universal distributions. They will also be expected to understand the proofs of all the results seen in class, and be able to argue with mathematical rigor.

Prerequisites:

MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) or MATH-UA 221 Honors Calculus I with a grade of C or better and/or the equivalent.

Description:

Techniques for counting and enumeration including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph-theoretic problems.

Prerequisites:

MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) with a grade of C or better and/or the equivalent.

Description:

Divisibility theory and prime numbers. Linear and quadratic congruences. The classical number-theoretic functions. Continued fractions. Diophantine equations.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors), and an introductory course in probability or statistics, MATH-UA 233 Theory of Probability, MATH-UA 235 Probability and Statistics, ECON-UA 18 Statistics, ECON-UA 20 Analytical Statistics, STAT-UB 14 Intro Theory of Probability, STAT-UB 103 Statistics for Business Control and Regression/Forecasting Models or equivalent) with a grade of C+ or better.

Description:

Introduction to the mathematics of finance. Topics include: Linear programming with application pricing and quadratic. Interest rates and present value. Basic probability: random walks, central limit theorem, Brownian motion, lognormal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) with a grade of C or better or permission of the instructor.

Description:

Formulation and analysis of mathematical models. Mathematical tool include dimensional analysis, optimization, simulation, probability, and elementary differential equations. Applications to biology, sports, economics, and other areas of science. The necessary mathematical and scientific background will be developed as needed. Students will participate in formulating models as well as in analyzing them.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 213 Mathematics for Economics III (for economics majors) with a grade of C or higher and MATH-UA 140 Linear Algebra with a grade of B- or higher or MATH-UA 148 Honors Linear Algebra with a grade of C or higher, or equivalents. 

Description:

In numerical analysis one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course gives an introduction to this subject for mathematics majors. Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method.

Prerequisites:

MATH-UA 123 Calculus III OR MATH-UA 129 Honors Calculus III OR MATH-UA 213 Math for Economics III (for Economics majors) with a grade of C or better and/or the equivalent, AND MATH-UA 140 Linear Algebra OR MATH-UA 148 Honors Linear Algebra with a grade of C or better and/or the equivalent. 

Description:

Prerequisites:

MATH-UA 121 Calculus I or MATH-UA 212 Math for Economics II (for Economics majors) and BIOL-UA 11 Principles of Biology I or permission of the instructor.

Description:

Intended primarily for premedical students with interest and ability in mathematics. Topics of medical importance using mathematics as a tool: control of the heart, optimal principles in the lung, cell membranes, electrophysiology, countercurrent exchange in the kidney, acid-base balance, muscle, cardiac catheterization, computer diagnosis. Material from the physical sciences and mathematics is introduced as needed and developed within the course.

Prerequisites:

MATH-UA 255 Mathematics in Medicine and Biology, or permission of the instructor. Familiarity with a programming language is recommended. The language used in the course will be MATLAB, but prior experience with MATLAB is not required.

Description:

Introduces students to the use of computer simulation as a tool for investigating biological phenomena. The course requirement is to construct three computer models during the semester, to report on results to the class, and to hand in a writeup describing each project. These projects can be done individually, or as part of a team. Topics discussed in class are the circulation of the blood, gas exchange in the lung, electrophysiology of neurons and neural networks, the renal countercurrent mechanism, cross-bridge dynamics in muscle, and the dynamics of epidemic and endemic diseases. Projects are normally chosen from this list, but may be chosen otherwise by students with other interests.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

Description:

A first course in ordinary differential equations, including analytical solution methods, elementary numerical methods, and modeling. Topics to be covered include: first-order equations including integrating factors; second-order equations including variation of parameters; series solutions; elementary numerical methods including Euler's methods, Runge-Kutta methods, and error analysis; Laplace transforms; systems of linear equations; boundary-value problems. Some optional topics to be chosen at the instructor's discretion include: nonlinear dynamics including phase-plane description; elementary partial differential equations and Fourier series.

Prerequisites:

MATH-UA 262 Ordinary Differential Equations OR MATH-UA 268 Honors Ordinary Differential Equations with a grade of C or better or the equivalent.

Description:

Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Non-linear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.

Prerequisites:

MATH-UA 122 Calculus II or MATH-UA 212 Math for Economics II (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

Description:

Topics will include dynamics of maps and of first order and second-order differential equations: stability, bifurcations, limit cycles, dissection of systems with fast and slow time scales. Geometric viewpoint, including phase planes, will be stressed. Chaotic behavior will be introduced in the context of one-variable maps (the logistic), fractal sets, etc. Applications will be drawn from physics and biology. There will be homework and projects, and a few computer lab sessions (programming experience is not a prerequisite).

Prerequisites:

MATH-UA 328 Honors Analysis I with a grade of B+ or higher OR MATH-UA 325 Analysis with a grade of A- or higher.

Description:

This class will develop rigorously the basic theory of Ordinary Differential Equations (ODEs). Existence and uniqueness of solutions to ODEs are first investigated, for linear and nonlinear problems, set on the real line or the complex plane. More qualitative questions are then considered, about the behavior of the solutions, with possible prolongations to various topics in Dynamical Systems theory. Applications to Physics and Biology will appear naturally when discussing examples.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) plus one higher level course such as MATH-UA 140 Linear Algebra with the grade of C or better.

Description:

Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy's theorem and the Cauchy integral formula. Singularities, residues, and Laurent series. Fractional Linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow etc.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

Description:

This course is an introduction to rigorous analysis on the real line. Topics include: the real number system, sequences and series of numbers, functions of a real variable (continuity and differentiability), the Riemann integral, basic topological notions in a metric space, sequences and series of functions including Taylor and Fourier series.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of A- or better or the equivalent. Recommended: MATH-UA 129 Honors Calculus III and MATH-UA 148 Honors Linear Algebra with a grade of B+ or better or the equivalent.

Description:

This is an introduction to the rigorous treatment of the foundations of real analysis in one variable. It is based entirely on proofs. Students are expected to know what a mathematical proof is and are also expected to be able to read a proof before taking this class. Topics include: properties of the real number system, sequences, continuous functions, topology of the real line, compactness, derivatives, the Riemann integral, sequences of functions, uniform convergence, infinite series and Fourier series. Additional topics may include: Lebesgue measure and integral on the real line, metric spaces, and analysis on metric spaces.

Prerequisites:

A grade of C or better in MATH-UA 328 Honors Analysis I or grade of A in MATH-UA 325 Analysis in conjunction with permission by instructor, and MATH-UA 140 Linear Algebra with a grade of C or better or the equivalent.

Description:

This is a continuation of MATH-UA 328 Honors Analysis I. Topics include: metric spaces, differentiation of functions of several real variables, the implicit and inverse function theorems, Riemann integral on Rⁿ, Lebesgue measure on Rⁿ, the Lebesgue integral.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors), and MATH-UA 140 Linear Algebra with a grade of C or better and/or the equivalent. Additionally, it is suggested for students to have taken MATH-UA 325 Analysis as a prerequisite.

Description:

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, homomorphisms and quotient groups. Rings and quotient rings, Euclidean rings, polynomial rings. Fields, finite extensions.

Prerequisites:

MATH-UA 123 Calculus III or MATH-UA 213 Math for Economics III (for Economics majors) and MATH-UA 140 Linear Algebra with a grade of A- or better or the equivalent. Recommended: MATH-UA 129 Honors Calculus III and MATH-UA 148 Honors Linear Algebra with a grade of B+ or better or the equivalent.

Description:

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, group actions, homomorphisms and quotient groups, direct products, classification of finitely generated abelian groups, Sylow theorems. Rings, ideals and quotient rings, Euclidean rings, polynomial rings, unique factorization.

Prerequisites:

A grade of C or better in MATH-UA 348 Honors Algebra I, or grade of A in MATH-UA 343 Algebra in conjunction with permission by instructor.

Description:

Principle ideal domains, polynomial rings in several variables, unique factorization domains. Fields, finite extensions, constructions with ruler and compass, Galois theory, solvability by radicals.

Prerequisites:

MATH-UA 325 Analysis OR MATH-UA 328 Honors Analysis I with a grade of C or higher or permission of the department.

Description:

Set-theoretic preliminaries. Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy groups.

Prerequisites:

MATH-UA 123 Calculus III, MATH-UA 129 Honors Calculus III OR MATH-UA 133 Math for Economics III with a grade of C or higher AND MATH-UA 140 Linear algebra OR MATH-UA 148 Honors Linear Algebra with a grade of C or higher. Recommended: MATH-UA 325 Analysis or MATH-UA 328 Honors Analysis I.

Description:

The geometry of curves and surfaces in Euclidean space. Frenet formulas, the isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the Gauss-Bonnet Theorem.

Prerequisites:

Honors standing or approval of the director of the honors program.

Prerequisite varies according to topic.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Honors standing or approval of the director of the honors program.

Prerequisite varies according to topic.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Prerequisite varies according to topic.

Description:

Please see Albert for course topic and description.

Prerequisites:

Prerequisite varies according to topic.

Description:

Please see Albert for course topic and description.

Prerequisites:

Honors standing or approval of the director of the honors program.

Prerequisite varies according to topic.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Honors standing or approval of the director of the honors program.

Prerequisite varies according to topic.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Permission of the department. Student must be a declared Math major, have a math GPA of 3.5 and an overall GPA of 3.0, and have at least 50% of the Math major courses completed.

Description:

To register for this course a student must complete the Enrollment Request Form and have the approval of the Director of Undergraduate Studies.

Prerequisites:

Permission of the department. Student must be a declared Math major, have a math GPA of 3.5 and an overall GPA of 3.0, and have at least 50% of the Math major courses completed.

Description:

To register for this course a student must complete the Enrollment Request Form and have the approval of the Director of Undergraduate Studies.

Prerequisites:

Permission of the department.

Description:

To register for this course a student must complete an application form for Independent Study and have the approval of a faculty sponsor and the Director of Undergraduate Studies.

Prerequisites:

Permission of the department.

Description:

To register for this course a student must complete an application form for Independent Study and have the approval of a faculty sponsor and the Director of Undergraduate Studies.

Prerequisites:

Diagnostic exam.

Corequisites:

EX-UY 1

Notes:

Credit for this course may not be used to satisfy the minimum credit requirement for graduation.

Description:

This course covers: foundations of algebra, exponents, multiplication of algebraic expressions, factoring algebraic expressions, working with algebraic fractions, proportionality, rates of change, equations of lines, completing squares, the quadratic formula, solving equations, systems of linear equations, inequalities, domain and range of functions, exponential and logarithmic functions, compositions of functions, transformations of functions, right triangles, trigonometry of triangles.

Prerequisites:

Diagnostic Exam or a grade of B or better in MA-UY 914.

Corequisites:

EX-UY 1

Description:

This course covers: Library of Functions, functions of one variable. Limits, derivatives of functions defined by graphs, tables and formulas, differentiation rules for power, polynomial, exponential and logarithmic functions, derivatives of trigonometric functions, the product and quotient rules, the chain rule, applications of the chain rule, maxima and minima, optimization. The definite integral, the Fundamental Theorem of Calculus and interpretations, theorems about definite integrals, anti-derivatives. MA-UY 1324 is for students who wish to take MA-UY 1024 but need more review of precalculus. MA-UY 1324 covers the same material as MA-UY 1024 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisites:

MA-UY 1024 or a grade of B or better in MA-UY 1324. 

Corequisites:

EX-UY 1

Description:

This course covers techniques of integration, introduction to ordinary differential equations, improper integrals, numerical methods of integration, applications of integration, sequences, series, power series, approximations of functions via Taylor polynomials, Taylor series, functions of two variables, graphs of functions of two variables, contour diagrams, linear functions, functions of three variables. MA-UY 1424 is for students who wish to take MA-UY 1124 but need more review of precalculus. MA-UY 1424 covers the same material as MA-UY 1124 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisites:

Diagnostic Exam or MA-UY 914.

Corequisites:

EX-UY 1

Description:

This course covers: Library of Functions, functions of one variable. Limits, derivatives of functions defined by graphs, tables and formulas, differentiation rules for power, polynomial, exponential and logarithmic functions, derivatives of trigonometric functions, the product and quotient rules, the chain rule, applications of the chain rule, maxima and minima, optimization. The definite integral, the Fundamental Theorem of Calculus and interpretations, theorems about definite integrals, anti-derivatives. MA-UY 1324 is for students who wish to take MA-UY 1024 but need more review of precalculus. MA-UY 1324 covers the same material as MA-UY 1024 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisites:

MA-UY 1324.

Corequisites:

EX-UY 1

Description:

This course covers techniques of integration, introduction to ordinary differential equations, improper integrals, numerical methods of integration, applications of integration, sequences, series, power series, approximations of functions via Taylor polynomials, Taylor series, functions of two variables, graphs of functions of two variables, contour diagrams, linear functions, functions of three variables. MA-UY 1424 is for students who wish to take MA-UY 1124 but need more review of precalculus. MA-UY 1424 covers the same material as MA-UY 1124 but with more contact hours per week, incorporating a full discussion of the required precalculus topics.

Prerequisites:

MA-UY 1124 or MA-UY 1424.

Notes:

Not open to students who have taken MA-UY 3044 or MA-UY 3054 or MA-UY 4204 or MA-UY 4254.

Description:

MA-UY 2034 is an introduction to ordinary differential equations and linear algebra. The course develops the techniques for the analytic and numeric solutions of ordinary differential equations (and systems) that are widely used in modern engineering and science. Linear algebra is used as a tool for solving systems of linear equations as well as for understanding the structure of solutions to linear (systems) of differential equations. Topics covered include the fundamental concepts of linear algebra such as Gaussian elimination, matrix theory, linear transformations, vector spaces, subspaces, basis, eigenvectors, eigenvalues and the diagonalization of matrices, as well as the techniques for the analytic and numeric solutions of ordinary differential equations (and systems) that commonly appear in modern engineering and science.

Prerequisites:

MA-UY 1124 or MA-UY 1424.

Description:

Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorems of Gauss and Stokes.

Prerequisites:

MA-UY 1124 or MA-UY 1424.

Notes:

Not open to math majors or students who have taken or will take MA-UY 2054 or MA-UY 2414 or MA-UY 3014 or MA-UY 3514.

Description:

An introductory course to probability and statistics. It affords the student some acquaintance with both probability and statistics in a single term. Topics in Probability include mathematical treatment of chance; combinatorics; binomial, Poisson, and Gaussian distributions; the Central Limit Theorem and the normal approximation. Topics in Statistics include sampling distributions of sample mean and sample variance; normal, t-, and Chi-square distributions; confidence intervals; testing of hypotheses; least squares regression model. Applications to scientific, industrial, and financial data are integrated into the course.

Prerequisites:

Math Diagnostic Exam or MA-UY 914 (minimum calculus level required). Prerequisite for Shanghai students: MATH-SHU 110.

Notes:

This course and CS-GY 6003 cannot both be taken for credit.

Description:

Logic, proofs, set theory, functions, relations, asymptotic notation, recurrences, modeling computation, graph theory.

Prerequisites:

None.

Notes:

This course is open to the following majors only: BIMS, IDM, STS, SUE. Not open to math majors or students who have taken or will take MA-UY 2054 or MA-UY 2224 or MA-UY 3014 or MA-UY 3514 or ECE-UY 2233 or equivalent. 

Description:

We are inundated by data, but data alone do not translate into useful information. Statistics provides the means for organizing, summarizing, and therefore better analyzing data so that we can understand what the data tell us about critical questions.  If one collects data then understanding how to use statistical methods is critical, but it is also necessary to understand and interpret all the information we consume on a daily basis.  This course provides these basic statistical approaches and techniques.  This course may not be acceptable as a substitute for any other Probability and Statistics course.

Prerequisites:

(MA-UY 1124 or MA-UY 1424) with a grade of A- or better OR a 5 on the AP Calculus BC Exam and Department Permission.

Description:

Similar to MA-UY 2114 Calculus III, but at a faster pace and deeper level. Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes. Students pursuing an honors mathematics degree are especially encouraged to consider this course.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Notes:

Not open to students who have taken or will take MA-UY 2224, ECE-UY 2233, or MA-UY 3514.

Description:

An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, the Central Limit Theorem and Laws of Large Numbers, Markov Chains, and basic stochastic processes.

Prerequisites:

A grade of C or better in MA-UY 1024 or MA-UY 1324.

Notes:

Not open to students who have taken or will take MA-UY 2034, MA-UY 3054, or MA-UY 3113.

Description:

Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer's rule.  Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms. Restricted to Tandon math and CS majors and students with a permission code from the math department. Fulfills linear algebra requirement for the BS Math and BS CS degrees.

Prerequisites:

A grade of A- or better in MA-UY 1024 or MA-UY 1324.

Notes:

Not open to students who have taken or will take MA-UY 2034, MA-UY 3044, or MA-UY 3113.

Description:

This honors section of Linear Algebra is intended for well-prepared students who have already developed some mathematical maturity. Its scope will include the usual Linear Algebra (MA-UY 3044) syllabus; however, this class will move faster, covering additional topics and going deeper. Vector spaces, linear dependence, basis and dimension, matrices, determinants, solving linear equations, eigenvalues and eigenvectors, quadratic forms, applications such as optimization or linear regression.

Prerequisites:

(MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Notes:

Not open to math majors or students who have taken or will take MA-UY 4434.

Description:

This course provides a deeper understanding of topics introduced in MA-UY 2012 and MA-UY 2034 and continues the development of those topics, while also covering functions of a Complex Variable. Topics covered include: The Gram-Schmidt process, inner product spaces and applications, singular value decomposition, LU decomposition. Derivatives and Cauchy-Riemann equations, integrals and Cauchy integral theorem. Power and Laurent Series, residue theory.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Description:

Prerequisites:

A grade of B+ or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054) and MA-UY 2314

Notes:

Not open to students who have taken or will take MA-UY 2224, ECE-UY 2233, or MA-UY 3014.

Description:

The aim of this class is to introduce students to probability theory, with a greater emphasis on rigor, more material, and a faster pace than the Theory of Probability/Applied Probability class. The material will include discrete and continuous probability, and the most fundamental limit theorems (law of large numbers and Central Limit Theorem). Students will be made familiar with the classical models, computations on densities, and convergence to universal distributions. They will also be expected to understand the proofs of all the results seen in class, and be able to argue with mathematical rigor.

Prerequisites:

PH-UY 2023 and MA-UY 2114 

Corequisites: 
PH-UY 2033 and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054) 

Description:

First course of two-semester lecture sequence in mathematical physics for undergraduate students in physics and engineering. Line, surface and volume integrals, gradient, divergence, and curl. Cylindrical and spherical coordinate systems. Tensors and tensor transformations. The Dirac delta function, and integrals and derivatives of the delta function. Functions of complex variables, analytic functions, and the residue theorem. Fourier series, integrals, and transforms. 

Prerequisites:

A grade of C or better in MA-UY 1124 or MA-UY 1424.

Description:

Divisibility and prime numbers. Linear and quadratic congruences. The classical number-theoretic functions. Continued fractions. Diophantine equations.

Prerequisites:

A grade of C or better in (MA-UY 4614 or MA-UY 4644) and (MA-UY 3044 or MA-UY 3054 or MA-UY 3113), or permission of instructor.

Notes:

Cannot receive credit for both MA-UY 4044 and MA-UY 4054.

Description:

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, homomorphisms and quotient groups. Rings and quotient rings, Euclidean rings, polynomial rings. Fields, finite extensions.

Prerequisites:

A grade of B or better in (MA-UY 4614 or MA-UY 4644) and (MA-UY 3044 or MA-UY 3054 or MA-UY 3113) or instructor permission.

Notes:

Cannot receive credit for both MA-UY 4044 and MA-UY 4054.

Description:

Introduction to abstract algebraic structures, including groups, rings, and fields. Sets and relations. Congruences and unique factorization of integers. Groups, permutation groups, group actions, homomorphisms and quotient groups, direct products, classification of finitely generated abelian groups, Sylow theorems. Rings, ideals and quotient rings, Euclidean rings, polynomial rings, unique factorization.

Prerequisites:

A grade of C or better in MA-UY 4054 or (a grade of A in MA-UY 4044 and instructor permission).

Description:

Fields, finite extensions, constructions with ruler and compass, Galois theory, solvability by radicals.

Prerequisites:

MA-UY 3014 or MA-UY 3514.

Notes:

Not open to students who have taken or will take MA-UY 2224.

Description:

An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences. Use of Matlab for doing computations of the statistical measures listed above.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 3044 or MA-UY 3054 or MA-UY 3113).

Notes:

Not open to students who have taken or will take MA-UY 2034 or MA-UY 4254.

Description:

A first course in ordinary differential equations, including analytical solution methods, elementary numerical methods, and modeling. Topics to be covered include: first-order equations including integrating factors; second-order equations including variation of parameters; series solutions; elementary numerical methods including Euler's methods, Runge-Kutta methods, and error analysis; Laplace transforms; systems of linear equations; boundary-value problems. Restricted to Tandon math majors and students with a permission code from the math department. Fulfills ordinary differential equations requirement for the BS Math degree.

Prerequisites:

A grade of A- or better in MA-UY 4614 Applied Analysis OR a grade of B+ or better in MA-UY 4644 Honors Analysis I.

Notes: Not open to students who have taken or will take MA-UY 2034 or MA-UY 4204.

Description:

This class will develop rigorously the basic theory of Ordinary Differential Equations (ODEs).
Existence and uniqueness of solutions to ODEs are first investigated, for linear and nonlinear problems, set on the real line or the complex plane. More qualitative questions are then considered, about the behavior of the solutions, with possible prolongations to various topics in Dynamical Systems theory. Applications to Physics and Biology will appear naturally when discussing examples.

Prerequisites:

A grade of C or better in MA-UY 1124 or MA-UY 1424.

Description:

Techniques for counting and enumeration including generating functions, the principle of inclusion and exclusion, and Polya counting. Graph theory. Modern algorithms and data structures for graph-theoretic problems.

Prerequisites:

A grade of C+ or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2054 or MA-UY 2224 or MA-UY 2414 or MA-UY 3014 or MA-UY 3514 or MA-UY 4114).

Description:

Introduction to the mathematics of finance. Topics include: Linear programming with application pricing and quadratic. Interest rates and present value. Basic probability: random walks, central limit theorem, Brownian motion, lognormal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization.

Prerequisites:

MA-UY 2034 or MA-UY 4204 or MA-UY 4254

Description:

Modeling of physical processes. Classification of equations. Formulation and treatment of boundary- and initial-value problems. Green’s functions. Maximum principle. Separation of variables. Fourier series and integrals. Quasilinear first-order equations and characteristics. D’Alembert solution of wave equation. Conservation laws and shock waves.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 3044 or MA-UY 3054 or MA-UY 3113).

Description:

In numerical analysis one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course gives an introduction to this subject for mathematics majors. Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Notes:

Not open to students who have taken MA-UY 3113.

Description:

A first course in complex analysis, with a focus on applications. Topics to be covered include the complex plane, analytic functions, complex differentiation, the Cauchy-Riemann equations, branch cuts, contour integration, the residue theorem, conformal mapping, applications to potential theory and fluid flow.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Description:

Formulation and analysis of mathematical models. Mathematical tools include dimensional analysis, optimization, simulation, probability, and elementary differential equations. Applications to biology, sports, economics, and other areas of science. The necessary mathematical and scientific background will be developed as needed. Students participate in formulating models as well as in analyzing them.

Prerequisites:

A grade of C or better in (MA-UY 1124 or MA-UY 1424) and (MA-UY 3044 or MA-UY 3054 or MA-UY 3113).

Description:

Topics will include dynamics of maps and of first order and second-order differential equations, stability, bifurcations, limit cycles, dissection of systems with fast and slow time scales.  Geometric viewpoint, including phase planes, will be stressed. Chaotic behavior will be introduced in the context of one-variable maps (the logistic), fractal sets, etc. Applications will be drawn from physics and biology. There will be homework and projects, and a few computer lab sessions (programming experience is not a prerequisite).

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Notes:

Cannot receive credit for both MA-UY 4614 and MA-UY 4644.

Description:

Limits of real and complex sequences and series; topology of metric spaces; continuity and differentiability of functions; definition, properties, and approximations of Riemann integrals; convergence of sequences and series of functions; Fourier series and other orthogonal systems of functions, approximations theorems.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Notes:

Cannot receive credit for both MA-UY 4614 and MA-UY 4644.

Description:

This is an introduction to the rigorous treatment of the foundations of real analysis in one variable. It is based entirely on proofs. Students are expected to know what a mathematical proof is and are also expected to be able to read a proof before taking this class. Topics include: properties of the real number system, sequences, continuous functions, topology of the real line, compactness, derivatives, the Riemann integral, sequences of functions, uniform convergence, infinite series and Fourier series. Additional topics may include: Lebesgue measure and integral on the real line, metric spaces, and analysis on metric spaces.

Prerequisites:

A grade of C or better in MA-UY 4644 or a grade of A in MA-UY 4614 in conjunction with permission by instructor.

Description:

This is a continuation of MA-UY 4644 Honors Analysis I. Topics include: metric spaces, differentiation of functions of several real variables, the implicit and inverse function theorems, Riemann integral on Rn, Lebesgue measure on Rn, the Lebesgue integral.

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 3044 or MA-UY 3054).

Description:

The geometry of curves and surfaces in Euclidean space. Frenet formulas, the isoperimetric inequality, local theory of surfaces in Euclidean space, first and second fundamental forms. Gaussian and mean curvature, isometries, geodesics, parallelism, the Gauss-Bonnet Theorem.

Prerequisites:

A grade of C or better in (MA-UY 4614 or MA-UY 4644) 

Description:

Set-theoretic preliminaries. Metric spaces, topological spaces, compactness, connectedness, covering spaces, and homotopy groups.

Prerequisites:

Prerequisite varies according to topic. Department Consent Required for Enrollment.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Prerequisite varies according to topic. Department Consent Required for Enrollment.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Prerequisite varies according to topic. Department Consent Required for Enrollment.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Prerequisite varies according to topic. Department Consent Required for Enrollment.

Description:

A lecture/seminar course on advanced topics. Topics vary yearly and are updated from time to time. Detailed course descriptions are available during preregistration.

Prerequisites:

Departmental adviser’s approval.

Notes:

This course is repeatable for credit.

Description:

In this course, students read, study and investigate selected topics in mathematics. Students discuss and present problems.