Undergraduate Course Descriptions

Prerequisites:

High school math 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.

This course was formerly titled "Precalculus" and then "Algebra and Calculus". The course title was last updated to "Algebra, Trigonometry and Functions" for the start of the Fall 2022 semester.

Prerequisites:

At least one (1) of the following prerequisites:

1. Completion of MATH-UA 009 Algebra, Trigonometry and Functions with a grade of C or higher
2. SAT score of 670 or higher on mathematics portion
3. ACT/ACTE Math score of 30 or higher
4. Valid AP Score:
   • AP Precalculus score of 4 or higher
   • AP Calculus AB score of 3 or higher
   • AP Calculus BC score of 3 or higher
5. A Level Maths score of C or higher 
   • Students who took A Level Further Maths should contact the Math Department
6. AS Level Maths score of B or higher
7. IB Mathematics exam result from 2021 - 2027 
   • IB Analysis and Approaches HL score of 5 or higher
   • IB Applications and Interpretations HL score of 5 or higher
   • IB Analysis and Approaches SL score of 7
8. IB Mathematics exam result from 2014 - 2020
   • IB Mathematics HL score of 5 or higher
   • IB Mathematics SL score of 6 or higher
   • IB Mathematical Studies SL score of 7
9. Passing Calculus/MFE I placement exam

Description:

This course is a one-semester introduction to discrete mathematics with an emphasis on the understanding, composition and critiquing of mathematical proofs. At the semester's conclusion, the successful student will be able to:

• write clear mathematical statements using standard notation and terminology.
• understand and execute a variety of proof techniques (contradiction, induction, etc.).
• show fluency in the language of basic set theory and Boolean logic.
• understand the basic theorems and their implications in a variety of (discrete) fields including:
  • function theory
  • group theory
  • number theory
  • graph theory

Prerequisites:

At least one (1) of the following prerequisites:

1. Completion of MATH-UA 009 Algebra, Trigonometry and Functions with a grade of C or higher
2. SAT score of 670 or higher on mathematics portion
3. ACT/ACTE Math score of 30 or higher
4. Valid AP Score:
   • AP Precalculus score of 4 or higher
   • AP Calculus AB score of 3 or higher
   • AP Calculus BC score of 3 or higher
5. A Level Maths score of C or higher 
   • Students who took A Level Further Maths should contact the Math Department
6. AS Level Maths score of B or higher
7. IB Mathematics exam result from 2021 - 2027 
   • IB Analysis and Approaches HL score of 5 or higher
   • IB Applications and Interpretations HL score of 5 or higher
   • IB Analysis and Approaches SL score of 7
8. IB Mathematics exam result from 2014 - 2020
   • IB Mathematics HL score of 5 or higher
   • IB Mathematics SL score of 6 or higher
   • IB Mathematical Studies SL score of 7
9. Passing Calculus/MFE I placement exam

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.

Note: Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements.

Prerequisites:

At least one (1) of the following prerequisites:

• Passing MATH-UA 121 Calculus I with a grade of C or better
• Valid AP score:
  • AP Calculus AB score of 4 or higher
  • AP Calculus BC score of 4 or higher
• A Level Maths score of B or higher
  • Students who took A Level Further Maths should contact the Math Department
• IB Mathematics exam result from 2021 - 2027 
  • IB Analysis and Approaches HL score of 6 or higher
  • IB Applications and Interpretations HL score of 6 or higher
• IB Mathematics exam result from 2014 - 2020
  • IB Mathematics HL score of 6 or higher
• Passing Calculus II placement exam

Description:

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

Note: Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements. The Math Department encourages students and advisors to contact us first if there are any questions regarding which sequence a student should register for.

Prerequisites:

At least one (1) of the following prerequisites:

• Passing MATH-UA 122 Calculus II with a grade of C or higher 
• AP Calculus BC score of 5
• IB Mathematics exam result from 2021 - 2027
  • Analysis and Approaches HL score of 7
• IB Mathematics exam result from 2014 - 2020
  • Further Mathematics HL score of 6 or higher
• Passing Calculus III placement exam

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.

Students cannot take both MATH-UA 123 and MATH-UA 129.

Note:  Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements. The Math Department encourages students and advisors to contact us first if there are any questions regarding which sequence a student should register for.

Prerequisites:

At least one (1) of the following prerequisites:

• Passing MATH-UA 122 Calculus II with a grade of A- or higher 
  • A grade of "Pass/Fail" or an "Incomplete" do NOT satisfy the prerequisite
• AP Calculus BC score of 5
• IB Mathematics exam result from 2021 - 2027
  • Analysis and Approaches HL score of 7
• IB Mathematics exam result from 2014 - 2020
  • Further Mathematics HL score of 6 or higher
• Passing Calculus III placement exam

Note: Students who took an A Level Further Maths exam should contact the Math Department with a copy of their official exam score report. The student's name and exam scores must be clearly visible.  

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.

Students cannot take both MATH-UA 123 and MATH-UA 129. 

Note: Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements. The Math Department encourages students and advisors to contact us first if there are any questions regarding which sequence a student should register for.

Prerequisites:

At least one (1) of the following prerequisites:

1. Completion of MATH-UA 009 Algebra, Trigonometry and Functions with a grade of C or higher
2. SAT score of 670 or higher on mathematics portion 
3. ACT/ACTE Math score of 30 or higher 
4. Valid AP Score:
   • AP Precalculus score of 4 or higher
   • AP Calculus AB score of 3 or higher
   • AP Calculus BC score of 3 or higher
5. A Level Maths score of C or higher 
   • Students who took A Level Further Maths should contact the Math Department
6. AS Level Maths score of B or higher
7. IB Mathematics exam result from 2021 - 2027 
   • IB Analysis and Approaches HL score of 5 or higher
   • IB Applications and Interpretations HL score of 5 or higher
   • IB Analysis and Approaches SL score of 7
8. IB Mathematics exam result from 2014 - 2020
   • IB Mathematics HL score of 5 or higher
   • IB Mathematics SL score of 6 or higher
   • IB Mathematical Studies SL score of 7
9. Passing Calculus/MFE I placement exam

Description:

This course is only open to Economics Majors and prospective Economics Majors. If an Economics Major decides to double or joint major in Math, the Math for Economics sequence of courses will replace the Calculus requirement. 

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; series, including geometric and Taylor series; ordinary differential equations; matrix algebra; eigenvalues; and (possibly) dynamic optimization and multivariable integration.

Note: Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements. The Math Department encourages students and advisors to contact us first if there are any questions regarding which sequence a student should register for.

This course was formerly numbered MATH-UA 211. The course number was updated to MATH-UA 131 for the start of the Fall 2022 semester. The course content remains the same, only the course number has changed. 

Prerequisites:

At least one (1) of the following prerequisites:

• Completion of MATH-UA 131 Math for Economics I with a grade of C or higher
• Passing Math for Economics II placement exam

Description:

This course is only open to Economics Majors and prospective Economics Majors. If an Economics Major decides to double or joint major in Math, the Math for Economics sequence of courses will replace the Calculus requirement. 

By the end of Math for Economics II, students should have a complete understanding of optimization and should be able to apply the Lagrange multipliers approach to constrained optimization problems. They will also learn to solve systems of equations using linear algebra. Time permitting, they will also be exposed to the principal methods of dynamic analysis of economic processes, and introductory concepts and results of integration and differential equations. A student should be able to find solutions of elementary differential equations and analyze their stability.

Note: Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements. The Math Department encourages students and advisors to contact us first if there are any questions regarding which sequence a student should register for.

This course was formerly numbered MATH-UA 212. The course number was updated to MATH-UA 132 for the start of the Fall 2022 semester. The course content remains the same, only the course number has changed. 

Prerequisites:

At least one (1) of the following prerequisites:

• Completion of MATH-UA 132 Math for Economics II with a grade of C or higher
  • A grade of "Pass/Fail" or an "Incomplete" do NOT satisfy the prerequisite
• Passing Math for Economics III placement exam

Description:

This course is only open to Economics Majors and prospective Economics Majors. If an Economics Major decides to double or joint major in Math, the Math for Economics sequence of courses will replace the Calculus requirement. 

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-livel mathematics courses.

Note: Students cannot mix-and-match, combine, or double-count between the Calculus and Math for Economics sequences. Switching back and forth between the Calculus and Math for Economics sequences is not permitted. Students may count only the Calculus sequence or only the Math for Economics sequence towards their degree requirements. The Math Department encourages students and advisors to contact us first if there are any questions regarding which sequence a student should register for.

This course was formerly numbered MATH-UA 213. The course number was updated to MATH-UA 133 for the start of the Fall 2022 semester. The course content remains the same, only the course number has changed. 

Prerequisites:

At least one (1) of the following prerequisites:

1. Completion of MATH-UA 009 Algebra, Trigonometry and Functions with a grade of C or higher
2. SAT score of 670 or higher on mathematics portion
3. ACT/ACTE Math score of 30 or higher
4. Valid AP Score:
   • AP Precalculus score of 4 or higher
   • AP Calculus AB score of 3 or higher
   • AP Calculus BC score of 3 or higher
5. A Level Maths score of C or higher 
   • Students who took A Level Further Maths should contact the Math Department
6. AS Level Maths score of B or higher
7. IB Mathematics exam result from 2021 - 2027 
   • IB Analysis and Approaches HL score of 5 or higher
   • IB Applications and Interpretations HL score of 5 or higher
   • IB Analysis and Approaches SL score of 7
8. IB Mathematics exam result from 2014 - 2020
   • IB Mathematics HL score of 5 or higher
   • IB Mathematics SL score of 6 or higher
   • IB Mathematical Studies SL score of 7
9. Passing Calculus/MFE I placement exam

Description:

Linear algebra is an area of mathematics devoted to the study of structure-preserving operators on special sets (linear operators on vector spaces). Linear algebra is a cornerstone of any mathematics curriculum for two very important (and related) reasons:

1. The theory of linear algebra is well understood and so a first step in many areas of applied mathematics is to reduce the problem into one in linear algebra.
2. The spaces and operations studied in the subject are commonplace in many different areas of mathematics, science, and engineering.

Over the semester we will study many topics that form a central part of the language of modern science. The successful student will be able to:

• Formulate, solve, apply, and interpret systems of linear equations in several variables;
• Compute with and classify matrices;
• Master the fundamental concepts of abstract vector spaces;
• Decompose linear transformations and analyze their spectra (eigenvectors and eigenvalues);
• Utilize length and orthogonality in each of the above contexts;
• Apply orthogonal projection to optimization (least-squares) problems;
• Explore other topics (as time permits).

The material we take up in this course has applications in physics, chemistry, biology, environmental science, astronomy, economics, statistics, and just about everything else. We want you to leave the course not only with computational ability, but with the ability to use these notions in their natural scientific contexts, and with an appreciation of their mathematical beauty and power.

Students cannot take both MATH-UA 140 and MATH-UA 148.

Prerequisites:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 121 Calculus I or MATH-UA 132 Math for Economics II
2. 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:

At least one (1) of the following prerequisites:

1. Completion of MATH-UA 009 Algebra, Trigonometry and Functions with a grade of C or higher
2. SAT score of 670 or higher on mathematics portion
3. ACT/ACTE Math score of 30 or higher
4. Valid AP Score:
   • AP Precalculus score of 4 or higher
   • AP Calculus AB score of 3 or higher
   • AP Calculus BC score of 3 or higher
5. A Level Maths score of C or higher 
   • Students who took A Level Further Maths should contact the Math Department
6. AS Level Maths score of B or higher
7. IB Mathematics exam result from 2021 - 2027 
   • IB Analysis and Approaches HL score of 5 or higher
   • IB Applications and Interpretations HL score of 5 or higher
   • IB Analysis and Approaches SL score of 7
8. IB Mathematics exam result from 2014 - 2020
   • IB Mathematics HL score of 5 or higher
   • IB Mathematics SL score of 6 or higher
   • IB Mathematical Studies SL score of 7
9. Passing Calculus/MFE I placement exam

Description:

Prerequisites:

A grade of B+ or higher in MATH-UA 122 Calculus II or MATH-UA 132 Math for Economics II.

Notes: Completion of MATH-UA 123 Calculus III or MATH-UA 133 Math for Economics III  is preferred. 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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 133 Math for Economics III
2. 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:

Students must have a declared major at the Math Department to be able to register for MATH-UA 232.

Non-math majors will not be able to register for MATH-UA 232 and should register for PHIL-UA 73 instead.

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

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 higher in MATH-UA 123 or MATH-UA 133 is strongly recommended.

Anti-requisites:

• MATH-UA 235 Probability & Statistics
• MATH-UA 238 Honors Theory of Probability

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.

Students cannot take MATH-UA 233 if they have taken MATH-UA 235 or 238.

Prerequisites:

MATH-UA 233 Theory of Probability or MATH-UA 238 Honors Theory of Probability with a grade of C or higher.

Anti-requisites: MATH-UA 235 Probability & 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.

Students cannot take MATH-UA 234 if they have taken MATH-UA 235. 

Prerequisites:

A grade of C or higher in MATH-UA 122 Calculus II or MATH-UA 132 Math for Economics II.

Anti-requisites:

• MATH-UA 233 Theory of Probability
• MATH-UA 234 Mathematical Statistics
• MATH-UA 238 Honors Theory of Probability

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.

Students cannot take MATH-UA 235 if they have taken MATH-UA 233, 234 or 238.

Prerequisites:

Students must earn grades of B+ or higher in the following three prerequisite courses:

1. MATH-UA 120 Discrete Mathematics
2. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
3. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Note: 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.

Anti-requisites:

• MATH-UA 233 Theory of Probability
• MATH-UA 235 Probability & Statistics

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.

Students cannot take MATH-UA 238 if they have taken MATH-UA 233 or 235.

Prerequisites:

A grade of C or higher in MATH-UA 122 Calculus II or MATH-UA 132 Math for Economics II. 

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 higher in MATH-UA 122 Calculus II or MATH-UA 132 Math for Economics II. 

Description:

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

Prerequisites:

Students must earn grades of C+ or higher in the following three prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra
3. An introductory course in probability or statistics. Acceptable courses include:
   • MATH-UA 233 Theory of Probability
   • MATH-UA 234 Mathematical Statistics
   • MATH-UA 235 Probability & Statistics
   • MATH-UA 238 Honors Theory of Probability
   • ECON-UA 18 Statistics
   • ECON-UA 20 Analytical Statistics
   • STAT-UB 103 Statistics for Business Control and Regression Models

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Anti-requisite: MATH-UA 258 Honors Numerical Analysis

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.

Students cannot take both MATH-UA 252 and 258.

Prerequisites:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Description:

Prerequisites:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 121 Calculus I or MATH-UA 132 Math for Economics II
2. BIOL-UA 11 Principles of Biology I or permission from 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:

A grade of C or higher in MATH-UA 255 Mathematics in Medicine and Biology, or permission from the instructor.

Note: Familiarity with a programming language is recommended, but not required. The course uses 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:

Students must earn grades of A- or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 133 Math for Economics III
   • Alternative: MATH-UA 129 Honors Calculus III with a grade of B+ or higher
2. MATH-UA 140 Linear Algebra
   • Alternative: MATH-UA 148 Honors Linear Algebra with a grade of B+ or higher

Notes: Programming experience is strongly recommended (e.g. Julia, Matlab, or NumPy), but not required (there is a programming component to this course).

Anti-requisite: MATH-UA 252 Numerical Analysis

Description:

Covers the analysis of numerical algorithms which are ubiquitously used to solve problems throughout mathematics, physics, engineering, finance, and the life sciences. Topics include: algorithms for solving nonlinear equations; optimization; finding eigenvalues/eigenvectors of matrices; computing matrix factorizations and performing linear regressions; function interpolation, approximation, and integration; basic signal processing using the Fast Fourier Transform; Monte Carlo simulation. An introduction to programming will be provided as it is an integral part of numerical analysis, but students should feel quite comfortable programming on their own (or be exceptionally willing to learn along the way).

Students cannot take both MATH-UA 252 and 258.

Prerequisites:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Anti-requisite: MATH-UA 268 Honors Ordinary Differential Equations

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.

Students cannot take both MATH-UA 262 and 268.

Prerequisites:

A grade of C or higher in MATH-UA 262 Ordinary Differential Equations or MATH-UA 268 Honors Ordinary Differential Equations. 

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 122 Calculus II or MATH-UA 132 Math for Economics II
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

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 B+ or higher in MATH-UA 328 Honors Analysis I
  • Alternative: a grade of A- or higher in MATH-UA 325 Analysis

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Anti-requisite: MATH-UA 328 Honors Analysis I

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.

Students cannot take both MATH-UA 325 and MATH-UA 328.

Prerequisites:

Students must earn grades of A- or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 133 Math for Economics III
   • Alternative: MATH-UA 129 Honors Calculus III with a grade of B+ or higher
2. MATH-UA 140 Linear Algebra
   • Alternative: MATH-UA 148 Honors Linear Algebra with a grade of B+ or higher

Anti-requisite: MATH-UA 325 Analysis

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.

Students cannot take both MATH-UA 325 and MATH-UA 328. 

Prerequisites:

• A grade of C or higher in MATH-UA 328 Honors Analysis I
  • Alternative: a grade of A in MATH-UA 325 Analysis and receive permission from the Honors Analysis II instructor

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:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Note: It is strongly recommended, but not required, that students complete MATH-UA 325 Analysis before registering for MATH-UA 343 Algebra. 

Anti-requisite: MATH-UA 348 Honors Algebra I

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.

Students cannot take both MATH-UA 343 and MATH-UA 348.

Prerequisites:

Students must earn grades of A- or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 133 Math for Economics III
   • Alternative: MATH-UA 129 Honors Calculus III with a grade of B+ or higher
2. MATH-UA 140 Linear Algebra
   • Alternative: MATH-UA 148 Honors Linear Algebra with a grade of B+ or higher

Notes: It is strongly recommended, but not required, that students complete MATH-UA 325 Analysis or MATH-UA 328 Honors Analysis I before registering for MATH-UA 348 Honors Algebra I.

Anti-requisite: MATH-UA 343 Algebra

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.

Students cannot take both MATH-UA 343 and MATH-UA 348.

Prerequisites:

• A grade of C or higher in MATH-UA 348 Honors Algebra I
  • Alternative: a grade of A in MATH-UA 343 Algebra and receive permission from the Honors Algebra II 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:

A grade of C or higher in MATH-UA 325 Analysis or MATH-UA 328 Honors Analysis I. 

Description:

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

Prerequisites:

Students must earn grades of C or higher in the following two prerequisite courses:

1. MATH-UA 123 Calculus III or MATH-UA 129 Honors Calculus III or MATH-UA 133 Math for Economics III
2. MATH-UA 140 Linear Algebra or MATH-UA 148 Honors Linear Algebra

Note: It is strongly recommended, but not required, that students have completed MATH-UA 325 Analysis or MATH-UA 328 Honors Analysis I before registering for MATH-UA 377 Differential Geometry. 

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:

Pre-Fall 2024: Diagnostic Exam or a grade of B or better in MA-UY 914.
Fall 2024+: 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:

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.

*Formerly MA-UY 3044*

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:

Pre-Fall 2024: MA-UY 1024 or a grade of B or better in MA-UY 1324. 
Fall 2024+: MA-UY 1024 or 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 1044 (formerly 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 or ECE-UY 2233.

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 IDM students only. 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.

This course does not count towards degree if student has already taken MA-UY 2224 or MA-UY 2054.

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 1044 (formerly 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 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 1044 (formerly 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 1044 (formerly 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 1044 (formerly 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 1044 (formerly 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 1044 (formerly 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 or MA-UY 2514)

Corequisites: 
PH-UY 2033 and (MA-UY 2034 or MA-UY 1044 (formerly 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:

Effective Spring 2025: A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 1044 (formerly 3044) or MA-UY 3054 or MA-UY 3113). Additionally, it is suggested for students to have taken MA-UY 4614 or MA-UY 4644 as a prerequisite.

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:

Effective Spring 2025: (A grade of A- or better in MA-UY 2114 or a grade or B+ or better in MA-UY 2514) and (a grade of A- or better in MA-UY 1044 (formerly 3044) or MA-UY 3113 or a grade of B+ or better in MA-UY 3054). Additionally, it is suggested for students to have taken MA-UY 4614 or MA-UY 4644 as a prerequisite.

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 1044 (formerly 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 1044 (formerly 3044) or MA-UY 3054 or MA-UY 3113).

Note: Not open to students who have taken MA-UY 4524.

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 1044 (formerly 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 1044 (formerly 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 1044 (formerly 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:

(Grade of A- or above in MA-UY 2114 OR grade of B+ or above in MA-UY 2514) AND ((grade of A- or above in MA-UY 1044 (formerly 3044) OR MA-UY 3113) OR grade of B+ or above in MA-UY 3054) AND programming experience strongly recommended (e.g. julia, Matlab, or numpy) but not required (there is a programming component to this course).

Notes:

Not open to students who have taken MA-UY 4424.

Description:

This course will cover the analysis of numerical algorithms which are ubiquitously used to solve problems throughout mathematics, physics, engineering, finance, and the life sciences. Topics include: algorithms for solving nonlinear equations; optimization; finding eigenvalues/eigenvectors of matrices; computing matrix factorizations and performing linear regressions; function interpolation, approximation, and integration; basic signal processing using the Fast Fourier Transform; Monte Carlo simulation. An introduction to programming will be provided as it is an integral part of numerical analysis, but students should feel quite comfortable programming on their own (or be exceptionally willing to learn along the way).

Prerequisites:

A grade of C or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 1044 (formerly 3044) or MA-UY 3054) and Junior level standing or above.

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 A- or or better in (MA-UY 2114 or MA-UY 2514) and (MA-UY 2034 or MA-UY 1044 (formerly 3044) or MA-UY 3054) and Junior level standing or above. Recommended: MA-UY 2514 Honors Calculus III and MA-UY 3054 Honors Linear Algebra with a grade of B+ or better.

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 1044 (formerly 3044) or MA-UY 3054 or MA-UY 3113).

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.